1. Time-dependent covariates and further remarks on likelihood construction
Presenter Li,Yin
Nov. 24

2. 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

3. 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. 

4. Cox regression model for uis_small

5. 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).

6. 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.

7. 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;

8. Including time dependent covariates in the cox model (continued)

9. 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 non-proportional predictors.

10. 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.

11. 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.

12. 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.

13. 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

14. Under random censoring

15. For any t>0, let the history be
Then, the likelihood can be constructed as a product of the conditional terms.

16. Let be the set of labels associated with the individuals failing in ,and is the set of labels associated with the individuals censored in

18. 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.

19. Likelihood construction for internal covariates
The probability contribution of the interval is written as
Where has failure, censoring and covariates information up to time t; and has only the covariates information.