Univariate Models

AalenJohansenFitter

class lifelines.fitters.aalen_johansen_fitter.AalenJohansenFitter(jitter_level=0.0001, seed=None, alpha=0.05, calculate_variance=True)

Bases: lifelines.fitters.UnivariateFitter

Class for fitting the Aalen-Johansen estimate for the cumulative incidence function in a competing risks framework. Treating competing risks as censoring can result in over-estimated cumulative density functions. Using the Kaplan Meier estimator with competing risks as censored is akin to estimating the cumulative density if all competing risks had been prevented.

Aalen-Johansen cannot deal with tied times. We can get around this by randomly jittering the event times slightly. This will be done automatically and generates a warning.

Parameters:

• alpha (float, option (default=0.05)) – The alpha value associated with the confidence intervals.
• jitter_level (float, option (default=0.00001)) – If tied event times are detected, event times are randomly changed by this factor.
• seed (int, option (default=None)) – To produce replicate results with tied event times, the numpy.random.seed can be specified in the function.
• calculate_variance (bool, option (default=True)) – By default, AalenJohansenFitter calculates the variance and corresponding confidence intervals. Due to how the variance is calculated, the variance must be calculated for each event time individually. This is computationally intensive. For some procedures, like bootstrapping, the variance is not necessary. To reduce computation time during these procedures, calculate_variance can be set to False to skip the variance calculation.

Example

>>> from lifelines import AalenJohansenFitter
>>> from lifelines.datasets import load_waltons
>>> T, E = load_waltons()['T'], load_waltons()['E']
>>> ajf = AalenJohansenFitter(calculate_variance=True)
>>> ajf.fit(T, E, event_of_interest=1)
>>> ajf.cumulative_density_
>>> ajf.plot()

References

If you are interested in learning more, we recommend the following open-access paper; Edwards JK, Hester LL, Gokhale M, Lesko CR. Methodologic Issues When Estimating Risks in Pharmacoepidemiology. Curr Epidemiol Rep. 2016;3(4):285-296.

conditional_time_to_event_

Return a DataFrame, with index equal to survival_function_’s index, that estimates the median duration remaining until the death event, given survival up until time t. For example, if an individual exists until age 1, their expected life remaining given they lived to time 1 might be 9 years.

Returns:conditional_time_to_ Return type:DataFrame

cumulative_density_at_times(times, label=None)

cumulative_hazard_at_times(times, label=None)

divide(other)

Divide the {0} of two {1} objects.

Parameters:other (same object as self) Returns: Return type:DataFrame

fit(durations, event_observed, event_of_interest, timeline=None, entry=None, label='AJ_estimate', alpha=None, ci_labels=None, weights=None)

Parameters:

• durations (an array or pd.Series of length n – duration of subject was observed for)
• event_observed (an array, or pd.Series, of length n. Integer indicator of distinct events. Must be) – only positive integers, where 0 indicates censoring.
• event_of_interest (integer – indicator for event of interest. All other integers are considered competing events) – Ex) event_observed contains 0, 1, 2 where 0:censored, 1:lung cancer, and 2:death. If event_of_interest=1, then death (2) is considered a competing event. The returned cumulative incidence function corresponds to risk of lung cancer
• timeline (return the best estimate at the values in timelines (positively increasing))
• entry (an array, or pd.Series, of length n – relative time when a subject entered the study. This is) – useful for left-truncated (not left-censored) observations. If None, all members of the population were born at time 0.
• label (a string to name the column of the estimate.)
• alpha (the alpha value in the confidence intervals. Overrides the initializing) – alpha for this call to fit only.
• ci_labels (add custom column names to the generated confidence intervals) – as a length-2 list: [<lower-bound name>, <upper-bound name>]. Default: <label>_lower_<1-alpha/2>
• weights (n array, or pd.Series, of length n, if providing a weighted dataset. For example, instead) – of providing every subject as a single element of durations and event_observed, one could weigh subject differently.

Returns:

self – self, with new properties like cumulative_incidence_.

Return type:

AalenJohansenFitter

fit_right_censoring(*args, **kwargs)

Alias for fit

See also

fit

hazard_at_times(times, label=None)

median_

Deprecated, use .median_survival_time_

Return the unique time point, t, such that S(t) = 0.5. This is the “half-life” of the population, and a robust summary statistic for the population, if it exists.

median_survival_time_

Return the unique time point, t, such that S(t) = 0.5. This is the “half-life” of the population, and a robust summary statistic for the population, if it exists.

percentile(p)

Return the unique time point, t, such that S(t) = p.

For known parametric models, this should be overwritten by something more accurate.

plot(**kwargs)

Plots a pretty figure of the model

Matplotlib plot arguments can be passed in inside the kwargs, plus

Parameters:

• show_censors (bool) – place markers at censorship events. Default: False

• censor_styles (dict) – If show_censors, this dictionary will be passed into the plot call.

• ci_alpha (float) – the transparency level of the confidence interval. Default: 0.3

• ci_force_lines (bool) – force the confidence intervals to be line plots (versus default shaded areas). Default: False

• ci_show (bool) – show confidence intervals. Default: True

• ci_legend (bool) – if ci_force_lines is True, this is a boolean flag to add the lines’ labels to the legend. Default: False

• at_risk_counts (bool) – show group sizes at time points. See function add_at_risk_counts for details. Default: False

• loc (slice) – specify a time-based subsection of the curves to plot, ex:

  >>> model.plot(loc=slice(0.,10.))
  

  will plot the time values between t=0. and t=10.

• iloc (slice) – specify a location-based subsection of the curves to plot, ex:

  >>> model.plot(iloc=slice(0,10))
  

  will plot the first 10 time points.

• invert_y_axis (bool) – boolean to invert the y-axis, useful to show cumulative graphs instead of survival graphs. (Deprecated, use plot_cumulative_density())

Returns:

a pyplot axis object

Return type:

ax

plot_cumulative_density(**kwargs)

plot_cumulative_hazard(**kwargs)

plot_hazard(**kwargs)

plot_survival_function(**kwargs)

predict(times, interpolate=False)

Predict the {0} at certain point in time. Uses a linear interpolation if points in time are not in the index.

Parameters:

• times (scalar, or array) – a scalar or an array of times to predict the value of {0} at.
• interpolate (boolean, optional (default=False)) – for methods that produce a stepwise solution (Kaplan-Meier, Nelson-Aalen, etc), turning this to True will use an linear interpolation method to provide a more “smooth” answer.

Returns:

predictions

Return type:

a scalar if time is a scalar, a numpy array if time in an array.

subtract(other)

Subtract the {0} of two {1} objects.

Parameters:other (same object as self) Returns: Return type:DataFrame

survival_function_at_times(times, label=None)

BreslowFlemingHarringtonFitter

class lifelines.fitters.breslow_fleming_harrington_fitter.BreslowFlemingHarringtonFitter(alpha=0.05)

Bases: lifelines.fitters.UnivariateFitter

Class for fitting the Breslow-Fleming-Harrington estimate for the survival function. This estimator is a biased estimator of the survival function but is more stable when the population is small and there are too few early truncation times, it may happen that is the number of patients at risk and the number of deaths is the same.

Mathematically, the NAF estimator is the negative logarithm of the BFH estimator.

BreslowFlemingHarringtonFitter(alpha=0.05)

Parameters:alpha (float, optional (default=0.05)) – The alpha value associated with the confidence intervals.

conditional_time_to_event_

Return a DataFrame, with index equal to survival_function_’s index, that estimates the median duration remaining until the death event, given survival up until time t. For example, if an individual exists until age 1, their expected life remaining given they lived to time 1 might be 9 years.

Returns:conditional_time_to_ Return type:DataFrame

cumulative_density_at_times(times, label=None)

cumulative_hazard_at_times(times, label=None)

divide(other)

Divide the {0} of two {1} objects.

Parameters:other (same object as self) Returns: Return type:DataFrame

fit(durations, event_observed=None, timeline=None, entry=None, label='BFH_estimate', alpha=None, ci_labels=None, weights=None)

Parameters:

• durations (an array, or pd.Series, of length n) – duration subject was observed for
• timeline – return the best estimate at the values in timelines (positively increasing)
• event_observed (an array, or pd.Series, of length n) – True if the the death was observed, False if the event was lost (right-censored). Defaults all True if event_observed==None
• entry (an array, or pd.Series, of length n) – relative time when a subject entered the study. This is useful for left-truncated observations, i.e the birth event was not observed. If None, defaults to all 0 (all birth events observed.)
• label (string) – a string to name the column of the estimate.
• alpha (float, optional (default=0.05)) – the alpha value in the confidence intervals. Overrides the initializing alpha for this call to fit only.
• ci_labels (iterable) – add custom column names to the generated confidence intervals as a length-2 list: [<lower-bound name>, <upper-bound name>]. Default: <label>_lower_<alpha>

Returns:

Return type:

self, with new properties like survival_function_.

fit_right_censoring(*args, **kwargs)

Alias for fit

See also

fit

hazard_at_times(times, label=None)

median_

Deprecated, use .median_survival_time_

Return the unique time point, t, such that S(t) = 0.5. This is the “half-life” of the population, and a robust summary statistic for the population, if it exists.

median_survival_time_

Return the unique time point, t, such that S(t) = 0.5. This is the “half-life” of the population, and a robust summary statistic for the population, if it exists.

percentile(p)

Return the unique time point, t, such that S(t) = p.

For known parametric models, this should be overwritten by something more accurate.

plot(**kwargs)

Plots a pretty figure of the model

Matplotlib plot arguments can be passed in inside the kwargs, plus

Parameters:

• show_censors (bool) – place markers at censorship events. Default: False

• censor_styles (dict) – If show_censors, this dictionary will be passed into the plot call.

• ci_alpha (float) – the transparency level of the confidence interval. Default: 0.3

• ci_force_lines (bool) – force the confidence intervals to be line plots (versus default shaded areas). Default: False

• ci_show (bool) – show confidence intervals. Default: True

• ci_legend (bool) – if ci_force_lines is True, this is a boolean flag to add the lines’ labels to the legend. Default: False

• at_risk_counts (bool) – show group sizes at time points. See function add_at_risk_counts for details. Default: False

• loc (slice) – specify a time-based subsection of the curves to plot, ex:

  >>> model.plot(loc=slice(0.,10.))
  

  will plot the time values between t=0. and t=10.

• iloc (slice) – specify a location-based subsection of the curves to plot, ex:

  >>> model.plot(iloc=slice(0,10))
  

  will plot the first 10 time points.

• invert_y_axis (bool) – boolean to invert the y-axis, useful to show cumulative graphs instead of survival graphs. (Deprecated, use plot_cumulative_density())

Returns:

a pyplot axis object

Return type:

ax

plot_cumulative_density(**kwargs)

plot_cumulative_hazard(**kwargs)

plot_hazard(**kwargs)

plot_survival_function(**kwargs)

predict(times, interpolate=False)

Predict the {0} at certain point in time. Uses a linear interpolation if points in time are not in the index.

Parameters:

• times (scalar, or array) – a scalar or an array of times to predict the value of {0} at.
• interpolate (boolean, optional (default=False)) – for methods that produce a stepwise solution (Kaplan-Meier, Nelson-Aalen, etc), turning this to True will use an linear interpolation method to provide a more “smooth” answer.

Returns:

predictions

Return type:

a scalar if time is a scalar, a numpy array if time in an array.

subtract(other)

Subtract the {0} of two {1} objects.

Parameters:other (same object as self) Returns: Return type:DataFrame

survival_function_at_times(times, label=None)

Return a Pandas series of the predicted survival value at specific times

Parameters:times (iterable or float) Returns: Return type:pd.Series

ExponentialFitter

class lifelines.fitters.exponential_fitter.ExponentialFitter(*args, **kwargs)

Bases: lifelines.fitters.KnownModelParametericUnivariateFitter

This class implements an Exponential model for univariate data. The model has parameterized form:

\[S(t) = \exp\left(\frac{-t}{\lambda}\right), \lambda >0\]

which implies the cumulative hazard rate is

\[H(t) = \frac{t}{\lambda}\]

and the hazard rate is:

\[h(t) = \frac{1}{\lambda}\]

After calling the .fit method, you have access to properties like: survival_function_, lambda_, cumulative_hazard_ A summary of the fit is available with the method print_summary()

Parameters:alpha (float, optional (default=0.05)) – the level in the confidence intervals.

Important

The parameterization of this model changed in lifelines 0.19.0. Previously, the cumulative hazard looked like \(\lambda t\). The parameterization is now the reciprocal of \(\lambda\).

cumulative_hazard_

The estimated cumulative hazard (with custom timeline if provided)

Type:DataFrame

confidence_interval_cumulative_hazard_

The lower and upper confidence intervals for the cumulative hazard

Type:DataFrame

hazard_

The estimated hazard (with custom timeline if provided)

Type:DataFrame

confidence_interval_hazard_

The lower and upper confidence intervals for the hazard

Type:DataFrame

survival_function_

The estimated survival function (with custom timeline if provided)

Type:DataFrame

confidence_interval_survival_function_

The lower and upper confidence intervals for the survival function

Type:DataFrame

variance_matrix_

The variance matrix of the coefficients

Type:numpy array

median_

The median time to event

Type:float

lambda_

The fitted parameter in the model

Type:float

durations

The durations provided

Type:array

event_observed

The event_observed variable provided

Type:array

timeline

The time line to use for plotting and indexing

Type:array

entry

The entry array provided, or None

Type:array or None

cumumlative_density_

The estimated cumulative density function (with custom timeline if provided)

Type:DataFrame

confidence_interval_cumulative_density_

The lower and upper confidence intervals for the cumulative density

Type:DataFrame

conditional_time_to_event_

Return a DataFrame, with index equal to survival_function_’s index, that estimates the median duration remaining until the death event, given survival up until time t. For example, if an individual exists until age 1, their expected life remaining given they lived to time 1 might be 9 years.

Returns:conditional_time_to_ Return type:DataFrame

confidence_interval_

The confidence interval of the cumulative hazard. This is an alias for confidence_interval_cumulative_hazard_.

confidence_interval_cumulative_density_

The lower and upper confidence intervals for the cumulative density

confidence_interval_cumulative_hazard_

The confidence interval of the cumulative hazard. This is an alias for confidence_interval_.

confidence_interval_hazard_

The confidence interval of the hazard.

confidence_interval_survival_function_

The lower and upper confidence intervals for the survival function

cumulative_density_at_times(times, label=None)

Return a Pandas series of the predicted cumulative density function (1-survival function) at specific times.

Parameters:

• times (iterable or float) – values to return the survival function at.
• label (string, optional) – Rename the series returned. Useful for plotting.

Returns:

Return type:

pd.Series

cumulative_hazard_at_times(times, label=None)

Return a Pandas series of the predicted cumulative hazard value at specific times.

Parameters:

• times (iterable or float) – values to return the cumulative hazard at.
• label (string, optional) – Rename the series returned. Useful for plotting.

Returns:

Return type:

pd.Series

divide(other)

Divide the {0} of two {1} objects.

Parameters:other (same object as self) Returns: Return type:DataFrame

event_table

fit(durations, event_observed=None, timeline=None, label=None, alpha=None, ci_labels=None, show_progress=False, entry=None, weights=None, left_censorship=False, initial_point=None)

Parameters:

• durations (an array, or pd.Series) – length n, duration subject was observed for
• event_observed (numpy array or pd.Series, optional) – length n, True if the the death was observed, False if the event was lost (right-censored). Defaults all True if event_observed==None
• timeline (list, optional) – return the estimate at the values in timeline (positively increasing)
• label (string, optional) – a string to name the column of the estimate.
• alpha (float, optional) – the alpha value in the confidence intervals. Overrides the initializing alpha for this call to fit only.
• ci_labels (list, optional) – add custom column names to the generated confidence intervals as a length-2 list: [<lower-bound name>, <upper-bound name>]. Default: <label>_lower_<alpha>
• show_progress (boolean, optional) – since this is an iterative fitting algorithm, switching this to True will display some iteration details.
• entry (an array, or pd.Series, of length n) – relative time when a subject entered the study. This is useful for left-truncated (not left-censored) observations. If None, all members of the population entered study when they were “born”: time zero.
• weights (an array, or pd.Series, of length n) – integer weights per observation
• initial_point ((d,) numpy array, optional) – initialize the starting point of the iterative algorithm. Default is the zero vector.

Returns:

self with new properties like cumulative_hazard_, survival_function_

Return type:

self

fit_interval_censoring(lower_bound, upper_bound, event_observed=None, timeline=None, label=None, alpha=None, ci_labels=None, show_progress=False, entry=None, weights=None, initial_point=None)

Fit the model to an interval censored dataset.

Parameters:

• lower_bound (an array, or pd.Series) – length n, the start of the period the subject experienced the event in.
• upper_bound (an array, or pd.Series) – length n, the end of the period the subject experienced the event in. If the value is equal to the corresponding value in lower_bound, then the individual’s event was observed (not censored).
• event_observed (numpy array or pd.Series, optional) – length n, if left optional, infer from lower_bound and upper_cound (if lower_bound==upper_bound then event observed, if lower_bound < upper_bound, then event censored)
• timeline (list, optional) – return the estimate at the values in timeline (positively increasing)
• label (string, optional) – a string to name the column of the estimate.
• alpha (float, optional) – the alpha value in the confidence intervals. Overrides the initializing alpha for this call to fit only.
• ci_labels (list, optional) – add custom column names to the generated confidence intervals as a length-2 list: [<lower-bound name>, <upper-bound name>]. Default: <label>_lower_<alpha>
• show_progress (boolean, optional) – since this is an iterative fitting algorithm, switching this to True will display some iteration details.
• entry (an array, or pd.Series, of length n) – relative time when a subject entered the study. This is useful for left-truncated (not left-censored) observations. If None, all members of the population entered study when they were “born”: time zero.
• weights (an array, or pd.Series, of length n) – integer weights per observation
• initial_point ((d,) numpy array, optional) – initialize the starting point of the iterative algorithm. Default is the zero vector.

Returns:

self with new properties like cumulative_hazard_, survival_function_

Return type:

self

fit_left_censoring(durations, event_observed=None, timeline=None, label=None, alpha=None, ci_labels=None, show_progress=False, entry=None, weights=None, initial_point=None)

Fit the model to a left-censored dataset

Parameters:

• durations (an array, or pd.Series) – length n, duration subject was observed for
• event_observed (numpy array or pd.Series, optional) – length n, True if the the death was observed, False if the event was lost (right-censored). Defaults all True if event_observed==None
• timeline (list, optional) – return the estimate at the values in timeline (positively increasing)
• label (string, optional) – a string to name the column of the estimate.
• alpha (float, optional) – the alpha value in the confidence intervals. Overrides the initializing alpha for this call to fit only.
• ci_labels (list, optional) – add custom column names to the generated confidence intervals as a length-2 list: [<lower-bound name>, <upper-bound name>]. Default: <label>_lower_<alpha>
• show_progress (boolean, optional) – since this is an iterative fitting algorithm, switching this to True will display some iteration details.
• entry (an array, or pd.Series, of length n) – relative time when a subject entered the study. This is useful for left-truncated (not left-censored) observations. If None, all members of the population entered study when they were “born”: time zero.
• weights (an array, or pd.Series, of length n) – integer weights per observation
• initial_point ((d,) numpy array, optional) – initialize the starting point of the iterative algorithm. Default is the zero vector.

Returns:

self with new properties like cumulative_hazard_, survival_function_

Return type:

self

fit_right_censoring(*args, **kwargs)

Alias for fit

See also

fit

hazard_at_times(times, label=None)

Return a Pandas series of the predicted hazard at specific times.

Parameters:

• times (iterable or float) – values to return the hazard at.
• label (string, optional) – Rename the series returned. Useful for plotting.

Returns:

Return type:

pd.Series

median_

Deprecated, use .median_survival_time_

Return the unique time point, t, such that S(t) = 0.5. This is the “half-life” of the population, and a robust summary statistic for the population, if it exists.

median_survival_time_

Return the unique time point, t, such that S(t) = 0.5. This is the “half-life” of the population, and a robust summary statistic for the population, if it exists.

percentile(p)

Return the unique time point, t, such that S(t) = p.

For known parametric models, this should be overwritten by something more accurate.

plot(**kwargs)

Produce a pretty-plot of the estimate.

plot_cumulative_density(**kwargs)

plot_cumulative_hazard(**kwargs)

plot_hazard(**kwargs)

plot_survival_function(**kwargs)

predict(times, interpolate=False)

Predict the {0} at certain point in time. Uses a linear interpolation if points in time are not in the index.

Parameters:

• times (scalar, or array) – a scalar or an array of times to predict the value of {0} at.
• interpolate (boolean, optional (default=False)) – for methods that produce a stepwise solution (Kaplan-Meier, Nelson-Aalen, etc), turning this to True will use an linear interpolation method to provide a more “smooth” answer.

Returns:

predictions

Return type:

a scalar if time is a scalar, a numpy array if time in an array.

print_summary(decimals=2, **kwargs)

Print summary statistics describing the fit, the coefficients, and the error bounds.

Parameters:

• decimals (int, optional (default=2)) – specify the number of decimal places to show
• kwargs – print additional metadata in the output (useful to provide model names, dataset names, etc.) when comparing multiple outputs.

subtract(other)

Subtract the {0} of two {1} objects.

Parameters:other (same object as self) Returns: Return type:DataFrame

summary

Summary statistics describing the fit.

Returns:df – Contains columns coef, exp(coef), se(coef), z, p, lower, upper Return type:pd.DataFrame

See also

print_summary

survival_function_at_times(times, label=None)

Return a Pandas series of the predicted survival value at specific times.

Parameters:

• times (iterable or float) – values to return the survival function at.
• label (string, optional) – Rename the series returned. Useful for plotting.

Returns:

Return type:

pd.Series

KaplanMeierFitter

class lifelines.fitters.kaplan_meier_fitter.KaplanMeierFitter(alpha=0.05)

Bases: lifelines.fitters.UnivariateFitter

Class for fitting the Kaplan-Meier estimate for the survival function.

Parameters:alpha (float, option (default=0.05)) – The alpha value associated with the confidence intervals.

Examples

>>> from lifelines import KaplanMeierFitter
>>> from lifelines.datasets import load_waltons
>>> waltons = load_waltons()
>>> kmf = KaplanMeierFitter()
>>> kmf.fit(waltons['T'], waltons['E'])
>>> kmf.plot()

survival_function_

The estimated survival function (with custom timeline if provided)

Type:DataFrame

median_

The estimated median time to event. np.inf if doesn’t exist.

