Survival Analysis

In this lab, we perform survival analyses on three separate data sets. In Section~\ref{brain.cancer.sec} we analyze the BrainCancer data that was first described in Section~\ref{sec:KM}. In Section~\ref{time.to.pub.sec}, we examine the Publication data from Section~\ref{sec:pub}. Finally, Section~\ref{call.center.sec} explores a simulated call-center data set.

We begin by importing some of our libraries at this top level. This makes the code more readable, as scanning the first few lines of the notebook tell us what libraries are used in this notebook.

from matplotlib.pyplot import subplots
import numpy as np
import pandas as pd
from ISLP.models import ModelSpec as MS
from ISLP import load_data

We also collect the new imports needed for this lab.

from lifelines import \
     (KaplanMeierFitter,
      CoxPHFitter)
from lifelines.statistics import \
     (logrank_test,
      multivariate_logrank_test)
from ISLP.survival import sim_time

Brain Cancer Data

We begin with the BrainCancer data set, contained in the ISLP package.

BrainCancer = load_data('BrainCancer')
BrainCancer.columns

Index(['sex', 'diagnosis', 'loc', 'ki', 'gtv', 'stereo', 'status', 'time'], dtype='object')

The rows index the 88 patients, while the 8 columns contain the predictors and outcome variables. We first briefly examine the data.

BrainCancer['sex'].value_counts()

sex
Female    45
Male      43
Name: count, dtype: int64

BrainCancer['diagnosis'].value_counts()

diagnosis
Meningioma    42
HG glioma     22
Other         14
LG glioma      9
Name: count, dtype: int64

BrainCancer['status'].value_counts()

status
0    53
1    35
Name: count, dtype: int64

Before beginning an analysis, it is important to know how the status variable has been coded. Most software uses the convention that a status of 1 indicates an uncensored observation (often death), and a status of 0 indicates a censored observation. But some scientists might use the opposite coding. For the BrainCancer data set 35 patients died before the end of the study, so we are using the conventional coding.

To begin the analysis, we re-create the Kaplan-Meier survival curve shown in Figure~\ref{fig:survbrain}. The main package we will use for survival analysis is lifelines. The variable time corresponds to \(y_i\), the time to the \(i\)th event (either censoring or death). The first argument to km.fit is the event time, and the second argument is the censoring variable, with a 1 indicating an observed failure time. The plot() method produces a survival curve with pointwise confidence intervals. By default, these are 90% confidence intervals, but this can be changed by setting the alpha argument to one minus the desired confidence level.

fig, ax = subplots(figsize=(8,8))
km = KaplanMeierFitter()
km_brain = km.fit(BrainCancer['time'], BrainCancer['status'])
km_brain.plot(label='Kaplan Meier estimate', ax=ax)

<Axes: xlabel='timeline'>

Next we create Kaplan-Meier survival curves that are stratified by sex, in order to reproduce Figure~\ref{fig:survbrain2}. We do this using the groupby() method of a dataframe. This method returns a generator that can be iterated over in the for loop. In this case, the items in the for loop are 2-tuples representing the groups: the first entry is the value of the grouping column sex while the second value is the dataframe consisting of all rows in the dataframe matching that value of sex. We will want to use this data below in the log-rank test, hence we store this information in the dictionary by_sex. Finally, we have also used the notion of string interpolation to automatically label the different lines in the plot. String interpolation is a powerful technique to format strings — Python has many ways to facilitate such operations.

fig, ax = subplots(figsize=(8,8))
by_sex = {}
for sex, df in BrainCancer.groupby('sex'):
    by_sex[sex] = df
    km_sex = km.fit(df['time'], df['status'])
    km_sex.plot(label='Sex=%s' % sex, ax=ax)

As discussed in Section~\ref{sec:logrank}, we can perform a log-rank test to compare the survival of males to females. We use the logrank_test() function from the lifelines.statistics module. The first two arguments are the event times, with the second denoting the corresponding (optional) censoring indicators.

logrank_test(by_sex['Male']['time'],
             by_sex['Female']['time'],
             by_sex['Male']['status'],
             by_sex['Female']['status'])

t_0	-1
null_distribution	chi squared
degrees_of_freedom	1
test_name	logrank_test

	test_statistic	p	-log2(p)
0	1.44	0.23	2.12

The resulting \(p\)-value is \(0.23\), indicating no evidence of a difference in survival between the two sexes.

