# Comparing Hazard Rate Ratios for different outcomes on the same sample

Suppose I have survival data coming from a cohort study. The outcome of interest is death and a subject can die either because of cancer or because of other causes. The two outcomes under study are therefore mutually exclusive.

Now, suppose that I'm interested in assessing whether the magnitude of the associations between the exposure (for simplicity let's imagine it's dichotomous) and the outcomes is different for the two outcomes considered. The measure of association could be, for example, the Hazard Rate Ratio.

I have found this paper by Lunn and McNeil that provides a possible solution to this problem. For those of you who don't have access to jstor or don't want to read it, here's the idea behind it:

I duplicate my data (since in this case I have 2 possible outcomes), and create an indicator variable $\delta_i$ which represents the failure type for the subject $i$ (let's say $\delta_i=0$ for cancer death and $\delta_i=1$ for other cause of death). This means that each subject is included twice in my dataset, for example:

subject time status failure_type exposure
i       t_i  1      0            e_i
i       t_i  0      1            e_i


as the 2 outcomes are mutually exclusive, for each subject, the failure_type vector can be either $(0,0)'$ (censored) or $(0,1)'$ (died because of other causes) or $(1,0)'$ (died because of cancer). The time to event/censoring (time) and the exposure (exposure) are the same for the two records of each subject.

At this point I fit a Cox regression model (using a robust variance/covariance estimator): $$\lambda_i(t) = \lambda_0(t)\exp(\beta_0*failuretype_i+\beta_1*exposure_i+\beta_2*failuretype_i*exposure_i)$$ and test $\hat{\beta_2}$ against the null $H_0: \beta_2 = 0$. (Of course I can also relax the assumption of proportionality of the hazards by stratifying over failure_type, but the bottom line doesn't change).

Do you know of other possible ways of performing this analysis? I've read a paper where the authors fit two different models (one for each cause of death) and then use an Hotelling test to compare the two HRRs, but they give no references or additional information and this method seems unsound to me.

• Why aren't you using regular competing risks? – Max Gordon Apr 17 '12 at 9:48
• @MaxGordon What do you mean by regular competing risk? Anyway, this is a competing risk analysis. – boscovich Apr 18 '12 at 9:37
• I was thinking of why you're not using Gray's model. I've found the Cox model slightly inconvenient for CRR. I've only used R and not Stata for CRR but it's fairly straightforward. – Max Gordon Apr 18 '12 at 9:45
• @MaxGordon: ok, I see. IMO, here the problem with using the approach by Fine & Gray is that you'd fit 2 different models for the 2 outcomes, thus obtaining 2 $\log(HRR)$ estimates for the exposure variable (one from each model). Now, how do you compare the 2 $\log(HRR)$ estimates? – boscovich Apr 18 '12 at 10:14
• Well, you're kind of comparing apples and pears, I don't think having them in the same analysis is a great idea. The thing that crr allows is that the HR are true to the actual frequencies and therefore you can compare them. From what I've read you need to look at both outcomes in your data analysis but it's usually suggested to be separated into two regressions. I guess the hard part is displaying it to the reader without getting it too messy – Max Gordon Apr 18 '12 at 11:15