1. Lecture 19: Competing Risk Regression

2. When Competing Risks? Recall censoring assumption:
Event times and censoring times are independent
If this is questionable, then competing risks is likely more appropriate
But… must be able to distinguish the other events/risks

3. Competing Risks Data
Subject can fail from any of K events types, but only the earliest failure time can be observed.
As in the non-competing risks setting, observations take the form of (T, δ).
T is the minimum of t1, t2, … tK
δ is 1, 2, … k if failed
Can also have 0 if no failure has yet occurred
Z are the covariates we are interested in

4. Examples of Types of Observations

5. Examples of Types of Observations

6. Summarizing Competing Risks
For a population, three approaches
Kaplan Meier: “net”
Cumulative incidence: “crude”
Conditional probability
Cumulative incidence most appropriate (and most commonly used) for most settings of CR
However, each provides it own potentially useful information

7. Recall The Issue
We’ve already discussed this for estimation of the survival distribution.
In the case of Kaplan-Meier analysis recall
Assumes that the event of interest is the only risk acting on the population
Censors all other events
i.e. treats all other events the same as LTFU or drop-out

8. Competing Risks
For estimation in a population, it is usually shown using ‘cumulative incidence’ instead of survival
Let’s review how CR approach differs from KM approach…

9. Recall: KM Survival
Estimate of survival for event r at time ti

10. “Net”: KM Cumulative Incidence (CI)
Estimate of cumulative incidence at time ti
(The sum of the CI current incidence rate plus the previous incidence rate)

11. Cumulative Incidence Approach
Estimating CI:
For t > ti Or
Alternatively it can be written as:

12. How does the cumulative incidence (CI) differ from Kaplan Meier (KM)…

13. Comparison KM CI

14. Why Competing Risk Regression?
Understand the effect of therapy on different subgroups
Allow us to target interventions to those most likely to benefit
Allow us to summarize the absolute failure via estimation of cause-specific failure probabilties

15. Cause-Specific Cumulative Hazard
Chief quantity in competing risks setting is the cause-specific hazard function lk
This also can be used to define the cause-specific cumulative hazard
Overall cumulative hazard function is

16. Cumultative Incidence
How does this all relate to our earlier discussion of estimating overall incidence?
We are still interested in cumulative incidence

17. Cumultative Incidence
Sum of the K cumulative incidence functions is the probability of failure from any cause…
NOTE! The cumulative probability of event k in the presence of competing risks is often miscalculated as

18. Competing Risk Model Recall a subject can fail from one of K events
This means that we have partial information on all events
A subject who experience the kth event at time ti is know to have survived to time ti for all other competing events.

19. Competing Risk Model
There are two basic approaches to competing risk regression
Modeling cause-specific hazard
Modeling cumulative incidence
Both are analogous to the Cox proportional hazards model

20. Modeling Cause Specific Hazard
Model denoted as
Function of some unspecified baseline hazard function for kth cause
Also function of covariate vector Z and regression coefficients b

21. BMT Example
Recall our BMT data examining the association between time to event disease type
ALL, AML low risk, AML high risk
Other factors: FAB class, donor/patient characteristics, Waiting time, platelet recovery,
Originally we had considered time to relapse or death as a single event.
What if we wanted to examine factors for each?

22. Cause Specific Hazard Models
library(survival); library(Kmsurv) ### Cause-specific hazard model ### data(bmt) colnames(bmt)<-c("dgroup","TTD","DFS","dead","relapse","Either","tAGVHD", "AGvHD","tCGVHD", "CGvHD","tPR","PR","PtAge","DonAge","PtSex","DonSex", "PtCMV","DonCVM", "TTTrans","FAB","Hosp","MTX") ### Either Death or Relapse rreg.cox<-coxph(Surv(DFS, Either)~factor(dgroup) + FAB + PtAge + DonAge + PtAge*DonAge, data=bmt) #Relapse Model rreg.cox<-coxph(Surv(DFS, relapse)~factor(dgroup) + FAB + PtAge + DonAge + PtAge*DonAge,

