1. Event History Analysis 6
Sociology 8811 Lecture 20
Copyright © 2007 by Evan Schofer
Do not copy or distribute without permission

2. Announcements Paper Assignment #2 handed out previously Class topic:
Due April 26
Class topic:
More details on Cox model diagnostics
Next class: parametric models, AFT models (if time)…
Then – new topics!

3. Review: Cox Model Basics
Choosing how to deal with ties
Several algorithms available
Efron method is better than Breslow (default)
Exact marginal is even better – but time consuming
The baseline hazard rate
Can be computed in stata…
Plus, you can estimate the hazard rates at particular values of variables
The proportional hazard assumption
We’ll continue discussing several methods…

4. Proportional Hazard Assumption
Key assumption: Proportional hazards
Estimated Hazard ratios are proportional over time
i.e., Estimates of a hazard ratio do NOT vary over time
Example: Effect of “abstinence” program on sexual behavior
Issue: Do abstinence programs lower the rate in a consistent manner across time?
Or, perhaps the rate is lower initially… but then the rate jumps back up (maybe even exceeds the control group).
Groups are assumed to have “parallel” hazards
Rather than rates that diverge, converge (or cross).

5. Proportional Hazard Assumption
Strategies:
1. Visually examine raw hazard plots for sub-groups in your data
Watch for non-parallel trends
A simple, crude method… but often identifies big violations

6. Proportional Hazard Assumption
Visual examination of raw hazard rate
Parallel trends in hazard rate indicate proportionality

7. Proportional Hazard Assumption
2. Plot –ln(-ln(survival plot)) versus ln(time) across values of X variables
What stata calls “stphplot”
Parallel lines indicate proportional hazards
Again, convergence and divergence (or crossing) indicates violation
A less-common approach: compare observed survivor plot to predicted values (for different values of X)
What stata calls “stcoxkm”
If observed are similar to predicted, assumption is not likely to be violated.

8. Proportional Hazard Assumption
-ln(-ln(survivor)) vs. ln(time) – “stphplot”
Convergence suggests violation of proportional hazard assumption
(But, I’ve seen worse!)

9. Proportional Hazard Assumption
Cox estimate vs. observed KM – “stcoxkm”
Predicted differs from observed for countries in West

10. Proportional Hazard Assumption
3. Piecewise Models
Piecewise = break model up into pieces (by time)
Ex: Split analysis in to “early” vs “late” time
If coefficients vary in different time periods, hazards are not proportional
Example:
stcox var1 var2 var3 if _t < 10
stcox var1 var2 var3 if _t >= 10
Look for large changes in coefficients!

11. Proportional Hazard Assumption
In a piecewise model, coefficients would differ in non-proportional models
Non-Proportional
Proportional
Early
Late
Here, the effect is the same in both time periods
Here, the effect is negative in the early period and positive in the late period

12. Piecewise Models Look at coefficients at 2 (or more) spans of time
EARLY
. stcox gdp degradation education democracy ngo ingo if year < 1985, robust nohr
          _t |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
         gdp |           1.05   0.294       
 degradation |          -1.80   0.072       
   education |         -0.59   0.552       
   democracy |           0.87   0.382        
         ngo |           2.18   0.029        
        ingo |           1.39   0.165       
Note: Effect of ngo is larger in early period
LATE
. stcox gdp degradation education democracy ngo ingo if year >= 1985, robust nohr
          _t |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
         gdp |            1.39   0.165        
 degradation |         -2.38   0.017      
   education |           0.99   0.323       
   democracy |           2.59   0.010        
         ngo |           1.20   0.229       
        ingo |         -0.54   0.590       

13. Proportional Hazard Assumption
4. Tests based on re-estimating model
Try including time interactions in your model
Recall: Interactions – effect of A on C varies with B
If effect of variable X on hazard rate (or ratio) varies with time, then hazards aren’t proportional
Recall example: Abstinence programs
Perhaps abstinence programs have a big effect initially, but the effect diminishes (or reverses) later on

14. Proportional Hazard Assumption
Red = Abstinence group; green = control
Positive time
interaction
No time interaction
In non-proportional case, the effect of abstinence programs varies across time

15. Proportional Hazard Assumption
Strategy: Create variables that reflect the interaction of X variables with time
Significant effects of time interactions indicate non-proportional hazard
Fortunately, inclusion of the interaction term in the model corrects the problem.
Issue: X variables can interact with time in multiple ways…
Linearly
With “log time” or time squared
With time dummies
You may have to try a range of things…