Next, we use the CoxPHFitter() estimator from lifelines to fit Cox proportional hazards models. To begin, we consider a model that uses sex as the only predictor.

coxph = CoxPHFitter # shorthand
sex_df = BrainCancer[['time', 'status', 'sex']]
model_df = MS(['time', 'status', 'sex'],
              intercept=False).fit_transform(sex_df)
cox_fit = coxph().fit(model_df,
                      'time',
                      'status')
cox_fit.summary[['coef', 'se(coef)', 'p']]

	coef	se(coef)	p
covariate
sex[Male]	0.407668	0.342004	0.233262

The first argument to fit should be a data frame containing at least the event time (the second argument time in this case), as well as an optional censoring variable (the argument status in this case). Note also that the Cox model does not include an intercept, which is why we used the intercept=False argument to ModelSpec above. The summary() method delivers many columns; we chose to abbreviate its output here. It is possible to obtain the likelihood ratio test comparing this model to the one with no features as follows:

cox_fit.log_likelihood_ratio_test()

null_distribution	chi squared
degrees_freedom	1
test_name	log-likelihood ratio test

	test_statistic	p	-log2(p)
0	1.44	0.23	2.12

Regardless of which test we use, we see that there is no clear evidence for a difference in survival between males and females. As we learned in this chapter, the score test from the Cox model is exactly equal to the log rank test statistic!

Now we fit a model that makes use of additional predictors. We first note that one of our diagnosis values is missing, hence we drop that observation before continuing.

cleaned = BrainCancer.dropna()
all_MS = MS(cleaned.columns, intercept=False)
all_df = all_MS.fit_transform(cleaned)
fit_all = coxph().fit(all_df,
                      'time',
                      'status')
fit_all.summary[['coef', 'se(coef)', 'p']]

	coef	se(coef)	p
covariate
sex[Male]	0.183748	0.360358	0.610119
diagnosis[LG glioma]	-1.239530	0.579555	0.032455
diagnosis[Meningioma]	-2.154566	0.450524	0.000002
diagnosis[Other]	-1.268870	0.617672	0.039949
loc[Supratentorial]	0.441195	0.703669	0.530665
ki	-0.054955	0.018314	0.002693
gtv	0.034293	0.022333	0.124661
stereo[SRT]	0.177778	0.601578	0.767597

The diagnosis variable has been coded so that the baseline corresponds to HG glioma. The results indicate that the risk associated with HG glioma is more than eight times (i.e. \(e^{2.15}=8.62\)) the risk associated with meningioma. In other words, after adjusting for the other predictors, patients with HG glioma have much worse survival compared to those with meningioma. In addition, larger values of the Karnofsky index, ki, are associated with lower risk, i.e. longer survival.

Finally, we plot estimated survival curves for each diagnosis category, adjusting for the other predictors. To make these plots, we set the values of the other predictors equal to the mean for quantitative variables and equal to the mode for categorical. To do this, we use the apply() method across rows (i.e. axis=0) with a function representative that checks if a column is categorical or not.

levels = cleaned['diagnosis'].unique()
def representative(series):
    if hasattr(series.dtype, 'categories'):
        return pd.Series.mode(series)
    else:
        return series.mean()
modal_data = cleaned.apply(representative, axis=0)

We make four copies of the column means and assign the diagnosis column to be the four different diagnoses.