Type:float

confidence_interval_

The lower and upper confidence intervals for the survival function. An alias of confidence_interval_survival_function_

Type:DataFrame

confidence_interval_survival_function_

The lower and upper confidence intervals for the survival function. An alias of confidence_interval_

Type:DataFrame

cumumlative_density_

The estimated cumulative density function (with custom timeline if provided)

Type:DataFrame

confidence_interval_cumulative_density_

The lower and upper confidence intervals for the cumulative density

Type:DataFrame

durations

The durations provided

Type:array

event_observed

The event_observed variable provided

Type:array

timeline

The time line to use for plotting and indexing

Type:array

entry

The entry array provided, or None

Type:array or None

event_table

A summary of the life table

Type:DataFrame

conditional_time_to_event_

Return a DataFrame, with index equal to survival_function_’s index, that estimates the median duration remaining until the death event, given survival up until time t. For example, if an individual exists until age 1, their expected life remaining given they lived to time 1 might be 9 years.

Returns:conditional_time_to_ Return type:DataFrame

cumulative_density_at_times(times, label=None)

Return a Pandas series of the predicted cumulative density at specific times

Parameters:times (iterable or float) Returns: Return type:pd.Series

cumulative_hazard_at_times(times, label=None)

divide(other)

Divide the {0} of two {1} objects.

Parameters:other (same object as self) Returns: Return type:DataFrame

fit(durations, event_observed=None, timeline=None, entry=None, label='KM_estimate', left_censorship=False, alpha=None, ci_labels=None, weights=None)

Fit the model to a right-censored dataset

Parameters:

• durations (an array, list, pd.DataFrame or pd.Series) – length n – duration subject was observed for
• event_observed (an array, list, pd.DataFrame, or pd.Series, optional) – True if the the death was observed, False if the event was lost (right-censored). Defaults all True if event_observed==None
• timeline (an array, list, pd.DataFrame, or pd.Series, optional) – return the best estimate at the values in timelines (postively increasing)
• entry (an array, list, pd.DataFrame, or pd.Series, optional) – relative time when a subject entered the study. This is useful for left-truncated (not left-censored) observations. If None, all members of the population entered study when they were “born”.
• label (string, optional) – a string to name the column of the estimate.
• alpha (float, optional) – the alpha value in the confidence intervals. Overrides the initializing alpha for this call to fit only.
• left_censorship (bool, optional (default=False)) – Deprecated, use fit_left_censoring
• ci_labels (tuple, optional) – add custom column names to the generated confidence intervals as a length-2 list: [<lower-bound name>, <upper-bound name>]. Default: <label>_lower_<1-alpha/2>
• weights (an array, list, pd.DataFrame, or pd.Series, optional) – if providing a weighted dataset. For example, instead of providing every subject as a single element of durations and event_observed, one could weigh subject differently.

Returns:

self – self with new properties like survival_function_, plot(), median

Return type:

KaplanMeierFitter

fit_left_censoring(durations, event_observed=None, timeline=None, entry=None, label='KM_estimate', alpha=None, ci_labels=None, weights=None)

Fit the model to a left-censored dataset

Parameters:

• durations (an array, list, pd.DataFrame or pd.Series) – length n – duration subject was observed for
• event_observed (an array, list, pd.DataFrame, or pd.Series, optional) – True if the the death was observed, False if the event was lost (right-censored). Defaults all True if event_observed==None
• timeline (an array, list, pd.DataFrame, or pd.Series, optional) – return the best estimate at the values in timelines (postively increasing)
• entry (an array, list, pd.DataFrame, or pd.Series, optional) – relative time when a subject entered the study. This is useful for left-truncated (not left-censored) observations. If None, all members of the population entered study when they were “born”.
• label (string, optional) – a string to name the column of the estimate.
• alpha (float, optional) – the alpha value in the confidence intervals. Overrides the initializing alpha for this call to fit only.
• left_censorship (bool, optional (default=False)) – Deprecated, use fit_left_censoring
• ci_labels (tuple, optional) – add custom column names to the generated confidence intervals as a length-2 list: [<lower-bound name>, <upper-bound name>]. Default: <label>_lower_<1-alpha/2>
• weights (an array, list, pd.DataFrame, or pd.Series, optional) – if providing a weighted dataset. For example, instead of providing every subject as a single element of durations and event_observed, one could weigh subject differently.

Returns:

self – self with new properties like survival_function_, plot(), median

Return type:

KaplanMeierFitter

fit_right_censoring(*args, **kwargs)

Alias for fit

See also

fit

hazard_at_times(times, label=None)

median_

Deprecated, use .median_survival_time_

Return the unique time point, t, such that S(t) = 0.5. This is the “half-life” of the population, and a robust summary statistic for the population, if it exists.

median_survival_time_

Return the unique time point, t, such that S(t) = 0.5. This is the “half-life” of the population, and a robust summary statistic for the population, if it exists.

percentile(p)

Return the unique time point, t, such that S(t) = p.

For known parametric models, this should be overwritten by something more accurate.

plot(**kwargs)

Plots a pretty figure of the model

Matplotlib plot arguments can be passed in inside the kwargs, plus

Parameters:

• show_censors (bool) – place markers at censorship events. Default: False

• censor_styles (dict) – If show_censors, this dictionary will be passed into the plot call.

• ci_alpha (float) – the transparency level of the confidence interval. Default: 0.3

• ci_force_lines (bool) – force the confidence intervals to be line plots (versus default shaded areas). Default: False

• ci_show (bool) – show confidence intervals. Default: True

• ci_legend (bool) – if ci_force_lines is True, this is a boolean flag to add the lines’ labels to the legend. Default: False

• at_risk_counts (bool) – show group sizes at time points. See function add_at_risk_counts for details. Default: False

• loc (slice) – specify a time-based subsection of the curves to plot, ex:

  >>> model.plot(loc=slice(0.,10.))
  

  will plot the time values between t=0. and t=10.

• iloc (slice) – specify a location-based subsection of the curves to plot, ex:

  >>> model.plot(iloc=slice(0,10))
  

  will plot the first 10 time points.

• invert_y_axis (bool) – boolean to invert the y-axis, useful to show cumulative graphs instead of survival graphs. (Deprecated, use plot_cumulative_density())

Returns:

a pyplot axis object

Return type:

ax

plot_cumulative_density(**kwargs)

Plots a pretty figure of {0}.{1}

Matplotlib plot arguments can be passed in inside the kwargs, plus

Parameters:

• show_censors (bool) – place markers at censorship events. Default: False

• censor_styles (bool) – If show_censors, this dictionary will be passed into the plot call.

• ci_alpha (bool) – the transparency level of the confidence interval. Default: 0.3

• ci_force_lines (bool) – force the confidence intervals to be line plots (versus default shaded areas). Default: False

• ci_show (bool) – show confidence intervals. Default: True

• ci_legend (bool) – if ci_force_lines is True, this is a boolean flag to add the lines’ labels to the legend. Default: False

• at_risk_counts (bool) – show group sizes at time points. See function add_at_risk_counts for details. Default: False

• loc (slice) – specify a time-based subsection of the curves to plot, ex:

  >>> model.plot(loc=slice(0.,10.))
  

  will plot the time values between t=0. and t=10.

• iloc (slice) – specify a location-based subsection of the curves to plot, ex:

  >>> model.plot(iloc=slice(0,10))
  

  will plot the first 10 time points.

• invert_y_axis (bool) – boolean to invert the y-axis, useful to show cumulative graphs instead of survival graphs. (Deprecated, use plot_cumulative_density())

Returns:

a pyplot axis object

Return type:

ax

plot_cumulative_hazard(**kwargs)

plot_hazard(**kwargs)

plot_loglogs(*args, **kwargs)

Plot \(\log(S(t))\) against \(\log(t)\). Same arguments as .plot.

plot_survival_function(**kwargs)

Alias of plot

predict(times, interpolate=False)

Predict the {0} at certain point in time. Uses a linear interpolation if points in time are not in the index.

Parameters:

• times (scalar, or array) – a scalar or an array of times to predict the value of {0} at.
• interpolate (boolean, optional (default=False)) – for methods that produce a stepwise solution (Kaplan-Meier, Nelson-Aalen, etc), turning this to True will use an linear interpolation method to provide a more “smooth” answer.

Returns:

predictions

Return type:

a scalar if time is a scalar, a numpy array if time in an array.

subtract(other)

Subtract the {0} of two {1} objects.

Parameters:other (same object as self) Returns: Return type:DataFrame

survival_function_at_times(times, label=None)

Return a Pandas series of the predicted survival value at specific times

Parameters:times (iterable or float) Returns: Return type:pd.Series

LogLogisticFitter

class lifelines.fitters.log_logistic_fitter.LogLogisticFitter(*args, **kwargs)

Bases: lifelines.fitters.KnownModelParametericUnivariateFitter

This class implements a Log-Logistic model for univariate data. The model has parameterized form:

\[S(t) = \left(1 + \left(\frac{t}{\alpha}\right)^{\beta}\right)^{-1}, \alpha > 0, \beta > 0,\]

and the hazard rate is:

\[h(t) = \frac{\left(\frac{\beta}{\alpha}\right)\left(\frac{t}{\alpha}\right) ^ {\beta-1}}{\left(1 + \left(\frac{t}{\alpha}\right)^{\beta}\right)}\]

and the cumulative hazard is:

\[H(t) = \log\left(\left(\frac{t}{\alpha}\right) ^ {\beta} + 1\right)\]

After calling the .fit method, you have access to properties like: cumulative_hazard_, plot, survival_function_, alpha_ and beta_. A summary of the fit is available with the method ‘print_summary()’

Parameters:alpha (float, optional (default=0.05)) – the level in the confidence intervals.

Examples

>>> from lifelines import LogLogisticFitter
>>> from lifelines.datasets import load_waltons
>>> waltons = load_waltons()
>>> llf = LogLogisticFitter()
>>> llf.fit(waltons['T'], waltons['E'])
>>> llf.plot()
>>> print(llf.alpha_)

cumulative_hazard_

The estimated cumulative hazard (with custom timeline if provided)

Type:DataFrame

hazard_

The estimated hazard (with custom timeline if provided)

Type:DataFrame

survival_function_

The estimated survival function (with custom timeline if provided)

Type:DataFrame

cumumlative_density_

The estimated cumulative density function (with custom timeline if provided)

Type:DataFrame

variance_matrix_

The variance matrix of the coefficients

Type:numpy array

median_

The median time to event

Type:float

alpha_

The fitted parameter in the model

Type:float

beta_

The fitted parameter in the model

Type:float

durations

The durations provided

Type:array

event_observed

The event_observed variable provided

Type:array

timeline

The time line to use for plotting and indexing

Type:array

entry

The entry array provided, or None

Type:array or None

conditional_time_to_event_

Return a DataFrame, with index equal to survival_function_’s index, that estimates the median duration remaining until the death event, given survival up until time t. For example, if an individual exists until age 1, their expected life remaining given they lived to time 1 might be 9 years.

Returns:conditional_time_to_ Return type:DataFrame

confidence_interval_

The confidence interval of the cumulative hazard. This is an alias for confidence_interval_cumulative_hazard_.

confidence_interval_cumulative_density_

The lower and upper confidence intervals for the cumulative density

confidence_interval_cumulative_hazard_

The confidence interval of the cumulative hazard. This is an alias for confidence_interval_.

confidence_interval_hazard_

The confidence interval of the hazard.

confidence_interval_survival_function_

The lower and upper confidence intervals for the survival function

cumulative_density_at_times(times, label=None)

Return a Pandas series of the predicted cumulative density function (1-survival function) at specific times.

Parameters:

• times (iterable or float) – values to return the survival function at.
• label (string, optional) – Rename the series returned. Useful for plotting.

Returns:

Return type:

pd.Series

cumulative_hazard_at_times(times, label=None)

Return a Pandas series of the predicted cumulative hazard value at specific times.

Parameters:

• times (iterable or float) – values to return the cumulative hazard at.
• label (string, optional) – Rename the series returned. Useful for plotting.

Returns:

Return type:

pd.Series

divide(other)

Divide the {0} of two {1} objects.

Parameters:other (same object as self) Returns: Return type:DataFrame

event_table

fit(durations, event_observed=None, timeline=None, label=None, alpha=None, ci_labels=None, show_progress=False, entry=None, weights=None, left_censorship=False, initial_point=None)

Parameters:

• durations (an array, or pd.Series) – length n, duration subject was observed for
• event_observed (numpy array or pd.Series, optional) – length n, True if the the death was observed, False if the event was lost (right-censored). Defaults all True if event_observed==None
• timeline (list, optional) – return the estimate at the values in timeline (positively increasing)
• label (string, optional) – a string to name the column of the estimate.
• alpha (float, optional) – the alpha value in the confidence intervals. Overrides the initializing alpha for this call to fit only.
• ci_labels (list, optional) – add custom column names to the generated confidence intervals as a length-2 list: [<lower-bound name>, <upper-bound name>]. Default: <label>_lower_<alpha>
• show_progress (boolean, optional) – since this is an iterative fitting algorithm, switching this to True will display some iteration details.
• entry (an array, or pd.Series, of length n) – relative time when a subject entered the study. This is useful for left-truncated (not left-censored) observations. If None, all members of the population entered study when they were “born”: time zero.
• weights (an array, or pd.Series, of length n) – integer weights per observation
• initial_point ((d,) numpy array, optional) – initialize the starting point of the iterative algorithm. Default is the zero vector.

Returns:

self with new properties like cumulative_hazard_, survival_function_

Return type:

self

fit_interval_censoring(lower_bound, upper_bound, event_observed=None, timeline=None, label=None, alpha=None, ci_labels=None, show_progress=False, entry=None, weights=None, initial_point=None)

Fit the model to an interval censored dataset.

Parameters:

• lower_bound (an array, or pd.Series) – length n, the start of the period the subject experienced the event in.
• upper_bound (an array, or pd.Series) – length n, the end of the period the subject experienced the event in. If the value is equal to the corresponding value in lower_bound, then the individual’s event was observed (not censored).
• event_observed (numpy array or pd.Series, optional) – length n, if left optional, infer from lower_bound and upper_cound (if lower_bound==upper_bound then event observed, if lower_bound < upper_bound, then event censored)
• timeline (list, optional) – return the estimate at the values in timeline (positively increasing)
• label (string, optional) – a string to name the column of the estimate.
• alpha (float, optional) – the alpha value in the confidence intervals. Overrides the initializing alpha for this call to fit only.
• ci_labels (list, optional) – add custom column names to the generated confidence intervals as a length-2 list: [<lower-bound name>, <upper-bound name>]. Default: <label>_lower_<alpha>
• show_progress (boolean, optional) – since this is an iterative fitting algorithm, switching this to True will display some iteration details.
• entry (an array, or pd.Series, of length n) – relative time when a subject entered the study. This is useful for left-truncated (not left-censored) observations. If None, all members of the population entered study when they were “born”: time zero.
• weights (an array, or pd.Series, of length n) – integer weights per observation
• initial_point ((d,) numpy array, optional) – initialize the starting point of the iterative algorithm. Default is the zero vector.

Returns:

self with new properties like cumulative_hazard_, survival_function_

Return type:

self

fit_left_censoring(durations, event_observed=None, timeline=None, label=None, alpha=None, ci_labels=None, show_progress=False, entry=None, weights=None, initial_point=None)

Fit the model to a left-censored dataset

Parameters:

• durations (an array, or pd.Series) – length n, duration subject was observed for
• event_observed (numpy array or pd.Series, optional) – length n, True if the the death was observed, False if the event was lost (right-censored). Defaults all True if event_observed==None
• timeline (list, optional) – return the estimate at the values in timeline (positively increasing)
• label (string, optional) – a string to name the column of the estimate.
• alpha (float, optional) – the alpha value in the confidence intervals. Overrides the initializing alpha for this call to fit only.
• ci_labels (list, optional) – add custom column names to the generated confidence intervals as a length-2 list: [<lower-bound name>, <upper-bound name>]. Default: <label>_lower_<alpha>
• show_progress (boolean, optional) – since this is an iterative fitting algorithm, switching this to True will display some iteration details.
• entry (an array, or pd.Series, of length n) – relative time when a subject entered the study. This is useful for left-truncated (not left-censored) observations. If None, all members of the population entered study when they were “born”: time zero.
• weights (an array, or pd.Series, of length n) – integer weights per observation
• initial_point ((d,) numpy array, optional) – initialize the starting point of the iterative algorithm. Default is the zero vector.

Returns:

self with new properties like cumulative_hazard_, survival_function_

Return type:

self

fit_right_censoring(*args, **kwargs)

Alias for fit

See also

fit

hazard_at_times(times, label=None)

Return a Pandas series of the predicted hazard at specific times.

Parameters:

• times (iterable or float) – values to return the hazard at.
• label (string, optional) – Rename the series returned. Useful for plotting.

Returns:

Return type:

pd.Series

median_

Deprecated, use .median_survival_time_

Return the unique time point, t, such that S(t) = 0.5. This is the “half-life” of the population, and a robust summary statistic for the population, if it exists.

median_survival_time_

Return the unique time point, t, such that S(t) = 0.5. This is the “half-life” of the population, and a robust summary statistic for the population, if it exists.

percentile(p)

Return the unique time point, t, such that S(t) = p.

For known parametric models, this should be overwritten by something more accurate.

plot(**kwargs)

Produce a pretty-plot of the estimate.

plot_cumulative_density(**kwargs)

plot_cumulative_hazard(**kwargs)

plot_hazard(**kwargs)

plot_survival_function(**kwargs)

predict(times, interpolate=False)

Predict the {0} at certain point in time. Uses a linear interpolation if points in time are not in the index.

Parameters:

• times (scalar, or array) – a scalar or an array of times to predict the value of {0} at.
• interpolate (boolean, optional (default=False)) – for methods that produce a stepwise solution (Kaplan-Meier, Nelson-Aalen, etc), turning this to True will use an linear interpolation method to provide a more “smooth” answer.

Returns:

predictions

Return type:

a scalar if time is a scalar, a numpy array if time in an array.

print_summary(decimals=2, **kwargs)

Print summary statistics describing the fit, the coefficients, and the error bounds.

Parameters:

• decimals (int, optional (default=2)) – specify the number of decimal places to show
• kwargs – print additional metadata in the output (useful to provide model names, dataset names, etc.) when comparing multiple outputs.

subtract(other)

Subtract the {0} of two {1} objects.

Parameters:other (same object as self) Returns: Return type:DataFrame

summary

Summary statistics describing the fit.

Returns:df – Contains columns coef, exp(coef), se(coef), z, p, lower, upper Return type:pd.DataFrame

See also

print_summary

survival_function_at_times(times, label=None)

Return a Pandas series of the predicted survival value at specific times.

Parameters:

• times (iterable or float) – values to return the survival function at.
• label (string, optional) – Rename the series returned. Useful for plotting.

Returns:

Return type:

pd.Series

LogNormalFitter

class lifelines.fitters.log_normal_fitter.LogNormalFitter(*args, **kwargs)

Bases: lifelines.fitters.KnownModelParametericUnivariateFitter

This class implements an Log Normal model for univariate data. The model has parameterized form:

\[S(t) = 1 - \Phi\left(\frac{\log(t) - \mu}{\sigma}\right), \;\; \sigma >0\]

where \(\Phi\) is the CDF of a standard normal random variable. This implies the cumulative hazard rate is

\[H(t) = -\log\left(1 - \Phi\left(\frac{\log(t) - \mu}{\sigma}\right)\right)\]

After calling the .fit method, you have access to properties like: survival_function_, mu_, sigma_. A summary of the fit is available with the method print_summary()

Parameters:alpha (float, optional (default=0.05)) – the level in the confidence intervals.

cumulative_hazard_

The estimated cumulative hazard (with custom timeline if provided)

Type:DataFrame

hazard_

The estimated hazard (with custom timeline if provided)

Type:DataFrame

survival_function_

The estimated survival function (with custom timeline if provided)

Type:DataFrame

cumumlative_density_

The estimated cumulative density function (with custom timeline if provided)

Type:DataFrame

variance_matrix_

The variance matrix of the coefficients

Type:numpy array

median_

The median time to event

Type:float

mu_

The fitted parameter in the model

Type:float

sigma_

The fitted parameter in the model

Type:float

durations

The durations provided

Type:array

event_observed

The event_observed variable provided

Type:array

timeline

The time line to use for plotting and indexing

Type:array

entry

The entry array provided, or None

Type:array or None

conditional_time_to_event_

Return a DataFrame, with index equal to survival_function_’s index, that estimates the median duration remaining until the death event, given survival up until time t. For example, if an individual exists until age 1, their expected life remaining given they lived to time 1 might be 9 years.

Returns:conditional_time_to_ Return type:DataFrame

confidence_interval_

The confidence interval of the cumulative hazard. This is an alias for confidence_interval_cumulative_hazard_.

confidence_interval_cumulative_density_

The lower and upper confidence intervals for the cumulative density

confidence_interval_cumulative_hazard_

The confidence interval of the cumulative hazard. This is an alias for confidence_interval_.

confidence_interval_hazard_

The confidence interval of the hazard.

confidence_interval_survival_function_

The lower and upper confidence intervals for the survival function

cumulative_density_at_times(times, label=None)

Return a Pandas series of the predicted cumulative density function (1-survival function) at specific times.

Parameters:

• times (iterable or float) – values to return the survival function at.
• label (string, optional) – Rename the series returned. Useful for plotting.

Returns:

Return type:

pd.Series

cumulative_hazard_at_times(times, label=None)

Return a Pandas series of the predicted cumulative hazard value at specific times.

Parameters:

• times (iterable or float) – values to return the cumulative hazard at.
• label (string, optional) – Rename the series returned. Useful for plotting.

Returns:

Return type:

pd.Series

divide(other)

Divide the {0} of two {1} objects.

Parameters:other (same object as self) Returns: Return type:DataFrame

event_table

fit(durations, event_observed=None, timeline=None, label=None, alpha=None, ci_labels=None, show_progress=False, entry=None, weights=None, left_censorship=False, initial_point=None)

Parameters:

• durations (an array, or pd.Series) – length n, duration subject was observed for
• event_observed (numpy array or pd.Series, optional) – length n, True if the the death was observed, False if the event was lost (right-censored). Defaults all True if event_observed==None
• timeline (list, optional) – return the estimate at the values in timeline (positively increasing)
• label (string, optional) – a string to name the column of the estimate.
• alpha (float, optional) – the alpha value in the confidence intervals. Overrides the initializing alpha for this call to fit only.
• ci_labels (list, optional) – add custom column names to the generated confidence intervals as a length-2 list: [<lower-bound name>, <upper-bound name>]. Default: <label>_lower_<alpha>
• show_progress (boolean, optional) – since this is an iterative fitting algorithm, switching this to True will display some iteration details.
• entry (an array, or pd.Series, of length n) – relative time when a subject entered the study. This is useful for left-truncated (not left-censored) observations. If None, all members of the population entered study when they were “born”: time zero.
• weights (an array, or pd.Series, of length n) – integer weights per observation
• initial_point ((d,) numpy array, optional) – initialize the starting point of the iterative algorithm. Default is the zero vector.

