1. 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.

2. 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

3. 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.

4. 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.

5. (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.

6. (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.

7. Plot of the deviance residuals against linear predictor

8. (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)

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

10. 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.

11. -------Lowest------ ------Highest------
SAS output
Approx. delta-betas for manner Extreme Observations
Lowest Highest------
Value Obs Value Obs
Approx. LDi Lowest Highest------ E E E E E
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 …………

12. 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)

13. 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

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

15. 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

16. Checking PH assumption for manner
It is roughly parallel.

17. 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.

18. 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:

19. 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

20. 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,

21. 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