# p-value of Covariate differs between Regression and ANCOVA

I'm using SPSS to run a regression analysis in order to predict navigational performance (continuous dependent variable) from self-assessment scores (continuous predictor), sex and age group (both categorical predictors). I'm also interested in the interaction effects of (score x sex) and (score x age group). Categorial predictors were coded with 0 and 1 and all predictors were centered before computing the interaction terms.

Since the residual plots indicated heterogenity of variances (heteroscedasticity), I wanted to perform a Levene's test on the model. In order to do so I used the "Univariate" dialogue in SPSS and specified the same (custom) model as in my regression analysis:

• Main effects: score, age group, sex.
• Interaction effects: score x age group, score x sex. In my understanding, regression analysis and AN(C)OVA are essentially the same linear models. However, using the "Univariate" option in SPSS with scores as a covariate I receive results differing from the regression analysis:

Sum of squares, R² and F-value of the whole model are identical between regression analysis and ANCOVA. p-values for the categorical predictors and interaction terms are identical as well.

The only difference I see is in the p-value of the main effect of score:

• Entering score as factor in the regression model: p = .182
• Entering score as covariate: p = .009

I'm just wondering how does this happen? Shouldn't both models be analoguous?

• This looks like a matter of being two-sided for the first p-value and one-sided for the second. – Dave Dec 9 '19 at 21:21

The models are indeed the same overall model, as shown by the identical R2, F, and sums of squares for the overall model. You could obtain the same results in UNIANOVA as in REGRESSION if you were to substitute all the variables you actually entered in REGRESSION as covariates in UNIANOVA.

I expect that not only is the regression coefficient and test for the score effect different in UNIANOVA, but also the constant or intercept coefficient and test result.

When you have interactions in your model, any effects contained in those interactions don't have unique definitions. In this situation, the intercept is contained in the main effects for the two categorical factors, and the score main effect is contained in both of the interaction effects involving it.

What I mean by contained is a formal definition that means that an effect A is contained in an effect B when:

1. Both A and B have the same covariate effects in them.
2. B has all the factor effects contained in A, if any, plus at least one more.

Note that the intercept is considered a factor effect for these purposes, and is contained in any pure factor effects. A covariate like your score is contained in any interactions involving score and at least one factor, but no other covariates.

This discussion is pretty much mathematical, but there's also a more general sense in which it doesn't make sense to look at a main effect of one variable when it's contained in interactions with any other variables, because the whole point of interaction is that it means that the effect of a predictor varies depending on the levels of any other predictors with which it interacts. See for example discussions of John Nelder's marginality principle (e.g., Wikipedia Principle of marginality).