Transcript Document

Time-dependent covariates and
further remarks on likelihood
construction
Presenter Li,Yin
Nov. 24
Tests of proportionality for cox regrssion
1 Kaplan-Meier curves
Works best for time fixed covariates
with few levels.
2 Including time dependent covariates in
the cox model
3 Tests and graphs based on the
Schoenfeld residuals
Data set for example
 The goal of the UIS data is to model time until return to drug use .
 The patients were randomly assigned to two different sites (site=0 is site A
and site=1 is site B).
 Herco indicates heroine or cocaine use in the past three months (herco=1
indicates heroine and cocaine use, herco=2 indicates either heroine or
cocaine use and herco=3 indicates neither heroine nor cocaine use).
 ndrugtx indicates the number of previous drug treatments.
Cox regression model for uis_small
Kaplan-Meier curves
 It is very easy to create the graphs in SAS using proc lifetest. The
plot option in the model statement lets you specify both the survival
function versus time as well as the log(-log(survival) versus log(time).
Tests and graphs based on the Schoenfeld residuals
 Testing the time dependent covariates is equivalent to testing for a
non-zero slope in a generalized linear regression of the scaled
Schoenfeld residuals on functions of time. A non-zero slope is an
indication of a violation of the proportional hazard assumption.
Including time dependent covariates in the cox model
 Generate the time dependent covariates by creating interactions of the
predictors and a function of survival time and include in the model. If
any of the time dependent covariates are significant then those
predictors are not proportional.
proc phreg data=uis;
model time*censor(0) = age race treat site
agesite aget racet treatt sitet;
aget = age*log(time);
racet = race*log(time);
treatt = treat*log(time);
sitet = site*log(time);
proportionality_test: test aget, racet, treat, sitet;
run;
Including time dependent covariates in the cox model
(continued)
When the proportional hazard assumption is violated
 If one of the predictors was not proportional there are various
solutions to consider.
 We can change from using a semi-parametric Cox regression
model to using a parametric regression model.
 Another solution is to include the time-dependent variable for the
non-proportional predictors.
 Finally, we can use a model where we stratify on the nonproportional predictors.
Two types of time dependent covariates
External
Internal
 A covariate that is not external is called internal.
 Internal covariates typically arise as time-dependent measurements taken on an
individual study subject, the path of which is affected by the survival status.
Examples
External covariates
 1st type: fixed or time independent covariates
 2nd type: defined covariates
eg. Stress factors under control of the experimenter
that is to be
varied in a predetermined way
eg. Age of an individual in a trial of long duration
 3rd type: ancillary covariates
eg. Measures of pollutions as a predictor for the
frequency of asthma attacks.
Examples
Internal covariates
 Eg. Measures of a patient’s general condition take
values of 0,1,2,3,4. 0 is assigned to z(t) for dead and 4 is
for no disease; and 1,2,3 represent levels of decreasing
disability.
 Eg. Measures of immune status such as white blood
count as a predictor for examining the effect of
immunotherapy on the failure rate in cancer.
Likelihood construction
Problem setup:
n individuals start on test at t=0;
The risk of failure at time t:
where x is a vector of fixed basic covariates measured
in advance and theta is a vector of parameters.
The data for the ith individual are
Under random censoring
For any t>0, let the history be
Then, the likelihood can be constructed as a
product of the conditional terms.
 Let
be the set of labels
associated with the individuals failing
in
,and
is the set of labels
associated with the individuals censored
in
.
 The first factor on the right side arises from the failure
information
• The remaining factor arises from the censoring
information
If this term depends on the
, the censoring
mechanism is said to be informative, and otherwise
noninformative.
•
Likelihood construction for internal covariates
 The probability contribution of the interval
written as
Where
has failure, censoring and covariates information
up to time t; and
has only the
covariates information.
is