# Solved – How to test significant interaction effect

If I have a model, lets say y = a + b1male + b2large + b3medium + b4malelarge + b5malemedium where male is dummy coded 1 for male and 0 for female and large and female are dummies (also 0,1 coded) for company size with small as reference categorie, how can I test whether the effect of male varies significantly with company size?

I presume that the t-test for the coeffs tells me the significance of the effect of male on the respective dummy-category compared to the referece category. So e.g. the effect of male varies significantly between large and small companies (=p-value for b4). But since the t-test is always compared to the reference group, how can I test "in general" whether the effect of male varies with company size?

My intuitive approach would be, to run a hierarchical regression with the interactions included in the second block. So if the change in R² is significant, this would mean that the interaction of at least one categorie of the dummy is significant. Is this correct?

And how can I figure out, which categories show sig. differences. Thats particularly hard if I have a large number of categories of one dummy. Do I have to run regressions and change manually the reference category?

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If for some reason you do need to use categories, then you can analyze your linear regression as an ANOVA to get the type of output you desire. In R, for example, you can run a linear regression with the `lm()` function, then wrap that output in `anova()` to present the results in a way that tests "in general" for significance of a multi-category factor or its interactions. You will, however, need to pay attention to the way that variance is partitioned, as explained on this page. The default for `anova()` in R is to use Type I sums of squares, hierarchically associating as much variance as possible with each main effect in the order the effects were specified in the model, then proceeding similarly with interactions. I think that's what you had in mind in the third paragraph of your question. Some statistical software instead uses Type III sums of squares.
Tests of differences among different categories of a predictor variable are better done in a systematically parallel way rather than resetting the reference category as you propose. You need to take into account the multiple testing of hypotheses if you do not have just a few pre-specified hypotheses to test. For example, the `TukeyHSD()` function in R provides a way to do this if the numbers of cases in the different categories are not too imbalanced. `TukeyHSD()` expects input from an `aov()` rather than an `lm()` analysis in R, but `aov()` is simply a particular wrapper for the underlying functionality of `lm()`. ANOVA and linear models are essentially just different ways of thinking about the same underlying structure.