# WeibullAFTFitter¶

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

Bases: lifelines.fitters.ParametericAFTRegressionFitter, lifelines.fitters.mixins.ProportionalHazardMixin

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, $$\rho(y) = \exp\left(\alpha_0 + \alpha_1 y_1 + ... + \alpha_m y_m \right)$$,

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

With no covariates, the Weibull model’s parameters has the following interpretations: The $$\lambda$$ (scale) parameter has an applicable interpretation: it represent the time when 37% of the population has died. The $$\rho$$ (shape) parameter controls if the cumulative hazard (see below) is convex or concave, representing accelerating or decelerating hazards.

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 if applicable. penalizer (float or array, optional (default=0.0)) – the penalizer coefficient to the size of the coefficients. See l1_ratio. Must be equal to or greater than 0. Alternatively, penalizer is an array equal in size to the number of parameters, with penalty coefficients for specific variables. For example, penalizer=0.01 * np.ones(p) is the same as penalizer=0.01 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: DataFrame
standard_errors_

the standard errors of the estimates

Type: Series
score_

the concordance index of the model.

Type: float
AIC_
BIC_
check_assumptions(training_df: pandas.core.frame.DataFrame, advice: bool = True, show_plots: bool = False, p_value_threshold: float = 0.01, plot_n_bootstraps: int = 15, columns: Optional[List[str]] = None) → 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 (bool, optional) – display advice as output to the user’s screen show_plots (bool, 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. A list of list of axes objects.

Examples

from lifelines.datasets import load_rossi
from lifelines import CoxPHFitter

cph = CoxPHFitter().fit(rossi, 'week', 'arrest')

axes = cph.check_assumptions(rossi, show_plots=True)


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

compute_followup_hazard_ratios(training_df: pandas.core.frame.DataFrame, followup_times: Iterable[T_co]) → pandas.core.frame.DataFrame

Recompute the hazard ratio at different follow-up times (lifelines handles accounting for updated censoring and updated durations). This is useful because we need to remember that the hazard ratio is actually a weighted-average of period-specific hazard ratios.

Parameters: training_df (pd.DataFrame) – The same dataframe used to train the model followup_times (Iterable) – a list/array of follow-up times to recompute the hazard ratio at.
compute_residuals(training_dataframe: pandas.core.frame.DataFrame, kind: str) → pandas.core.frame.DataFrame

Compute the residuals the model.

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

Notes

• 'scaled_schoenfeld': lifelines does not add the coefficients to the final results, but R does when you call residuals(c, "scaledsch")
concordance_index_

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 concordance_index_ is a measure of the predictive accuracy of the fitted model onto the training dataset.

fit(df, duration_col, event_col=None, ancillary=None, fit_intercept=None, show_progress=False, timeline=None, weights_col=None, robust=False, initial_point=None, entry_col=None, formula: str = None, fit_options: Optional[dict] = None) → ParametericAFTRegressionFitter

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 (bool, optional (default=False)) – since the fitter is iterative, show convergence diagnostics. Useful if convergence is failing. formula (string) – Use an R-style formula for modeling the dataset. See formula syntax: https://matthewwardrop.github.io/formulaic/basic/grammar/ If a formula is not provided, all variables in the dataframe are used (minus those used for other purposes like event_col, etc.)
ancillary: None, boolean, str, 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. If str, should be a formula
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: bool, 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: string
specify a column in the DataFrame that denotes any late-entries (left truncation) that occurred. See the docs on left truncation
fit_options: dict, optional
pass kwargs into the underlying minimization algorithm, like tol, etc.
Returns: 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)
aft.print_summary()
aft.predict_median(df)

fit_intercept = False
fit_interval_censoring(df, lower_bound_col, upper_bound_col, event_col=None, ancillary=None, fit_intercept=None, show_progress=False, timeline=None, weights_col=None, robust=False, initial_point=None, entry_col=None, formula=None, fit_options: Optional[dict] = None) → ParametericAFTRegressionFitter

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) formula (string) – Use an R-style formula for modeling the dataset. See formula syntax: https://matthewwardrop.github.io/formulaic/basic/grammar/ If a formula is not provided, all variables in the dataframe are used (minus those used for other purposes like event_col, etc.) ancillary (None, boolean, str, 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. If str, should be a formula fit_intercept (bool, optional) – If true, add a constant column to the regression. Overrides value set in class instantiation. show_progress (bool, 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 (bool, 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 (str) – specify a column in the DataFrame that denotes any late-entries (left truncation) that occurred. See the docs on left truncation fit_options (dict, optional) – pass kwargs into the underlying minimization algorithm, like tol, etc. 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)
aft.print_summary()
aft.predict_median(df)

fit_left_censoring(df, duration_col: str = None, event_col: str = None, ancillary=None, fit_intercept=None, show_progress=False, timeline=None, weights_col=None, robust=False, initial_point=None, entry_col=None, formula: str = None, fit_options: Optional[dict] = None) → ParametericAFTRegressionFitter

