Though there are multiple tests performed in a regression model, you need not adjust for multiple comparisons. In a regression model, there is a null model test that is generally performed to check if the null model (model with no predictors) is better than the model you've specified. If you are able to reject the null hypothesis for that test, you can proceed with interpreting coefficients that you've got without worrying about multiple comparisons problem. If you cannot reject the null, you should ignore any further interpretations you can make from the coefficients. However, if you are performing an exploratory analyses and testing for differences for multiple different characteristics between two group-pairs using t-tests etc., you may need to adjust for multiple comparisons.
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