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I have a very basic interaction model with this setup:

Y = $\beta_0$ + $\beta_1$ Var_1 + $\beta_2$ Var_2 + $\beta_3$ Var_1 * Var_2 + Controls + $\epsilon$

  • Y = Continuous variable
  • Var_1 = Continuous variable
  • Var_2 = Indicator variable (i.e., 1 or 0)

In this model, $\beta_3$ is significant--but joint test of $\beta_1$ + $\beta_3$ (i.e., when Var_2 = 1) is insignificant.

How do I interpret the interaction term when the total effect isn't significant?

Edit:

My post has been identified as a potential duplicate, but I don't think that this is the case because those links appear to be concerned with the interpretation of main effects when an interaction effect makes the main effects insignificant.

I'm interested in the interpretation of a significant interaction effect when the total effect (i.e., $\beta_1$ + $\beta_3$) is insignificant.

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  • $\begingroup$ Can you explain a bit further what your scientific hypothesis is? $\endgroup$
    – mdewey
    Dec 30, 2016 at 12:47
  • $\begingroup$ Why would significance (or lack thereof) of an estimate have any effect whatsoever on its interpretation? Are you asking about what the coefficients mean in the model or how to react to the associated p-values? $\endgroup$
    – whuber ♦
    Dec 30, 2016 at 15:49
  • $\begingroup$ This doesn't seem like quite a duplicate to me. $\endgroup$
    – gung - Reinstate Monica
    Dec 30, 2016 at 19:18
  • $\begingroup$ @mdewey: I think that this helps answer my question. I need to think about what each comparison means. B1 + B3 indicates the total effect of the condition where Var2 = 1 on Y. B3 tells me that there's an incremental difference between B1 and Y in the presence of Var2. $\endgroup$
    – Tots
    Dec 30, 2016 at 19:35
  • $\begingroup$ @whuber: I was wanting to know how to interpret each coefficient in the presence of each of the associated p-values. I think that I have it now. The interaction says that there's a significant difference in the relation between Var1 and Y in the presence of Var2, but then the total effect indicates that the total effect isn't different from zero (i.e., B1 + B3 is insignificant). $\endgroup$
    – Tots
    Dec 30, 2016 at 19:37

2 Answers 2

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It's worth considering what an interaction is. The existence of an interaction means that the relationship between one variable and the response varies as a function of the other variable. It doesn't mean that the relationship differs from 0 at any particular level (although it must somewhere). Consider the simplest case: a balanced design with two categorical variables with two levels each. Let's say that the data within each combination of levels is normally distributed and with the same variance. Let's further stipulate that the means are as listed below, and the standard errors on the slopes are 0.3: 

        Var 2
Var 1   A     B
    A   0    -0.5
    B  -0.5   0

Now, in a model where level A is taken as the reference level, the interaction would be significant, but neither main effect would be. Moreover, the linear contrast of Var 1 + interaction would mostly cancel itself out, because the effects move in opposite directions and the standard error of the sum of the two slopes would increase.

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It just says that effect of Var_1 on Y vanish (disappear) when Var_2=1. Indeed because b1 is significant it means that when Var_2=0 then marginal effect of Var_1 on Y is b1. However when Var_2=1 then marginal effect is b1+b3 that is statistically insignificant differ from zero (as you have tested).

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