16. Proportional Hazard Assumption
Red = Abstinence group; green = control
Linear time interaction
Effect grows consistently over time
Try “Abstinence*time”
Interaction with time-period… Effect differs early vs. late Try “Abstinence*DLate”

17. Proportional Hazard Assumption
5. Grambsch & Therneau test
Ex: Stata “estat phtest”
Test for non-zero slope of Schoenfeld residuals vs time
Implies log hazard ratio function = proportional
Can be applied to general model, or for each variable
stcox gdp degradation education democracy ngo ingo, robust nohr scaledsch(sca*) schoenfeld(sch*)
. estat phtest
Test of proportional hazards assumption
Time: Time
| chi df Prob>chi2
global test |
Significant chi-square indicates violation of proportional hazard assumption

18. Proportional Hazard Assumption
Variable-by-variable test “estat phtest”:
Note: Certain variables are especially problematic…
. estat phtest, detail
Test of proportional hazards assumption
Time: Time
| rho chi df Prob>chi2
gdp |
degradation |
education |
democracy |
ngo |
ingo |
global test |

19. Proportional Hazard Assumption
Notes on estat phtest :
1. Requires that you calculate “schoenfeld residuals” when you run the original cox model
And, if you want a test for each variable, you must also request scaled schoenfeld residuals
2. Test is based on identifying non-zero time trend… but how should we characterize time?
Options: normal/linear time, log time, time dummies, etc
Results may differ depending on your choice
Ex: estat phtest, log – specifies “log time”
Plot of smoothed Schoenfeld residuals can indicate best way to characterize time
Linear trend (not a curve) indicates that time is characterized OK
Ex: estat phtest, plot(ngo) OR estat phtest, log plot(ngo)

20. Proportional Hazard Assumption
What if the assumption is violated?
1. Improve model specification
Add time interactions to address nonproportionality
Ex: If high democracies are not proportional to low democracies, try adding “highdemoc*time”
Variables can be interacted with linear time, log time, time dummies, etc., to address the issue
2. Model groups separately
Split sample along variables that are non-proportional.

21. Proportional Hazard Assumption
What if the assumption is violated?
3. Use a stratified Cox model
Allows a different baseline hazard for each group
But, you can’t estimate effect of stratifying variable!
Ex: stcox var1 var2 var3, strata(Dhighdemoc)
4. Use a piecewise model
Split time into chunks… in which PH assumption is met
Requires sufficient sample size in all time periods!

22. Proportional Hazard Assumption
What if the assumption is violated?
5. Live with it (but temper your conclusions)
Violation of proportional hazard assumption tends to:
Overestimate the effect of variables whose hazard ratios are increasing over time
And, underestimate those whose hazard ratios are decreasing
However, Allison points out: Cox model is reasonably robust
Other issues (e.g., model misspecification) are bigger issues

23. Cox Model: Residuals
OLS regression: Residuals = difference between predicted value of Y and observed
Y-hat – Yi
EHA: Residuals are more complicated
You could compute predicted failure minus observed…
But, what about censored cases? What is observed?
There are a number of different ways to calculate residuals… each with different properties.

24. Cox Model: Residuals – Summary
From Cleves et al. (2004) An Introduction to Survival Analysis Using Stata, p. 184:
1. Cox-Snell residuals
… are useful for assessing overall model fit
2. Martingale residuals
Are useful in determining the functional form of the covariates to be included in the model
3. Schoenfeld residuals (scaled & unscaled), score residuals, and efficient score residuals
Are useful for checking & testing the proportional hazard assumption, examining leverage points, and identifying outliers
4. Deviance residuals
Are useful fin examining model accuracy and identifying outliers.

25. Martingale/deviance Residuals Outliers
Martingale residuals: difference over time of observed failures minus expected failures
Feature: range from +1 to –infinity
Deviance residuals = martingale residuals that are rescaled to be symmetric around zero
Easier to interpret
Extreme martingale or deviance residuals may indicate outliers
Plot residuals vs. time, case number, IVs, etc.
Or simply sort data by residuals & list the cases.

