Love does not come by demanding from others, but it is a self initiation.
Survival Analysis
1
Survival Analysis
Semiparametric Proportional
Hazards Regression (Part III)
2
Survival Analysis
Hypothesis Tests for the
Regression Coefficients

Does the entire set of variables contribute
significantly to the prediction of
survivorship? (global test)

Does the addition of a group variables
contribute significantly to the prediction of
survivorship over and above that achieved
by other variables? (local test)
3
Survival Analysis
Three Tests
They are all likelihood-based tests:
Likelihood Ratio (LR) Test
 Wald Test
 Score Test

4
Survival Analysis
Three Tests
Asymptotically equivalent
 Approximately low-order Taylor series
expansion of each other
 LR test considered most reliable and
Wald test the least

5
Survival Analysis
Global Tests


6
Overall test for a model containing p
covariates
H0: b1 = b2 = ... = bp = 0
Survival Analysis
Global Tests
7
Survival Analysis
Global Tests
8
Survival Analysis
Local Tests

Tests for the additional contribution of
a group of covariates

Suppose X1,…,Xp are included in the
model already and Xp+1,…,Xq are yet
included
9
Survival Analysis
Local Tests
10
Survival Analysis
Local Tests
Only one: likelihood ratio test
 The statistics -2logPLn(MPLE) is a
measure of “amount” of collected
information; the smaller the better.
 It sometimes inappropriately referred
to as a deviance; it does not measure
deviation from the saturated model
(the model which is prefect fit to the
data)

11
Survival Analysis
12
Survival Analysis
Example: PBC

Consider the following models:
LR test stat = 2.027; DF = 2; p-value =0.3630
 conclusion?
13
Survival Analysis
Estimation of Survival
Function
To estimate S(y|X), the baseline
survival function S0(y) must be
estimated first.
 Two estimates:

Breslow estimate
 Kalbfleisch-Prentice estimate

14
Survival Analysis
Breslow Estimate
15
Survival Analysis
Kalbfleisch-Prentice
Estimate

An estimate of h0(y) was derived by
Kalbfleisch and Prentice using an approach
based on the method of maximum
likelihood.

Reference: Kalbfleisc, J.D. and Prentice,
R.L. (1973). Marginal likelihoods based on
Cox’s regression and life model. Biometrika,
60, 267-278
16
Survival Analysis
Example: PBC
17
Survival Analysis
Estimation of the Median
Survival Time
18
Survival Analysis
19
Survival Analysis
Example: PBC

The estimated median survival time for 60year-old males treated with DPCA is 2105
days (=5.76 years) with an approximate
95% C. I. (970.86,3239.14).

The estimated median survival time for 40year-old males treated with DPCA is 3584
days (=9.81 years) with an approximate
95% C. I. (2492.109, 4675.891).
20
Survival Analysis
Assessment of Model
Adequacy
Model-based inferences depend
completely on the fitted statistical
model  validity of these inferences
depends on the adequacy of the
model
 The evaluation of model adequacy are
often based on quantities known as
residuals

21
Survival Analysis
Residuals for Cox Models

Four major residuals:
Cox-Snell residuals (to check for overall fit)
 Martingale residuals (to identify functional
forms and assess PH assumption)
 Deviance residuals (to identify outliers)
 Schoenfeld residuals (to assess PH
assumption)

22
Survival Analysis
Cox-Snell Residuals
23
Survival Analysis
24
Survival Analysis
Limitations




Do not indicate the type of departure when the
plot is not linear.
Do not take into account (heaving) censoring.
The exponential distribution for the residuals
holds only when the actual parameter values
are used.
Crowley & Storer (1983, JASA 78, 277-281)
showed empirically that the plot is ineffective
at assessing overall model adequacy; can
only identify a very poor fit.
25
Survival Analysis
Martingale Residuals
Martingale residuals are a transformation of Cox-Snell residuals.
26
Survival Analysis
Martingale Residuals

Martingale residuals are useful for
exploring the correct functional form
for the effect of a covariate.

Example: PBC
27
Survival Analysis
Martingale Residuals
1.
Fit a full (or final) model.
2.
Plot the martingale residuals against
each ordinal covariate separately.
3.
Superimpose a scatterplot smooth
(such as LOESS) to see the
functional form for the covariate.
28
Survival Analysis
29
Survival Analysis
30
Survival Analysis
Martingale Residuals

Example: PBC
The covariates are now modified to
be: Age, log(bili), and other
categorical variables.

The simple method may fail when
covariates are correlated.
31
Survival Analysis
Deviance Residuals

Martingale residuals are a
transformation of Cox-Snell residuals

Deviance residuals are a
transformation of martingale residuals.
32
Survival Analysis
Deviance Residuals

Deviance residuals can be used like
residuals from OLS regression:
They follow approximately the standard
normal distribution when censoring is light
(<25%)

Can help to identify outliers (subjects with
poor fit):


33
Large positive value  died too soon
Large negative value  lived too long
Survival Analysis
Example: PBC
34
Survival Analysis
Schoenfeld Residuals
35
Survival Analysis
Assessing the Proportional
Hazards Assumption



The main function of Schoenfeld residuals
is to detect possible departures from the
proportional hazards (PH) assumption.
The plot of Schoenfeld residual against
survival time (or its rank) should show a
random scatter of points centered on 0
A time-dependent pattern is evidence
against the PH assumption.
Ref: Schoenfeld, D. (1982). Partial residauls for the
proportional hazards regression model. Biometrika, Vol.
69, P. 239-241
36
Survival Analysis
Scaled schoenfeld residuals
37
Survival Analysis
Assessing the Proportional
Hazards Assumption



38
Scaled Schoenfeld residuals is popular than
the un-scaled ones to detect possible
departures from the proportional hazards
(PH) assumption. (SAS uses this.)
A time-dependent pattern is evidence
against the PH assumption.
Most of tests for PH are tests for zero
slopes in a linear regression of scaled Sch.
residuals on chosen functions of times.
Survival Analysis
Example: PBC
39
Survival Analysis
Example: PBC
40
Survival Analysis
Example: PBC
41
Survival Analysis
Assessing the Proportional
Hazards Assumption
By empirical score
process/simulations
 In SAS: add a statement
ASSESS PH/ RESAMPLE;
 A p-value will be given to assess the
significance level of deviation from the
proportional hazards assumption

42
Survival Analysis
Strategies for Non-proportionality




Stratify the covariates with non-proportional
effects
 No test for the effect of a stratification
factor (so only for nuisance covariates)
 How to categorize a numerical covariate?
Partition the time axis
Add a time-dependent covariate
Use a different model (such as AFT model)
43
Survival Analysis
The End
Good Luck for Finals!!
44
Survival Analysis