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I have data with about 700 observations total. There are 9 subgroups in my sample with sizes ranging from 54-96.

I want to use linear regression to explore how 8 independent variables might predict my outcome of interest, and expect patterns may be different for each of my subgroups. I don't have enough power to run a regression for any of my subgroups (I was reading that rule of thumb for regression sample size was at least N>104+k). But what if I kept my whole sample together instead of stratifying, and then did 8 different models with interactions with subgroup*each predictor? Would interaction help me explore the same general question that isn't possible with stratification due to insufficient sample size?

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The interactions with the class assume that all the groups respond equally to the independent variables, but that they have a different intercept ($\beta_0$). If there is a difference in level but not in response to the regressors, then the interactions will work great. Otherwise they won't be the best option. Also, that rule of thumb is not quite right. To fit a model, you need to have more observations than variables ($N > k$) and the number required to fit a "good" model grows exponentially with $k$. But you can always try the stratified modeling and see if the coefficients are significant.

Just for some fun learning, search "the curse of dimensionality". That will give you some insight on the $N$ vs $k$ balance.

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