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. formula (string) – Use an R-style formula for modeling the dataset. See formula syntax: https://matthewwardrop.github.io/formulaic/basic/grammar/ If a formula is not provided, all variables in the dataframe are used (minus those used for other purposes like event_col, etc.) ancillary (None, boolean, str, 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. If str, should be a formula fit_intercept (bool, optional) – If true, add a constant column to the regression. Overrides value set in class instantiation. show_progress (bool, 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 (bool, 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 (str) – specify a column in the DataFrame that denotes any late-entries (left truncation) that occurred. See the docs on left truncation fit_options (dict, optional) – pass kwargs into the underlying minimization algorithm, like tol, etc. self 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)
aft.print_summary()
aft.predict_median(df)

fit_right_censoring(*args, **kwargs)

Alias for fit

force_no_intercept = False
hazard_ratios_
log_likelihood_ratio_test() → StatisticalResult

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, ax=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 ax – the matplotlib axis that be edited. matplotlib axis
plot_covariate_groups(*args, **kwargs)

Deprecated as of v0.25.0. Use plot_partial_effects_on_outcome instead.

plot_partial_effects_on_outcome(covariates, values, plot_baseline=True, times=None, y='survival_function', ax=None, **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. times (iterable) – pass in a times to plot kwargs – pass in additional plotting commands ax – the matplotlib axis that be edited. matplotlib axis, or list of axis’

Examples

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

# multiple variables at once
wf.plot_partial_effects_on_outcome(['prio', 'paro'], values=[[0, 0], [5, 0], [10, 0], [0, 1], [5, 1], [10, 1]], cmap='coolwarm', y="hazard")

predict_cumulative_hazard(df, *, ancillary=None, times=None, conditional_after=None) → pandas.core.frame.DataFrame

Predict the cumulative hazard for the individuals.

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. ancillary – supply an dataframe to regress ancillary parameters against, if necessary. 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 (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 $$T | T > s$$. 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.
predict_expectation(df: pandas.core.frame.DataFrame, ancillary: Optional[pandas.core.frame.DataFrame] = None) → pandas.core.series.Series

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 (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. the expected lifetimes for the individuals. DataFrame
predict_hazard(df, *, ancillary=None, times=None, conditional_after=None) → pandas.core.frame.DataFrame

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, Series, 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 – supply an dataframe to regress ancillary parameters against, if necessary. 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) – Not implemented yet
predict_median(df, *, ancillary=None, conditional_after=None) → pandas.core.frame.DataFrame

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. ancillary – supply an dataframe to regress ancillary parameters against, if necessary. conditional_after (iterable, optional) – Must be equal is size to df.shape (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 $$T | T > s$$. 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.
predict_percentile(df: pandas.core.frame.DataFrame, *, ancillary: Optional[pandas.core.frame.DataFrame] = None, p: float = 0.5, conditional_after: Optional[autograd.numpy.numpy_wrapper.array] = None) → pandas.core.series.Series

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 (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. percentiles DataFrame
predict_survival_function(df, times=None, conditional_after=None, ancillary=None) → pandas.core.frame.DataFrame

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 – supply an dataframe to regress ancillary parameters against, if necessary. 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 (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 $$T | T > s$$. 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.
print_summary(decimals: int = 2, style: Optional[str] = None, columns: Optional[list] = None, **kwargs) → None

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 style (string) – {html, ascii, latex} columns – only display a subset of summary columns. Default all. kwargs – print additional metadata in the output (useful to provide model names, dataset names, etc.) when comparing multiple outputs.
regressors = None
score(df: pandas.core.frame.DataFrame, scoring_method: str = 'log_likelihood') → float

Score the data in df on the fitted model. With default scoring method, returns the _average log-likelihood_.

Parameters: df (DataFrame) – the dataframe with duration col, event col, etc. scoring_method (str) – one of {‘log_likelihood’, ‘concordance_index’} log_likelihood: returns the average unpenalized log-likelihood. concordance_index: returns the concordance-index

Examples

from lifelines import WeibullAFTFitter
from lifelines.datasets import load_rossi


strata = None
summary