Returns:

self with new properties like cumulative_hazard_, survival_function_

Return type:

self

fit_interval_censoring(lower_bound, upper_bound, event_observed=None, timeline=None, label=None, alpha=None, ci_labels=None, show_progress=False, entry=None, weights=None, initial_point=None)

Fit the model to an interval censored dataset.

Parameters:

• lower_bound (an array, or pd.Series) – length n, the start of the period the subject experienced the event in.
• upper_bound (an array, or pd.Series) – length n, the end of the period the subject experienced the event in. If the value is equal to the corresponding value in lower_bound, then the individual’s event was observed (not censored).
• event_observed (numpy array or pd.Series, optional) – length n, if left optional, infer from lower_bound and upper_cound (if lower_bound==upper_bound then event observed, if lower_bound < upper_bound, then event censored)
• timeline (list, optional) – return the estimate at the values in timeline (positively increasing)
• label (string, optional) – a string to name the column of the estimate.
• alpha (float, optional) – the alpha value in the confidence intervals. Overrides the initializing alpha for this call to fit only.
• ci_labels (list, optional) – add custom column names to the generated confidence intervals as a length-2 list: [<lower-bound name>, <upper-bound name>]. Default: <label>_lower_<alpha>
• show_progress (boolean, optional) – since this is an iterative fitting algorithm, switching this to True will display some iteration details.
• entry (an array, or pd.Series, of length n) – relative time when a subject entered the study. This is useful for left-truncated (not left-censored) observations. If None, all members of the population entered study when they were “born”: time zero.
• weights (an array, or pd.Series, of length n) – integer weights per observation
• initial_point ((d,) numpy array, optional) – initialize the starting point of the iterative algorithm. Default is the zero vector.

Returns:

self with new properties like cumulative_hazard_, survival_function_

Return type:

self

fit_left_censoring(durations, event_observed=None, timeline=None, label=None, alpha=None, ci_labels=None, show_progress=False, entry=None, weights=None, initial_point=None)

Fit the model to a left-censored dataset

Parameters:

• durations (an array, or pd.Series) – length n, duration subject was observed for
• event_observed (numpy array or pd.Series, optional) – length n, True if the the death was observed, False if the event was lost (right-censored). Defaults all True if event_observed==None
• timeline (list, optional) – return the estimate at the values in timeline (positively increasing)
• label (string, optional) – a string to name the column of the estimate.
• alpha (float, optional) – the alpha value in the confidence intervals. Overrides the initializing alpha for this call to fit only.
• ci_labels (list, optional) – add custom column names to the generated confidence intervals as a length-2 list: [<lower-bound name>, <upper-bound name>]. Default: <label>_lower_<alpha>
• show_progress (boolean, optional) – since this is an iterative fitting algorithm, switching this to True will display some iteration details.
• entry (an array, or pd.Series, of length n) – relative time when a subject entered the study. This is useful for left-truncated (not left-censored) observations. If None, all members of the population entered study when they were “born”: time zero.
• weights (an array, or pd.Series, of length n) – integer weights per observation
• initial_point ((d,) numpy array, optional) – initialize the starting point of the iterative algorithm. Default is the zero vector.

Returns:

self with new properties like cumulative_hazard_, survival_function_

Return type:

self

fit_right_censoring(*args, **kwargs)

Alias for fit

See also

fit

hazard_at_times(times, label=None)

Return a Pandas series of the predicted hazard at specific times.

Parameters:

• times (iterable or float) – values to return the hazard at.
• label (string, optional) – Rename the series returned. Useful for plotting.

Returns:

Return type:

pd.Series

median_

Deprecated, use .median_survival_time_

Return the unique time point, t, such that S(t) = 0.5. This is the “half-life” of the population, and a robust summary statistic for the population, if it exists.

median_survival_time_

Return the unique time point, t, such that S(t) = 0.5. This is the “half-life” of the population, and a robust summary statistic for the population, if it exists.

percentile(p)

Return the unique time point, t, such that S(t) = p.

For known parametric models, this should be overwritten by something more accurate.

plot(**kwargs)

Produce a pretty-plot of the estimate.

plot_cumulative_density(**kwargs)

plot_cumulative_hazard(**kwargs)

plot_hazard(**kwargs)

plot_survival_function(**kwargs)

predict(times, interpolate=False)

Predict the {0} at certain point in time. Uses a linear interpolation if points in time are not in the index.

Parameters:

• times (scalar, or array) – a scalar or an array of times to predict the value of {0} at.
• interpolate (boolean, optional (default=False)) – for methods that produce a stepwise solution (Kaplan-Meier, Nelson-Aalen, etc), turning this to True will use an linear interpolation method to provide a more “smooth” answer.

Returns:

predictions

Return type:

a scalar if time is a scalar, a numpy array if time in an array.

print_summary(decimals=2, **kwargs)

Print summary statistics describing the fit, the coefficients, and the error bounds.

Parameters:

• decimals (int, optional (default=2)) – specify the number of decimal places to show
• kwargs – print additional metadata in the output (useful to provide model names, dataset names, etc.) when comparing multiple outputs.

subtract(other)

Subtract the {0} of two {1} objects.

Parameters:other (same object as self) Returns: Return type:DataFrame

summary

Summary statistics describing the fit.

Returns:df – Contains columns coef, exp(coef), se(coef), z, p, lower, upper Return type:pd.DataFrame

See also

print_summary

survival_function_at_times(times, label=None)

Return a Pandas series of the predicted survival value at specific times.

Parameters:

• times (iterable or float) – values to return the survival function at.
• label (string, optional) – Rename the series returned. Useful for plotting.

Returns:

Return type:

pd.Series

NelsonAalenFitter

class lifelines.fitters.nelson_aalen_fitter.NelsonAalenFitter(alpha=0.05, nelson_aalen_smoothing=True)

Bases: lifelines.fitters.UnivariateFitter

Class for fitting the Nelson-Aalen estimate for the cumulative hazard.

NelsonAalenFitter(alpha=0.05, nelson_aalen_smoothing=True)

Parameters:

• alpha (float, optional (default=0.05)) – The alpha value associated with the confidence intervals.
• nelson_aalen_smoothing (bool, optional) – If the event times are naturally discrete (like discrete years, minutes, etc.) then it is advisable to turn this parameter to False. See [1], pg.84.

Notes

[1] Aalen, O., Borgan, O., Gjessing, H., 2008. Survival and Event History Analysis

cumulative_hazard_

The estimated cumulative hazard (with custom timeline if provided)

Type:DataFrame

confidence_interval_

The lower and upper confidence intervals for the cumulative hazard

Type:DataFrame

durations

The durations provided

Type:array

event_observed

The event_observed variable provided

Type:array

timeline

The time line to use for plotting and indexing

Type:array

entry

The entry array provided, or None

Type:array or None

event_table

A summary of the life table

Type:DataFrame

conditional_time_to_event_

Return a DataFrame, with index equal to survival_function_’s index, that estimates the median duration remaining until the death event, given survival up until time t. For example, if an individual exists until age 1, their expected life remaining given they lived to time 1 might be 9 years.

Returns:conditional_time_to_ Return type:DataFrame

cumulative_density_at_times(times, label=None)

cumulative_hazard_at_times(times, label=None)

divide(other)

Divide the {0} of two {1} objects.

Parameters:other (same object as self) Returns: Return type:DataFrame

fit(durations, event_observed=None, timeline=None, entry=None, label='NA_estimate', alpha=None, ci_labels=None, weights=None)

Parameters:

• durations (an array, or pd.Series, of length n) – duration subject was observed for
• timeline (iterable) – return the best estimate at the values in timelines (positively increasing)
• event_observed (an array, or pd.Series, of length n) – True if the the death was observed, False if the event was lost (right-censored). Defaults all True if event_observed==None
• entry (an array, or pd.Series, of length n) – relative time when a subject entered the study. This is useful for left-truncated observations, i.e the birth event was not observed. If None, defaults to all 0 (all birth events observed.)
• label (string) – a string to name the column of the estimate.
• alpha (float) – the alpha value in the confidence intervals. Overrides the initializing alpha for this call to fit only.
• ci_labels (iterable) – add custom column names to the generated confidence intervals as a length-2 list: [<lower-bound name>, <upper-bound name>]. Default: <label>_lower_<1-alpha/2>
• weights (n array, or pd.Series, of length n) – if providing a weighted dataset. For example, instead of providing every subject as a single element of durations and event_observed, one could weigh subject differently.

Returns:

Return type:

self, with new properties like cumulative_hazard_.

fit_right_censoring(*args, **kwargs)

Alias for fit

See also

fit

hazard_at_times(times, label=None)

median_

Deprecated, use .median_survival_time_

Return the unique time point, t, such that S(t) = 0.5. This is the “half-life” of the population, and a robust summary statistic for the population, if it exists.

median_survival_time_

Return the unique time point, t, such that S(t) = 0.5. This is the “half-life” of the population, and a robust summary statistic for the population, if it exists.

percentile(p)

Return the unique time point, t, such that S(t) = p.

For known parametric models, this should be overwritten by something more accurate.

plot(**kwargs)

Plots a pretty figure of the model

Matplotlib plot arguments can be passed in inside the kwargs, plus

Parameters:

• show_censors (bool) – place markers at censorship events. Default: False

• censor_styles (dict) – If show_censors, this dictionary will be passed into the plot call.

• ci_alpha (float) – the transparency level of the confidence interval. Default: 0.3

• ci_force_lines (bool) – force the confidence intervals to be line plots (versus default shaded areas). Default: False

• ci_show (bool) – show confidence intervals. Default: True

• ci_legend (bool) – if ci_force_lines is True, this is a boolean flag to add the lines’ labels to the legend. Default: False

• at_risk_counts (bool) – show group sizes at time points. See function add_at_risk_counts for details. Default: False

• loc (slice) – specify a time-based subsection of the curves to plot, ex:

  >>> model.plot(loc=slice(0.,10.))
  

  will plot the time values between t=0. and t=10.

• iloc (slice) – specify a location-based subsection of the curves to plot, ex:

  >>> model.plot(iloc=slice(0,10))
  

  will plot the first 10 time points.

• invert_y_axis (bool) – boolean to invert the y-axis, useful to show cumulative graphs instead of survival graphs. (Deprecated, use plot_cumulative_density())

Returns:

a pyplot axis object

Return type:

ax

plot_cumulative_density(**kwargs)

plot_cumulative_hazard(**kwargs)

plot_hazard(**kwargs)

plot_survival_function(**kwargs)

predict(times, interpolate=False)

Predict the {0} at certain point in time. Uses a linear interpolation if points in time are not in the index.

Parameters:

• times (scalar, or array) – a scalar or an array of times to predict the value of {0} at.
• interpolate (boolean, optional (default=False)) – for methods that produce a stepwise solution (Kaplan-Meier, Nelson-Aalen, etc), turning this to True will use an linear interpolation method to provide a more “smooth” answer.

Returns:

predictions

Return type:

a scalar if time is a scalar, a numpy array if time in an array.

smoothed_hazard_(bandwidth)

Parameters:bandwidth (float) – the bandwith used in the Epanechnikov kernel. Returns:a DataFrame of the smoothed hazard Return type:DataFrame

smoothed_hazard_confidence_intervals_(bandwidth, hazard_=None)

Parameters:

• bandwidth (float) – the bandwidth to use in the Epanechnikov kernel. > 0
• hazard_ (numpy array) – a computed (n,) numpy array of estimated hazard rates. If none, uses smoothed_hazard_

subtract(other)

Subtract the {0} of two {1} objects.

Parameters:other (same object as self) Returns: Return type:DataFrame

survival_function_at_times(times, label=None)

PiecewiseExponentialFitter

class lifelines.fitters.piecewise_exponential_fitter.PiecewiseExponentialFitter(breakpoints, *args, **kwargs)

Bases: lifelines.fitters.KnownModelParametericUnivariateFitter

This class implements an Piecewise Exponential model for univariate data. The model has parameterized hazard rate:

\[\begin{split}h(t) = \begin{cases} 1/\lambda_0, & \text{if $t \le \tau_0$} \\ 1/\lambda_1 & \text{if $\tau_0 < t \le \tau_1$} \\ 1/\lambda_2 & \text{if $\tau_1 < t \le \tau_2$} \\ ... \end{cases}\end{split}\]

You specify the breakpoints, \(\tau_i\), and lifelines will find the optional values for the parameters.

After calling the .fit method, you have access to properties like: survival_function_, plot, cumulative_hazard_ A summary of the fit is available with the method print_summary()

Parameters:

• breakpoints (list) – a list of times when a new exponential model is constructed.
• alpha (float, optional (default=0.05)) – the level in the confidence intervals.

Important

The parameterization of this model changed in lifelines 0.19.1. Previously, the cumulative hazard looked like \(\lambda_i t\). The parameterization is now the reciprocal of \(\lambda_i\).

cumulative_hazard_

The estimated cumulative hazard (with custom timeline if provided)

Type:DataFrame

hazard_

The estimated hazard (with custom timeline if provided)

Type:DataFrame

survival_function_

The estimated survival function (with custom timeline if provided)

Type:DataFrame

cumumlative_density_

The estimated cumulative density function (with custom timeline if provided)

Type:DataFrame

variance_matrix_

The variance matrix of the coefficients

Type:numpy array

median_

The median time to event

Type:float

lambda_i_

The fitted parameter in the model, for i = 0, 1 … n-1 breakpoints

Type:float

durations

The durations provided

Type:array

event_observed

The event_observed variable provided

Type:array

timeline

The time line to use for plotting and indexing

Type:array

entry

The entry array provided, or None

Type:array or None

breakpoints

The provided breakpoints

Type:array

conditional_time_to_event_

Return a DataFrame, with index equal to survival_function_’s index, that estimates the median duration remaining until the death event, given survival up until time t. For example, if an individual exists until age 1, their expected life remaining given they lived to time 1 might be 9 years.

Returns:conditional_time_to_ Return type:DataFrame

confidence_interval_

The confidence interval of the cumulative hazard. This is an alias for confidence_interval_cumulative_hazard_.

confidence_interval_cumulative_density_

The lower and upper confidence intervals for the cumulative density

confidence_interval_cumulative_hazard_

The confidence interval of the cumulative hazard. This is an alias for confidence_interval_.

confidence_interval_hazard_

The confidence interval of the hazard.

confidence_interval_survival_function_

The lower and upper confidence intervals for the survival function

cumulative_density_at_times(times, label=None)

Return a Pandas series of the predicted cumulative density function (1-survival function) at specific times.

Parameters:

• times (iterable or float) – values to return the survival function at.
• label (string, optional) – Rename the series returned. Useful for plotting.

Returns:

Return type:

pd.Series

cumulative_hazard_at_times(times, label=None)

Return a Pandas series of the predicted cumulative hazard value at specific times.

Parameters:

• times (iterable or float) – values to return the cumulative hazard at.
• label (string, optional) – Rename the series returned. Useful for plotting.

Returns:

Return type:

pd.Series

divide(other)

Divide the {0} of two {1} objects.

Parameters:other (same object as self) Returns: Return type:DataFrame

event_table

fit(durations, event_observed=None, timeline=None, label=None, alpha=None, ci_labels=None, show_progress=False, entry=None, weights=None, left_censorship=False, initial_point=None)

Parameters:

• durations (an array, or pd.Series) – length n, duration subject was observed for
• event_observed (numpy array or pd.Series, optional) – length n, True if the the death was observed, False if the event was lost (right-censored). Defaults all True if event_observed==None
• timeline (list, optional) – return the estimate at the values in timeline (positively increasing)
• label (string, optional) – a string to name the column of the estimate.
• alpha (float, optional) – the alpha value in the confidence intervals. Overrides the initializing alpha for this call to fit only.
• ci_labels (list, optional) – add custom column names to the generated confidence intervals as a length-2 list: [<lower-bound name>, <upper-bound name>]. Default: <label>_lower_<alpha>
• show_progress (boolean, optional) – since this is an iterative fitting algorithm, switching this to True will display some iteration details.
• entry (an array, or pd.Series, of length n) – relative time when a subject entered the study. This is useful for left-truncated (not left-censored) observations. If None, all members of the population entered study when they were “born”: time zero.
• weights (an array, or pd.Series, of length n) – integer weights per observation
• initial_point ((d,) numpy array, optional) – initialize the starting point of the iterative algorithm. Default is the zero vector.

Returns:

self with new properties like cumulative_hazard_, survival_function_

Return type:

self

fit_interval_censoring(lower_bound, upper_bound, event_observed=None, timeline=None, label=None, alpha=None, ci_labels=None, show_progress=False, entry=None, weights=None, initial_point=None)

Fit the model to an interval censored dataset.

Parameters:

• lower_bound (an array, or pd.Series) – length n, the start of the period the subject experienced the event in.
• upper_bound (an array, or pd.Series) – length n, the end of the period the subject experienced the event in. If the value is equal to the corresponding value in lower_bound, then the individual’s event was observed (not censored).
• event_observed (numpy array or pd.Series, optional) – length n, if left optional, infer from lower_bound and upper_cound (if lower_bound==upper_bound then event observed, if lower_bound < upper_bound, then event censored)
• timeline (list, optional) – return the estimate at the values in timeline (positively increasing)
• label (string, optional) – a string to name the column of the estimate.
• alpha (float, optional) – the alpha value in the confidence intervals. Overrides the initializing alpha for this call to fit only.
• ci_labels (list, optional) – add custom column names to the generated confidence intervals as a length-2 list: [<lower-bound name>, <upper-bound name>]. Default: <label>_lower_<alpha>
• show_progress (boolean, optional) – since this is an iterative fitting algorithm, switching this to True will display some iteration details.
• entry (an array, or pd.Series, of length n) – relative time when a subject entered the study. This is useful for left-truncated (not left-censored) observations. If None, all members of the population entered study when they were “born”: time zero.
• weights (an array, or pd.Series, of length n) – integer weights per observation
• initial_point ((d,) numpy array, optional) – initialize the starting point of the iterative algorithm. Default is the zero vector.

Returns:

self with new properties like cumulative_hazard_, survival_function_

Return type:

self

fit_left_censoring(durations, event_observed=None, timeline=None, label=None, alpha=None, ci_labels=None, show_progress=False, entry=None, weights=None, initial_point=None)

Fit the model to a left-censored dataset

Parameters:

• durations (an array, or pd.Series) – length n, duration subject was observed for
• event_observed (numpy array or pd.Series, optional) – length n, True if the the death was observed, False if the event was lost (right-censored). Defaults all True if event_observed==None
• timeline (list, optional) – return the estimate at the values in timeline (positively increasing)
• label (string, optional) – a string to name the column of the estimate.
• alpha (float, optional) – the alpha value in the confidence intervals. Overrides the initializing alpha for this call to fit only.
• ci_labels (list, optional) – add custom column names to the generated confidence intervals as a length-2 list: [<lower-bound name>, <upper-bound name>]. Default: <label>_lower_<alpha>
• show_progress (boolean, optional) – since this is an iterative fitting algorithm, switching this to True will display some iteration details.
• entry (an array, or pd.Series, of length n) – relative time when a subject entered the study. This is useful for left-truncated (not left-censored) observations. If None, all members of the population entered study when they were “born”: time zero.
• weights (an array, or pd.Series, of length n) – integer weights per observation
• initial_point ((d,) numpy array, optional) – initialize the starting point of the iterative algorithm. Default is the zero vector.

Returns:

self with new properties like cumulative_hazard_, survival_function_

Return type:

self

fit_right_censoring(*args, **kwargs)

Alias for fit

See also

fit

hazard_at_times(times, label=None)

Return a Pandas series of the predicted hazard at specific times.

Parameters:

• times (iterable or float) – values to return the hazard at.
• label (string, optional) – Rename the series returned. Useful for plotting.

Returns:

Return type:

pd.Series

median_

Deprecated, use .median_survival_time_

Return the unique time point, t, such that S(t) = 0.5. This is the “half-life” of the population, and a robust summary statistic for the population, if it exists.

median_survival_time_

Return the unique time point, t, such that S(t) = 0.5. This is the “half-life” of the population, and a robust summary statistic for the population, if it exists.

percentile(p)

Return the unique time point, t, such that S(t) = p.

For known parametric models, this should be overwritten by something more accurate.

plot(**kwargs)

Produce a pretty-plot of the estimate.

plot_cumulative_density(**kwargs)

plot_cumulative_hazard(**kwargs)

plot_hazard(**kwargs)

plot_survival_function(**kwargs)

predict(times, interpolate=False)

Predict the {0} at certain point in time. Uses a linear interpolation if points in time are not in the index.

Parameters:

• times (scalar, or array) – a scalar or an array of times to predict the value of {0} at.
• interpolate (boolean, optional (default=False)) – for methods that produce a stepwise solution (Kaplan-Meier, Nelson-Aalen, etc), turning this to True will use an linear interpolation method to provide a more “smooth” answer.

Returns:

predictions

Return type:

a scalar if time is a scalar, a numpy array if time in an array.

print_summary(decimals=2, **kwargs)

Print summary statistics describing the fit, the coefficients, and the error bounds.

Parameters:

• decimals (int, optional (default=2)) – specify the number of decimal places to show
• kwargs – print additional metadata in the output (useful to provide model names, dataset names, etc.) when comparing multiple outputs.

subtract(other)

Subtract the {0} of two {1} objects.

Parameters:other (same object as self) Returns: Return type:DataFrame

summary

Summary statistics describing the fit.

Returns:df – Contains columns coef, exp(coef), se(coef), z, p, lower, upper Return type:pd.DataFrame

See also

print_summary

survival_function_at_times(times, label=None)

Return a Pandas series of the predicted survival value at specific times.

Parameters:

• times (iterable or float) – values to return the survival function at.
• label (string, optional) – Rename the series returned. Useful for plotting.

