http://i.imgur.com/EOowdSD.png

Survival Regression

Often we have additional data aside from the duration, and if applicable any censorships that occurred. In the regime dataset, we have the type of government the political leader was part of, the country they were head of, and the year they were elected. Can we use this data in survival analysis?

Yes, the technique is called survival regression – the name implies we regress covariates (e.g., year elected, country, etc.) against a another variable – in this case durations and lifetimes. Similar to the logic in the first part of this tutorial, we cannot use traditional methods like linear regression.

There are two popular competing techniques in survival regression: Cox’s model and Aalen’s additive model. Both models attempt to represent the hazard rate \(\lambda(t | x)\) as a function of \(t\) and some covariates \(x\). In Cox’s model, the relationship is defined:

\[\lambda(t | x) = b_0(t)\exp\left( b_1x_1 + ... + b_dx_d\right)\]

On the other hand, Aalen’s additive model assumes the following form:

\[\lambda(t | x) = b_0(t) + b_1(t)x_1 + ... + b_d(t)x_d\]

Cox’s Proportional Hazard model

Lifelines has an implementation of the Cox propotional hazards regression model (implemented in R under coxph). The idea behind the model is that the log-hazard of an individual is a linear function of their static covariates and a population-level baseline hazard that changes over time. Mathematically:

\[\lambda(t | x) = \overbrace{b_0(t)}^{\text{baseline}}\underbrace{\exp \overbrace{\left(\sum_{i=1}^n b_i x_i \right)}^{\text{log-partial hazard}}}_ {\text{partial hazard}}\]

Note a few facts about this model: the only time component is in the baseline hazard, \(b_0(t)\). In the above product, the partial hazard is a time-invariant scalar factor that only increases or decreases the baseline hazard. Thus a changes in covariates will only increase or decrease this baseline hazard.

Lifelines implementation

The implementation of the Cox model in lifelines, called CoxPHFitter has a similar API to AalensAdditiveFitter. Like R, it has a print_summary function that prints a tabular view of coefficients and related stats.

This example data is from the paper here, avaible as load_rossi in lifelines.

from lifelines.datasets import load_rossi
from lifelines import CoxPHFitter

rossi_dataset = load_rossi()
cph = CoxPHFitter()
cph.fit(rossi_dataset, duration_col='week', event_col='arrest', show_progress=True)

cph.print_summary()  # access the results using cph.summary

"""
n=432, number of events=114

        coef  exp(coef)  se(coef)       z      p  lower 0.95  upper 0.95
fin  -0.3790     0.6845    0.1914 -1.9806 0.0476     -0.7542     -0.0039   *
age  -0.0572     0.9444    0.0220 -2.6042 0.0092     -0.1003     -0.0142  **
race  0.3141     1.3691    0.3080  1.0198 0.3078     -0.2897      0.9180
wexp -0.1511     0.8597    0.2121 -0.7124 0.4762     -0.5670      0.2647
mar  -0.4328     0.6487    0.3818 -1.1335 0.2570     -1.1813      0.3157
paro -0.0850     0.9185    0.1957 -0.4341 0.6642     -0.4687      0.2988
prio  0.0911     1.0954    0.0286  3.1824 0.0015      0.0350      0.1472  **
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Concordance = 0.640
Likelihood ratio test = 33.266 on 7 df, p=0.00002
"""

To access the coefficients and the baseline hazard directly, you can use cph.hazards_ and cph.baseline_hazard_ respectively.

Convergence

Fitting the Cox model to the data involves using gradient descent. Lifelines takes extra effort to help with convergence. If you wish to see the fitting, there is a show_progress parameter in CoxPHFitter.fit function. For further help, see Problems with convergence in the Cox Proportional Hazard Model.

After fitting, the value of the maximum log-likelihood this available using cph._log_likelihood. Similarly, the score and Hessian matrix are available under _score_ and _hessian_ respectively. The _hessian_ can be used the find the covariance matrix of the coefficients.

