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I've adjusted a linear model: $Y = \beta_{0} + \beta_{1}X_{1}+\beta_{2}X_{2}+\beta_{3}X_{3}+\beta_{4}X_{1}X_{2}+\beta_{5}X_{2}X_{3}+\beta_{4}X_{1}X_{3}$. All coefficients are significant (p<2e-15) but I've observed $\beta_{4}$ which is related to the interaction between $X_{1}$ and $X_{4}$ has a pretty low value itself. I was wondering if I can say that regardless significant it has no effect.

Besides, can is a low value of interaction coefficient an evidence of no need of interaction? Are there any kind of difference in the interpretation of low level main effect or low level interaction effect?

Thanks in advance!

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    $\begingroup$ If you say it is significant, then presumably you are rejecting the hypothesis that it has zero effect. But if the suggested magnitude is very small you might want to say it does not have a substantial effect $\endgroup$ – Henry May 30 at 1:23
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$\beta_4$ seems to be the coefficient for the interaction between $X_1$ and $X_3$, not between $X_1$ and $X_4$ as stated in the question.

Anyway, putting that asside, when you have a small p-value it is telling you that the probability of observing these data, or data more extreme, IF the null hypothesis is true (that the true parameter estimate is actuall zero) is small.

It is good that you are paying attention to the effect size. It is telling you that allthough the interaction is statistically significant, it is clinically/practically small. If it is so small to be meaningless in the context of your study, then the statistical significance is largely irrelevant.

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