Returns:

Return type:

pd.Series

WeibullFitter

class lifelines.fitters.weibull_fitter.WeibullFitter(*args, **kwargs)

Bases: lifelines.fitters.KnownModelParametericUnivariateFitter

This class implements a Weibull model for univariate data. The model has parameterized form:

\[S(t) = \exp\left(-\left(\frac{t}{\lambda}\right)^\rho\right), \lambda > 0, \rho > 0,\]

which implies the cumulative hazard rate is

\[H(t) = \left(\frac{t}{\lambda}\right)^\rho,\]

and the hazard rate is:

\[h(t) = \frac{\rho}{\lambda}\left(\frac{t}{\lambda}\right)^{\rho-1}\]

After calling the .fit method, you have access to properties like: cumulative_hazard_, survival_function_, lambda_ and rho_. A summary of the fit is available with the method print_summary().

Parameters:alpha (float, optional (default=0.05)) – the level in the confidence intervals.

Important

The parameterization of this model changed in lifelines 0.19.0. Previously, the cumulative hazard looked like \((\lambda t)^\rho\). The parameterization is now the reciprocal of \(\lambda\).

Examples

>>> from lifelines import WeibullFitter
>>> from lifelines.datasets import load_waltons
>>> waltons = load_waltons()
>>> wbf = WeibullFitter()
>>> wbf.fit(waltons['T'], waltons['E'])
>>> wbf.plot()
>>> print(wbf.lambda_)

cumulative_hazard_

The estimated cumulative hazard (with custom timeline if provided)

Type:DataFrame

hazard_

The estimated hazard (with custom timeline if provided)

Type:DataFrame

survival_function_

The estimated survival function (with custom timeline if provided)

Type:DataFrame

cumumlative_density_

The estimated cumulative density function (with custom timeline if provided)

Type:DataFrame

variance_matrix_

The variance matrix of the coefficients

Type:numpy array

median_

The median time to event

Type:float

lambda_

The fitted parameter in the model

Type:float

rho_

The fitted parameter in the model

Type:float

durations

The durations provided

Type:array

event_observed

The event_observed variable provided

Type:array

timeline

The time line to use for plotting and indexing

Type:array

entry

The entry array provided, or None

Type:array or None

See also

Looking

conditional_time_to_event_

Return a DataFrame, with index equal to survival_function_’s index, that estimates the median duration remaining until the death event, given survival up until time t. For example, if an individual exists until age 1, their expected life remaining given they lived to time 1 might be 9 years.

Returns:conditional_time_to_ Return type:DataFrame

confidence_interval_

The confidence interval of the cumulative hazard. This is an alias for confidence_interval_cumulative_hazard_.

confidence_interval_cumulative_density_

The lower and upper confidence intervals for the cumulative density

confidence_interval_cumulative_hazard_

The confidence interval of the cumulative hazard. This is an alias for confidence_interval_.

confidence_interval_hazard_

The confidence interval of the hazard.

confidence_interval_survival_function_

The lower and upper confidence intervals for the survival function

cumulative_density_at_times(times, label=None)

Return a Pandas series of the predicted cumulative density function (1-survival function) at specific times.

Parameters:

• times (iterable or float) – values to return the survival function at.
• label (string, optional) – Rename the series returned. Useful for plotting.

Returns:

Return type:

pd.Series

cumulative_hazard_at_times(times, label=None)

Return a Pandas series of the predicted cumulative hazard value at specific times.

Parameters:

• times (iterable or float) – values to return the cumulative hazard at.
• label (string, optional) – Rename the series returned. Useful for plotting.

Returns:

Return type:

pd.Series

divide(other)

Divide the {0} of two {1} objects.

Parameters:other (same object as self) Returns: Return type:DataFrame

event_table

fit(durations, event_observed=None, timeline=None, label=None, alpha=None, ci_labels=None, show_progress=False, entry=None, weights=None, left_censorship=False, initial_point=None)

Parameters:

• durations (an array, or pd.Series) – length n, duration subject was observed for
• event_observed (numpy array or pd.Series, optional) – length n, True if the the death was observed, False if the event was lost (right-censored). Defaults all True if event_observed==None
• timeline (list, optional) – return the estimate at the values in timeline (positively increasing)
• label (string, optional) – a string to name the column of the estimate.
• alpha (float, optional) – the alpha value in the confidence intervals. Overrides the initializing alpha for this call to fit only.
• ci_labels (list, optional) – add custom column names to the generated confidence intervals as a length-2 list: [<lower-bound name>, <upper-bound name>]. Default: <label>_lower_<alpha>
• show_progress (boolean, optional) – since this is an iterative fitting algorithm, switching this to True will display some iteration details.
• entry (an array, or pd.Series, of length n) – relative time when a subject entered the study. This is useful for left-truncated (not left-censored) observations. If None, all members of the population entered study when they were “born”: time zero.
• weights (an array, or pd.Series, of length n) – integer weights per observation
• initial_point ((d,) numpy array, optional) – initialize the starting point of the iterative algorithm. Default is the zero vector.

Returns:

self with new properties like cumulative_hazard_, survival_function_

Return type:

self

fit_interval_censoring(lower_bound, upper_bound, event_observed=None, timeline=None, label=None, alpha=None, ci_labels=None, show_progress=False, entry=None, weights=None, initial_point=None)

Fit the model to an interval censored dataset.

Parameters:

• lower_bound (an array, or pd.Series) – length n, the start of the period the subject experienced the event in.
• upper_bound (an array, or pd.Series) – length n, the end of the period the subject experienced the event in. If the value is equal to the corresponding value in lower_bound, then the individual’s event was observed (not censored).
• event_observed (numpy array or pd.Series, optional) – length n, if left optional, infer from lower_bound and upper_cound (if lower_bound==upper_bound then event observed, if lower_bound < upper_bound, then event censored)
• timeline (list, optional) – return the estimate at the values in timeline (positively increasing)
• label (string, optional) – a string to name the column of the estimate.
• alpha (float, optional) – the alpha value in the confidence intervals. Overrides the initializing alpha for this call to fit only.
• ci_labels (list, optional) – add custom column names to the generated confidence intervals as a length-2 list: [<lower-bound name>, <upper-bound name>]. Default: <label>_lower_<alpha>
• show_progress (boolean, optional) – since this is an iterative fitting algorithm, switching this to True will display some iteration details.
• entry (an array, or pd.Series, of length n) – relative time when a subject entered the study. This is useful for left-truncated (not left-censored) observations. If None, all members of the population entered study when they were “born”: time zero.
• weights (an array, or pd.Series, of length n) – integer weights per observation
• initial_point ((d,) numpy array, optional) – initialize the starting point of the iterative algorithm. Default is the zero vector.

Returns:

self with new properties like cumulative_hazard_, survival_function_

Return type:

self

fit_left_censoring(durations, event_observed=None, timeline=None, label=None, alpha=None, ci_labels=None, show_progress=False, entry=None, weights=None, initial_point=None)

Fit the model to a left-censored dataset

Parameters:

• durations (an array, or pd.Series) – length n, duration subject was observed for
• event_observed (numpy array or pd.Series, optional) – length n, True if the the death was observed, False if the event was lost (right-censored). Defaults all True if event_observed==None
• timeline (list, optional) – return the estimate at the values in timeline (positively increasing)
• label (string, optional) – a string to name the column of the estimate.
• alpha (float, optional) – the alpha value in the confidence intervals. Overrides the initializing alpha for this call to fit only.
• ci_labels (list, optional) – add custom column names to the generated confidence intervals as a length-2 list: [<lower-bound name>, <upper-bound name>]. Default: <label>_lower_<alpha>
• show_progress (boolean, optional) – since this is an iterative fitting algorithm, switching this to True will display some iteration details.
• entry (an array, or pd.Series, of length n) – relative time when a subject entered the study. This is useful for left-truncated (not left-censored) observations. If None, all members of the population entered study when they were “born”: time zero.
• weights (an array, or pd.Series, of length n) – integer weights per observation
• initial_point ((d,) numpy array, optional) – initialize the starting point of the iterative algorithm. Default is the zero vector.

Returns:

self with new properties like cumulative_hazard_, survival_function_

Return type:

self

fit_right_censoring(*args, **kwargs)

Alias for fit

See also

fit

hazard_at_times(times, label=None)

Return a Pandas series of the predicted hazard at specific times.

Parameters:

• times (iterable or float) – values to return the hazard at.
• label (string, optional) – Rename the series returned. Useful for plotting.

Returns:

Return type:

pd.Series

median_

Deprecated, use .median_survival_time_

Return the unique time point, t, such that S(t) = 0.5. This is the “half-life” of the population, and a robust summary statistic for the population, if it exists.

median_survival_time_

Return the unique time point, t, such that S(t) = 0.5. This is the “half-life” of the population, and a robust summary statistic for the population, if it exists.

percentile(p)

Return the unique time point, t, such that S(t) = p.

For known parametric models, this should be overwritten by something more accurate.

plot(**kwargs)

Produce a pretty-plot of the estimate.

plot_cumulative_density(**kwargs)

plot_cumulative_hazard(**kwargs)

plot_hazard(**kwargs)

plot_survival_function(**kwargs)

predict(times, interpolate=False)

Predict the {0} at certain point in time. Uses a linear interpolation if points in time are not in the index.

Parameters:

• times (scalar, or array) – a scalar or an array of times to predict the value of {0} at.
• interpolate (boolean, optional (default=False)) – for methods that produce a stepwise solution (Kaplan-Meier, Nelson-Aalen, etc), turning this to True will use an linear interpolation method to provide a more “smooth” answer.

Returns:

predictions

Return type:

a scalar if time is a scalar, a numpy array if time in an array.

print_summary(decimals=2, **kwargs)

Print summary statistics describing the fit, the coefficients, and the error bounds.

Parameters:

• decimals (int, optional (default=2)) – specify the number of decimal places to show
• kwargs – print additional metadata in the output (useful to provide model names, dataset names, etc.) when comparing multiple outputs.

subtract(other)

Subtract the {0} of two {1} objects.

Parameters:other (same object as self) Returns: Return type:DataFrame

summary

Summary statistics describing the fit.

Returns:df – Contains columns coef, exp(coef), se(coef), z, p, lower, upper Return type:pd.DataFrame

See also

print_summary

survival_function_at_times(times, label=None)

Return a Pandas series of the predicted survival value at specific times.

Parameters:

• times (iterable or float) – values to return the survival function at.
• label (string, optional) – Rename the series returned. Useful for plotting.

Returns:

Return type:

pd.Series

GeneralizedGammaFitter

class lifelines.fitters.generalized_gamma_fitter.GeneralizedGammaFitter(*args, **kwargs)

Bases: lifelines.fitters.KnownModelParametericUnivariateFitter

This class implements a Generalized Gamma model for univariate data. The model has parameterized form:

The survival function is:

\[\begin{split}S(t)=\left\{ \begin{array}{} 1-{{\Gamma}_{RL}}\left( \tfrac{1}{{{\lambda }^{2}}};\tfrac{{{e}^{\lambda \left( \tfrac{\text{ln}(t)-\mu }{\sigma } \right)}}}{{{\lambda }^{2}}} \right)\text{ if }\lambda >0 \\ {{\Gamma}_{RL}}\left( \tfrac{1}{{{\lambda }^{2}}};\tfrac{{{e}^{\lambda \left( \tfrac{\text{ln}(t)-\mu }{\sigma } \right)}}}{{{\lambda }^{2}}} \right)\text{ if }\lambda < 0 \\ \end{array} \right.\,\!\end{split}\]

where \(\Gamma_{RL}\) is the regularized lower incomplete Gamma function.

This model has the Exponential, Weibull, Gamma and Log-Normal as sub-models, and thus can be used as a way to test which model to use:

1. When \(\lambda = 1\) and \(\sigma = 1\), then the data is Exponential.
2. When \(\lambda = 1\) then the data is Weibull.
3. When \(\sigma = \lambda\) then the data is Gamma.
4. When \(\lambda = 0\) then the data is Log-Normal.
5. When \(\lambda = -1\) then the data is Inverse-Weibull.
6. When \(-\sigma = \lambda\) then the data is Inverse-Gamma.

After calling the .fit method, you have access to properties like: cumulative_hazard_, survival_function_, A summary of the fit is available with the method print_summary().

Important

The parameterization implemented has \(\log\sigma\), thus there is a ln_sigma_ in the output. Exponentiate this parameter to recover \(\sigma\).

Important

This model is experimental. It’s API may change in the future. Also, it’s convergence is not very stable.

Parameters:alpha (float, optional (default=0.05)) – the level in the confidence intervals.

Examples

>>> from lifelines import GeneralizedGammaFitter
>>> from lifelines.datasets import load_waltons
>>> waltons = load_waltons()
>>> ggf = GeneralizedGammaFitter()
>>> ggf.fit(waltons['T'], waltons['E'])
>>> ggf.plot()
>>> ggf.summary

cumulative_hazard_

The estimated cumulative hazard (with custom timeline if provided)

Type:DataFrame

hazard_

The estimated hazard (with custom timeline if provided)

Type:DataFrame

survival_function_

The estimated survival function (with custom timeline if provided)

Type:DataFrame

cumumlative_density_

The estimated cumulative density function (with custom timeline if provided)

Type:DataFrame

variance_matrix_

The variance matrix of the coefficients

Type:numpy array

median_

The median time to event

Type:float

lambda_

The fitted parameter in the model

Type:float

rho_

The fitted parameter in the model

Type:float

alpha_

The fitted parameter in the model

Type:float

durations

The durations provided

Type:array

event_observed

The event_observed variable provided

Type:array

timeline

The time line to use for plotting and indexing

Type:array

entry

The entry array provided, or None

Type:array or None

conditional_time_to_event_

Return a DataFrame, with index equal to survival_function_’s index, that estimates the median duration remaining until the death event, given survival up until time t. For example, if an individual exists until age 1, their expected life remaining given they lived to time 1 might be 9 years.

Returns:conditional_time_to_ Return type:DataFrame

confidence_interval_

The confidence interval of the cumulative hazard. This is an alias for confidence_interval_cumulative_hazard_.

confidence_interval_cumulative_density_

The lower and upper confidence intervals for the cumulative density

confidence_interval_cumulative_hazard_

The confidence interval of the cumulative hazard. This is an alias for confidence_interval_.

confidence_interval_hazard_

The confidence interval of the hazard.

confidence_interval_survival_function_

The lower and upper confidence intervals for the survival function

cumulative_density_at_times(times, label=None)

Return a Pandas series of the predicted cumulative density function (1-survival function) at specific times.

Parameters:

• times (iterable or float) – values to return the survival function at.
• label (string, optional) – Rename the series returned. Useful for plotting.

Returns:

Return type:

pd.Series

cumulative_hazard_at_times(times, label=None)

Return a Pandas series of the predicted cumulative hazard value at specific times.

Parameters:

• times (iterable or float) – values to return the cumulative hazard at.
• label (string, optional) – Rename the series returned. Useful for plotting.

Returns:

Return type:

pd.Series

divide(other)

Divide the {0} of two {1} objects.

Parameters:other (same object as self) Returns: Return type:DataFrame

event_table

fit(durations, event_observed=None, timeline=None, label=None, alpha=None, ci_labels=None, show_progress=False, entry=None, weights=None, left_censorship=False, initial_point=None)

Parameters:

• durations (an array, or pd.Series) – length n, duration subject was observed for
• event_observed (numpy array or pd.Series, optional) – length n, True if the the death was observed, False if the event was lost (right-censored). Defaults all True if event_observed==None
• timeline (list, optional) – return the estimate at the values in timeline (positively increasing)
• label (string, optional) – a string to name the column of the estimate.
• alpha (float, optional) – the alpha value in the confidence intervals. Overrides the initializing alpha for this call to fit only.
• ci_labels (list, optional) – add custom column names to the generated confidence intervals as a length-2 list: [<lower-bound name>, <upper-bound name>]. Default: <label>_lower_<alpha>
• show_progress (boolean, optional) – since this is an iterative fitting algorithm, switching this to True will display some iteration details.
• entry (an array, or pd.Series, of length n) – relative time when a subject entered the study. This is useful for left-truncated (not left-censored) observations. If None, all members of the population entered study when they were “born”: time zero.
• weights (an array, or pd.Series, of length n) – integer weights per observation
• initial_point ((d,) numpy array, optional) – initialize the starting point of the iterative algorithm. Default is the zero vector.

Returns:

self with new properties like cumulative_hazard_, survival_function_

Return type:

self

fit_interval_censoring(lower_bound, upper_bound, event_observed=None, timeline=None, label=None, alpha=None, ci_labels=None, show_progress=False, entry=None, weights=None, initial_point=None)

Fit the model to an interval censored dataset.

Parameters:

• lower_bound (an array, or pd.Series) – length n, the start of the period the subject experienced the event in.
• upper_bound (an array, or pd.Series) – length n, the end of the period the subject experienced the event in. If the value is equal to the corresponding value in lower_bound, then the individual’s event was observed (not censored).
• event_observed (numpy array or pd.Series, optional) – length n, if left optional, infer from lower_bound and upper_cound (if lower_bound==upper_bound then event observed, if lower_bound < upper_bound, then event censored)
• timeline (list, optional) – return the estimate at the values in timeline (positively increasing)
• label (string, optional) – a string to name the column of the estimate.
• alpha (float, optional) – the alpha value in the confidence intervals. Overrides the initializing alpha for this call to fit only.
• ci_labels (list, optional) – add custom column names to the generated confidence intervals as a length-2 list: [<lower-bound name>, <upper-bound name>]. Default: <label>_lower_<alpha>
• show_progress (boolean, optional) – since this is an iterative fitting algorithm, switching this to True will display some iteration details.
• entry (an array, or pd.Series, of length n) – relative time when a subject entered the study. This is useful for left-truncated (not left-censored) observations. If None, all members of the population entered study when they were “born”: time zero.
• weights (an array, or pd.Series, of length n) – integer weights per observation
• initial_point ((d,) numpy array, optional) – initialize the starting point of the iterative algorithm. Default is the zero vector.

Returns:

self with new properties like cumulative_hazard_, survival_function_

Return type:

self

fit_left_censoring(durations, event_observed=None, timeline=None, label=None, alpha=None, ci_labels=None, show_progress=False, entry=None, weights=None, initial_point=None)

Fit the model to a left-censored dataset

Parameters:

• durations (an array, or pd.Series) – length n, duration subject was observed for
• event_observed (numpy array or pd.Series, optional) – length n, True if the the death was observed, False if the event was lost (right-censored). Defaults all True if event_observed==None
• timeline (list, optional) – return the estimate at the values in timeline (positively increasing)
• label (string, optional) – a string to name the column of the estimate.
• alpha (float, optional) – the alpha value in the confidence intervals. Overrides the initializing alpha for this call to fit only.
• ci_labels (list, optional) – add custom column names to the generated confidence intervals as a length-2 list: [<lower-bound name>, <upper-bound name>]. Default: <label>_lower_<alpha>
• show_progress (boolean, optional) – since this is an iterative fitting algorithm, switching this to True will display some iteration details.
• entry (an array, or pd.Series, of length n) – relative time when a subject entered the study. This is useful for left-truncated (not left-censored) observations. If None, all members of the population entered study when they were “born”: time zero.
• weights (an array, or pd.Series, of length n) – integer weights per observation
• initial_point ((d,) numpy array, optional) – initialize the starting point of the iterative algorithm. Default is the zero vector.

Returns:

self with new properties like cumulative_hazard_, survival_function_

Return type:

self

fit_right_censoring(*args, **kwargs)

Alias for fit

See also

fit

hazard_at_times(times, label=None)

Return a Pandas series of the predicted hazard at specific times.

Parameters:

• times (iterable or float) – values to return the hazard at.
• label (string, optional) – Rename the series returned. Useful for plotting.

Returns:

Return type:

pd.Series

median_

Deprecated, use .median_survival_time_

Return the unique time point, t, such that S(t) = 0.5. This is the “half-life” of the population, and a robust summary statistic for the population, if it exists.

median_survival_time_

Return the unique time point, t, such that S(t) = 0.5. This is the “half-life” of the population, and a robust summary statistic for the population, if it exists.

percentile(p)

Return the unique time point, t, such that S(t) = p.

For known parametric models, this should be overwritten by something more accurate.

plot(**kwargs)

Produce a pretty-plot of the estimate.

plot_cumulative_density(**kwargs)

plot_cumulative_hazard(**kwargs)

plot_hazard(**kwargs)

plot_survival_function(**kwargs)

predict(times, interpolate=False)

Predict the {0} at certain point in time. Uses a linear interpolation if points in time are not in the index.

Parameters:

• times (scalar, or array) – a scalar or an array of times to predict the value of {0} at.
• interpolate (boolean, optional (default=False)) – for methods that produce a stepwise solution (Kaplan-Meier, Nelson-Aalen, etc), turning this to True will use an linear interpolation method to provide a more “smooth” answer.

Returns:

predictions

Return type:

a scalar if time is a scalar, a numpy array if time in an array.

print_summary(decimals=2, **kwargs)

Print summary statistics describing the fit, the coefficients, and the error bounds.

Parameters:

• decimals (int, optional (default=2)) – specify the number of decimal places to show
• kwargs – print additional metadata in the output (useful to provide model names, dataset names, etc.) when comparing multiple outputs.

subtract(other)

Subtract the {0} of two {1} objects.

Parameters:other (same object as self) Returns: Return type:DataFrame

summary

Summary statistics describing the fit.

Returns:df – Contains columns coef, exp(coef), se(coef), z, p, lower, upper Return type:pd.DataFrame

See also

print_summary

survival_function_at_times(times, label=None)

Return a Pandas series of the predicted survival value at specific times.

Parameters:

• times (iterable or float) – values to return the survival function at.
• label (string, optional) – Rename the series returned. Useful for plotting.

Returns:

Return type:

pd.Series

Regression Models

AalenAdditiveFitter

class lifelines.fitters.aalen_additive_fitter.AalenAdditiveFitter(fit_intercept=True, alpha=0.05, coef_penalizer=0.0, smoothing_penalizer=0.0)

Bases: lifelines.fitters.BaseFitter

This class fits the regression model:

\[h(t|x) = b_0(t) + b_1(t) x_1 + ... + b_N(t) x_N\]

that is, the hazard rate is a linear function of the covariates with time-varying coefficients. This implementation assumes non-time-varying covariates, see TODO: name

Note

This class was rewritten in lifelines 0.17.0 to focus solely on static datasets. There is no guarantee of backwards compatibility.

