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In my data set, I have two primary predictors A and B. I also have the outcome Y.
I have created four multivariate models that include other covariates.

  • Model 1 includes A and covariates,

  • Model 2 includes B and covariates,

  • Model 3 includes both A and B and covariates, and

  • Model 4 includes A and B and covariates and an A**B interaction term.

    In Model 1, A is highly associated with Y (Beta=0.037, p<.0001).
    In Model 2, B is highly associated with Y (Beta=0.234, P<.0001).
    In Model 3, the parameter estimates of both A and B shift closer to 0 and significance decreases (A Beta=0.029, p=.0003; B Beta=0.157, p=.009).
    In Model 4, significance and magnitude of both predictors are the highest out of all models (A Beta=0.237, p<.0001; B Beta=0.399, p<.0001; A*B interaction Beta=-0.066, p<.0001). . I also know that A and B are correlated (Spearman's r=0.484, p>.0001). I am looking for assistance in interpretation of these results.
    I hypothesize that there may some collinearity in Model 3 without the interaction term.

Does the interaction term serve to diminish that? In general, I am unsure why significance decreases in Model 3 and increases in Model 4.

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1 Answer 1

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In regression there is a danger of omitted-variable bias. In linear regression, if you omit a predictor that is associated both with outcome and with a predictor that is in the model, your results will be biased.

With A and B highly correlated, your results are what you thus might expect. Model 1 attributes to A both its own contribution to outcome and a portion of what B might be contributing on its own, due to the correlation between A and B. Model 2 performs similarly with respect to B.

Model 3 includes both A and B. I suspect that if you look at the variance-covariance matrix of the coefficient estimates (often hidden from view but eventually accessible with a function applied to the model, like vcov() in R), there will be a high-magnitude correlation between their coefficients. I also suspect that predictions based on Model 3 would be superior to those based on either Model 1 or Model 2, even if the individual coefficients for A and B appear to be less "significant."

The significant interaction in Model 4 shows that neither A nor B contribute separately to outcome; the effect of one depends on the level of the other. As the interaction term is associated both with outcome and with each of A and B, its omission from Model 3 would be consistent with omitted-variable bias in Model 3.

One warning on a point that frequently leads to confusion. In Model 4 with the interaction, the individual coefficients for each of A and B represent the situation when the other member of the interaction is at a particular level: 0 for a continuous predictor, or (often) the reference level of a categorical predictor. Re-centering or re-referencing a predictor thus can affect the coefficient of other predictors with which it shares interaction terms.

As the standard test for "significance" is whether a value differs from 0, adding an interaction term can thus make the individual coefficient for one or both of its components appear to be "non-significant." That's an illusion, based on a fairly meaningless type of significance test. When there is an interaction term, proper tests of significance must take into account all of the terms that include a predictor, its interactions as well as its "own" coefficient.

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