# 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. Jul 29, 2019 at 5:23

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.
• @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.