Parameters:

• fit_intercept (bool, optional (default: True)) – If False, do not attach an intercept (column of ones) to the covariate matrix. The intercept, \(b_0(t)\) acts as a baseline hazard.
• alpha (float, optional (default=0.05)) – the level in the confidence intervals.
• coef_penalizer (float, optional (default: 0)) – Attach a L2 penalizer to the size of the coefficients during regression. This improves stability of the estimates and controls for high correlation between covariates. For example, this shrinks the absolute value of \(c_{i,t}\).
• smoothing_penalizer (float, optional (default: 0)) – Attach a L2 penalizer to difference between adjacent (over time) coefficients. For example, this shrinks the absolute value of \(c_{i,t} - c_{i,t+1}\).

cumulative_hazards_

The estimated cumulative hazard

Type:DataFrame

hazards_

The estimated hazards

Type:DataFrame

confidence_intervals_

The lower and upper confidence intervals for the cumulative hazard

Type:DataFrame

durations

The durations provided

Type:array

event_observed

The event_observed variable provided

Type:array

weights

The event_observed variable provided

Type:array

fit(df, duration_col, event_col=None, weights_col=None, show_progress=False)

Parameters:

Fit the Aalen Additive model to a dataset.

Parameters:

• df (DataFrame) – a Pandas DataFrame with necessary columns duration_col and event_col (see below), covariates columns, and special columns (weights). duration_col refers to the lifetimes of the subjects. event_col refers to whether the ‘death’ events was observed: 1 if observed, 0 else (censored).
• duration_col (string) – the name of the column in DataFrame that contains the subjects’ lifetimes.
• event_col (string, optional) – the name of the column in DataFrame that contains the subjects’ death observation. If left as None, assume all individuals are uncensored.
• weights_col (string, optional) – an optional column in the DataFrame, df, that denotes the weight per subject. This column is expelled and not used as a covariate, but as a weight in the final regression. Default weight is 1. This can be used for case-weights. For example, a weight of 2 means there were two subjects with identical observations. This can be used for sampling weights.
• show_progress (boolean, optional (default=False)) – Since the fitter is iterative, show iteration number.

Returns:

self – self with additional new properties: cumulative_hazards_, etc.

Return type:

AalenAdditiveFitter

Examples

>>> from lifelines import AalenAdditiveFitter
>>>
>>> df = pd.DataFrame({
>>>     'T': [5, 3, 9, 8, 7, 4, 4, 3, 2, 5, 6, 7],
>>>     'E': [1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0],
>>>     'var': [0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2],
>>>     'age': [4, 3, 9, 8, 7, 4, 4, 3, 2, 5, 6, 7],
>>> })
>>>
>>> aaf = AalenAdditiveFitter()
>>> aaf.fit(df, 'T', 'E')
>>> aaf.predict_median(df)
>>> aaf.print_summary()

fit_right_censoring(*args, **kwargs)

Alias for fit

See also

fit

plot(columns=None, loc=None, iloc=None, **kwargs)

” A wrapper around plotting. Matplotlib plot arguments can be passed in, plus:

Parameters:

• columns (string or list-like, optional) – If not empty, plot a subset of columns from the cumulative_hazards_. Default all.

• loc

• iloc (slice, optional) –

  specify a location-based subsection of the curves to plot, ex:

  .plot(iloc=slice(0,10)) will plot the first 10 time points.

predict_cumulative_hazard(X)

Returns the hazard rates for the individuals

Parameters:X (a (n,d) covariate numpy array or DataFrame. If a DataFrame, columns) – can be in any order. If a numpy array, columns must be in the same order as the training data.

predict_expectation(X)

Compute the expected lifetime, E[T], using covariates X.

Parameters:

• X (a (n,d) covariate numpy array or DataFrame) – If a DataFrame, columns can be in any order. If a numpy array, columns must be in the same order as the training data.
• Returns the expected lifetimes for the individuals

predict_median(X)

Parameters:

• X (a (n,d) covariate numpy array or DataFrame) – If a DataFrame, columns can be in any order. If a numpy array, columns must be in the same order as the training data.
• Returns the median lifetimes for the individuals

predict_percentile(X, p=0.5)

Returns the median lifetimes for the individuals. http://stats.stackexchange.com/questions/102986/percentile-loss-functions

Parameters:

• X (a (n,d) covariate numpy array or DataFrame) – If a DataFrame, columns can be in any order. If a numpy array, columns must be in the same order as the training data.
• p (float) – default: 0.5

predict_survival_function(X, times=None)

Returns the survival functions for the individuals

Parameters:

• X (a (n,d) covariate numpy array or DataFrame) – If a DataFrame, columns can be in any order. If a numpy array, columns must be in the same order as the training data.
• times – Not implemented yet

print_summary(decimals=2, **kwargs)

Print summary statistics describing the fit, the coefficients, and the error bounds.

Parameters:

• decimals (int, optional (default=2)) – specify the number of decimal places to show
• kwargs – print additional meta data in the output (useful to provide model names, dataset names, etc.) when comparing multiple outputs.

score_

The concordance score (also known as the c-index) of the fit. The c-index is a generalization of the ROC AUC to survival data, including censorships.

For this purpose, the score_ is a measure of the predictive accuracy of the fitted model onto the training dataset. It’s analogous to the R^2 in linear models.

smoothed_hazards_(bandwidth=1)

Using the epanechnikov kernel to smooth the hazard function, with sigma/bandwidth

summary

Summary statistics describing the fit.

Returns:df Return type:DataFrame

CoxTimeVaryingFitter

class lifelines.fitters.cox_time_varying_fitter.CoxTimeVaryingFitter(alpha=0.05, penalizer=0.0, strata=None)

Bases: lifelines.fitters.BaseFitter

This class implements fitting Cox’s time-varying proportional hazard model:

\[h(t|x(t)) = h_0(t)\exp(x(t)'\beta)\]

Parameters:

• alpha (float, optional (default=0.05)) – the level in the confidence intervals.
• penalizer (float, optional) – the coefficient of an L2 penalizer in the regression

params_

The estimated coefficients. Changed in version 0.22.0: use to be .hazards_

Type:Series

hazard_ratios_

The exp(coefficients)

Type:Series

confidence_intervals_

The lower and upper confidence intervals for the hazard coefficients

Type:DataFrame

event_observed

The event_observed variable provided

Type:Series

weights

The event_observed variable provided

Type:Series

variance_matrix_

The variance matrix of the coefficients

Type:numpy array

strata

the strata provided

Type:list

standard_errors_

the standard errors of the estimates

Type:Series

baseline_cumulative_hazard_

Type:DataFrame

baseline_survival_

Type:DataFrame

fit(df, id_col, event_col, start_col='start', stop_col='stop', weights_col=None, show_progress=False, step_size=None, robust=False, strata=None, initial_point=None)

Fit the Cox Proportional Hazard model to a time varying dataset. Tied survival times are handled using Efron’s tie-method.

Parameters:

• df (DataFrame) – a Pandas DataFrame with necessary columns duration_col and event_col, plus other covariates. duration_col refers to the lifetimes of the subjects. event_col refers to whether the ‘death’ events was observed: 1 if observed, 0 else (censored).
• id_col (string) – A subject could have multiple rows in the DataFrame. This column contains the unique identifier per subject.
• event_col (string) – the column in DataFrame that contains the subjects’ death observation. If left as None, assume all individuals are non-censored.
• start_col (string) – the column that contains the start of a subject’s time period.
• stop_col (string) – the column that contains the end of a subject’s time period.
• weights_col (string, optional) – the column that contains (possibly time-varying) weight of each subject-period row.
• show_progress (since the fitter is iterative, show convergence) – diagnostics.
• robust (boolean, optional (default: True)) – Compute the robust errors using the Huber sandwich estimator, aka Wei-Lin estimate. This does not handle ties, so if there are high number of ties, results may significantly differ. See “The Robust Inference for the Cox Proportional Hazards Model”, Journal of the American Statistical Association, Vol. 84, No. 408 (Dec., 1989), pp. 1074- 1078
• step_size (float, optional) – set an initial step size for the fitting algorithm.
• strata (list or string, optional) – specify a column or list of columns n to use in stratification. This is useful if a categorical covariate does not obey the proportional hazard assumption. This is used similar to the strata expression in R. See http://courses.washington.edu/b515/l17.pdf.
• initial_point ((d,) numpy array, optional) – initialize the starting point of the iterative algorithm. Default is the zero vector.

Returns:

self – self, with additional properties like hazards_ and print_summary

Return type:

CoxTimeVaryingFitter

fit_right_censoring(*args, **kwargs)

Alias for fit

See also

fit

log_likelihood_ratio_test()

This function computes the likelihood ratio test for the Cox model. We compare the existing model (with all the covariates) to the trivial model of no covariates.

Conveniently, we can actually use CoxPHFitter class to do most of the work.

plot(columns=None, **errorbar_kwargs)

Produces a visual representation of the coefficients, including their standard errors and magnitudes.

Parameters:

• columns (list, optional) – specify a subset of the columns to plot
• errorbar_kwargs – pass in additional plotting commands to matplotlib errorbar command

Returns:

ax – the matplotlib axis that be edited.

Return type:

matplotlib axis

predict_log_partial_hazard(X)

This is equivalent to R’s linear.predictors. Returns the log of the partial hazard for the individuals, partial since the baseline hazard is not included. Equal to :math:`(x - bar{x})’beta `

Parameters:X (numpy array or DataFrame) – a (n,d) covariate numpy array or DataFrame. If a DataFrame, columns can be in any order. If a numpy array, columns must be in the same order as the training data. Returns: Return type:DataFrame

Note

If X is a DataFrame, the order of the columns do not matter. But if X is an array, then the column ordering is assumed to be the same as the training dataset.

predict_partial_hazard(X)

Returns the partial hazard for the individuals, partial since the baseline hazard is not included. Equal to \(\exp{(x - \bar{x})'\beta }\)

Parameters:X (numpy array or DataFrame) – a (n,d) covariate numpy array or DataFrame. If a DataFrame, columns can be in any order. If a numpy array, columns must be in the same order as the training data. Returns: Return type:DataFrame

Note

If X is a DataFrame, the order of the columns do not matter. But if X is an array, then the column ordering is assumed to be the same as the training dataset.

print_summary(decimals=2, **kwargs)

Print summary statistics describing the fit, the coefficients, and the error bounds.

Parameters:

• decimals (int, optional (default=2)) – specify the number of decimal places to show
• kwargs – print additional meta data in the output (useful to provide model names, dataset names, etc.) when comparing multiple outputs.

summary

Summary statistics describing the fit. Set alpha property in the object before calling.

Returns:df – Contains columns coef, np.exp(coef), se(coef), z, p, lower, upper Return type:DataFrame

CoxPHFitter

class lifelines.fitters.coxph_fitter.CoxPHFitter(alpha=0.05, tie_method='Efron', penalizer=0.0, strata=None)

Bases: lifelines.fitters.BaseFitter

This class implements fitting Cox’s proportional hazard model:

\[h(t|x) = h_0(t) \exp((x - \overline{x})' \beta)\]

Parameters:

• alpha (float, optional (default=0.05)) – the level in the confidence intervals.
• tie_method (string, optional) – specify how the fitter should deal with ties. Currently only ‘Efron’ is available.
• penalizer (float, optional (default=0.0)) – Attach an L2 penalizer to the size of the coefficients during regression. This improves stability of the estimates and controls for high correlation between covariates. For example, this shrinks the absolute value of \(\beta_i\). The penalty is \(\frac{1}{2} \text{penalizer} ||\beta||^2\).
• strata (list, optional) – specify a list of columns to use in stratification. This is useful if a categorical covariate does not obey the proportional hazard assumption. This is used similar to the strata expression in R. See http://courses.washington.edu/b515/l17.pdf.

Examples

>>> from lifelines.datasets import load_rossi
>>> from lifelines import CoxPHFitter
>>> rossi = load_rossi()
>>> cph = CoxPHFitter()
>>> cph.fit(rossi, 'week', 'arrest')
>>> cph.print_summary()

params_

The estimated coefficients. Changed in version 0.22.0: use to be .hazards_

Type:Series

hazard_ratios_

The exp(coefficients)

Type:Series

confidence_intervals_

The lower and upper confidence intervals for the hazard coefficients

Type:DataFrame

durations

The durations provided

Type:Series

event_observed

The event_observed variable provided

Type:Series

weights

The event_observed variable provided

Type:Series

variance_matrix_

The variance matrix of the coefficients

Type:numpy array

strata

the strata provided

Type:list

standard_errors_

the standard errors of the estimates

Type:Series

score_

the concordance index of the model.

Type:float

baseline_hazard_

Type:DataFrame

baseline_cumulative_hazard_

Type:DataFrame

baseline_survival_

Type:DataFrame

check_assumptions(training_df, advice=True, show_plots=False, p_value_threshold=0.01, plot_n_bootstraps=10, columns=None)

Use this function to test the proportional hazards assumption. See usage example at https://lifelines.readthedocs.io/en/latest/jupyter_notebooks/Proportional%20hazard%20assumption.html

Parameters:

• training_df (DataFrame) – the original DataFrame used in the call to fit(...) or a sub-sampled version.
• advice (boolean, optional) – display advice as output to the user’s screen
• show_plots (boolean, optional) – display plots of the scaled schoenfeld residuals and loess curves. This is an eyeball test for violations. This will slow down the function significantly.
• p_value_threshold (float, optional) – the threshold to use to alert the user of violations. See note below.
• plot_n_bootstraps – in the plots displayed, also display plot_n_bootstraps bootstrapped loess curves. This will slow down the function significantly.
• columns (list, optional) – specify a subset of columns to test.

Examples

>>> from lifelines.datasets import load_rossi
>>> from lifelines import CoxPHFitter
>>>
>>> rossi = load_rossi()
>>> cph = CoxPHFitter().fit(rossi, 'week', 'arrest')
>>>
>>> cph.check_assumptions(rossi)

Notes

The p_value_threshold is arbitrarily set at 0.01. Under the null, some covariates will be below the threshold (i.e. by chance). This is compounded when there are many covariates.

Similarly, when there are lots of observations, even minor deviances from the proportional hazard assumption will be flagged.

With that in mind, it’s best to use a combination of statistical tests and eyeball tests to determine the most serious violations.

References

section 5 in https://socialsciences.mcmaster.ca/jfox/Books/Companion/appendices/Appendix-Cox-Regression.pdf, http://www.mwsug.org/proceedings/2006/stats/MWSUG-2006-SD08.pdf, http://eprints.lse.ac.uk/84988/1/06_ParkHendry2015-ReassessingSchoenfeldTests_Final.pdf

compute_residuals(training_dataframe, kind)

Parameters:

• training_dataframe (pandas DataFrame) – the same training DataFrame given in fit
• kind (string) – {‘schoenfeld’, ‘score’, ‘delta_beta’, ‘deviance’, ‘martingale’, ‘scaled_schoenfeld’}

fit(df, duration_col=None, event_col=None, show_progress=False, initial_point=None, strata=None, step_size=None, weights_col=None, cluster_col=None, robust=False, batch_mode=None)

Fit the Cox proportional hazard model to a dataset.

Parameters:

• df (DataFrame) – a Pandas DataFrame with necessary columns duration_col and event_col (see below), covariates columns, and special columns (weights, strata). duration_col refers to the lifetimes of the subjects. event_col refers to whether the ‘death’ events was observed: 1 if observed, 0 else (censored).
• duration_col (string) – the name of the column in DataFrame that contains the subjects’ lifetimes.
• event_col (string, optional) – the name of thecolumn in DataFrame that contains the subjects’ death observation. If left as None, assume all individuals are uncensored.
• weights_col (string, optional) – an optional column in the DataFrame, df, that denotes the weight per subject. This column is expelled and not used as a covariate, but as a weight in the final regression. Default weight is 1. This can be used for case-weights. For example, a weight of 2 means there were two subjects with identical observations. This can be used for sampling weights. In that case, use robust=True to get more accurate standard errors.
• show_progress (boolean, optional (default=False)) – since the fitter is iterative, show convergence diagnostics. Useful if convergence is failing.
• initial_point ((d,) numpy array, optional) – initialize the starting point of the iterative algorithm. Default is the zero vector.
• strata (list or string, optional) – specify a column or list of columns n to use in stratification. This is useful if a categorical covariate does not obey the proportional hazard assumption. This is used similar to the strata expression in R. See http://courses.washington.edu/b515/l17.pdf.
• step_size (float, optional) – set an initial step size for the fitting algorithm. Setting to 1.0 may improve performance, but could also hurt convergence.
• robust (boolean, optional (default=False)) – Compute the robust errors using the Huber sandwich estimator, aka Wei-Lin estimate. This does not handle ties, so if there are high number of ties, results may significantly differ. See “The Robust Inference for the Cox Proportional Hazards Model”, Journal of the American Statistical Association, Vol. 84, No. 408 (Dec., 1989), pp. 1074- 1078
• cluster_col (string, optional) – specifies what column has unique identifiers for clustering covariances. Using this forces the sandwich estimator (robust variance estimator) to be used.
• batch_mode (bool, optional) – enabling batch_mode can be faster for datasets with a large number of ties. If left as None, lifelines will choose the best option.

Returns:

self – self with additional new properties: print_summary, hazards_, confidence_intervals_, baseline_survival_, etc.

Return type:

CoxPHFitter

Note

Tied survival times are handled using Efron’s tie-method.

Examples

>>> from lifelines import CoxPHFitter
>>>
>>> df = pd.DataFrame({
>>>     'T': [5, 3, 9, 8, 7, 4, 4, 3, 2, 5, 6, 7],
>>>     'E': [1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0],
>>>     'var': [0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2],
>>>     'age': [4, 3, 9, 8, 7, 4, 4, 3, 2, 5, 6, 7],
>>> })
>>>
>>> cph = CoxPHFitter()
>>> cph.fit(df, 'T', 'E')
>>> cph.print_summary()
>>> cph.predict_median(df)

>>> from lifelines import CoxPHFitter
>>>
>>> df = pd.DataFrame({
>>>     'T': [5, 3, 9, 8, 7, 4, 4, 3, 2, 5, 6, 7],
>>>     'E': [1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0],
>>>     'var': [0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2],
>>>     'weights': [1.1, 0.5, 2.0, 1.6, 1.2, 4.3, 1.4, 4.5, 3.0, 3.2, 0.4, 6.2],
>>>     'month': [10, 3, 9, 8, 7, 4, 4, 3, 2, 5, 6, 7],
>>>     'age': [4, 3, 9, 8, 7, 4, 4, 3, 2, 5, 6, 7],
>>> })
>>>
>>> cph = CoxPHFitter()
>>> cph.fit(df, 'T', 'E', strata=['month', 'age'], robust=True, weights_col='weights')
>>> cph.print_summary()
>>> cph.predict_median(df)

fit_right_censoring(*args, **kwargs)

Alias for fit

See also

fit

log_likelihood_ratio_test()

This function computes the likelihood ratio test for the Cox model. We compare the existing model (with all the covariates) to the trivial model of no covariates.

plot(columns=None, hazard_ratios=False, **errorbar_kwargs)

Produces a visual representation of the coefficients (i.e. log hazard ratios), including their standard errors and magnitudes.

Parameters:

• columns (list, optional) – specify a subset of the columns to plot
• hazard_ratios (bool, optional) – by default, plot will present the log-hazard ratios (the coefficients). However, by turning this flag to True, the hazard ratios are presented instead.
• errorbar_kwargs – pass in additional plotting commands to matplotlib errorbar command

Examples

>>> from lifelines import datasets, CoxPHFitter
>>> rossi = datasets.load_rossi()
>>> cph = CoxPHFitter().fit(rossi, 'week', 'arrest')
>>> cph.plot(hazard_ratios=True)

Returns:ax – the matplotlib axis that be edited. Return type:matplotlib axis

plot_covariate_groups(covariates, values, plot_baseline=True, **kwargs)

Produces a plot comparing the baseline survival curve of the model versus what happens when a covariate(s) is varied over values in a group. This is useful to compare subjects’ survival as we vary covariate(s), all else being held equal. The baseline survival curve is equal to the predicted survival curve at all average values in the original dataset.

Parameters:

• covariates (string or list) – a string (or list of strings) of the covariate(s) in the original dataset that we wish to vary.
• values (1d or 2d iterable) – an iterable of the specific values we wish the covariate(s) to take on.
• plot_baseline (bool) – also display the baseline survival, defined as the survival at the mean of the original dataset.
• kwargs – pass in additional plotting commands.

Returns:

ax – the matplotlib axis that be edited.

Return type:

matplotlib axis, or list of axis’

Examples

>>> from lifelines import datasets, CoxPHFitter
>>> rossi = datasets.load_rossi()
>>> cph = CoxPHFitter().fit(rossi, 'week', 'arrest')
>>> cph.plot_covariate_groups('prio', values=np.arange(0, 15, 3), cmap='coolwarm')

>>> # multiple variables at once
>>> cph.plot_covariate_groups(['prio', 'paro'], values=[
>>>  [0,  0],
>>>  [5,  0],
>>>  [10, 0],
>>>  [0,  1],
>>>  [5,  1],
>>>  [10, 1]
>>> ], cmap='coolwarm')

>>> # if you have categorical variables, you can do the following to see the
>>> # effect of all the categories on one plot.
>>> cph.plot_covariate_groups(['dummy1', 'dummy2', 'dummy3'], values=[[1, 0, 0], [0, 1, 0], [0, 0, 1]])
>>> # same as:
>>> cph.plot_covariate_groups(['dummy1', 'dummy2', 'dummy3'], values=np.eye(3))

predict_cumulative_hazard(X, times=None, conditional_after=None)

Parameters:

• X (numpy array or DataFrame) – a (n,d) covariate numpy array or DataFrame. If a DataFrame, columns can be in any order. If a numpy array, columns must be in the same order as the training data.
• times (iterable, optional) – an iterable of increasing times to predict the cumulative hazard at. Default is the set of all durations (observed and unobserved). Uses a linear interpolation if points in time are not in the index.
• conditional_after (iterable, optional) – Must be equal is size to X.shape[0] (denoted n above). An iterable (array, list, series) of possibly non-zero values that represent how long the subject has already lived for. Ex: if \(T\) is the unknown event time, then this represents :math`T | T > s`. This is useful for knowing the remaining hazard/survival of censored subjects. The new timeline is the remaining duration of the subject, i.e. reset back to starting at 0.

Returns:

cumulative_hazard_ – the cumulative hazard of individuals over the timeline

Return type:

DataFrame

predict_expectation(X)

Compute the expected lifetime, \(E[T]\), using covariates X. This algorithm to compute the expectation is to use the fact that \(E[T] = \int_0^\inf P(T > t) dt = \int_0^\inf S(t) dt\). To compute the integral, we use the trapizoidal rule to approximate the integral.

