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

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

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

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)

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

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

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

7. Global Tests

8. Global Tests

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

10. Local Tests

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)

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

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

15. Breslow Estimate

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, 267-278

17. Example: PBC

18. Estimation of the Median Survival Time

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. (970.86,3239.14). • 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. (2492.109, 4675.891).

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

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)

23. Cox-Snell Residuals

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, 277-281) showed empirically that the plot is ineffective at assessing overall model adequacy.

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

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

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.

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.

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

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

34. Example: PBC

35. Schoenfeld Residuals

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. 239-241

37. Scaled schoenfeld residuals

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.

39. Example: PBC

40. Example: PBC

41. Example: PBC

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)

43. The End Good Luck for Finals!!