Goodness of fit and prediction

After fitting, you may want to know how “good” of a fit your model was to the data. Aside from traditional approaches, two methods the author has found useful is to 1. look at the concordance-index (see below section on Model Selection in Survival Regression), available as cph.score_ or in the print_summary and 2. compare spread between the baseline survival function vs the Kaplan Meier survival function (Why? a small spread between these two curves means that the impact of the exponential in the Cox model does very little, whereas a large spread means most of the changes in individual hazard can be attributed to the exponential term). For example, the first figure below is a good fit, and the second figure is a much weaker fit.

_images/goodfit.png _images/badfit.png

After fitting, you can use use the suite of prediction methods (similar to Aalen’s additive model): .predict_partial_hazard, .predict_survival_function, etc.

X = rossi_dataset.drop(["week", "arrest"], axis=1)
cph.predict_partial_hazard(X)
cph.predict_survival_function(X)

Plotting the coefficients

With a fitted model, an altervative way to view the coefficients and their ranges is to use the plot method.

from lifelines.datasets import load_rossi
from lifelines import CoxPHFitter

rossi_dataset = load_rossi()
cph = CoxPHFitter()
cph.fit(rossi_dataset, duration_col='week', event_col='arrest', show_progress=True)

cph.plot()
_images/coxph_plot.png

Plotting the effect of varying a covariate

After fitting, we can plot what the survival curves look like as we vary a single covarite while holding everything else equal. This is useful to understand the impact of a covariate, given the model. To do this, we use the plot_covariate_groups method and give it the covariate of interest, and the values to display.

from lifelines.datasets import load_rossi
from lifelines import CoxPHFitter

rossi_dataset = load_rossi()
cph = CoxPHFitter()
cph.fit(rossi_dataset, duration_col='week', event_col='arrest', show_progress=True)

cph.plot_covariate_groups('prio', [0, 5, 10, 15])
_images/coxph_plot_covarite_groups.png

Checking the proportional hazards assumption

A quick and visual way to check the proportional hazards assumption of a variable is to plot the survival curves segmented by the values of the variable. If the survival curves are the same “shape” and differ only by a constant factor, then the assumption holds. A more clear way to see this is to plot what’s called the logs curve: the loglogs (-log(survival curve)) vs log(time). If the curves are parallel (and hence do not cross each other), then it’s likely the variable satisfies the assumption. If the curves do cross, likely you’ll have to “stratify” the variable (see next section). In lifelines, the KaplanMeierFitter object has a .plot_loglogs function for this purpose.

The following is the loglogs curves of two variables in our regime dataset. The first is the democracy type, which does have (close to) parallel lines, hence satisfies our assumption:

from lifelines.datasets import load_dd
from lifelines import KaplanMeierFitter

data = load_dd()

democracy_0 = data.loc[data['democracy'] == 'Non-democracy']
democracy_1 = data.loc[data['democracy'] == 'Democracy']

kmf0 = KaplanMeierFitter()
kmf0.fit(democracy_0['duration'], event_observed=democracy_0['observed'])

kmf1 = KaplanMeierFitter()
kmf1.fit(democracy_1['duration'], event_observed=democracy_1['observed'])

fig, axes = plt.subplots()
kmf0.plot_loglogs(ax=axes)
kmf1.plot_loglogs(ax=axes)

axes.legend(['Non-democracy', 'Democracy'])

plt.show()
_images/lls_democracy.png

The second variable is the regime type, and this variable does not follow the proportional hazards assumption.

