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I am currently carrying out a research where I have two categorical predictors with two levels each. The research basically consists of presenting participants with one of four profiles of a target, and asking them to rate this target on several indicators. The independent variables are the name of the target (T. or L.) and the personality traits of this target (C. or A.). Participants view one target name which displays one of two types of traits (so either T.C, T.A, L.C or L.A). My main hypothesis is of an interaction effect, which would show that personality traits have a different effect depending on the name of the target. So I expect, for example, that personality trait C would lead target T to be judged more negatively, but that personality trait A would lead target L to be judged more negatively. This judgment is made on three separate measures. This question would be answered by the interaction effect between my two independent variables.

However, the question that I am asking myself is how can I tell which target has been judged more negatively, between T (with traits C) and L (with traits A).

The idea that I have right now is that if I find this interaction effect coupled with a main effect of the name of the target (showing that, for example T is judged more negatively than L, regardless of personality traits), then this would show that T(C) is judged more negatively than L(A).

I am not 100% sure of this idea, and would love to hear other people's perspective on this.

(P.S. I have already considered just comparing means with a post hoc test, but this option is not ideal for my research question because I would then be comparing between personality traits as opposed to within traits. Comparing T(C) and L(A) as opposed to knowing if T(C) is judged more negatively compared to L(C) than L(A) compared to T(A). Not sure if this makes sense but it is not easy to explain while being concise).

Thank you in advance to anyone who might be able to help !

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Your focus on the interaction term is correct for testing your main hypothesis, but the presence of the interaction term makes what you seem to desire impossible. When there is an interaction between two predictors there are no longer any uniquely defined "main effects." The association of target with the outcomes then depends on the trait that was displayed for the target in the trial, and vice-versa. There's no way around that.

the question that I am asking myself is how can I tell which target has been judged more negatively, between T (with traits C) and L (with traits A).

That particular comparison necessarily is "comparing between personality traits as opposed to within traits." To see if "T(C) is judged more negatively than L(A)" you compare their individual estimates with a post-hoc test.

With your interaction model you can similarly examine any linear combinations of coefficients that you want. You can examine the combinations of regression coefficients that provide estimates of T(C) - L(C), or T(A) - L(A), or T(C) - T(A), or L(C) - L(A), or differences among those types of differences, if any such comparisons helps to evaluate one of your hypotheses. But the interaction means that if you want to "tell which target has been judged more negatively, between T (with traits C) and L (with traits A)," then you necessarily are "comparing between personality traits." The particular idea you have in mind, to see if T is judged more negatively than L with either set of traits, could be evaluated via T(C) - L(C) and T(A) - L(A). Nevertheless, you could still have a significant interaction, your main interest, and not have both of those differences be significant in the same direction.

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Thank you very much for your answer EdM! If I understand correctly, your point is that main effects are not meaningful when an interaction is present. I assumed that they might be in some cases (like in my case, or in others I have read about https://www.theanalysisfactor.com/interpret-main-effects-interaction/)

Maybe I should have included this from the beginning, but this is the pattern of results that I expect to have. In this graph, (from my admittedly limited knowledge of data analysis) I identify an interaction effect between both predictors, as well as main effect of the name of the target (the average of the green line is lower than the average of the blue line). And this pattern of means would in my mind confirm my prediction that T(C) is judged more negatively than L(A). And I guess I assume that the statistical translation of this pattern of means would be interaction + main effect of name.

I guess I will keep in my back pocket the idea of just using a post-hoc test to compare means since as you said it will easily answer my question and is regularly used for such questions.

enter image description here

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  • $\begingroup$ In your plot the green and blue lines cross. In the linked post, this is just as the case "When interactions do make main effects nonsensical", and not as the case "When interactions don’t affect main effects". $\endgroup$
    – dipetkov
    Commented May 5, 2022 at 6:31
  • $\begingroup$ If you read the post again, they argue that the main effect is meaningful when one group's mean is always higher than the other group's mean. For every value on the x-axis. $\endgroup$
    – dipetkov
    Commented May 5, 2022 at 6:32
  • $\begingroup$ Thank you for your answer as well ! I understand that, I just added this link as a way to show that in some cases main effect can be intersting coupled with interactions. I was assuming this could be the case for my question as well (a different case). I still don't quit understand why this pattern of result that I expect could not be shown via the interaction + main effect since that is what is seen in the graph. But from what I gather from the responses to this post, it might not be accurate. $\endgroup$
    – ERMM
    Commented May 5, 2022 at 12:09
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    $\begingroup$ You might also be interested in learning about contrasts (ie comparisons), which you can do easily in R with the emmeans packages. It has lots of vignettes as well. $\endgroup$
    – dipetkov
    Commented May 5, 2022 at 13:20
  • $\begingroup$ The "main effects" are meaningful; it's just that their values depend on how interacting predictors are coded and thus they aren't uniquely defined. For any model and its specific predictor coding , however, the "main effects" are meaningful in the context of that model. They are needed to do the types of comparisons/contrasts that you need. I second the recommendation of @dipetkov that you look into the emmeans package; once you get the hang of it it makes post-hoc comparisons pretty simple. $\endgroup$
    – EdM
    Commented May 6, 2022 at 18:26

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