# Inappropriate use of a constant covariate in repeated measures ANCOVA

When a covariate which only varies between subjects (e.g. age) is in a repeated measures ANCOVA it is termed a *constant* covariate. A constant covariate has no effect on the *main effect* of any repeated measures factor e.g. time. It can, however, influence the w subs factor by being present in a w subs factor by covariate interaction e.g. age by time. A repeated measures ANCOVA assumes there is *no* covariate by w subs factor interaction but some packages such as SPSS automatically fit such a term. If there is *no* covariate by time interaction then the interaction term may be dropped from the model using, for example, the *custom model* option in the SPSS GLM procedure. This can also be achieved using *SPSS MANOVA* OR *SPSS MIXED* syntax although the latter requires a reformatting of the data.

There is some debate about what to do if there is such an interaction. Some authors advocate dropping the w subs factor by subject interaction and interpreting the main effect of the within subjects factor with no interaction present even if the interaction is statistically significant (see here). This approach is supported by SPSS (See here.) Others suggest ignoring the main within subjects effect altogether and explaining the interaction.

It is a good idea to *centre* the covariate (by subtracting the overall sample mean from each data point) before entering it into the repeated measures ANCOVA (Van Breuken and Van Dijk, 2007). Tabachnick and Fidell (2007) and Aiken and West (1991) also advocate centering covariates if they are involved in an interaction. This has the advantage over using raw covariate values of correctly leaving unaltered the within subjects factor sum of squares in the presence of the covariate. In some earlier versions of SPSS (V8.0 and below) covariates were, by default, centred before being placed in a repeated measures ANCOVA. The error sum of squares for the within subjects factor (ignoring the covariate) is divided into a sum of squares for the factor by covariate interaction and a remainder term. (See here)

If the interaction term is not statistically significant it may be dropped from the model using procedures mentioned above. On the other hand if the interaction term is statistically significant then it would appear useful to interpret the covariate by factor interaction by, for example, splitting the covariate into two groups and analysing the within subjects factor in the two groups separately.

If the interaction is not a crossed interaction and, therefore, the differences between the levels of a within subjects factor are *statistically significant* and in the *same* direction and differ *only* in the *magnitude* of these differences then one might consider also quoting the main effect of the w subs factor (after dropping the interaction term from the model) which would represent w subs factor level differences averaged across the covariate values.

Reference

Aiken LS and West SG (1991) Multiple regression: testing an interpreting interactions. Sage:Newbury Park, CA.

Van Breukelen GJP and Van Dijk KRA (2007) Use of covariates in randomized controlled trials. *Journal of the International Neuropsychological Society* **13** 903-904.

Tabachnick BG and Fidell LS (2007) Using multivariate statistics. Fifth Edition. Pearson International: Boston.