_images/lls_regime_type.png

Stratification

Sometimes a covariate may not obey the proportional hazard assumption. In this case, we can allow a factor without estimating its effect to be adjusted. To specify categorical variables to be used in stratification, we define them in the call to fit:

from lifelines.datasets import load_rossi
from lifelines import CoxPHFitter

rossi_dataset = load_rossi()

cph.fit(rossi_dataset, 'week', event_col='arrest', strata=['race'], show_progress=True)

cph.print_summary()  # access the results using cph.summary

"""
n=432, number of events=114

        coef  exp(coef)  se(coef)       z      p  lower 0.95  upper 0.95
fin  -0.3775     0.6856    0.1913 -1.9731 0.0485     -0.7525     -0.0024   *
age  -0.0573     0.9443    0.0220 -2.6081 0.0091     -0.1004     -0.0142  **
wexp -0.1435     0.8664    0.2127 -0.6746 0.4999     -0.5603      0.2734
mar  -0.4419     0.6428    0.3820 -1.1570 0.2473     -1.1907      0.3068
paro -0.0839     0.9196    0.1958 -0.4283 0.6684     -0.4677      0.3000
prio  0.0919     1.0962    0.0287  3.1985 0.0014      0.0356      0.1482  **
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Concordance = 0.638
Likelihood ratio test = 109.634 on 6 df, p=0.00000
"""

Aalen’s Additive model

Warning

This implementation is still experimental.

The estimator to fit unknown coefficients in Aalen’s additive model is located in estimators under AalenAdditiveFitter. For this exercise, we will use the regime dataset and include the categorical variables un_continent_name (eg: Asia, North America,…), the regime type (e.g., monarchy, civilian,…) and the year the regime started in, start_year.

Aalen’s additive model typically does not estimate the individual \(b_i(t)\) but instead estimates \(\int_0^t b_i(s) \; ds\) (similar to the estimate of the hazard rate using NelsonAalenFitter above). This is important to keep in mind when analyzing the output. .. code:: python

from lifelines import AalenAdditiveFitter from lifelines.datasets import load_dd

data = load_dd() data.head()

ctryname cowcode2 politycode un_region_name un_continent_name ehead leaderspellreg democracy regime start_year duration observed
Afghanistan 700 700 Southern Asia Asia Mohammad Zahir Shah Mohammad Zahir Shah.Afghanistan.1946.1952.Mona... Non-democracy Monarchy 1946 7 1
Afghanistan 700 700 Southern Asia Asia Sardar Mohammad Daoud Sardar Mohammad Daoud.Afghanistan.1953.1962.Ci... Non-democracy Civilian Dict 1953 10 1
Afghanistan 700 700 Southern Asia Asia Mohammad Zahir Shah Mohammad Zahir Shah.Afghanistan.1963.1972.Mona... Non-democracy Monarchy 1963 10 1
Afghanistan 700 700 Southern Asia Asia Sardar Mohammad Daoud Sardar Mohammad Daoud.Afghanistan.1973.1977.Ci... Non-democracy Civilian Dict 1973 5 0
Afghanistan 700 700 Southern Asia Asia Nur Mohammad Taraki Nur Mohammad Taraki.Afghanistan.1978.1978.Civi... Non-democracy Civilian Dict 1978 1 0

5 rows × 12 columns

I’m using the lovely library patsy here to create a covariance matrix from my original dataframe.

import patsy
X = patsy.dmatrix('un_continent_name + regime + start_year', data, return_type='dataframe')
X = X.rename(columns={'Intercept': 'baseline'})
X.columns.tolist()
['baseline',
 'un_continent_name[T.Americas]',
 'un_continent_name[T.Asia]',
 'un_continent_name[T.Europe]',
 'un_continent_name[T.Oceania]',
 'regime[T.Military Dict]',
 'regime[T.Mixed Dem]',
 'regime[T.Monarchy]',
 'regime[T.Parliamentary Dem]',
 'regime[T.Presidential Dem]',
 'start_year']

We have also included the coef_penalizer option. During the estimation, a linear regression is computed at each step. Often the regression can be unstable (due to high co-linearity or small sample sizes) – adding a penalizer term controls the stability. I recommend always starting with a small penalizer term – if the estimates still appear to be too unstable, try increasing it.

