Without interaction terms in a model, coefficients of lower order terms represents partial regression weights (i.e. how much influence does one parameter have assuming all other parameters are held constant). With interaction terms in a model, these same coefficients now represents conditional regression weights (i.e. how much influence does one parameter have when all other parameters are zero) and interaction terms* represent how much does one parameter's influence change when the value of another parameter changes. (Citation)

This means that by including interaction terms you lose the partial regression weights. Is there ever a situation where I can justify not including interaction terms by saying 'I wanted to know how much influence a parameter had when all other parameters where held constant, so I had to exclude the interaction terms'?

*This is the interpretation for second degree interaction terms, but my question does not change if you include higher degree interaction terms.

Edit to provide example: Lets say you have enough data to make a linear model where the parameters are a bunch of characteristics of female wolves and the response is fitness. Three of these parameters are how dark the wolves' coat is (C), the weight above the average (W), and the age of the wolves (A). You have three wolves available to you; a male and 2 females. The females are the same age, but one has a darker coat and the other is larger. If presented with both females, which will the male prefer?

If you do not include interaction terms, this is a simple question to answer. A is constant, so if C has a larger coefficient than W than the darker wolf should be selected, and if the coefficient of W is larger than the reverse should happen. In other words, if the benefit gained by a dark coat is greater than the benefit gained by having a larger mother, than the male should pick the darker wolf, and the reverse should happen if the benefits are reversed.

If you include interaction terms, I do not see how you can answer the question. The coefficients of C and W only mean something if all other parameters are 0, which they are not in this case. The interaction term CxW tells you how the influence of C changes if W changes or vice versa, but that is irrelevant to the question you are asking.

  • $\begingroup$ The edit you have made substantially changes the question. I suggest making a separate post about this example. Namely, the answer to your example is that both a model with and without interaction can perfectly predict which wolf has a higher chance of being selected. $\endgroup$ – Frans Rodenburg Feb 16 at 13:29
  • $\begingroup$ Can you explain what you mean by "changes the question"? My intention by adding the example was to give a situation where the question would be applicable. A model without interaction terms can answer the question, but a model with interaction terms cannot. I do not see how this example is not an application of the question. $\endgroup$ – E Tam Feb 16 at 13:43
  • $\begingroup$ Using the interaction, you can compute the effect of coat color at any value of age. Therefore, it seems your question is not whether you can use a main effects model, but rather how you can use an interaction model to obtain predictions of the effect of one variable, given the other. These are both good questions and both on-topic here, but they are very different questions. That is why I suggest you remove the example and turn that into a new question. $\endgroup$ – Frans Rodenburg Feb 16 at 13:49

You have given proper justification in your question. If you want to know the effect of one with the rest held constant, use a main effects model.

The principle of marginality states that main effects do not exist in the presence of an interaction, so beware that this is only valid if there is indeed only a negligible interaction effect.

Note that the estimate is then the effect of one explanatory variable with the rest fixed at $x_j = 0$. If you want to know the effect of one variable averaged over the other variables, you can use estimated marginal means. See for example the emmeans package.

  • $\begingroup$ Said another way, when the model you pre-specified contains interaction terms, the weights of variables that interact with other variables are functions of those other variables and it is not very fruitful to try to hide that fact.Form the needed contrasts to estimate effects at your choice of settings of interacting factors. You can also get simultaneous confidence regions for these contrasts. $\endgroup$ – Frank Harrell Feb 16 at 11:39

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