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I am approaching competing risk analysis and I have several questions regarding how to account for what I call the "ceiling" effect and how to interpret the results. I'll try to be structured as much as I can.

I will use the mgus2 dataset from survival with some simulated columns to expose my issues.

library(survival)
library(tidycmprsk)
library(ggsurvfit)
library(dplyr)

set.seed(85)
mgus_new <- survival::mgus2 %>% mutate(infection=rbinom(length(mgus2$death), size=1, prob=0.08)) #Create a new event
mgus_new <- mgus_new %>% rowwise() %>% mutate(time_infection=sample(0:futime, n(), replace=TRUE)) #Simulate time to new event
mgus_new <- mgus_new %>% mutate(time_infection=ifelse(infection==0,futime,time_infection)) #Simulate censorship time for new event

mgus_new <- mgus_new %>% mutate(time = ifelse(infection==0, futime, time_infection))
event <- with(mgus_new, ifelse(infection==0, 2*death, 1))
mgus_new$event <- factor(event, 0:2, labels=c("Censor","Infection","Death"))

So in this dataset, I have 3 events: pstat, infection and death. The first two are not competing (i.e., one can occur also if the other has already occurred, because follow-up won't stop).

Question 1: It is right to perform competing risk analysis only with death (i.e., performing two different analyses, infection vs. death and pstat vs. death)? I am not sure that performing an overall competing risk analysis considering all the three events would be appropriate, given that two of these can occur together (one would only look at the first occurring).

Anyway, this is how went further.

cif <- cuminc(Surv(time, event) ~ sex, data=mgus_new) 

cif %>% ggcuminc(outcome=c("Infection","Death"))

And this is the output:

CIF Plot

Next, I will report the results of the Fine-Gray and Cox regressions, using sex as a covariate, and Infection as the outcome of interest:

Fine-Gray:

crr(Surv(time, event) ~ sex, data=mgus_new)
#> 
#> ── crr() ───────────────────────────────────────────────────────────────────────
#> • Call Surv(time, event) ~ sex
#> • Failure type of interest "Infection"
#> 
#> Variable   Coef     SE      HR     95% CI       p-value    
#> sexM       -0.072   0.197   0.93   0.63, 1.37   0.72

Cox-Regression (for infection):

coxph(Surv(time_infection, infection) ~ sex, data=mgus_new)
#> Call:
#> coxph(formula = Surv(time_infection, infection) ~ sex, data = mgus_new)
#> 
#>         coef exp(coef) se(coef)     z     p
#> sexM 0.02389   1.02418  0.19750 0.121 0.904
#> 
#> Likelihood ratio test=0.01  on 1 df, p=0.9037
#> n= 1384, number of events= 103

As we can see, the two results are somewhat different, although both far from being statistically significant. We know that in this context, the Cox Regression estimate the cause-specific hazard, while the Fine-Gray model in this context estimate the effect of sex on the "relative-incidence" of Infection (or in other word, the subdistribution hazard).

Question 2: Assuming that my interpretation of both Cox and Fine-Gray model in this context is correct (and obviously, if not, please correct me), the question is: how can we account for what I call the ceiling effect, when evaluating infection?

In other words, we can see from the plot that males (blue line) and females (red line) had a very similar cumulative incidence of infection; however, most death rate was higher in males, for most of the follow-up. The incidence of infection in males, therefore, may have been "underestimated" by the fact that there were less males at risk for developing infection at each time, since more have died. To me, it doesn't not seem that neither Cox, nor Fine-Gray model account for this potential confounding, and I am wondering whether a) I am mis-interpreting something, b) there are other way to account for this.

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Question 1

I am not sure that performing an overall competing risk analysis considering all the three events would be appropriate, given that two of these can occur together...

I'm somewhat skeptical of modeling non-fatal, potentially repeating events like infection as competing with death. Even in the examples from the R competing risks vignette, death can occur along with the pstat event (occurrence of plasma-cell malignancy), too. That data set is helpful for illustrating modeling principles, but a multi-state model as illustrated (for a simple case) at the bottom right of Figure 1 of the vignette is probably more appropriate in practice with things like infection.

Putting that issue aside, it's a question of what you want to model: what's meaningful, based on your understanding of the subject matter?

Do you actually care about one of the event types? I suspect that patients in the mgus2 data set had infections that just weren't recorded, as they weren't of primary interest to the investigators.

If it helps to think about two or more events of interest as strictly competing, then model them that way. If not, then a multi-state model would be better.

Question 2

As you note, Cox cause-specific modeling removes those experiencing one event from the risk set for other events at later times. Thus the hazards associated with one event type are intertwined with the hazards of other event types. I think that's pretty much what you call the "ceiling effect": "there were less males at risk for developing infection at each time, since more have died."

The Fine-Gray subdistribution was designed to avoid your "ceiling effect." Those who experience one of the competing events are actually included (in a particular way) in the risk sets for the other events at times after their own events. The Fine-Gray approach thus allows separate regression modeling of each competing risk with respect to covariates in a way that cause-specific modeling can't. The downside is some difficulty in translating back to event probabilities, with some circumstances leading to sums of event probabilities exceeding 1. This page provides some more details, with links to the literature.

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  • $\begingroup$ Thank you for your reply. As per the infection, just suppose this event is meaningful and was fully registered in the dataset - this is to simplify the example. Also, I think the point here is that the "competitiveness" of the events is not bi-directional (i.e., infection cannot occur after death, but death can occur after infection). So I was asking how to interpret this situation and what is the better way of dealing with it. $\endgroup$ Sep 26, 2022 at 11:56
  • $\begingroup$ also, as death can occur both after infection and pstat, would you perform two different Fine-Gray regression (one for death/infection pair, one other for the death/pstat pair) in this context? $\endgroup$ Sep 26, 2022 at 11:59
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    $\begingroup$ @user89547235 I think that Fine-Gray analysis is best reserved for truly competing terminal states, like death from different causes. That's the situation shown in the bottom left of Fig. 1 of the R competing risks vignette. Otherwise, a multi-state model like that in the bottom right of that Figure could describe the data better. If Fine-Gray is appropriate, you don't have to do the comparisons pairwise. In R, if the event marker is a multi-level factor with the lowest level being censoring, you can set up the data and analyze all event types at once. That uses all of your data efficiently. $\endgroup$
    – EdM
    Sep 26, 2022 at 13:07

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