aaf = AalenAdditiveFitter(coef_penalizer=1.0)

An instance of AalenAdditiveFitter includes a fit method that performs the inference on the coefficients. This method accepts a pandas DataFrame: each row is an individual and columns are the covariates and two individual columns: a duration column and a boolean event occurred column (where event occurred refers to the event of interest - expulsion from government in this case)

X['T'] = data['duration']
X['E'] = data['observed']
aaf.fit(X, 'T', event_col='E')

After fitting, the instance exposes a cumulative_hazards_ DataFrame containing the estimates of \(\int_0^t b_i(s) \; ds\):

aaf.cumulative_hazards_.head()
un_continent_name[Africa] un_continent_name[Americas] un_continent_name[Asia] un_continent_name[Europe] un_continent_name[Oceania] regime[T.Military Dict] regime[T.Mixed Dem] regime[T.Monarchy] regime[T.Parliamentary Dem] regime[T.Presidential Dem] start_year baseline
-0.051595 -0.082406 0.010666 0.154493 -0.060438 0.075333 0.086274 -0.133938 0.048077 0.127171 0.000116 -0.029280
-0.014713 -0.039471 0.095668 0.194251 -0.092696 0.115033 0.358702 -0.226233 0.168783 0.121862 0.000053 0.143039
0.007389 -0.064758 0.115121 0.170549 0.069371 0.161490 0.677347 -0.271183 0.328483 0.146234 0.000004 0.297672
-0.058418 0.011399 0.091784 0.205824 0.125722 0.220028 0.932674 -0.294900 0.365604 0.422617 0.000002 0.376311
-0.099282 0.106641 0.112083 0.150708 0.091900 0.241575 1.123860 -0.391103 0.536185 0.743913 0.000057 0.362049

5 rows × 12 columns

AalenAdditiveFitter also has built in plotting:

aaf.plot(columns=['regime[T.Presidential Dem]', 'baseline', 'un_continent_name[T.Europe]'], iloc=slice(1,15))
_images/survival_regression_aaf.png

Regression is most interesting if we use it on data we have not yet seen, i.e., prediction! We can use what we have learned to predict individual hazard rates, survival functions, and median survival time. The dataset we are using is available up until 2008, so let’s use this data to predict the duration of former Canadian Prime Minister Stephen Harper.

ix = (data['ctryname'] == 'Canada') & (data['start_year'] == 2006)
harper = X.loc[ix]
print("Harper's unique data point:")
print(harper)
Harper's unique data point
array([[    0.,     0.,     1.,     0.,     0.,     0.,     0.,     1.,
            0.,     0.,  2003.]])
ax = plt.subplot(2,1,1)
aaf.predict_cumulative_hazard(harper).plot(ax=ax)

ax = plt.subplot(2,1,2)
aaf.predict_survival_function(harper).plot(ax=ax);
_images/survival_regression_harper.png

Warning

Because of the nature of the model, estimated survival functions of individuals can increase. This is an expected artifact of Aalen’s additive model.

Cox’s Time Varying Proportional Hazard model

Warning

This implementation is still experimental.

Often an individual will have a covariate change over time. An example of this is hospital patients who enter the study and, at some future time, may recieve a heart transplant. We would like to know the effect of the transplant, but we cannot condition on whether they recieved the transplant naively. Consider that if patients needed to wait at least 1 year before getting a transplant, then everyone who dies before that year is considered as a non-transplant patient, and hence this would overestimate the hazard of not recieving a transplant.

We can incorporate changes over time into our survival analysis by using a modification of the Cox model above. The general mathematical description is:

\[\lambda(t | x) = \overbrace{b_0(t)}^{\text{baseline}}\underbrace{\exp \overbrace{\left(\sum_{i=1}^n \beta_i x_i(t) \right)}^{\text{log-partial hazard}}}_ {\text{partial hazard}}\]

Note the time-varying \(x_i(t)\) to denote that covariates can change over time. This model is implemented in lifelines as CoxTimeVaryingFitter. The dataset schema required is different than previous models, so we will spend some time describing this.