Caution

However, if the survival function doesn’t converge to 0, the the expectation is really infinity and the returned values are meaningless/too large. In that case, using predict_median or predict_percentile would be better.

Parameters:X (numpy array or DataFrame) – a (n,d) covariate numpy array or DataFrame. If a DataFrame, columns can be in any order. If a numpy array, columns must be in the same order as the training data. Returns:expectations Return type:DataFrame

Notes

If X is a DataFrame, the order of the columns do not matter. But if X is an array, then the column ordering is assumed to be the same as the training dataset.

See also

predict_median(), predict_percentile()

predict_log_partial_hazard(X)

This is equivalent to R’s linear.predictors. Returns the log of the partial hazard for the individuals, partial since the baseline hazard is not included. Equal to \((x - \text{mean}(x_{\text{train}})) \beta\)

Parameters:X (numpy array or DataFrame) – a (n,d) covariate numpy array or DataFrame. If a DataFrame, columns can be in any order. If a numpy array, columns must be in the same order as the training data. Returns:log_partial_hazard Return type:DataFrame

Notes

If X is a DataFrame, the order of the columns do not matter. But if X is an array, then the column ordering is assumed to be the same as the training dataset.

predict_median(X, conditional_after=None)

Predict the median lifetimes for the individuals. If the survival curve of an individual does not cross 0.5, then the result is infinity.

Parameters:X (numpy array or DataFrame) – a (n,d) covariate numpy array or DataFrame. If a DataFrame, columns can be in any order. If a numpy array, columns must be in the same order as the training data. Returns:percentiles – the median lifetimes for the individuals. If the survival curve of an individual does not cross 0.5, then the result is infinity. Return type:DataFrame

See also

predict_percentile()

predict_partial_hazard(X)

Parameters:X (numpy array or DataFrame) – a (n,d) covariate numpy array or DataFrame. If a DataFrame, columns can be in any order. If a numpy array, columns must be in the same order as the training data. Returns:partial_hazard – Returns the partial hazard for the individuals, partial since the baseline hazard is not included. Equal to \(\exp{(x - mean(x_{train}))'\beta}\) Return type:DataFrame

Notes

If X is a DataFrame, the order of the columns do not matter. But if X is an array, then the column ordering is assumed to be the same as the training dataset.

predict_percentile(X, p=0.5, conditional_after=None)

Returns the median lifetimes for the individuals, by default. If the survival curve of an individual does not cross 0.5, then the result is infinity. http://stats.stackexchange.com/questions/102986/percentile-loss-functions

Parameters:

• X (numpy array or DataFrame) – a (n,d) covariate numpy array or DataFrame. If a DataFrame, columns can be in any order. If a numpy array, columns must be in the same order as the training data.
• p (float, optional (default=0.5)) – the percentile, must be between 0 and 1.
• conditional_after (iterable, optional) – Must be equal is size to X.shape[0] (denoted n above). An iterable (array, list, series) of possibly non-zero values that represent how long the subject has already lived for. Ex: if \(T\) is the unknown event time, then this represents :math`T | T > s`. This is useful for knowing the remaining hazard/survival of censored subjects. The new timeline is the remaining duration of the subject, i.e. normalized back to starting at 0.

Returns:

percentiles

Return type:

DataFrame

See also

predict_median()

predict_survival_function(X, times=None, conditional_after=None)

Predict the survival function for individuals, given their covariates. This assumes that the individual just entered the study (that is, we do not condition on how long they have already lived for.)

Parameters:

• X (numpy array or DataFrame) – a (n,d) covariate numpy array or DataFrame. If a DataFrame, columns can be in any order. If a numpy array, columns must be in the same order as the training data.
• times (iterable, optional) – an iterable of increasing times to predict the cumulative hazard at. Default is the set of all durations (observed and unobserved). Uses a linear interpolation if points in time are not in the index.
• conditional_after (iterable, optional) – Must be equal is size to X.shape[0] (denoted n above). An iterable (array, list, series) of possibly non-zero values that represent how long the subject has already lived for. Ex: if \(T\) is the unknown event time, then this represents :math`T | T > s`. This is useful for knowing the remaining hazard/survival of censored subjects. The new timeline is the remaining duration of the subject, i.e. normalized back to starting at 0.

Returns:

survival_function – the survival probabilities of individuals over the timeline

Return type:

DataFrame

print_summary(decimals=2, **kwargs)

Print summary statistics describing the fit, the coefficients, and the error bounds.

Parameters:

• decimals (int, optional (default=2)) – specify the number of decimal places to show
• kwargs – print additional metadata in the output (useful to provide model names, dataset names, etc.) when comparing multiple outputs.

score_

The concordance score (also known as the c-index) of the fit. The c-index is a generalization of the ROC AUC to survival data, including censorships.

For this purpose, the score_ is a measure of the predictive accuracy of the fitted model onto the training dataset.

References

https://stats.stackexchange.com/questions/133817/stratified-concordance-index-survivalsurvconcordance

summary

Summary statistics describing the fit. Set alpha property in the object before calling.

Returns:df – Contains columns coef, np.exp(coef), se(coef), z, p, lower, upper Return type:DataFrame

WeibullAFTFitter

class lifelines.fitters.weibull_aft_fitter.WeibullAFTFitter(alpha=0.05, penalizer=0.0, l1_ratio=0.0, fit_intercept=True, model_ancillary=False)

Bases: lifelines.fitters.ParametericAFTRegressionFitter

This class implements a Weibull AFT model. The model has parameterized form, with \(\lambda(x) = \exp\left(\beta_0 + \beta_1x_1 + ... + \beta_n x_n \right)\), and optionally, \(\rho(y) = \exp\left(\alpha_0 + \alpha_1 y_1 + ... + \alpha_m y_m \right)\),

\[S(t; x, y) = \exp\left(-\left(\frac{t}{\lambda(x)}\right)^{\rho(y)}\right),\]

which implies the cumulative hazard rate is

\[H(t; x, y) = \left(\frac{t}{\lambda(x)} \right)^{\rho(y)},\]

After calling the .fit method, you have access to properties like: params_, print_summary(). A summary of the fit is available with the method print_summary().

Parameters:

• alpha (float, optional (default=0.05)) – the level in the confidence intervals.
• fit_intercept (boolean, optional (default=True)) – Allow lifelines to add an intercept column of 1s to df, and ancillary_df if applicable.
• penalizer (float, optional (default=0.0)) – the penalizer coefficient to the size of the coefficients. See l1_ratio. Must be equal to or greater than 0.
• l1_ratio (float, optional (default=0.0)) – how much of the penalizer should be attributed to an l1 penalty (otherwise an l2 penalty). The penalty function looks like penalizer * l1_ratio * ||w||_1 + 0.5 * penalizer * (1 - l1_ratio) * ||w||^2_2
• model_ancillary (optional (default=False)) – set the model instance to always model the ancillary parameter with the supplied Dataframe. This is useful for grid-search optimization.

params_

The estimated coefficients

Type:DataFrame

confidence_intervals_

The lower and upper confidence intervals for the coefficients

Type:DataFrame

durations

The event_observed variable provided

Type:Series

event_observed

The event_observed variable provided

Type:Series

weights

The event_observed variable provided

Type:Series

variance_matrix_

The variance matrix of the coefficients

Type:numpy array

standard_errors_

the standard errors of the estimates

Type:Series

score_

the concordance index of the model.

Type:float

fit(df, duration_col, event_col=None, ancillary_df=None, fit_intercept=None, show_progress=False, timeline=None, weights_col=None, robust=False, initial_point=None, entry_col=None)

Fit the accelerated failure time model to a right-censored dataset.

Parameters:

• df (DataFrame) – a Pandas DataFrame with necessary columns duration_col and event_col (see below), covariates columns, and special columns (weights). duration_col refers to the lifetimes of the subjects. event_col refers to whether the ‘death’ events was observed: 1 if observed, 0 else (censored).
• duration_col (string) – the name of the column in DataFrame that contains the subjects’ lifetimes.
• event_col (string, optional) – the name of the column in DataFrame that contains the subjects’ death observation. If left as None, assume all individuals are uncensored.
• show_progress (boolean, optional (default=False)) – since the fitter is iterative, show convergence diagnostics. Useful if convergence is failing.
• ancillary_df (None, boolean, or DataFrame, optional (default=None)) – Choose to model the ancillary parameters. If None or False, explicitly do not fit the ancillary parameters using any covariates. If True, model the ancillary parameters with the same covariates as df. If DataFrame, provide covariates to model the ancillary parameters. Must be the same row count as df.
• fit_intercept (bool, optional) – If true, add a constant column to the regression. Overrides value set in class instantiation.
• timeline (array, optional) – Specify a timeline that will be used for plotting and prediction
• weights_col (string) – the column in DataFrame that specifies weights per observation.
• robust (boolean, optional (default=False)) – Compute the robust errors using the Huber sandwich estimator.
• initial_point ((d,) numpy array, optional) – initialize the starting point of the iterative algorithm. Default is the zero vector.
• entry_col (specify a column in the DataFrame that denotes any late-entries (left truncation) that occurred. See) – the docs on left truncation

Returns:

Return type:

self with additional new properties print_summary, params_, confidence_intervals_ and more

Examples

>>> from lifelines import WeibullAFTFitter, LogNormalAFTFitter, LogLogisticAFTFitter
>>>
>>> df = pd.DataFrame({
>>>     'T': [5, 3, 9, 8, 7, 4, 4, 3, 2, 5, 6, 7],
>>>     'E': [1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0],
>>>     'var': [0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2],
>>>     'age': [4, 3, 9, 8, 7, 4, 4, 3, 2, 5, 6, 7],
>>> })
>>>
>>> aft = WeibullAFTFitter()
>>> aft.fit(df, 'T', 'E')
>>> aft.print_summary()
>>> aft.predict_median(df)
>>>
>>> aft = WeibullAFTFitter()
>>> aft.fit(df, 'T', 'E', ancillary_df=df)
>>> aft.print_summary()
>>> aft.predict_median(df)

fit_interval_censoring(df, lower_bound_col, upper_bound_col, event_col=None, ancillary_df=None, fit_intercept=None, show_progress=False, timeline=None, weights_col=None, robust=False, initial_point=None, entry_col=None)

Fit the accelerated failure time model to a interval-censored dataset.

Parameters:

• df (DataFrame) – a Pandas DataFrame with necessary columns lower_bound_col, upper_bound_col (see below), and any other covariates or weights.
• lower_bound_col (string) – the name of the column in DataFrame that contains the subjects’ left-most observation.
• upper_bound_col (string) – the name of the column in DataFrame that contains the subjects’ right-most observation. Values can be np.inf (and should be if the subject is right-censored).
• event_col (string, optional) – the name of the column in DataFrame that contains the subjects’ death observation. If left as None, will be inferred from the start and stop columns (lower_bound==upper_bound means uncensored)
• ancillary_df (None, boolean, or DataFrame, optional (default=None)) – Choose to model the ancillary parameters. If None or False, explicitly do not fit the ancillary parameters using any covariates. If True, model the ancillary parameters with the same covariates as df. If DataFrame, provide covariates to model the ancillary parameters. Must be the same row count as df.
• fit_intercept (bool, optional) – If true, add a constant column to the regression. Overrides value set in class instantiation.
• show_progress (boolean, optional (default=False)) – since the fitter is iterative, show convergence diagnostics. Useful if convergence is failing.
• timeline (array, optional) – Specify a timeline that will be used for plotting and prediction
• weights_col (string) – the column in DataFrame that specifies weights per observation.
• robust (boolean, optional (default=False)) – Compute the robust errors using the Huber sandwich estimator.
• initial_point ((d,) numpy array, optional) – initialize the starting point of the iterative algorithm. Default is the zero vector.
• entry_col (specify a column in the DataFrame that denotes any late-entries (left truncation) that occurred. See) – the docs on left truncation

Returns:

Return type:

self with additional new properties print_summary, params_, confidence_intervals_ and more

Examples

>>> from lifelines import WeibullAFTFitter, LogNormalAFTFitter, LogLogisticAFTFitter
>>>
>>> df = pd.DataFrame({
>>>     'start': [5, 3, 9, 8, 7, 4, 4, 3, 2, 5, 6, 7],
>>>     'stop':  [5, 3, 9, 8, 7, 4, 8, 5, 2, 5, 6, np.inf],  # this last subject is right-censored.
>>>     'E':     [1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0],
>>>     'var': [0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2],
>>>     'age': [4, 3, 9, 8, 7, 4, 4, 3, 2, 5, 6, 7],
>>> })
>>>
>>> aft = WeibullAFTFitter()
>>> aft.fit_interval_censoring(df, 'start', 'stop', 'E')
>>> aft.print_summary()
>>> aft.predict_median(df)
>>>
>>> aft = WeibullAFTFitter()
>>> aft.fit_interval_censoring(df, 'start', 'stop', 'E', ancillary_df=df)
>>> aft.print_summary()
>>> aft.predict_median(df)

fit_left_censoring(df, duration_col=None, event_col=None, ancillary_df=None, fit_intercept=None, show_progress=False, timeline=None, weights_col=None, robust=False, initial_point=None, entry_col=None)

Fit the accelerated failure time model to a left-censored dataset.

Parameters:

• df (DataFrame) – a Pandas DataFrame with necessary columns duration_col and event_col (see below), covariates columns, and special columns (weights). duration_col refers to the lifetimes of the subjects. event_col refers to whether the ‘death’ events was observed: 1 if observed, 0 else (censored).
• duration_col (string) – the name of the column in DataFrame that contains the subjects’ lifetimes/measurements/etc. This column contains the (possibly) left-censored data.
• event_col (string, optional) – the name of the column in DataFrame that contains the subjects’ death observation. If left as None, assume all individuals are uncensored.
• ancillary_df (None, boolean, or DataFrame, optional (default=None)) – Choose to model the ancillary parameters. If None or False, explicitly do not fit the ancillary parameters using any covariates. If True, model the ancillary parameters with the same covariates as df. If DataFrame, provide covariates to model the ancillary parameters. Must be the same row count as df.
• fit_intercept (bool, optional) – If true, add a constant column to the regression. Overrides value set in class instantiation.
• show_progress (boolean, optional (default=False)) – since the fitter is iterative, show convergence diagnostics. Useful if convergence is failing.

timeline: array, optional

Specify a timeline that will be used for plotting and prediction

weights_col: string

the column in DataFrame that specifies weights per observation.

robust: boolean, optional (default=False)

Compute the robust errors using the Huber sandwich estimator.

initial_point: (d,) numpy array, optional

initialize the starting point of the iterative algorithm. Default is the zero vector.

entry_col: specify a column in the DataFrame that denotes any late-entries (left truncation) that occurred. See

the docs on left truncation

Returns: Return type:self with additional new properties print_summary, params_, confidence_intervals_ and more

Examples

>>> from lifelines import WeibullAFTFitter, LogNormalAFTFitter, LogLogisticAFTFitter
>>>
>>> df = pd.DataFrame({
>>>     'T': [5, 3, 9, 8, 7, 4, 4, 3, 2, 5, 6, 7],
>>>     'E': [1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0],
>>>     'var': [0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2],
>>>     'age': [4, 3, 9, 8, 7, 4, 4, 3, 2, 5, 6, 7],
>>> })
>>>
>>> aft = WeibullAFTFitter()
>>> aft.fit_left_censoring(df, 'T', 'E')
>>> aft.print_summary()
>>> aft.predict_median(df)
>>>
>>> aft = WeibullAFTFitter()
>>> aft.fit_left_censoring(df, 'T', 'E', ancillary_df=df)
>>> aft.print_summary()
>>> aft.predict_median(df)

fit_right_censoring(*args, **kwargs)

Alias for fit

See also

fit

log_likelihood_ratio_test()

This function computes the likelihood ratio test for the model. We compare the existing model (with all the covariates) to the trivial model of no covariates.

mean_survival_time_

The mean survival time of the average subject in the training dataset.

median_survival_time_

The median survival time of the average subject in the training dataset.

plot(columns=None, parameter=None, **errorbar_kwargs)

Produces a visual representation of the coefficients, including their standard errors and magnitudes.

Parameters:

• columns (list, optional) – specify a subset of the columns to plot
• errorbar_kwargs – pass in additional plotting commands to matplotlib errorbar command

Returns:

ax – the matplotlib axis that be edited.

Return type:

matplotlib axis

plot_covariate_groups(covariates, values, plot_baseline=True, **kwargs)

Produces a visual representation comparing the baseline survival curve of the model versus what happens when a covariate(s) is varied over values in a group. This is useful to compare subjects’ survival as we vary covariate(s), all else being held equal. The baseline survival curve is equal to the predicted survival curve at all average values in the original dataset.

Parameters:

• covariates (string or list) – a string (or list of strings) of the covariate in the original dataset that we wish to vary.
• values (1d or 2d iterable) – an iterable of the values we wish the covariate to take on.
• plot_baseline (bool) – also display the baseline survival, defined as the survival at the mean of the original dataset.
• kwargs – pass in additional plotting commands

Returns:

ax – the matplotlib axis that be edited.

Return type:

matplotlib axis, or list of axis’

Examples

>>> from lifelines import datasets, WeibullAFTFitter
>>> rossi = datasets.load_rossi()
>>> wf = WeibullAFTFitter().fit(rossi, 'week', 'arrest')
>>> wf.plot_covariate_groups('prio', values=np.arange(0, 15), cmap='coolwarm')

>>> # multiple variables at once
>>> wf.plot_covariate_groups(['prio', 'paro'], values=[[0, 0], [5, 0], [10, 0], [0, 1], [5, 1], [10, 1]], cmap='coolwarm')

>>> # if you have categorical variables, you can simply things:
>>> wf.plot_covariate_groups(['dummy1', 'dummy2', 'dummy3'], values=np.eye(3))

predict_cumulative_hazard(df, ancillary_df=None, times=None, conditional_after=None)

Predict the median lifetimes for the individuals. If the survival curve of an individual does not cross 0.5, then the result is infinity.

Parameters:

• df (DataFrame) – a (n,d) covariate numpy array or DataFrame. If a DataFrame, columns can be in any order. If a numpy array, columns must be in the same order as the training data.
• times (iterable, optional) – an iterable of increasing times to predict the cumulative hazard at. Default is the set of all durations (observed and unobserved).
• conditional_after (iterable, optional) – Must be equal is size to df.shape[0] (denoted n above). An iterable (array, list, series) of possibly non-zero values that represent how long the subject has already lived for. Ex: if \(T\) is the unknown event time, then this represents :math`T | T > s`. This is useful for knowing the remaining hazard/survival of censored subjects. The new timeline is the remaining duration of the subject, i.e. normalized back to starting at 0.

Returns:

the median lifetimes for the individuals. If the survival curve of an individual does not cross 0.5, then the result is infinity.

Return type:

DataFrame

See also

predict_percentile(), predict_expectation(), predict_survival_function()

predict_expectation(df, ancillary_df=None)

Predict the expectation of lifetimes, \(E[T | x]\).

Parameters:

• df (DataFrame) – a (n,d) DataFrame. If a DataFrame, columns can be in any order. If a numpy array, columns must be in the same order as the training data.
• ancillary_df (DataFrame, optional) – a (n,d) DataFrame. If a DataFrame, columns can be in any order. If a numpy array, columns must be in the same order as the training data.

Returns:

percentiles – the median lifetimes for the individuals. If the survival curve of an individual does not cross 0.5, then the result is infinity.

Return type:

DataFrame

See also

predict_median()

predict_median(df, ancillary_df=None, conditional_after=None)

Predict the median lifetimes for the individuals. If the survival curve of an individual does not cross 0.5, then the result is infinity.

Parameters:

• df (DataFrame) – a (n,d) covariate numpy array or DataFrame. If a DataFrame, columns can be in any order. If a numpy array, columns must be in the same order as the training data.
• conditional_after (iterable, optional) – Must be equal is size to df.shape[0] (denoted n above). An iterable (array, list, series) of possibly non-zero values that represent how long the subject has already lived for. Ex: if \(T\) is the unknown event time, then this represents :math`T | T > s`. This is useful for knowing the remaining hazard/survival of censored subjects. The new timeline is the remaining duration of the subject, i.e. normalized back to starting at 0.

Returns:

percentiles – the median lifetimes for the individuals. If the survival curve of an individual does not cross 0.5, then the result is infinity.

Return type:

DataFrame

See also

predict_percentile(), predict_expectation()

predict_percentile(df, ancillary_df=None, p=0.5, conditional_after=None)

Returns the median lifetimes for the individuals, by default. If the survival curve of an individual does not cross 0.5, then the result is infinity. http://stats.stackexchange.com/questions/102986/percentile-loss-functions

Parameters:

• df (DataFrame) – a (n,d) DataFrame. If a DataFrame, columns can be in any order. If a numpy array, columns must be in the same order as the training data.
• ancillary_df (DataFrame, optional) – a (n,d) DataFrame. If a DataFrame, columns can be in any order. If a numpy array, columns must be in the same order as the training data.
• p (float, optional (default=0.5)) – the percentile, must be between 0 and 1.

Returns:

percentiles

Return type:

DataFrame

See also

predict_median()

predict_survival_function(df, ancillary_df=None, times=None, conditional_after=None)

Predict the survival function for individuals, given their covariates. This assumes that the individual just entered the study (that is, we do not condition on how long they have already lived for.)

Parameters:

• X (numpy array or DataFrame) – a (n,d) covariate numpy array or DataFrame. If a DataFrame, columns can be in any order. If a numpy array, columns must be in the same order as the training data.
• ancillary_X (numpy array or DataFrame, optional) – a (n,d) covariate numpy array or DataFrame. If a DataFrame, columns can be in any order. If a numpy array, columns must be in the same order as the training data.
• times (iterable, optional) – an iterable of increasing times to predict the survival function at. Default is the set of all durations (observed and unobserved).
• conditional_after (iterable, optional) – Must be equal is size to df.shape[0] (denoted n above). An iterable (array, list, series) of possibly non-zero values that represent how long the subject has already lived for. Ex: if \(T\) is the unknown event time, then this represents :math`T | T > s`. This is useful for knowing the remaining hazard/survival of censored subjects. The new timeline is the remaining duration of the subject, i.e. normalized back to starting at 0.

Returns:

survival_function – the survival probabilities of individuals over the timeline

Return type:

DataFrame

print_summary(decimals=2, **kwargs)

Print summary statistics describing the fit, the coefficients, and the error bounds.

