# Between-subjects effect becomes non-signicant after centering covariate. Should I center or not?

I've used a general linear model function to run an ANCOVA, involving a categorical predictor (with two levels), a continuous predictor, and their interaction effect.

If I don't center the continuous predictor, I get a significant effect for the categorical predictor and a significant interaction effect. If I mean-center the covariate, the categorical predictor effect becomes non-significant but the other effects remain unaltered. Does anyone know why this happens? And which option is the correct one (to center or not to center?)?

Neither, or both, are correct.

You often cannot interpret the main effect when you have an interaction in there. The main effect is the difference between the groups when the covariate is equal to zero - if zero is meaningful, then it's interpretable. If you move where 0 is (by centering) you change the main effect of the between subjects factor. But the actual model doesn't change, you just shuffle the parameters about.

Usually a graph is sufficient to see what's going on. If you want to get more technical, you can calculate 'regions of significance', see: http://quantpsy.org/interact/

(Note that ANCOVA is multiple regression, which is why the pages refer to multiple regression.) Here's a picture, the lines of best fit for two groups are shown. If the intercept is at the left, then group 2 has a higher score. If the intercept is on the right, then group 1 has the higher score. If you center the covariate, then the two lines are at the same height, and there's no difference.

• Thank you for the helpful answer! I'm still wondering if I'm using the best (or the most applicable) analysis method to test my hypothesis. So depending on the type of experimental manipulation (level of categorical predictor), the continuous predictor (X) and the dependent variable (Y) have either a negative or a positive relationship (as hypothesized). I guess this is what one would call a cross-over interaction. X is never 0, so this would be a reason for centering it. A main effect of the categorical predictor would be cool, but it teeters on the threshold of statistical significance. Dec 17, 2014 at 19:57
• Yes, you're doing the best thing (I think). The main effect of the categorical predictor doesn't mean anything in the presence of the interaction though. They're not necessarily cross (as I've drawn them), but they'll meet at some point. Draw scatterplot, and then add lines of best fit for each group (I've forgotten how to do this in SPSS, but you should be able to Google it.) Dec 17, 2014 at 21:25
• (+1) This is perhaps a bit nit-picky, but the coefficient of main effect certainly means something, mathematically at least--it's the expected change in response due to the categorical predictor, when the interaction has been zeroed out (ie, is held at its mean value if centered, or held at exactly zero if uncentered). Whether a test on the main effect is scientifically meaningful depends on if there's any reason to single out "0" for the continuous variable. Dec 18, 2014 at 1:13
• Thank you Andrew for the clarification! I think there is no reason to single out "0" for the continuous variable; it's a psychometric variable that can never have the value 0. Only the effect sizes and their significance are of interest for my study. The only real worry that I have is that am I somehow distorting the results (the effect sizes) by not centering the continuous variable (as it never has and cannot have the value 0)? Dec 18, 2014 at 12:10
• @Jonna - you're not distorting the results, the results are the same whether you center the variable or add 1000 to the variable. But whatever you do, you need to be careful with interpretation. Dec 18, 2014 at 17:34

In your first regression the mean of the covariate is in both covariate itself and the intercept, and both are significant. In the regression notice what happens to the intercept: it must have changed quite a bit, because now the covariate's mean went into it.

UPDATE For instance, say the true process is $y=x+\varepsilon$, but you model it as $y=\beta_0+\beta_1z+\epsilon$ where $z=x-1$. If you estimate the model, the intercept should come as $\beta_0=1$, because $y=\beta_0-\beta_1+\beta_1x+\epsilon\equiv x+\varepsilon$ which will make $\beta_1=1$ and $\beta_0-1=1$

On the other hand, if you estimate the "correct" model, i.e. $y=\beta_0'+\beta_1'x+\varepsilon$, then the intercept must come as $\beta_0'=0$.

Concluding, I think the interaction is a red herring in this case. The real phenomenon is that when you subtract the mean from the main effects it will go into the intercept. Depending on the situation it may or may not change the significance of the intercept. It will definitely change its estimated value.

• Thanks for the helpful answer! Indeed, when I don't center the covariate the intercept is non-significant, but if I center it the intercept becomes significant. I'm not quite sure what to make of that, though! Dec 17, 2014 at 23:42
• @Jonna, see my update Dec 18, 2014 at 0:53