23. Death or Relapse Model
> summary(reg.cox) coxph(formula = Surv(DFS, relapse) ~ factor(dgroup)+FAB+PtAge+DonAge+PtAge*DonAge, data = bmt) n= 137, number of events= 83 coef exp(coef) se(coef) z Pr(>|z|) factor(dgroup) ** factor(dgroup) FAB ** PtAge * DonAge ** PtAge:DonAge *** Concordance= (se = ) Rsquare= (max possible= ) Likelihood ratio test= 32.8 on 6 df, p=1.144e-05 Wald test = on 6 df, p=1.039e-05 Score (logrank) test = on 6 df, p=3.078e-06

24. Relapse Specific Hazard Model
> summary(rreg.cox) coxph(formula = Surv(DFS, relapse) ~ factor(dgroup)+FAB+PtAge+DonAge+PtAge*DonAge, data = bmt) n= 137, number of events= 42 coef exp(coef) se(coef) z Pr(>|z|) factor(dgroup) ** factor(dgroup) FAB ** PtAge DonAge PtAge:DonAge Concordance= 0.75 (se = ) Rsquare= (max possible= ) Likelihood ratio test= on 6 df, p=2.361e-05 Wald test = on 6 df, p= Score (logrank) test = on 6 df, p=1.993e-05

25. Interpretation of Cause-Specific Model
Interpret the hazard ratio for two individuals the same as from a normal Cox model
However, there is a dependency between the failure types…
Problem
Effects seen in the model may reflect the influence of the competing events
As a result, covariate effects don’t necessarily pertain to the cumulative incidence of the kth event

26. Modeling Cumulative Incidence
Analogous to our general approach for a study population with competing risk, there is a model base on cumulative incidence.
Competing risk regression model (Fine and Gray)
Direct regression of the effect of covariates on cumulative incidence
Distinguish between patients who have had other events and those at risk for event of interest
Based on PHM approach

27. Modeling Cumulative Incidence
In this case the form of the model is
Where is referred to as the sub-distribution hazard
crude hazard from CIcr

28. Sub-Distribution Hazard
Expression for the sub-distribution hazard
Can also think of this as the hazard function for an improper random variable

29. Risk Set in CRR Model
Just as in the case of the Cox model, we need a likelihood expression for estimation of the model
Definition of the risk set slightly altered

30. Partial Likelihood
The expression for the partial Likelihood based on our newly defined risk set is
Partial log-likelihood

31. Estimation & Testing
We can maximize the partial log-likelihood in the same way we did with the Cox PHM
Additionally, Fine and Gray developed a score test to make inference about the regression coefficients in the model

32. What About Censoring
Up until now, we have been assuming we have “complete” data.
If we observe censoring we must change the definition of the risk set slightly…

33. Competing Risk Regression in R
Can implement the cause-specific hazard model using the coxph function in the survival library
The “cmprsk” package in R implements the Fine and Gray model we’ve just discussed
So back to our BMT example

34. “cmprsk” Library
crr: Fit the Fine and Gray model of the subdistribution functions in competing risk
ftime and fstatus: define survival data
cov1 and cov2: matrix of covariates
tf: functions of time for covariate in cov2
failcode: which event are you modeling?
Other functionality deals with estimating and comparing cumulative incidence across groups

35. Fitting CRR Model
### Fine and Gray Cumulative Incidence Model ### ###First we have to generate a single event type variable etime<-ifelse(bmt$relapse==1, bmt$DFS, bmt$TTD) etype<-ifelse(bmt$relapse==1, 1, 0) etype<-ifelse(bmt$dead==1 & bmt$relapse==0, 2, etype) dx2<-ifelse(bmt$dgroup==2, 1, 0) dx3<-ifelse(bmt$dgroup==3, 1, 0) ptdn.intx<-bmt$PtAge*bmt$DonAge fab<-bmt$FAB ptage<-bmt$PtAge dnage<-bmt$DonAge cov<-cbind(dx2, dx3, fab, ptage, dnage, ptdn.intx)