Dataset creation for time-varying regression

Lifelines requires that the dataset be in what is called the long format. This looks like one row per state change, including an ID, the left (exclusive) time point, and right (inclusive) time point. For example, the following dataset tracks three unique subjects.

id start stop group z event
1 0 8 1 0 False
2 0 5 0 0 False
2 5 8 0 1 True
3 0 3 1 0 False
3 3 12 1 1 True

5 rows × 6 columns

In the above dataset, start and stop denote the boundaries, id is the unique identifier per subject, and event denotes if the subject died at the end of that period. For example, subject ID 2 had variable z=0 up to and including the end of time period 5 (we can think that measurements happen at end of the time period), after which it was set to 1. Since event is 1 in that row, we conclude that the subject died at time 8,

This desired dataset can be built up from smaller datasets. To do this we can use some helper functions provided in lifelines. Typically, data will be in a format that looks like it comes out of a relational database. You may have a “base” table with ids, durations alive, and a censorsed flag, and possibly static covariates. Ex:

id duration event var1
1 10 True 0.1
2 12 False 0.5

2 rows × 4 columns

We will perform a light transform to this dataset to modify it into the “long” format.

from lifelines.utils import to_long_format

base_df = to_long_format(base_df, duration_col="duration")

The new dataset looks like:

id start stop var1 event
1 0 10 0.1 True
2 0 12 0.5 False

2 rows × 5 columns

You’ll also have secondary dataset that references future measurements. This could come in two “types”. The first is when you have a variable that changes over time (ex: administering varying medication over time, or taking a tempature over time). The second types is an event-based dataset: an event happens at some time in the future (ex: an organ transplant occurs, or an intervention). We will address this second type later. The first type of dataset may look something like:

Example:

id time var2
1 0 1.4
1 4 1.2
1 8 1.5
2 0 1.6

4 rows × 3 columns

where time is the duration from the entry event. Here we see subject 1 had a change in their var2 covariate at the end of time 4 and at the end of time 8. We can use add_covariate_to_timeline to fold the covariate dataset into the original dataset.

from lifelines.utils import add_covariate_to_timeline

df = add_covariate_to_timeline(base_df, cv, duration_col="time", id_col="id", event_col="event")
id start stop var1 var2 event
1 0 4 0.1 1.4 False
1 4 8 0.1 1.2 False
1 8 10 0.1 1.5 True
2 0 12 0.5 1.6 False

4 rows × 6 columns

From the above output, we can see that subject 1 changed state twice over the observation period, finally expiring at the end of time 10. Subject 2 was a censored case, and we lost track of them after time 2.

You may have multiple covariates you wish to add, so the above could be streamlined like so:

from lifelines.utils import add_covariate_to_timeline

df = base_df.pipe(add_covariate_to_timeline, cv1, duration_col="time", id_col="id", event_col="event")\
            .pipe(add_covariate_to_timeline, cv2, duration_col="time", id_col="id", event_col="event")\
            .pipe(add_covariate_to_timeline, cv3, duration_col="time", id_col="id", event_col="event")

If your dataset is of the second type, that is, event-based, your dataset may look something like the following, where values in the matrix denote times since the subject’s birth, and None or NaN represent the event not happening (subjects can be excluded if the event never occurred as well) :

print(event_df)


    id    E1
0   1     1.0
1   2     NaN
2   3     3.0
...

Initially, this can’t be added to our baseline dataframe. However, using utils.covariates_from_event_matrix we can convert a dataframe like this into one that can be easily added.

from lifelines.utils import covariates_from_event_matrix

cv = covariates_from_event_matrix(event_df, id_col="id")
print(cv)


event  id  duration  E1
0       1       1.0   1
1       3       3.0   1
...


base_df = add_covariate_to_timeline(base_df, cv, duration_col="time", id_col="id", event_col="E")

For an example of pulling datasets like this from a SQL-store, and other helper functions, see Example SQL queries and transformations to get time varying data.

