After gathering valuable feedback from previous questions and discussions, I have came up with the following question: Suppose that the goal is to detect effect differences across two groups, male vs. female for example. There are two ways to do it:
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running two separate regressions for the two groups, and employ Wald test to reject (or not) the null hypothesis $H_0$: $b_1-b_2=0$, where $b_1$ is the coefficient of one IV in male regression, and $b_2$ is the coefficient of the same IV in female regression.
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pool the two groups together, and run a joint model by including a gender dummy and an interaction term (IV*genderdummy). Then, the detection of the group effect will be based on the sign of interaction and the t-test for significance.
What if Ho is rejected in case (1), i.e. group difference is significant, but the coefficient of interaction term in case (2) is statistically insignificant, i.e. group difference is insignifant. Or vice versa, Ho is not rejected in case (1), and interaction term is significant in case (2). I have ended up with this outcome several times, and I was wondering what outcome would be more reliable, and what is the reason behind this contradiction.
Best Answer
The first model will fully interact gender with all other covariates in the model. Essentially, the effect of each covariate (b2, b3... bn). In the second model, the effect of gender is only interacted with your IV. So, assuming you have more covariates than just the IV and gender, this may drive somewhat different results.
If you just have the two covariates, there are documented occasions where the difference in maximization between the Wald test and the Likelihood ratio test lead to different answers (see more on the wikipedia).
In my own experience, I try to be guided by theory. If there is a dominant theory that suggests gender would interact with only the IV, but not the other covariates, I would go with the partial interaction.