36. CRR Model
> rreg.crr<-crr(etime, etype, cov, failcode=1) > summary(rreg.crr) Competing Risks Regression Call: crr(ftime = etime, fstatus = etype, cov1 = cov, failcode = 1) coef exp(coef) se(coef) z p-value dx dx fab ptage dnage ptdn.intx Num. cases = 137 Pseudo Log-likelihood = -187 Pseudo likelihood ratio test = 24.1 on 6 df,

37. CRR Model Results
exp(coef) exp(-coef) 2.5% 97.5% dx dx fab ptage dnage ptdn.intx Num. cases = 137 Pseudo Log-likelihood = -187 Pseudo likelihood ratio test = 24.1 on 6 df,

38. Additional Notes Can also include a matrix of time varying covariates
Can NOT use factor here… Must create dummy variables for all factor variables of interest in the data

39. Comparison of CHR and CRR
Cause-specific 
Competing Risks  b HR p
AML low risk
-1.841
0.16 (0.05, 0.50)
0.0016
-1.556
0.0048
0.21 (0.07, 0.62)
AML high risk
-0.579
0.56 (0.19, 1.62)
0.2800
-0.539
0.3100
0.58 (0.21, 1.65)
FAB
1.424
4.15 (1.78, 9.71)
0.0010
1.303
0.0030
3.68 (1.56, 8.70)
Patient Age
-0.038
0.96 (0.87, 1.07)
0.4700
-0.017
0.8000
0.98 (0.87, 1.12)
Donor Age
-0.084
0.92 (0.84, 1.00)
0.0580
-0.059
0.2300
0.94 (0.86, 1.04)
PatAge x DonAge
0.0024
1.00 (0.999, 1.005)
0.0990
0.0015
0.3900
1.00 (0.998, 1.005)

40. Cause-Specific vs CRR approach
Competing risks are truly independent
Cause-specific model provides valid estimates of the risk of each event
CRR model tends to be biased towards the null
If competing risks are dependent
Cause specific model

41. Which One to Use? It depends… The cause-specific approach
can give cause-specific hazards and CIFs.
However, we cannot examine covariate effects
Subdistribution approach
allows us to test covariate effects on the CIF
but subdistribution hazards are dicult to interpret and should be used with caution.

42. Which One to Use?
Cumulative incidence model more realistically models treatment effect in a population.
If we want real world probabilities of death then competing risks methodology should be used as opposed to standard survival analysis methods.
Allows us to separate the probability of death into different causes.

43. Model Selection for CRR
A nice feature of the CRR approach is that we can evaluate associations between covariates and our different events
This means we can conduct model selection to find a more parsimonious model
Use same approaches as before
P-values
AIC, BIC
Forward/backward selection

44. Example (forward using p-values)
#p-value approach, with forward selection dx2<-ifelse(bmt$dgroup==2, 1, 0); dx3<-ifelse(bmt$dgroup==3, 1, 0) ptage<-bmt$PtAge; dnage<-bmt$DonAge; ptdn.intx<-bmt$PtAge*bmt$DonAge fab<-bmt$FAB tttrans<-bmt$TTTrans mtx<-bmt$MTX ptsex<-bmt$PtSex; dnsex<-bmt$DonSex; sx.intx<-bmt$PtSex*bmt$DonSex h2<-ifelse(bmt$Hosp==2, 1, 0); h3<-ifelse(bmt$Hosp==3, 1, 0); h4<-ifelse(bmt$Hosp==4, 1, 0) cov1a<-cbind(dx2, dx3, fab) cov1b<-cbind(dx2, dx3, ptage, dnage, ptdn.intx) cov1c<-cbind(dx2, dx3, ptsex, dnsex, sx.intx) cov1d<-cbind(dx2, dx3, tttrans) cov1e<-cbind(dx2, dx3, mtx) cov1f<-cbind(dx2, dx3, h2,h3,h4) rreg.crra<-crr(etime, etype, cov1a, failcode=1) #p FAB = rreg.crrb<-crr(etime, etype, cov1b, failcode=1) #p Pt/Dn age = 0.80, 0.23, 0.39 rreg.crrc<-crr(etime, etype, cov1c, failcode=1) #p Pt/Dn sex = 0.36, 0.65, 0.69 rreg.crrd<-crr(etime, etype, cov1d, failcode=1) #p TTTrans = rreg.crre<-crr(etime, etype, cov1e, failcode=1 #p MTX = 0.73 rreg.crrf<-crr(etime, etype, cov1f, failcode=1) #p Hops = 0.94