26. Martingale & Deviance Residuals: Outliers
Stata code to identify outliers:
*run Cox Model, calculate martingale residuals
stcox var1 var2 var3, robust nohr mgale(mg)
* Creates variable “mg” which contains martingale residuals
* Next, compute deviance residuals using “predict”
predict dev, deviance
gen caseid = _n
* create plots of various types
scatter mg caseid
* Deviance residual plots are generally easier to interpret
scatter dev caseid
scatter dev caseid, mlabel(newname2)
scatter dev _t

27. Deviance Residuals Plot
Extreme values may be outliers
Here, no obvious outliers are visible

28. Efficient Score Residuals: Influential Cases
Procedure for identifying outliers using ESRs
It is possible to compute DFBETAs based on ESRs
DFBETA: Change in coefficient a variable’s coefficient due to a particular case in the analysis
Cases with big DFBETAS may be overly influential
Issue: Stata cannot automatically compute DFBETAS…
You have to compute them manually
Also, computation = limited to 800 cases (for “intercooled stata”)
Hopefully stata will improve this in the future.

29. ESRs: Influential Cases
Stata code to estimate DFBETAs:
* Run Cox model, request efficient score residuals
* Creates vars: esr1 to esr5 corresponding to vars listed in model
stcox gdp var1 var2 var3 var4, robust nohr esr(esr*)
* Create room for a matrix of up to 800 rows (for your cases)
set matsize 800
* Create esr matrix
mkmat esr1 esr2 esr3 esr4, matrix(esr)
* Multiply ESRs and Var/Cov matrix to estimate DFBETAs, save results
mat V=e(V)
mat Inf = esr*V
svmat Inf, names(s)
* Label estimates for subsequent plots
label var s1 "dfbeta – var 1"
label var s2 "dfbeta – var 2"
label var s3 "dfbeta – var 3"
label var s4 "dfbeta – var 4"
* Plot DFBETAs for each variable vs. time or case number
scatter s1 _t, yline(0) mlab(caseID) s(i)
scatter s1 casenumber, yline(0) mlab(caseID) s(i)
* Look for extreme values (for each IV – s1 to s4)

30. DFBETA Example DFBETA for NGOs (plotted by casenumber)
DFBETA value indicates that presence of Latvia changes NGO coefficient by standard deviations

31. Cox-Snell residuals: Model Fit
Cox-Snell residuals can be plotted to assess model fit
If model fits well, graph of integrated (cumulative) hazard conditional on Cox-Snell residuals vs. Cox-Snell residuals will fall on a line
Strategy in stata:
Run Cox model, request martingale residuals
Use “predict” to compute Cox-Snell residuals
Stset your data again, with Cox-Snell as time variable
Compute integrated hazard
Graph integrated hazard versus residuals.

32. Cox-Snell residuals: Model Fit
Cox-Snell residuals can be plotted to assess model fit
If model fits well, graph of integrated (cumulative) hazard conditional on Cox-Snell residuals vs. Cox-Snell residuals will fall on a line
Strategy in stata:
Run Cox model, request martingale residuals
Use “predict” to compute Cox-Snell residuals
Stset your data again, with Cox-Snell as time variable
Compute integrated hazard
Graph integrated hazard versus residuals.

33. Cox-Snell Model Fit Example
Cox-Snell Plot for Environmental Law data
This looks quite bad. Cumulative hazard should fall on the line… Instead, there is a sizable gap.
Note: Don’t worry about small deviations from the line at the right edge of the plot. There are typically few cases there…

34. Stratified Cox Models
Stratified models allow different baseline hazards for sub-groups in your data
But constrain all model coefficients to be the same across all groups
Useful if we think that some sub-groups have very different hazard curves over time
AND we aren’t really interested in differences across those groups – it is just a nuisance
Another option is to simply analyze groups separately
But, we lose sample size. Stratifying avoids that.

35. Cox Models for Grouped Data
Sometimes cases are not independent
Ex: Students in same class; people in same family
Two useful options:
1. Stata’s “cluster” command: Adjusts standard errors based on group membership
stcox var1 var2 var3, cluster(FamilyID)
2. Cox model with shared frailty
Another name for a random effects model
We’ll discuss this later in the course
Don’t confuse with non-shared frailty models!
stcox var1 var2 var3, shared(FamilyID)

36. Stata Note: TVC
Note: Stata has a special option for “time varying covariates”
You DO NOT need to use this!!!
It is designed for cases where you wish to specify the character of time variation (e.g., a rate of decay)
You can simply create time-varying variables in your dataset…
STATA will analyze it properly using the stcox command.