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Consider the following model: $$Y_i = \alpha + \beta X_{i1} + \gamma_1D_{i1} + \gamma_2{D_{i2}} + \delta_{11}(D_{i1}X_{i1}) + \delta_{i2}(D_{i2}X_{i1}) + E_i$$ where $X_{i1}$ is a continuous regressor, and $D_{i1}$ and $D_{i2}$ are dummy variables that takes values 0 and 1. Also, $E_i$ is normally distributed with mean 0 and equal variance for all $i$.

I was wondering whether it makes sense to test the significance of $\gamma_2$ for instance, by looking at the result of t-test from summary. Aren't you violating the principle of marginality if you test $H_0: \gamma_1 = 0$ against $H_1: \gamma_1 \neq 0$.

Vic Ryan
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The values of $\gamma_1$ and $\gamma_2$ in the your model with interactions represent their values when $X_1=0$. Their apparent "significance" (in terms of differences from 0) as reported by the usual summary will thus vary depending on how your continuous predictor $X_1$ has been centered. See this page for a similar, simpler situation worked through. So there isn't much point in testing either of them independently of their interaction terms, unless you have a specific hypothesis about their values when $X_1 = 0$ under your centering of that continuous predictor.

EdM
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  • If the significance is very dependent on how $X_1$ has been centered, does it mean that the t-test from "summary" is not very reliable? – Vic Ryan Mar 21 '21 at 09:24
  • @VicRyan the t-test is reliable, in terms of distinguishing whether the coefficient estimate differs significantly from a value of 0. What is confusing (not unreliable) is that the coefficient value itself for a predictor will change depending on how an interacting predictor is centered or has a reference level chosen. When there's an interaction you have to be careful to evaluate the "significance" of a predictor by considering all of the terms involving it, not just so-called (and poorly named) "main effect" coefficients like $\gamma_1$. – EdM Mar 21 '21 at 13:39
  • Do you have to be careful because of the principle of marginality? – Vic Ryan Mar 21 '21 at 19:24
  • @VicRyan the [principle of marginality](https://en.wikipedia.org/wiki/Principle_of_marginality) is why you shouldn't evaluate a "main effect" by itself when there is a true interaction. If you omit the interaction from the model, then the apparent "main effect" will represent an effect averaged/marginalized over the values of the interacting predictor that were in your sample. (That's most directly the "principle of marginality.") If you include the interaction term, then the "main effect" represents only the effect when the interacting predictor is at its reference or 0 level. – EdM Mar 22 '21 at 13:54