45. Example (forward using p-values)
#Final Model (choosing p<0.1) > rreg.crra<-crr(etime, etype, cov2a, failcode=1) > summary(rreg.crra) Competing Risks Regression Call: crr(ftime = etime, fstatus = etype, cov1 = cov2a, failcode = 1) coef exp(coef) se(coef) z p-value dx dx fab tttrans exp(coef) exp(-coef) 2.5% 97.5% dx dx fab tttrans Num. cases = 137 Pseudo Log-likelihood = -187 Pseudo likelihood ratio test = 25 on 4 df,

46. Automated Model Selection
There is an automated selection algorithm for the fine and Gray Model
Traditional selection criteria include
AIC = -2logLp + 2p
BIC = -2logLp + plogn
Alternative proposed in the literature
BICcr = -2logLp + plog(n*)

47. BICcr selection criteria
Should select a more parsimonious model than the AIC, and has a less stringent penalty than the BIC.
Provides a good compromise for working with the Fine and Gray model for competing risk data.

48. Crrstep function in R
R package called crrstep implements stepwise model selection for the Fine and Gray competing risks model.
Available selection criterion include AIC, BIC, or BICcr selection criteria for choosing covariates.

49. BMT Example
###Automated Approach library(crrstep) mAIC<-crrstep(etime~factor(dgroup)+FAB+TTTrans+MTX+PtSex+DonSex+PtAge+ DonAge+PtAge*DonAge, etype=etype, data=bmt, direction="forward", failcode=1, criterion = "AIC") mBIC<-crrstep(etime~factor(dgroup)+FAB+TTTrans+MTX+PtSex+DonSex+PtAge+ DonAge+PtAge*DonAge, etype=etype, data=bmt, direction="forward", failcode=1, criterion = "BIC") mBICcr<-crrstep(etime~factor(dgroup)+FAB+TTTrans+MTX+PtSex+DonSex+PtAge+ DonAge+PtAge*DonAge, etype=etype, data=bmt, direction="forward", failcode=1, criterion = "BICcr")

50. BMT Example (AIC)
> mAIC<-crrstep(etime~factor(dgroup)+FAB+TTTrans+MTX+PtSex+DonSex+PtAge+DonAge+PtAge*DonAge, etype=etype, data=bmt, direction="forward", failcode=1, criterion = "AIC") NULL AIC +FAB factor(dgroup) <none> PtSex TTTrans DonSex MTX PtAge:DonAge PtAge DonAge [1] "FAB"

51. BMT Example (AIC)
> mAIC<-crrstep(etime~factor(dgroup)+FAB+TTTrans+MTX+PtSex+DonSex+PtAge+DonAge+PtAge*DonAge, etype=etype, data=bmt, direction="forward", failcode=1, criterion = "AIC") … [1] "FAB" "factor(dgroup)" "TTTrans" AIC <none> DonSex PtSex PtAge:DonAge MTX DonAge PtAge

52. Comparison AIC to p-value approach
> mAIC $coefficients estimate std.error t-stat FAB factor(dgroup) factor(dgroup) TTTrans $log.likelihood [1] > summary(rreg.crra) Competing Risks Regression Call: crr(ftime = etime, fstatus = etype, cov1 = cov2a, failcode = 1) coef exp(coef) se(coef) z p-value dx dx fab tttrans Pseudo Log-likelihood = -187

53. Comparison of Three Criterion
> mAIC estimate std.error t-stat FAB factor(dgroup) factor(dgroup) TTTrans > mBIC FAB factor(dgroup) factor(dgroup) > mBICcr factor(dgroup)

54. One Final Note
Competing risk regression generally assumes that events are mutually exclusive
BMT data this isn’t true
We can’t really look at relapse as competing for death
Joint Frailty modeling offers an alternative and can be implemented using frailtypack
Idea is to model recurrent events jointly with some terminal event

55. Next Time
A little about sample size estimation and power