Cumulative sums

One additional flag on add_covariate_to_timeline that is of interest is the cumulative_sum flag. By default it is False, but turning it to True will perform a cumulative sum on the covariate before joining. This is useful if the covariates describe an incremental change, instead of a state update. For example, we may have measurements of drugs administered to a patient, and we want the covariate to reflect how much we have administered since the start. Event columns do make sense to cumulative sum as well. In contrast, a covariate to measure the temperature of the patient is a state update, and should not be summed. See Example cumulative total using and time-varying covariates to see an example of this.

Delaying time-varying covariates

add_covariate_to_timeline also has an option for delaying, or shifting, a covariate so it changes later than originally observed. One may ask, why should one delay a time-varying covariate? Here’s an example. Consider investigating the impact of smoking on mortality and available to us are time-varying observations of how many cigarettes are consumed each month. Unbeknownst to us, when a subject reaches critical illness levels, they are admitted to the hospital and their cigarette consumption drops to zero. Some expire while in hospital. If we used this dataset naively, we would see that not smoking leads to sudden death, and conversely, smoking helps your health! This is a case of reverse causation: the upcoming death event actually influences the covariates.

To handle this, you can delay the observations by time periods:

from lifelines.utils import covariates_from_event_matrix


base_df = add_covariate_to_timeline(base_df, cv, duration_col="time", id_col="id", event_col="E", delay=14)

Fitting the model

Once your dataset is in the correct orientation, we can use CoxTimeVaryingFitter to fit the model to your data. The method is similar to CoxPHFitter, expect we need to tell the fit about the additional time columns.

from lifelines import CoxTimeVaryingFitter

ctv = CoxTimeVaryingFitter()
ctv.fit(df, id_col="id", event_col="event", start_col="start", stop_col="stop", show_progress=True)
ctv.print_summary()
ctv.plot()

Short note on prediction

Unlike the other regression models, prediction in a time-varying setting is not trivial. To predict, we would need to know the covariates values beyond the observed times, but if we knew that, we would also know if the subject was still alive or not! However, it is still possible to compute the hazard values of subjects at known observations, the baseline cumulative hazard rate, and baseline survival function. So while CoxTimeVaryingFitter exposes prediction methods, there are logicial limitations to what these predictions mean.

Model Selection in Survival Regression

If censorship is present, it’s not appropriate to use a loss function like mean-squared-error or mean-absolute-loss. Instead, one measure is the concordance-index, also known as the c-index. This measure evaluates the accuracy of the ordering of predicted time. It is infact a generalization of AUC, another common loss function, and is interpreted similarly:

  • 0.5 is the expected result from random predictions,
  • 1.0 is perfect concordance and,
  • 0.0 is perfect anti-concordance (multiply predictions with -1 to get 1.0)

The measure is implemented in lifelines under lifelines.utils.concordance_index and accepts the actual times (along with any censorships) and the predicted times.

Cross Validation

Lifelines has an implementation of k-fold cross validation under lifelines.utils.k_fold_cross_validation. This function accepts an instance of a regression fitter (either CoxPHFitter of AalenAdditiveFitter), a dataset, plus k (the number of folds to perform, default 5). On each fold, it splits the data into a training set and a testing set fits itself on the training set and evaluates itself on the testing set (using the concordance measure).

from lifelines import CoxPHFitter
from lifelines.datasets import load_regression_dataset
from lifelines.utils import k_fold_cross_validation

regression_dataset = load_regression_dataset()
cph = CoxPHFitter()
scores = k_fold_cross_validation(cph, regression_dataset, 'T', event_col='E', k=3)
print(scores)
print(np.mean(scores))
print(np.std(scores))

#[ 0.5896  0.5358  0.5028]
# 0.542
# 0.035