11.8.1 Brain Cancer Data

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

In [3]:BrainCancer=load_data('BrainCancer')
BrainCancer.columns

Out[3]: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.

In [4]:BrainCancer['sex'].value_counts()

Out[4]:Female45
Male43
Name:sex,dtype:int64

In [5]:BrainCancer['diagnosis'].value_counts()

490 11. Survival Analysis and Censored Data

Out[5]:Meningioma42
HGglioma22
Other14
LGglioma9
Name:diagnosis,dtype:int64

In [6]:BrainCancer['status'].value_counts()

Out[6]:053
135
Name:status,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 11.2. The main package we will use for survival analysis is lifelines . The variable time corresponds to yi , the time to the i th lifelines 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 .plot() 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.

In [7]:fig,ax=subplots(figsize=(8,8))
km=KaplanMeierFitter()
km_brain=km.fit(BrainCancer['time'],BrainCancer['status'])
km_brain.plot(label='KaplanMeierestimate',ax=ax)

Next we create Kaplan-Meier survival curves that are stratified by sex , in order to reproduce Figure 11.3. We do this using the groupby() method of .groupby() 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 logrank test, hence we store this information in the dictionary by_sex . Finally, we have also used the notion of string interpolation to automatically label string the different lines in the plot. String interpolation is a powerful technique interpolation to format strings — Python has many ways to facilitate such operations.

In [8]:fig,ax=subplots(figsize=(8,8))
by_sex={}
forsex,dfinBrainCancer.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 11.4, we can perform a log-rank test to compare the survival of males to females. We use the logrank_test() function from logrank_

test()

11.8 Lab: Survival Analysis 491

the lifelines.statistics module. The first two arguments are the event times, with the second denoting the corresponding (optional) censoring indicators.

In [9]:logrank_test(by_sex['Male']['time'],
by_sex['Female']['time'],
by_sex['Male']['status'],
by_sex['Female']['status'])

Out[9]:

t_0-1
null_distributionchisquared
degrees_of_freedom1
test_namelogrank_test
test_statisticp-log2(p)
1.440.232.12

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 CoxPHFitter() proportional hazards models. To begin, we consider a model that uses sex as the only predictor.

In [10]: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']]

Out[10]:

coefse(coef)p
covariate
sex[Male]0.4076670.3420040.233263

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:

In [11]:cox_fit.log_likelihood_ratio_test()

Out[11]:null_distributionchisquared
degrees_freedom1
test_namelog-likelihoodratiotest
test_statisticp-log2(p)
1.440.232.12

492 11. Survival Analysis and Censored Data

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.

In [12]: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']]

Out[12]:

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.

In [13]:levels=cleaned['diagnosis'].unique()
defrepresentative(series):
ifhasattr(series.dtype,'categories'):
returnpd.Series.mode(series)
else:
returnseries.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.

In [14]:modal_df=pd.DataFrame(
[modal_data.iloc[0]for_inrange(len(levels))])
modal_df['diagnosis']=levels
modal_df

11.8 Lab: Survival Analysis 493

Out[14]:

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 .

In [15]:modal_X=all_MS.transform(modal_df)
modal_X.index=levels
modal_X

We can use the predict_survival_function() method to obtain the esti- .predict_ mated survival function.

survival_
function()

In [16]:predicted_survival=fit_all.predict_survival_function(modal_X)
predicted_survival

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.

In [17]:fig,ax=subplots(figsize=(8,8))
predicted_survival.plot(ax=ax);