Model Checking in the Proportional Hazard model When we get the data set, sometime we are more interesting in the hazards ration. For example, how is the new treatment or new drug compared to the standard one, so we fit the pH model.

Influential observations PH assumption Model adequacy Influential observations PH assumption Example: leaders data Total 472 observations Total 11 covariates fitted in the initial PH model Manner start age conflict loginc region After we fit the model ,we will ask: how is model fit? How to assess the model? a couple of things we need take care is checking

Assessing model adequacy Residuals checking: Cox-Snell (& modified) Martingale Deviance Schoenfeld Score First to checking the model overall fit , …. These are the widely used residuals, some of them we will talk more detail later.

Residual Plots (a) Cox-Snell residual plot Yielding a straight line passing origin with unity slope When we got the residuals, we can plot them. Some graph will be helpful. As we have learned, If the fitted cox model is correct, the residuals will be a straight line ……..the values rci have a unit exponential distribution.

(b) Index plot of Martingale, Deviance residuals take values between - and one in large samples uncorrelated with each other, with mean zero can be interpreted as the difference the observed and expected number of deaths in (0,ti), for ith individual.

(c) deviance residuals vs. risk score A transformation of martingale residuals More symmetrically distributed about zero The smaller residuals, the better fitted by the model Risk score: the linear predictor in Large negative values a lower than average risk of fail Large positive values a higher than average risk of fail Sometimes the plot of …. Is more helpful. As we know, the deviance residual is ………… the residuals have negative value when their survival time is less than expected.

Plot of the deviance residuals against linear predictor

(3) Checking the functional form of covariate Martingale residual from fitting the null model (contains no covariates) plot residuals vs. covariate of interest a straight line indicates a linear term is needed (but most time need using smoother)

Martingale residuals from null model vs start It shows a roughly straight line

Indentifying influential observations delta-beta statistics the difference in the parameter estimate between the all observations fit and the ith observation omitted from the fit assessing the influence of observations on a parameter estimate To identify the influential observation, there are two basic ways. One is using the statistics called Delta-bets……. An approximation to the change is based on the score residual.

-------Lowest------ ------Highest------ SAS output Approx. delta-betas for manner Extreme Observations -------Lowest------ ------Highest------ Value Obs Value Obs -0.0300321 342 0.0109460 180 -0.0263841 234 0.0121594 212 -0.0180988 428 0.0128754 403 -0.0168336 141 0.0141662 258 -0.0156685 247 0.0154732 286 Approx. LDi --------Lowest------- ------Highest------ 6.21925E-05 302 0.0838694 234 6.33683E-05 345 0.0973350 286 7.60771E-05 346 0.1086945 54 7.62382E-05 395 0.1152216 342 8.00260E-05 373 0.1250329 449 In SAS we can use the dfbeta and lrchange option to do the diagnostic analysis. A negative value that means removing this observation will decrease the hazard of fail relative to the baseline hazard. Positive will increase …………

Likelihood displacement (LD) The changes of the maximized log-likelihood if omitting the ith observation from the fit Assessing the influence of observations on the overall fit( the set of parameter estimates)

Treatment: Check the original data, corrected it if found any mistakes Make inferences based on both situations (full & reduced data), contrast results In recording or transcribing……………..sometimes we can’t confirm the data is valid

Testing PH assumption A crucial assumption when using Cox model The effect of covariates on the hazard rate are the same over time

Before fitting a Cox Model Grouping data according the level of one or more factors Plot Parallel curves if the hazard ratios are proportional across the different groups In SAS, strata option

Checking PH assumption for manner It is roughly parallel.

Checking PH assumption for region Since we include total 11 covariates in the model, this plot takes no account of the values of the others variables, this could be the survive time has been affected by other variants. There is little reason to suspect the assumption.

After fitting a Cox Model Plot weighted Schoenfeld residuals vs. time the time-varying coefficient at the ith failure time the estimate in the fitted Cox PH model When the data has some time-varying effect covariate, we can plot weighted schoenfeld residuals against time to detect such effect. Since the scoenfeld residuals has such properties:

Plot weighted Schoenfeld residuals vs. time (cont’s) Detect if some form of time dependency in particular covariate exist A horizontal line show the coefficient is constant, and PH assumption is valid

Adding a time-dependent covariate if hazard rate varies with time For example, a PH model with one covariate for ith observation assumes If the hazard ratio varies with time between two groups, we can include an interaction term to the model: The relative hazard is now , which depend on t. If the coefficient is significant, the model is no longer a PH model. The test of the hypothesis that =0 is a test of the PH assumption. If the PH assumption is fail,

Summary Assessing model fit: Influence diagnostics: Residuals & residual plots Functional form of the covariates Influence diagnostics: Delta-beta statistics LD statistics Testing PH assumption : Plot of log[-logS] vs. log(t) Plot of Schoenfeld residuals vs. t Adding a time-dependent variable