1. Love does not come by demanding from others, but it is a self initiation.
Survival Analysis

2. Semiparametric Proportional Hazards Regression (Part III)
Survival Analysis
Semiparametric Proportional Hazards Regression (Part III)
Survival Analysis

3. 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)
Survival Analysis

4. Three Tests They are all likelihood-based tests:
Likelihood Ratio (LR) Test
Wald Test
Score Test
Survival Analysis

5. Three Tests Asymptotically equivalent
Approximately low-order Taylor series expansion of each other
LR test considered most reliable and Wald test the least
Survival Analysis

6. Global Tests Overall test for a model containing p covariates
H0: b1 = b2 = ... = bp = 0
Survival Analysis

7. Global Tests
Survival Analysis

8. Global Tests
Survival Analysis

9. 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
Survival Analysis

10. Local Tests
Survival Analysis

11. 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)
Survival Analysis

12. Survival Analysis

13. Example: PBC Consider the following models:
LR test stat = 2.027; DF = 2; p-value =0.3630
 conclusion?
Survival Analysis

14. 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
Survival Analysis

15. Breslow Estimate
Survival Analysis

16. 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,
Survival Analysis

17. Example: PBC
Survival Analysis

18. Estimation of the Median Survival Time
Survival Analysis

19. Survival Analysis

20. Example: PBC
The estimated median survival time for 60-year-old males treated with DPCA is 2105 days (=5.76 years) with an approximate 95% C. I. (970.86, ).
The estimated median survival time for 40-year-old males treated with DPCA is 3584 days (=9.81 years) with an approximate 95% C. I. ( , ).
Survival Analysis

21. 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
Survival Analysis

22. Residuals for Cox Models
Four major residuals:
Cox-Snell residuals (to assess overall fitting)
Martingale residuals (to explore the functional form of each covariate)
Deviance residuals (to assess overall fitting and identify outliers)
Schoenfeld residuals (to assess PH assumption)
Survival Analysis

23. Cox-Snell Residuals
Survival Analysis

24. Survival Analysis

25. Limitations
Do not indicate the type of departure when the plot is not linear.
The exponential distribution for the residuals holds only when the actual parameter values are used.
Crowley & Storer (1983, JASA 78, ) showed empirically that the plot is ineffective at assessing overall model adequacy.
Survival Analysis

26. Martingale Residuals
Martingale residuals are a transformation of Cox-Snell residuals.
Survival Analysis

27. Martingale Residuals
Martingale residuals are useful for exploring the correct functional form for the effect of a (ordinal) covariate.
Example: PBC
Survival Analysis

28. Martingale Residuals Fit a full model.
Plot the martingale residuals against each ordinal covariate separately.
Superimpose a scatterplot smooth (such as LOESS) to see the functional form for the covariate.
Survival Analysis

29. Survival Analysis

30. Survival Analysis

31. 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.
Survival Analysis

32. Deviance Residuals
Martingale residuals are a transformation of Cox-Snell residuals
Deviance residuals are a transformation of martingale residuals.
Survival Analysis

33. 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):
Large positive value  died too soon
Large negative value  lived too long
Survival Analysis

34. Example: PBC
Survival Analysis

35. Schoenfeld Residuals
Survival Analysis

36. 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
Survival Analysis

37. Scaled schoenfeld residuals
Survival Analysis

38. Assessing the Proportional Hazards Assumption
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

39. Example: PBC
Survival Analysis

40. Example: PBC
Survival Analysis

41. Example: PBC
Survival Analysis

42. Strategies for Non-proportionality
Stratify the covariates with non-proportional effects
No test for the effect of a stratification factor
How to categorize a numerical covariate?
Partition the time axis
Use a different model (such as AFT model)
Survival Analysis

43. The End
Good Luck for Finals!!
Survival Analysis