modal_df = pd.DataFrame(
              [modal_data.iloc[0] for _ in range(len(levels))])
modal_df['diagnosis'] = levels
modal_df

	sex	diagnosis	loc	ki	gtv	stereo	status	time
0	Female	Meningioma	Supratentorial	80.91954	8.687011	SRT	0.402299	27.188621
0	Female	HG glioma	Supratentorial	80.91954	8.687011	SRT	0.402299	27.188621
0	Female	LG glioma	Supratentorial	80.91954	8.687011	SRT	0.402299	27.188621
0	Female	Other	Supratentorial	80.91954	8.687011	SRT	0.402299	27.188621

We then construct the model matrix based on the model specification all_MS used to fit the model, and name the rows according to the levels of diagnosis.

modal_X = all_MS.transform(modal_df)
modal_X.index = levels
modal_X

	sex[Male]	diagnosis[LG glioma]	diagnosis[Meningioma]	diagnosis[Other]	loc[Supratentorial]	ki	gtv	stereo[SRT]	status	time
Meningioma	0.0	0.0	1.0	0.0	1.0	80.91954	8.687011	1.0	0.402299	27.188621
HG glioma	0.0	0.0	0.0	0.0	1.0	80.91954	8.687011	1.0	0.402299	27.188621
LG glioma	0.0	1.0	0.0	0.0	1.0	80.91954	8.687011	1.0	0.402299	27.188621
Other	0.0	0.0	0.0	1.0	1.0	80.91954	8.687011	1.0	0.402299	27.188621

We can use the predict_survival_function() method to obtain the estimated survival function.

predicted_survival = fit_all.predict_survival_function(modal_X)
predicted_survival

	Meningioma	HG glioma	LG glioma	Other
0.07	0.997947	0.982430	0.994881	0.995029
1.18	0.997947	0.982430	0.994881	0.995029
1.41	0.995679	0.963342	0.989245	0.989555
1.54	0.995679	0.963342	0.989245	0.989555
2.03	0.995679	0.963342	0.989245	0.989555
...	...	...	...	...
65.02	0.688772	0.040136	0.394181	0.404936
67.38	0.688772	0.040136	0.394181	0.404936
73.74	0.688772	0.040136	0.394181	0.404936
78.75	0.688772	0.040136	0.394181	0.404936
82.56	0.688772	0.040136	0.394181	0.404936

85 rows × 4 columns

This returns a data frame, whose plot methods yields the different survival curves. To avoid clutter in the plots, we do not display confidence intervals.

fig, ax = subplots(figsize=(8, 8))
predicted_survival.plot(ax=ax);

Publication Data

The Publication data presented in Section~\ref{sec:pub} can be found in the ISLP package. We first reproduce Figure~\ref{fig:lauersurv} by plotting the Kaplan-Meier curves stratified on the posres variable, which records whether the study had a positive or negative result.

fig, ax = subplots(figsize=(8,8))
Publication = load_data('Publication')
by_result = {}
for result, df in Publication.groupby('posres'):
    by_result[result] = df
    km_result = km.fit(df['time'], df['status'])
    km_result.plot(label='Result=%d' % result, ax=ax)

As discussed previously, the \(p\)-values from fitting Cox’s proportional hazards model to the posres variable are quite large, providing no evidence of a difference in time-to-publication between studies with positive versus negative results.

posres_df = MS(['posres',
                'time',
                'status'],
                intercept=False).fit_transform(Publication)
posres_fit = coxph().fit(posres_df,
                         'time',
                         'status')
posres_fit.summary[['coef', 'se(coef)', 'p']]

	coef	se(coef)	p
covariate
posres	0.148076	0.161625	0.359579

However, the results change dramatically when we include other predictors in the model. Here we exclude the funding mechanism variable.

model = MS(Publication.columns.drop('mech'),
           intercept=False)
coxph().fit(model.fit_transform(Publication),
            'time',
            'status').summary[['coef', 'se(coef)', 'p']]

	coef	se(coef)	p
covariate
posres	0.570773	0.175960	1.179610e-03
multi	-0.040860	0.251194	8.707842e-01
clinend	0.546183	0.262000	3.709944e-02
sampsize	0.000005	0.000015	7.507005e-01
budget	0.004386	0.002465	7.515984e-02
impact	0.058318	0.006676	2.426306e-18

We see that there are a number of statistically significant variables, including whether the trial focused on a clinical endpoint, the impact of the study, and whether the study had positive or negative results.