Parameters:

• decimals (int, optional (default=2)) – specify the number of decimal places to show
• alpha (float or iterable) – specify confidence intervals to show
• kwargs – print additional metadata in the output (useful to provide model names, dataset names, etc.) when comparing multiple outputs.

score_

The concordance score (also known as the c-index) of the fit. The c-index is a generalization of the ROC AUC to survival data, including censorships.

For this purpose, the score_ is a measure of the predictive accuracy of the fitted model onto the training dataset.

summary

Summary statistics describing the fit.

Returns:df – Contains columns coef, np.exp(coef), se(coef), z, p, lower, upper Return type:DataFrame

LogNormalAFTFitter

class lifelines.fitters.log_normal_aft_fitter.LogNormalAFTFitter(alpha=0.05, penalizer=0.0, l1_ratio=0.0, fit_intercept=True, model_ancillary=False)

Bases: lifelines.fitters.ParametericAFTRegressionFitter

This class implements a Log-Normal AFT model. The model has parameterized form, with \(\mu(x) = \exp\left(a_0 + a_1x_1 + ... + a_n x_n \right)\), and optionally, \(\sigma(y) = \exp\left(b_0 + b_1 y_1 + ... + b_m y_m \right)\),

The cumulative hazard rate is

\[H(t; x, y) = -\log\left(1 - \Phi\left(\frac{\log(T) - \mu(x)}{\sigma(y)}\right)\right)\]

After calling the .fit method, you have access to properties like: params_, print_summary(). A summary of the fit is available with the method print_summary().

Parameters:

• alpha (float, optional (default=0.05)) – the level in the confidence intervals.
• fit_intercept (boolean, optional (default=True)) – Allow lifelines to add an intercept column of 1s to df, and ancillary_df if applicable.
• penalizer (float, optional (default=0.0)) – the penalizer coefficient to the size of the coefficients. See l1_ratio. Must be equal to or greater than 0.
• l1_ratio (float, optional (default=0.0)) – how much of the penalizer should be attributed to an l1 penalty (otherwise an l2 penalty). The penalty function looks like penalizer * l1_ratio * ||w||_1 + 0.5 * penalizer * (1 - l1_ratio) * ||w||^2_2
• model_ancillary (optional (default=False)) – set the model instance to always model the ancillary parameter with the supplied DataFrame. This is useful for grid-search optimization.

params_

The estimated coefficients

Type:DataFrame

confidence_intervals_

The lower and upper confidence intervals for the coefficients

Type:DataFrame

durations

The event_observed variable provided

Type:Series

event_observed

The event_observed variable provided

Type:Series

weights

The event_observed variable provided

Type:Series

variance_matrix_

The variance matrix of the coefficients

Type:numpy array

standard_errors_

the standard errors of the estimates

Type:Series

score_

the concordance index of the model.

Type:float

fit(df, duration_col, event_col=None, ancillary_df=None, fit_intercept=None, show_progress=False, timeline=None, weights_col=None, robust=False, initial_point=None, entry_col=None)

Fit the accelerated failure time model to a right-censored dataset.

Parameters:

• df (DataFrame) – a Pandas DataFrame with necessary columns duration_col and event_col (see below), covariates columns, and special columns (weights). duration_col refers to the lifetimes of the subjects. event_col refers to whether the ‘death’ events was observed: 1 if observed, 0 else (censored).
• duration_col (string) – the name of the column in DataFrame that contains the subjects’ lifetimes.
• event_col (string, optional) – the name of the column in DataFrame that contains the subjects’ death observation. If left as None, assume all individuals are uncensored.
• show_progress (boolean, optional (default=False)) – since the fitter is iterative, show convergence diagnostics. Useful if convergence is failing.
• ancillary_df (None, boolean, or DataFrame, optional (default=None)) – Choose to model the ancillary parameters. If None or False, explicitly do not fit the ancillary parameters using any covariates. If True, model the ancillary parameters with the same covariates as df. If DataFrame, provide covariates to model the ancillary parameters. Must be the same row count as df.
• fit_intercept (bool, optional) – If true, add a constant column to the regression. Overrides value set in class instantiation.
• timeline (array, optional) – Specify a timeline that will be used for plotting and prediction
• weights_col (string) – the column in DataFrame that specifies weights per observation.
• robust (boolean, optional (default=False)) – Compute the robust errors using the Huber sandwich estimator.
• initial_point ((d,) numpy array, optional) – initialize the starting point of the iterative algorithm. Default is the zero vector.
• entry_col (specify a column in the DataFrame that denotes any late-entries (left truncation) that occurred. See) – the docs on left truncation

Returns:

Return type:

self with additional new properties print_summary, params_, confidence_intervals_ and more

Examples

>>> from lifelines import WeibullAFTFitter, LogNormalAFTFitter, LogLogisticAFTFitter
>>>
>>> df = pd.DataFrame({
>>>     'T': [5, 3, 9, 8, 7, 4, 4, 3, 2, 5, 6, 7],
>>>     'E': [1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0],
>>>     'var': [0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2],
>>>     'age': [4, 3, 9, 8, 7, 4, 4, 3, 2, 5, 6, 7],
>>> })
>>>
>>> aft = WeibullAFTFitter()
>>> aft.fit(df, 'T', 'E')
>>> aft.print_summary()
>>> aft.predict_median(df)
>>>
>>> aft = WeibullAFTFitter()
>>> aft.fit(df, 'T', 'E', ancillary_df=df)
>>> aft.print_summary()
>>> aft.predict_median(df)

fit_interval_censoring(df, lower_bound_col, upper_bound_col, event_col=None, ancillary_df=None, fit_intercept=None, show_progress=False, timeline=None, weights_col=None, robust=False, initial_point=None, entry_col=None)

Fit the accelerated failure time model to a interval-censored dataset.

Parameters:

• df (DataFrame) – a Pandas DataFrame with necessary columns lower_bound_col, upper_bound_col (see below), and any other covariates or weights.
• lower_bound_col (string) – the name of the column in DataFrame that contains the subjects’ left-most observation.
• upper_bound_col (string) – the name of the column in DataFrame that contains the subjects’ right-most observation. Values can be np.inf (and should be if the subject is right-censored).
• event_col (string, optional) – the name of the column in DataFrame that contains the subjects’ death observation. If left as None, will be inferred from the start and stop columns (lower_bound==upper_bound means uncensored)
• ancillary_df (None, boolean, or DataFrame, optional (default=None)) – Choose to model the ancillary parameters. If None or False, explicitly do not fit the ancillary parameters using any covariates. If True, model the ancillary parameters with the same covariates as df. If DataFrame, provide covariates to model the ancillary parameters. Must be the same row count as df.
• fit_intercept (bool, optional) – If true, add a constant column to the regression. Overrides value set in class instantiation.
• show_progress (boolean, optional (default=False)) – since the fitter is iterative, show convergence diagnostics. Useful if convergence is failing.
• timeline (array, optional) – Specify a timeline that will be used for plotting and prediction
• weights_col (string) – the column in DataFrame that specifies weights per observation.
• robust (boolean, optional (default=False)) – Compute the robust errors using the Huber sandwich estimator.
• initial_point ((d,) numpy array, optional) – initialize the starting point of the iterative algorithm. Default is the zero vector.
• entry_col (specify a column in the DataFrame that denotes any late-entries (left truncation) that occurred. See) – the docs on left truncation

Returns:

Return type:

self with additional new properties print_summary, params_, confidence_intervals_ and more

Examples

>>> from lifelines import WeibullAFTFitter, LogNormalAFTFitter, LogLogisticAFTFitter
>>>
>>> df = pd.DataFrame({
>>>     'start': [5, 3, 9, 8, 7, 4, 4, 3, 2, 5, 6, 7],
>>>     'stop':  [5, 3, 9, 8, 7, 4, 8, 5, 2, 5, 6, np.inf],  # this last subject is right-censored.
>>>     'E':     [1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0],
>>>     'var': [0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2],
>>>     'age': [4, 3, 9, 8, 7, 4, 4, 3, 2, 5, 6, 7],
>>> })
>>>
>>> aft = WeibullAFTFitter()
>>> aft.fit_interval_censoring(df, 'start', 'stop', 'E')
>>> aft.print_summary()
>>> aft.predict_median(df)
>>>
>>> aft = WeibullAFTFitter()
>>> aft.fit_interval_censoring(df, 'start', 'stop', 'E', ancillary_df=df)
>>> aft.print_summary()
>>> aft.predict_median(df)

fit_left_censoring(df, duration_col=None, event_col=None, ancillary_df=None, fit_intercept=None, show_progress=False, timeline=None, weights_col=None, robust=False, initial_point=None, entry_col=None)

Fit the accelerated failure time model to a left-censored dataset.

Parameters:

• df (DataFrame) – a Pandas DataFrame with necessary columns duration_col and event_col (see below), covariates columns, and special columns (weights). duration_col refers to the lifetimes of the subjects. event_col refers to whether the ‘death’ events was observed: 1 if observed, 0 else (censored).
• duration_col (string) – the name of the column in DataFrame that contains the subjects’ lifetimes/measurements/etc. This column contains the (possibly) left-censored data.
• event_col (string, optional) – the name of the column in DataFrame that contains the subjects’ death observation. If left as None, assume all individuals are uncensored.
• ancillary_df (None, boolean, or DataFrame, optional (default=None)) – Choose to model the ancillary parameters. If None or False, explicitly do not fit the ancillary parameters using any covariates. If True, model the ancillary parameters with the same covariates as df. If DataFrame, provide covariates to model the ancillary parameters. Must be the same row count as df.
• fit_intercept (bool, optional) – If true, add a constant column to the regression. Overrides value set in class instantiation.
• show_progress (boolean, optional (default=False)) – since the fitter is iterative, show convergence diagnostics. Useful if convergence is failing.

timeline: array, optional

Specify a timeline that will be used for plotting and prediction

weights_col: string

the column in DataFrame that specifies weights per observation.

robust: boolean, optional (default=False)

Compute the robust errors using the Huber sandwich estimator.

initial_point: (d,) numpy array, optional

initialize the starting point of the iterative algorithm. Default is the zero vector.

entry_col: specify a column in the DataFrame that denotes any late-entries (left truncation) that occurred. See

the docs on left truncation

Returns: Return type:self with additional new properties print_summary, params_, confidence_intervals_ and more

Examples

>>> from lifelines import WeibullAFTFitter, LogNormalAFTFitter, LogLogisticAFTFitter
>>>
>>> df = pd.DataFrame({
>>>     'T': [5, 3, 9, 8, 7, 4, 4, 3, 2, 5, 6, 7],
>>>     'E': [1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0],
>>>     'var': [0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2],
>>>     'age': [4, 3, 9, 8, 7, 4, 4, 3, 2, 5, 6, 7],
>>> })
>>>
>>> aft = WeibullAFTFitter()
>>> aft.fit_left_censoring(df, 'T', 'E')
>>> aft.print_summary()
>>> aft.predict_median(df)
>>>
>>> aft = WeibullAFTFitter()
>>> aft.fit_left_censoring(df, 'T', 'E', ancillary_df=df)
>>> aft.print_summary()
>>> aft.predict_median(df)

fit_right_censoring(*args, **kwargs)

Alias for fit

See also

fit

log_likelihood_ratio_test()

This function computes the likelihood ratio test for the model. We compare the existing model (with all the covariates) to the trivial model of no covariates.

mean_survival_time_

The mean survival time of the average subject in the training dataset.

median_survival_time_

The median survival time of the average subject in the training dataset.

plot(columns=None, parameter=None, **errorbar_kwargs)

Produces a visual representation of the coefficients, including their standard errors and magnitudes.

Parameters:

• columns (list, optional) – specify a subset of the columns to plot
• errorbar_kwargs – pass in additional plotting commands to matplotlib errorbar command

Returns:

ax – the matplotlib axis that be edited.

Return type:

matplotlib axis

plot_covariate_groups(covariates, values, plot_baseline=True, **kwargs)

Produces a visual representation comparing the baseline survival curve of the model versus what happens when a covariate(s) is varied over values in a group. This is useful to compare subjects’ survival as we vary covariate(s), all else being held equal. The baseline survival curve is equal to the predicted survival curve at all average values in the original dataset.

Parameters:

• covariates (string or list) – a string (or list of strings) of the covariate in the original dataset that we wish to vary.
• values (1d or 2d iterable) – an iterable of the values we wish the covariate to take on.
• plot_baseline (bool) – also display the baseline survival, defined as the survival at the mean of the original dataset.
• kwargs – pass in additional plotting commands

Returns:

ax – the matplotlib axis that be edited.

Return type:

matplotlib axis, or list of axis’

Examples

>>> from lifelines import datasets, WeibullAFTFitter
>>> rossi = datasets.load_rossi()
>>> wf = WeibullAFTFitter().fit(rossi, 'week', 'arrest')
>>> wf.plot_covariate_groups('prio', values=np.arange(0, 15), cmap='coolwarm')

>>> # multiple variables at once
>>> wf.plot_covariate_groups(['prio', 'paro'], values=[[0, 0], [5, 0], [10, 0], [0, 1], [5, 1], [10, 1]], cmap='coolwarm')

>>> # if you have categorical variables, you can simply things:
>>> wf.plot_covariate_groups(['dummy1', 'dummy2', 'dummy3'], values=np.eye(3))

predict_cumulative_hazard(df, ancillary_df=None, times=None, conditional_after=None)

Predict the median lifetimes for the individuals. If the survival curve of an individual does not cross 0.5, then the result is infinity.

Parameters:

• df (DataFrame) – a (n,d) covariate numpy array or DataFrame. If a DataFrame, columns can be in any order. If a numpy array, columns must be in the same order as the training data.
• times (iterable, optional) – an iterable of increasing times to predict the cumulative hazard at. Default is the set of all durations (observed and unobserved).
• conditional_after (iterable, optional) – Must be equal is size to df.shape[0] (denoted n above). An iterable (array, list, series) of possibly non-zero values that represent how long the subject has already lived for. Ex: if \(T\) is the unknown event time, then this represents :math`T | T > s`. This is useful for knowing the remaining hazard/survival of censored subjects. The new timeline is the remaining duration of the subject, i.e. normalized back to starting at 0.

Returns:

the median lifetimes for the individuals. If the survival curve of an individual does not cross 0.5, then the result is infinity.

Return type:

DataFrame

See also

predict_percentile(), predict_expectation(), predict_survival_function()

predict_expectation(df, ancillary_df=None)

Predict the expectation of lifetimes, \(E[T | x]\).

Parameters:

• X (numpy array or DataFrame) – a (n,d) covariate numpy array or DataFrame. If a DataFrame, columns can be in any order. If a numpy array, columns must be in the same order as the training data.
• ancillary_X (numpy array or DataFrame, optional) – a (n,d) covariate numpy array or DataFrame. If a DataFrame, columns can be in any order. If a numpy array, columns must be in the same order as the training data.

Returns:

percentiles – the median lifetimes for the individuals. If the survival curve of an individual does not cross 0.5, then the result is infinity.

Return type:

DataFrame

See also

predict_median()

predict_median(df, ancillary_df=None, conditional_after=None)

Predict the median lifetimes for the individuals. If the survival curve of an individual does not cross 0.5, then the result is infinity.

Parameters:

• df (DataFrame) – a (n,d) covariate numpy array or DataFrame. If a DataFrame, columns can be in any order. If a numpy array, columns must be in the same order as the training data.
• conditional_after (iterable, optional) – Must be equal is size to df.shape[0] (denoted n above). An iterable (array, list, series) of possibly non-zero values that represent how long the subject has already lived for. Ex: if \(T\) is the unknown event time, then this represents :math`T | T > s`. This is useful for knowing the remaining hazard/survival of censored subjects. The new timeline is the remaining duration of the subject, i.e. normalized back to starting at 0.

Returns:

percentiles – the median lifetimes for the individuals. If the survival curve of an individual does not cross 0.5, then the result is infinity.

Return type:

DataFrame

See also

predict_percentile(), predict_expectation()

predict_percentile(df, ancillary_df=None, p=0.5, conditional_after=None)

Returns the median lifetimes for the individuals, by default. If the survival curve of an individual does not cross p, then the result is infinity. http://stats.stackexchange.com/questions/102986/percentile-loss-functions

Parameters:

• X (numpy array or DataFrame) – a (n,d) covariate numpy array or DataFrame. If a DataFrame, columns can be in any order. If a numpy array, columns must be in the same order as the training data.
• ancillary_X (numpy array or DataFrame, optional) – a (n,d) covariate numpy array or DataFrame. If a DataFrame, columns can be in any order. If a numpy array, columns must be in the same order as the training data.
• p (float, optional (default=0.5)) – the percentile, must be between 0 and 1.
• conditional_after (iterable, optional) – Must be equal is size to df.shape[0] (denoted n above). An iterable (array, list, series) of possibly non-zero values that represent how long the subject has already lived for. Ex: if \(T\) is the unknown event time, then this represents :math`T | T > s`. This is useful for knowing the remaining hazard/survival of censored subjects. The new timeline is the remaining duration of the subject, i.e. normalized back to starting at 0.

Returns:

percentiles

Return type:

DataFrame

See also

predict_median()

predict_survival_function(df, ancillary_df=None, times=None, conditional_after=None)

Predict the survival function for individuals, given their covariates. This assumes that the individual just entered the study (that is, we do not condition on how long they have already lived for.)

Parameters:

• X (numpy array or DataFrame) – a (n,d) covariate numpy array or DataFrame. If a DataFrame, columns can be in any order. If a numpy array, columns must be in the same order as the training data.
• ancillary_X (numpy array or DataFrame, optional) – a (n,d) covariate numpy array or DataFrame. If a DataFrame, columns can be in any order. If a numpy array, columns must be in the same order as the training data.
• times (iterable, optional) – an iterable of increasing times to predict the survival function at. Default is the set of all durations (observed and unobserved).
• conditional_after (iterable, optional) – Must be equal is size to df.shape[0] (denoted n above). An iterable (array, list, series) of possibly non-zero values that represent how long the subject has already lived for. Ex: if \(T\) is the unknown event time, then this represents :math`T | T > s`. This is useful for knowing the remaining hazard/survival of censored subjects. The new timeline is the remaining duration of the subject, i.e. normalized back to starting at 0.

Returns:

survival_function – the survival probabilities of individuals over the timeline

Return type:

DataFrame

print_summary(decimals=2, **kwargs)

Print summary statistics describing the fit, the coefficients, and the error bounds.

Parameters:

• decimals (int, optional (default=2)) – specify the number of decimal places to show
• alpha (float or iterable) – specify confidence intervals to show
• kwargs – print additional metadata in the output (useful to provide model names, dataset names, etc.) when comparing multiple outputs.

score_

The concordance score (also known as the c-index) of the fit. The c-index is a generalization of the ROC AUC to survival data, including censorships.

For this purpose, the score_ is a measure of the predictive accuracy of the fitted model onto the training dataset.

summary

Summary statistics describing the fit.

Returns:df – Contains columns coef, np.exp(coef), se(coef), z, p, lower, upper Return type:DataFrame

LogLogisticAFTFitter

class lifelines.fitters.log_logistic_aft_fitter.LogLogisticAFTFitter(alpha=0.05, penalizer=0.0, l1_ratio=0.0, fit_intercept=True, model_ancillary=False)

Bases: lifelines.fitters.ParametericAFTRegressionFitter

This class implements a Log-Logistic AFT model. The model has parameterized form, with \(\alpha(x) = \exp\left(a_0 + a_1x_1 + ... + a_n x_n \right)\), and optionally, \(\beta(y) = \exp\left(b_0 + b_1 y_1 + ... + b_m y_m \right)\),

The cumulative hazard rate is

\[H(t; x , y) = \log\left(1 + \left(\frac{t}{\alpha(x)}\right)^{\beta(y)}\right)\]

After calling the .fit method, you have access to properties like: params_, print_summary(). A summary of the fit is available with the method print_summary().

Parameters:

• alpha (float, optional (default=0.05)) – the level in the confidence intervals.
• fit_intercept (boolean, optional (default=True)) – Allow lifelines to add an intercept column of 1s to df, and ancillary_df if applicable.
• penalizer (float, optional (default=0.0)) – the penalizer coefficient to the size of the coefficients. See l1_ratio. Must be equal to or greater than 0.
• l1_ratio (float, optional (default=0.0)) – how much of the penalizer should be attributed to an l1 penalty (otherwise an l2 penalty). The penalty function looks like penalizer * l1_ratio * ||w||_1 + 0.5 * penalizer * (1 - l1_ratio) * ||w||^2_2
• model_ancillary (optional (default=False)) – set the model instance to always model the ancillary parameter with the supplied Dataframe. This is useful for grid-search optimization.

params_

The estimated coefficients

Type:DataFrame

confidence_intervals_

The lower and upper confidence intervals for the coefficients

Type:DataFrame

durations

The event_observed variable provided

Type:Series

event_observed

The event_observed variable provided

Type:Series

weights

The event_observed variable provided

Type:Series

variance_matrix_

The variance matrix of the coefficients

Type:numpy array

standard_errors_

the standard errors of the estimates

Type:Series

score_

the concordance index of the model.

Type:float

fit(df, duration_col, event_col=None, ancillary_df=None, fit_intercept=None, show_progress=False, timeline=None, weights_col=None, robust=False, initial_point=None, entry_col=None)

Fit the accelerated failure time model to a right-censored dataset.

