# Conducting a subgroup analysis with regression modeling

I'm conducting a survival analysis using Cox Proportional Hazards regression to identify prognostic factors for cancer patients. My covariates include information such as age, sex, tumor location etc. I hypothesize that there are sex differences when it comes to how tumor location affects survival, i.e. males with tumors in a certain region of the brain survival longer than females.

I have proposed using an interaction term between sex and tumor location (sex*location) to test for sex-location heterogeneity. This means there will be a $$\beta$$ with a corresponding p-value and confidence interval that we can use to assess whether sex heterogeneity exists.

My colleague has proposed an alternative solution. They suggest we stratify patients by sex into male and female cohorts and then fit separate Cox regression models (still including all other covariates). We'd then see if the hazard ratios for each covariate are different between models, and if so this would establish heterogeneity.

Which procedure is the proper method to conduct a subgroup analysis to investigate sex heterogeneity? I feel that creating separate male and female models is problematic as there's no way to evaluate whether the different hazard ratios from each model are "significant" (no way to address uncertainty about estimates). Is this reasoning correct? If so, are there any circumstances when creating separate male and female models would be appropriate?

• Generally, single model is better than stratified models. – user158565 Jul 29 '19 at 5:23

Your reasoning is correct - as already commented by @user158565, a single model is generally better, as you are able to obtain a proper CI for the heterogeneity term.

Another major difference is that the interaction model constrains the differences to that term, or the intercept, while the stratum-specific approach allows different effects for each of the terms. In other words, for a binary stratifying variable, a stratified model is similar to adding the interaction on every term, ~ 1 + sex + location + A + B + sex*location + sex*A + sex*B .... This is usually not the intent, and you might end up losing a large amount of precision for the effects where no interaction was needed.

• Could you explain why adding an interaction on every term could lose a large amount of precision when there is no effect there? Would you say its bad practice to add an interaction on every term when one suspects there may be sex heterogeneity but doesn't know which other terms sex interacts with? – Tomas Bencomo Jul 29 '19 at 14:31
• @TomasBencomo think of it like this: you're estimating each covariate's effect on twice smaller sample size now (only men or only women instead of the entire N). Frank Harrell and Andrew Gelman wrote a lot about this, e.g. fharrell.com/post/varyor statmodeling.stat.columbia.edu/2018/03/15/… – juod Jul 29 '19 at 21:09
• Does this twice smaller sample size affect the estimates of all coefficients or just the interaction term coefficient? I would think the smaller sample size only affects the interaction term coefficients, not the main effects. Say I don't have any clue which terms could have interactions - why not include interactions with everything, not expecting to find anything, and then focus on the main effects? Are you saying that the main effects will suffer from a loss in precision? – Tomas Bencomo Jul 29 '19 at 22:30
• @TomasBencomo say our model includes an interaction between a binary variable $B$ and any other variable $X$ as $Y = \beta_X X + \beta_{BX} BX$ + some intercepts. This is the same as $Y = \beta_0 X | b=0$, $Y= \beta_1 X | b=1$. In effect, you estimate each coefficient on part of your sample, and it's up to you whether this cost is acceptable. – juod Jul 30 '19 at 2:40