Call Center Data

In this section, we will simulate survival data using the relationship between cumulative hazard and the survival function explored in Exercise \ref{ex:all3}. Our simulated data will represent the observed wait times (in seconds) for 2,000 customers who have phoned a call center. In this context, censoring occurs if a customer hangs up before his or her call is answered.

There are three covariates: Operators (the number of call center operators available at the time of the call, which can range from \(5\) to \(15\)), Center (either A, B, or C), and Time of day (Morning, Afternoon, or Evening). We generate data for these covariates so that all possibilities are equally likely: for instance, morning, afternoon and evening calls are equally likely, and any number of operators from \(5\) to \(15\) is equally likely.

rng = np.random.default_rng(10)
N = 2000
Operators = rng.choice(np.arange(5, 16),
                       N,
                       replace=True)
Center = rng.choice(['A', 'B', 'C'],
                    N,
                    replace=True)
Time = rng.choice(['Morn.', 'After.', 'Even.'],
                   N,
                   replace=True)
D = pd.DataFrame({'Operators': Operators,
                  'Center': pd.Categorical(Center),
                  'Time': pd.Categorical(Time)})

We then build a model matrix (omitting the intercept)

model = MS(['Operators',
            'Center',
            'Time'],
           intercept=False)
X = model.fit_transform(D)

It is worthwhile to take a peek at the model matrix X, so that we can be sure that we understand how the variables have been coded. By default, the levels of categorical variables are sorted and, as usual, the first column of the one-hot encoding of the variable is dropped.

X[:5]

	Operators	Center[B]	Center[C]	Time[Even.]	Time[Morn.]
0	13	0.0	1.0	0.0	0.0
1	15	0.0	0.0	1.0	0.0
2	7	1.0	0.0	0.0	1.0
3	7	0.0	1.0	0.0	1.0
4	13	0.0	1.0	1.0	0.0

Next, we specify the coefficients and the hazard function.

true_beta = np.array([0.04, -0.3, 0, 0.2, -0.2])
true_linpred = X.dot(true_beta)
hazard = lambda t: 1e-5 * t

Here, we have set the coefficient associated with Operators to equal \(0.04\); in other words, each additional operator leads to a \(e^{0.04}=1.041\)-fold increase in the “risk” that the call will be answered, given the Center and Time covariates. This makes sense: the greater the number of operators at hand, the shorter the wait time! The coefficient associated with Center == B is \(-0.3\), and Center == A is treated as the baseline. This means that the risk of a call being answered at Center B is 0.74 times the risk that it will be answered at Center A; in other words, the wait times are a bit longer at Center B.

Recall from Section~\ref{Ch2-statlearn-lab:loading-data} the use of lambda for creating short functions on the fly. We use the function sim_time() from the ISLP.survival package. This function uses the relationship between the survival function and cumulative hazard \(S(t) = \exp(-H(t))\) and the specific form of the cumulative hazard function in the Cox model to simulate data based on values of the linear predictor true_linpred and the cumulative hazard. We need to provide the cumulative hazard function, which we do here.

cum_hazard = lambda t: 1e-5 * t**2 / 2

We are now ready to generate data under the Cox proportional hazards model. We truncate the maximum time to 1000 seconds to keep simulated wait times reasonable. The function sim_time() takes a linear predictor, a cumulative hazard function and a random number generator.

W = np.array([sim_time(l, cum_hazard, rng)
              for l in true_linpred])
D['Wait time'] = np.clip(W, 0, 1000)

We now simulate our censoring variable, for which we assume 90% of calls were answered (Failed==1) before the customer hung up (Failed==0).