Parameters:

• df (DataFrame) – a Pandas DataFrame with necessary columns duration_col and event_col (see below), covariates columns, and special columns (weights). duration_col refers to the lifetimes of the subjects. event_col refers to whether the ‘death’ events was observed: 1 if observed, 0 else (censored).
• duration_col (string) – the name of the column in DataFrame that contains the subjects’ lifetimes.
• event_col (string, optional) – the name of the column in DataFrame that contains the subjects’ death observation. If left as None, assume all individuals are uncensored.
• show_progress (boolean, optional (default=False)) – since the fitter is iterative, show convergence diagnostics. Useful if convergence is failing.
• ancillary_df (None, boolean, or DataFrame, optional (default=None)) – Choose to model the ancillary parameters. If None or False, explicitly do not fit the ancillary parameters using any covariates. If True, model the ancillary parameters with the same covariates as df. If DataFrame, provide covariates to model the ancillary parameters. Must be the same row count as df.
• fit_intercept (bool, optional) – If true, add a constant column to the regression. Overrides value set in class instantiation.
• timeline (array, optional) – Specify a timeline that will be used for plotting and prediction
• weights_col (string) – the column in DataFrame that specifies weights per observation.
• robust (boolean, optional (default=False)) – Compute the robust errors using the Huber sandwich estimator.
• initial_point ((d,) numpy array, optional) – initialize the starting point of the iterative algorithm. Default is the zero vector.
• entry_col (specify a column in the DataFrame that denotes any late-entries (left truncation) that occurred. See) – the docs on left truncation

Returns:

Return type:

self with additional new properties print_summary, params_, confidence_intervals_ and more

Examples

>>> from lifelines import WeibullAFTFitter, LogNormalAFTFitter, LogLogisticAFTFitter
>>>
>>> df = pd.DataFrame({
>>>     'T': [5, 3, 9, 8, 7, 4, 4, 3, 2, 5, 6, 7],
>>>     'E': [1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0],
>>>     'var': [0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2],
>>>     'age': [4, 3, 9, 8, 7, 4, 4, 3, 2, 5, 6, 7],
>>> })
>>>
>>> aft = WeibullAFTFitter()
>>> aft.fit(df, 'T', 'E')
>>> aft.print_summary()
>>> aft.predict_median(df)
>>>
>>> aft = WeibullAFTFitter()
>>> aft.fit(df, 'T', 'E', ancillary_df=df)
>>> aft.print_summary()
>>> aft.predict_median(df)

fit_interval_censoring(df, lower_bound_col, upper_bound_col, event_col=None, ancillary_df=None, fit_intercept=None, show_progress=False, timeline=None, weights_col=None, robust=False, initial_point=None, entry_col=None)

Fit the accelerated failure time model to a interval-censored dataset.

Parameters:

• df (DataFrame) – a Pandas DataFrame with necessary columns lower_bound_col, upper_bound_col (see below), and any other covariates or weights.
• lower_bound_col (string) – the name of the column in DataFrame that contains the subjects’ left-most observation.
• upper_bound_col (string) – the name of the column in DataFrame that contains the subjects’ right-most observation. Values can be np.inf (and should be if the subject is right-censored).
• event_col (string, optional) – the name of the column in DataFrame that contains the subjects’ death observation. If left as None, will be inferred from the start and stop columns (lower_bound==upper_bound means uncensored)
• ancillary_df (None, boolean, or DataFrame, optional (default=None)) – Choose to model the ancillary parameters. If None or False, explicitly do not fit the ancillary parameters using any covariates. If True, model the ancillary parameters with the same covariates as df. If DataFrame, provide covariates to model the ancillary parameters. Must be the same row count as df.
• fit_intercept (bool, optional) – If true, add a constant column to the regression. Overrides value set in class instantiation.
• show_progress (boolean, optional (default=False)) – since the fitter is iterative, show convergence diagnostics. Useful if convergence is failing.
• timeline (array, optional) – Specify a timeline that will be used for plotting and prediction
• weights_col (string) – the column in DataFrame that specifies weights per observation.
• robust (boolean, optional (default=False)) – Compute the robust errors using the Huber sandwich estimator.
• initial_point ((d,) numpy array, optional) – initialize the starting point of the iterative algorithm. Default is the zero vector.
• entry_col (specify a column in the DataFrame that denotes any late-entries (left truncation) that occurred. See) – the docs on left truncation

Returns:

Return type:

self with additional new properties print_summary, params_, confidence_intervals_ and more

Examples

>>> from lifelines import WeibullAFTFitter, LogNormalAFTFitter, LogLogisticAFTFitter
>>>
>>> df = pd.DataFrame({
>>>     'start': [5, 3, 9, 8, 7, 4, 4, 3, 2, 5, 6, 7],
>>>     'stop':  [5, 3, 9, 8, 7, 4, 8, 5, 2, 5, 6, np.inf],  # this last subject is right-censored.
>>>     'E':     [1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0],
>>>     'var': [0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2],
>>>     'age': [4, 3, 9, 8, 7, 4, 4, 3, 2, 5, 6, 7],
>>> })
>>>
>>> aft = WeibullAFTFitter()
>>> aft.fit_interval_censoring(df, 'start', 'stop', 'E')
>>> aft.print_summary()
>>> aft.predict_median(df)
>>>
>>> aft = WeibullAFTFitter()
>>> aft.fit_interval_censoring(df, 'start', 'stop', 'E', ancillary_df=df)
>>> aft.print_summary()
>>> aft.predict_median(df)

fit_left_censoring(df, duration_col=None, event_col=None, ancillary_df=None, fit_intercept=None, show_progress=False, timeline=None, weights_col=None, robust=False, initial_point=None, entry_col=None)

Fit the accelerated failure time model to a left-censored dataset.

Parameters:

• df (DataFrame) – a Pandas DataFrame with necessary columns duration_col and event_col (see below), covariates columns, and special columns (weights). duration_col refers to the lifetimes of the subjects. event_col refers to whether the ‘death’ events was observed: 1 if observed, 0 else (censored).
• duration_col (string) – the name of the column in DataFrame that contains the subjects’ lifetimes/measurements/etc. This column contains the (possibly) left-censored data.
• event_col (string, optional) – the name of the column in DataFrame that contains the subjects’ death observation. If left as None, assume all individuals are uncensored.
• ancillary_df (None, boolean, or DataFrame, optional (default=None)) – Choose to model the ancillary parameters. If None or False, explicitly do not fit the ancillary parameters using any covariates. If True, model the ancillary parameters with the same covariates as df. If DataFrame, provide covariates to model the ancillary parameters. Must be the same row count as df.
• fit_intercept (bool, optional) – If true, add a constant column to the regression. Overrides value set in class instantiation.
• show_progress (boolean, optional (default=False)) – since the fitter is iterative, show convergence diagnostics. Useful if convergence is failing.

timeline: array, optional

Specify a timeline that will be used for plotting and prediction

weights_col: string

the column in DataFrame that specifies weights per observation.

robust: boolean, optional (default=False)

Compute the robust errors using the Huber sandwich estimator.

initial_point: (d,) numpy array, optional

initialize the starting point of the iterative algorithm. Default is the zero vector.

entry_col: specify a column in the DataFrame that denotes any late-entries (left truncation) that occurred. See

the docs on left truncation

Returns: Return type:self with additional new properties print_summary, params_, confidence_intervals_ and more

Examples

>>> from lifelines import WeibullAFTFitter, LogNormalAFTFitter, LogLogisticAFTFitter
>>>
>>> df = pd.DataFrame({
>>>     'T': [5, 3, 9, 8, 7, 4, 4, 3, 2, 5, 6, 7],
>>>     'E': [1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0],
>>>     'var': [0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2],
>>>     'age': [4, 3, 9, 8, 7, 4, 4, 3, 2, 5, 6, 7],
>>> })
>>>
>>> aft = WeibullAFTFitter()
>>> aft.fit_left_censoring(df, 'T', 'E')
>>> aft.print_summary()
>>> aft.predict_median(df)
>>>
>>> aft = WeibullAFTFitter()
>>> aft.fit_left_censoring(df, 'T', 'E', ancillary_df=df)
>>> aft.print_summary()
>>> aft.predict_median(df)

fit_right_censoring(*args, **kwargs)

Alias for fit

See also

fit

log_likelihood_ratio_test()

This function computes the likelihood ratio test for the model. We compare the existing model (with all the covariates) to the trivial model of no covariates.

mean_survival_time_

The mean survival time of the average subject in the training dataset.

median_survival_time_

The median survival time of the average subject in the training dataset.

plot(columns=None, parameter=None, **errorbar_kwargs)

Produces a visual representation of the coefficients, including their standard errors and magnitudes.

Parameters:

• columns (list, optional) – specify a subset of the columns to plot
• errorbar_kwargs – pass in additional plotting commands to matplotlib errorbar command

Returns:

ax – the matplotlib axis that be edited.

Return type:

matplotlib axis

plot_covariate_groups(covariates, values, plot_baseline=True, **kwargs)

Produces a visual representation comparing the baseline survival curve of the model versus what happens when a covariate(s) is varied over values in a group. This is useful to compare subjects’ survival as we vary covariate(s), all else being held equal. The baseline survival curve is equal to the predicted survival curve at all average values in the original dataset.

Parameters:

• covariates (string or list) – a string (or list of strings) of the covariate in the original dataset that we wish to vary.
• values (1d or 2d iterable) – an iterable of the values we wish the covariate to take on.
• plot_baseline (bool) – also display the baseline survival, defined as the survival at the mean of the original dataset.
• kwargs – pass in additional plotting commands

Returns:

ax – the matplotlib axis that be edited.

Return type:

matplotlib axis, or list of axis’

Examples

>>> from lifelines import datasets, WeibullAFTFitter
>>> rossi = datasets.load_rossi()
>>> wf = WeibullAFTFitter().fit(rossi, 'week', 'arrest')
>>> wf.plot_covariate_groups('prio', values=np.arange(0, 15), cmap='coolwarm')

>>> # multiple variables at once
>>> wf.plot_covariate_groups(['prio', 'paro'], values=[[0, 0], [5, 0], [10, 0], [0, 1], [5, 1], [10, 1]], cmap='coolwarm')

>>> # if you have categorical variables, you can simply things:
>>> wf.plot_covariate_groups(['dummy1', 'dummy2', 'dummy3'], values=np.eye(3))

predict_cumulative_hazard(df, ancillary_df=None, times=None, conditional_after=None)

Predict the median lifetimes for the individuals. If the survival curve of an individual does not cross 0.5, then the result is infinity.

Parameters:

• df (DataFrame) – a (n,d) covariate numpy array or DataFrame. If a DataFrame, columns can be in any order. If a numpy array, columns must be in the same order as the training data.
• times (iterable, optional) – an iterable of increasing times to predict the cumulative hazard at. Default is the set of all durations (observed and unobserved).
• conditional_after (iterable, optional) – Must be equal is size to df.shape[0] (denoted n above). An iterable (array, list, series) of possibly non-zero values that represent how long the subject has already lived for. Ex: if \(T\) is the unknown event time, then this represents :math`T | T > s`. This is useful for knowing the remaining hazard/survival of censored subjects. The new timeline is the remaining duration of the subject, i.e. normalized back to starting at 0.

Returns:

the median lifetimes for the individuals. If the survival curve of an individual does not cross 0.5, then the result is infinity.

Return type:

DataFrame

See also

predict_percentile(), predict_expectation(), predict_survival_function()

predict_expectation(df, ancillary_df=None)

Predict the expectation of lifetimes, \(E[T | x]\).

Parameters:

• X (DataFrame) – a (n,d) DataFrame. If a DataFrame, columns can be in any order. If a numpy array, columns must be in the same order as the training data.
• ancillary_X (DataFrame, optional) – a (n,d) DataFrame. If a DataFrame, columns can be in any order. If a numpy array, columns must be in the same order as the training data.

Returns:

percentiles – the median lifetimes for the individuals. If the survival curve of an individual does not cross 0.5, then the result is infinity.

Return type:

DataFrame

See also

predict_median()

predict_median(df, ancillary_df=None, conditional_after=None)

Predict the median lifetimes for the individuals. If the survival curve of an individual does not cross 0.5, then the result is infinity.

Parameters:

• df (DataFrame) – a (n,d) covariate numpy array or DataFrame. If a DataFrame, columns can be in any order. If a numpy array, columns must be in the same order as the training data.
• conditional_after (iterable, optional) – Must be equal is size to df.shape[0] (denoted n above). An iterable (array, list, series) of possibly non-zero values that represent how long the subject has already lived for. Ex: if \(T\) is the unknown event time, then this represents :math`T | T > s`. This is useful for knowing the remaining hazard/survival of censored subjects. The new timeline is the remaining duration of the subject, i.e. normalized back to starting at 0.

Returns:

percentiles – the median lifetimes for the individuals. If the survival curve of an individual does not cross 0.5, then the result is infinity.

Return type:

DataFrame

See also

predict_percentile(), predict_expectation()

predict_percentile(df, ancillary_df=None, p=0.5, conditional_after=None)

Returns the median lifetimes for the individuals, by default. If the survival curve of an individual does not cross p, then the result is infinity. http://stats.stackexchange.com/questions/102986/percentile-loss-functions

Parameters:

• X (DataFrame) – a (n,d) DataFrame. If a DataFrame, columns can be in any order. If a numpy array, columns must be in the same order as the training data.
• ancillary_X (DataFrame, optional) – a (n,d) DataFrame. If a DataFrame, columns can be in any order. If a numpy array, columns must be in the same order as the training data.
• p (float, optional (default=0.5)) – the percentile, must be between 0 and 1.

Returns:

percentiles

Return type:

DataFrame

See also

predict_median()

predict_survival_function(df, ancillary_df=None, times=None, conditional_after=None)

Predict the survival function for individuals, given their covariates. This assumes that the individual just entered the study (that is, we do not condition on how long they have already lived for.)

Parameters:

• X (numpy array or DataFrame) – a (n,d) covariate numpy array or DataFrame. If a DataFrame, columns can be in any order. If a numpy array, columns must be in the same order as the training data.
• ancillary_X (numpy array or DataFrame, optional) – a (n,d) covariate numpy array or DataFrame. If a DataFrame, columns can be in any order. If a numpy array, columns must be in the same order as the training data.
• times (iterable, optional) – an iterable of increasing times to predict the survival function at. Default is the set of all durations (observed and unobserved).
• conditional_after (iterable, optional) – Must be equal is size to df.shape[0] (denoted n above). An iterable (array, list, series) of possibly non-zero values that represent how long the subject has already lived for. Ex: if \(T\) is the unknown event time, then this represents :math`T | T > s`. This is useful for knowing the remaining hazard/survival of censored subjects. The new timeline is the remaining duration of the subject, i.e. normalized back to starting at 0.

Returns:

survival_function – the survival probabilities of individuals over the timeline

Return type:

DataFrame

print_summary(decimals=2, **kwargs)

Print summary statistics describing the fit, the coefficients, and the error bounds.

Parameters:

• decimals (int, optional (default=2)) – specify the number of decimal places to show
• alpha (float or iterable) – specify confidence intervals to show
• kwargs – print additional metadata in the output (useful to provide model names, dataset names, etc.) when comparing multiple outputs.

score_

The concordance score (also known as the c-index) of the fit. The c-index is a generalization of the ROC AUC to survival data, including censorships.

For this purpose, the score_ is a measure of the predictive accuracy of the fitted model onto the training dataset.

summary

Summary statistics describing the fit.

Returns:df – Contains columns coef, np.exp(coef), se(coef), z, p, lower, upper Return type:DataFrame

PiecewiseExponentialRegressionFitter

class lifelines.fitters.piecewise_exponential_regression_fitter.PiecewiseExponentialRegressionFitter(breakpoints, alpha=0.05, penalizer=0.0)

Bases: lifelines.fitters.ParametricRegressionFitter

TODO: docs

Examples

See blog post <here https://dataorigami.net/blogs/napkin-folding/churn>_.

fit(df, duration_col, event_col=None, regressors=None, show_progress=False, timeline=None, weights_col=None, robust=False, initial_point=None, entry_col=None)

Fit the regression model to a right-censored dataset.

Parameters:

• df (DataFrame) – a Pandas DataFrame with necessary columns duration_col and event_col (see below), covariates columns, and special columns (weights). duration_col refers to the lifetimes of the subjects. event_col refers to whether the ‘death’ events was observed: 1 if observed, 0 else (censored).
• duration_col (string) – the name of the column in DataFrame that contains the subjects’ lifetimes.
• event_col (string, optional) – the name of the column in DataFrame that contains the subjects’ death observation. If left as None, assume all individuals are uncensored.
• show_progress (boolean, optional (default=False)) – since the fitter is iterative, show convergence diagnostics. Useful if convergence is failing.
• regressors (dict, optional) – a dictionary of parameter names -> list of column names that maps model parameters to a linear combination of variables. If left as None, all variables will be used for all parameters.
• timeline (array, optional) – Specify a timeline that will be used for plotting and prediction
• weights_col (string) – the column in DataFrame that specifies weights per observation.
• robust (boolean, optional (default=False)) – Compute the robust errors using the Huber sandwich estimator.
• initial_point ((d,) numpy array, optional) – initialize the starting point of the iterative algorithm. Default is the zero vector.
• entry_col (specify a column in the DataFrame that denotes any late-entries (left truncation) that occurred. See) – the docs on left truncation

Returns:

self with additional new properties

Return type:

print_summary, params_, confidence_intervals_ and more

fit_interval_censoring()

fit_left_censoring()

fit_right_censoring(*args, **kwargs)

Alias for fit

See also

fit

log_likelihood_ratio_test()

This function computes the likelihood ratio test for the model. We compare the existing model (with all the covariates) to the trivial model of no covariates.

mean_survival_time_

The mean survival time of the average subject in the training dataset.

median_survival_time_

The median survival time of the average subject in the training dataset.

plot(columns=None, parameter=None, **errorbar_kwargs)

Produces a visual representation of the coefficients, including their standard errors and magnitudes.

Parameters:

• columns (list, optional) – specify a subset of the columns to plot
• errorbar_kwargs – pass in additional plotting commands to matplotlib errorbar command

Returns:

ax – the matplotlib axis that be edited.

Return type:

matplotlib axis

plot_covariate_groups(covariates, values, plot_baseline=True, **kwargs)

Produces a plot comparing the baseline survival curve of the model versus what happens when a covariate(s) is varied over values in a group. This is useful to compare subjects’ survival as we vary covariate(s), all else being held equal. The baseline survival curve is equal to the predicted survival curve at all average values in the original dataset.

Parameters:

• covariates (string or list) – a string (or list of strings) of the covariate in the original dataset that we wish to vary.
• values (1d or 2d iterable) – an iterable of the values we wish the covariate to take on.
• plot_baseline (bool) – also display the baseline survival, defined as the survival at the mean of the original dataset.
• kwargs – pass in additional plotting commands

Returns:

ax – the matplotlib axis that be edited.

Return type:

matplotlib axis, or list of axis’

Examples

>>> from lifelines import datasets, WeibullAFTFitter
>>> rossi = datasets.load_rossi()
>>> wf = WeibullAFTFitter().fit(rossi, 'week', 'arrest')
>>> wf.plot_covariate_groups('prio', values=np.arange(0, 15, 3), cmap='coolwarm')

>>> # multiple variables at once
>>> wf.plot_covariate_groups(['prio', 'paro'], values=[[0, 0], [5, 0], [10, 0], [0, 1], [5, 1], [10, 1]], cmap='coolwarm')

>>> # if you have categorical variables, you can simply things:
>>> wf.plot_covariate_groups(['dummy1', 'dummy2', 'dummy3'], values=np.eye(3))

predict_cumulative_hazard(df, times=None, conditional_after=None)

Return the cumulative hazard rate of subjects in X at time points.

Parameters:

• X (numpy array or DataFrame) – a (n,d) covariate numpy array or DataFrame. If a DataFrame, columns can be in any order. If a numpy array, columns must be in the same order as the training data.
• times (iterable, optional) – an iterable of increasing times to predict the cumulative hazard at. Default is the set of all durations (observed and unobserved). Uses a linear interpolation if points in time are not in the index.

Returns:

cumulative_hazard_ – the cumulative hazard of individuals over the timeline

Return type:

DataFrame

predict_expectation(X)

Compute the expected lifetime, \(E[T]\), using covariates X. This algorithm to compute the expectation is to use the fact that \(E[T] = \int_0^\inf P(T > t) dt = \int_0^\inf S(t) dt\). To compute the integral, we use the trapizoidal rule to approximate the integral. .. caution:

However, if the survival function doesn't converge to 0, the the expectation is really infinity and the returned
values are meaningless/too large. In that case, using ``predict_median`` or ``predict_percentile`` would be better.

Parameters:X (numpy array or DataFrame) – a (n,d) covariate numpy array or DataFrame. If a DataFrame, columns can be in any order. If a numpy array, columns must be in the same order as the training data. Returns:expectations Return type:DataFrame

Notes

If X is a DataFrame, the order of the columns do not matter. But if X is an array, then the column ordering is assumed to be the same as the training dataset.

See also

predict_median(), predict_percentile()

predict_median(df, conditional_after=None)

Predict the median lifetimes for the individuals. If the survival curve of an individual does not cross 0.5, then the result is infinity.

Parameters:

• X (numpy array or DataFrame) – a (n,d) covariate numpy array or DataFrame. If a DataFrame, columns can be in any order. If a numpy array, columns must be in the same order as the training data.
• conditional_after (iterable, optional) – Must be equal is size to df.shape[0] (denoted n above). An iterable (array, list, series) of possibly non-zero values that represent how long the subject has already lived for. Ex: if \(T\) is the unknown event time, then this represents :math`T | T > s`. This is useful for knowing the remaining hazard/survival of censored subjects. The new timeline is the remaining duration of the subject, i.e. normalized back to starting at 0.

Returns:

percentiles – the median lifetimes for the individuals. If the survival curve of an individual does not cross 0.5, then the result is infinity.

Return type:

DataFrame

See also

predict_percentile(), predict_expectation()

predict_percentile(df, p=0.5, conditional_after=None)

predict_survival_function(df, times=None, conditional_after=None)

Predict the survival function for individuals, given their covariates. This assumes that the individual just entered the study (that is, we do not condition on how long they have already lived for.)

Parameters:

• df (DataFrame) – a (n,d) DataFrame. If a DataFrame, columns can be in any order. If a numpy array, columns must be in the same order as the training data.
• times (iterable, optional) – an iterable of increasing times to predict the cumulative hazard at. Default is the set of all durations (observed and unobserved). Uses a linear interpolation if points in time are not in the index.
• conditional_after (iterable, optional) – Must be equal is size to df.shape[0] (denoted n above). An iterable (array, list, series) of possibly non-zero values that represent how long the subject has already lived for. Ex: if \(T\) is the unknown event time, then this represents :math`T | T > s`. This is useful for knowing the remaining hazard/survival of censored subjects.

Returns:

survival_function – the survival probabilities of individuals over the timeline

Return type:

DataFrame

print_summary(decimals=2, **kwargs)

Print summary statistics describing the fit, the coefficients, and the error bounds.

Parameters:

• decimals (int, optional (default=2)) – specify the number of decimal places to show
• alpha (float or iterable) – specify confidence intervals to show
• kwargs – print additional metadata in the output (useful to provide model names, dataset names, etc.) when comparing multiple outputs.

score_

The concordance score (also known as the c-index) of the fit. The c-index is a generalization of the ROC AUC to survival data, including censorships.

For this purpose, the score_ is a measure of the predictive accuracy of the fitted model onto the training dataset.

summary

Summary statistics describing the fit.

Returns:df – Contains columns coef, np.exp(coef), se(coef), z, p, lower, upper Return type:DataFrame