# Subgroup analysis vs Cox model

I first heard of subgroup analysis which determines the subgroup that may benefit from treatment. It seems to me that it is just a hypothesis test of OR (or HR) between subgroups (child vs adult) upon Cox model. Is it appropriate to think it as simple as this?

Subgroup analyses are used to investigate whether treatment effect differs between some grouping such as sex or mutation status. Subgroups can also be continuous variables like age or blood pressure.

Here are two ways a treatment effect can differ between subgroups:

1. Reduced Effect - The treatment is more effective in one subgroup vs another. An example could be a drug that has a hazard ratio (HR) of .5 for women but HR=.8 for men. This would indicate the drug is more potent in women and men do not gain as much benefit from the drug

2. Reversed Effect - The drug benefits one subgroup but actually harms another. An example would be a drug with HR=.7 for women but HR=1.2 for men. Here the drug is decreasing the risk of death in women but increasing the risk of death in men.

These situations describe treatment effect heterogeneity, where different types of patients experience different effects from treatment. Because RCTs and the regression models used to analyze them estimate average treatment effect, these effects can be "hidden" due to averaging the effect between subgroups.

In regression models, including the Cox PH model, treatment effect heterogeneity is modeled with an interaction term. Consider the following regression model where $$X_1$$ is treatment status (drug=1, control=0) and $$X_2$$ is sex (male=1, female=0) and $$Y$$ is overall survival (OS)

$$Y = \beta_1 X_1 + \beta_2 X_2$$

Here $$\beta_1$$ is the drug's effect on OS and $$\beta_2$$ is the effect of sex on OS. The model is additive, making the assumption that treatment and sex independently affect survival. If the drug works better for women and doesn't work as well for men, this can be modeled by adding an interaction term $$\beta_3$$

$$Y = \beta_1 X_1 + \beta_2 X_2 + \beta_3 X_1 X_2$$

This model no longer assumes drug and sex are additive, but rather the effect of one variable depends on the other. This effect dependence is estimated by $$\beta_3$$. Of important note, if there is an interaction, you must interpret the drug effect for each group separately as $$\beta_1$$ doesn't represent the effect for each groups but only the average effect between groups.

When people conduct a subgroup analysis, they usually split their data into separate subgroups and then estimate a hazard ratio for each group separately. They then compare the hazard ratios between groups and determine whether there is a differential treatment effect.

The problem with this approach is that it's difficult to judge whether the effect difference is due to signal or noise. Using an interaction term helps with this issue because the interaction estimates the treatment effect difference ($$\beta_3$$) and let's us compute a confidence interval and p-value for this difference.

TLDR: Subgroup analyses are similar to a treatment*subgroup interaction term in a regression model. Using an interaction term helps quantify the uncertainty surrounding a differential treatment effect between subgroups.

Here is a good paper on subgroup analyses and interaction terms:

Detecting Moderator Effects Using Subgroup Analyses by Wang and Ware

• Nice answer (+1). Do be careful with the use of the word "strata" in the first paragraph, though, as it can raise a connotation of being a stratified analysis (with different baseline hazard functions) rather than a Cox model with all cases having the same baseline hazard (as in the interaction model you propose). Might want to use terminology like "whether treatment effect differs depending on the values of other covariates." A stratified analysis (interaction between $X_1$ and strata of $X_2$) is another way to proceed if you don't care about the effect of $X_2$ per se.
– EdM
Feb 13, 2020 at 16:04
• Thanks for the advice! I was trying to keep the discussion applicable to all regression models but since the question is about Cox models and you can have a stratified Cox model, better not stick with precise terminology. I've edited the answer to reflect this change. Feb 13, 2020 at 18:56