D['Failed'] = rng.choice([1, 0],
                         N,
                         p=[0.9, 0.1])
D[:5]

	Operators	Center	Time	Wait time	Failed
0	13	C	After.	525.064979	1
1	15	A	Even.	254.677835	1
2	7	B	Morn.	487.739224	1
3	7	C	Morn.	308.580292	1
4	13	C	Even.	154.174608	1

D['Failed'].mean()

0.9075

We now plot Kaplan-Meier survival curves. First, we stratify by Center.

fig, ax = subplots(figsize=(8,8))
by_center = {}
for center, df in D.groupby('Center'):
    by_center[center] = df
    km_center = km.fit(df['Wait time'], df['Failed'])
    km_center.plot(label='Center=%s' % center, ax=ax)
ax.set_title("Probability of Still Being on Hold")

Text(0.5, 1.0, 'Probability of Still Being on Hold')

Next, we stratify by Time.

fig, ax = subplots(figsize=(8,8))
by_time = {}
for time, df in D.groupby('Time'):
    by_time[time] = df
    km_time = km.fit(df['Wait time'], df['Failed'])
    km_time.plot(label='Time=%s' % time, ax=ax)
ax.set_title("Probability of Still Being on Hold")

Text(0.5, 1.0, 'Probability of Still Being on Hold')

It seems that calls at Call Center B take longer to be answered than calls at Centers A and C. Similarly, it appears that wait times are longest in the morning and shortest in the evening hours. We can use a log-rank test to determine whether these differences are statistically significant using the function multivariate_logrank_test().

multivariate_logrank_test(D['Wait time'],
                          D['Center'],
                          D['Failed'])

t_0	-1
null_distribution	chi squared
degrees_of_freedom	2
test_name	multivariate_logrank_test

	test_statistic	p	-log2(p)
0	20.30	<0.005	14.65

Next, we consider the effect of Time.

multivariate_logrank_test(D['Wait time'],
                          D['Time'],
                          D['Failed'])

t_0	-1
null_distribution	chi squared
degrees_of_freedom	2
test_name	multivariate_logrank_test

	test_statistic	p	-log2(p)
0	49.90	<0.005	35.99

As in the case of a categorical variable with 2 levels, these results are similar to the likelihood ratio test from the Cox proportional hazards model. First, we look at the results for Center.

X = MS(['Wait time',
        'Failed',
        'Center'],
        intercept=False).fit_transform(D)
F = coxph().fit(X, 'Wait time', 'Failed')
F.log_likelihood_ratio_test()

null_distribution	chi squared
degrees_freedom	2
test_name	log-likelihood ratio test

	test_statistic	p	-log2(p)
0	20.58	<0.005	14.85

Next, we look at the results for Time.

X = MS(['Wait time',
        'Failed',
        'Time'],
       intercept=False).fit_transform(D)
F = coxph().fit(X, 'Wait time', 'Failed')
F.log_likelihood_ratio_test()

null_distribution	chi squared
degrees_freedom	2
test_name	log-likelihood ratio test

	test_statistic	p	-log2(p)
0	48.12	<0.005	34.71

We find that differences between centers are highly significant, as are differences between times of day.

Finally, we fit Cox’s proportional hazards model to the data.

X = MS(D.columns,
       intercept=False).fit_transform(D)
fit_queuing = coxph().fit(
                  X,
                 'Wait time',
                 'Failed')
fit_queuing.summary[['coef', 'se(coef)', 'p']]

	coef	se(coef)	p
covariate
Operators	0.043934	0.007520	5.143589e-09
Center[B]	-0.236060	0.058113	4.864162e-05
Center[C]	0.012231	0.057518	8.316096e-01
Time[Even.]	0.268845	0.057797	3.294956e-06
Time[Morn.]	-0.148217	0.057334	9.733557e-03

The \(p\)-values for Center B and evening time are very small. It is also clear that the hazard — that is, the instantaneous risk that a call will be answered — increases with the number of operators. Since we generated the data ourselves, we know that the true coefficients for Operators, Center = B, Center = C, Time = Even. and Time = Morn. are \(0.04\), \(-0.3\), \(0\), \(0.2\), and \(-0.2\), respectively. The coefficient estimates from the fitted Cox model are fairly accurate.