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I've been told by my stats prof that running an ANCOVA relies on a series of assumptions, including that the the dependent variable and the covariate are linearly related. So, in SPSS the first thing to do before running ANCOVA is to check for bivariate correlation between them.

The question is, if that correlation is not significant... does this mean that you are NOT allowed to run the ANCOVA (statistically speaking)? Or does it simply indicate that what you get with it may not be very different from what you would get with a regular ANOVA?

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It is possible to have a significant covariate in an ANCOVA model that does not show a significant correlation with the response variable when only looking at those 2 variables (due to some relationship between the covariate and the other terms being analyzed in the ANCOVA variable). In some of these cases not including the covariate (regular ANOVA) can give very different answers from the correct ANCOVA. So I would suggest still doing the ANCOVA if there was some a priori reason to believe that there may be a relationship.

Generally when we warn people about the linear assumption in ANCOVA (also a parallel assumption) it is not to discourage ANCOVA, but rather to point out that if the relationship is non-linear (curved) then standard built-in ANCOVA routines will not capture the whole relationship and other routines may be needed (multiple regression with indicator variables, interactions, and polynomial or spline terms for example).

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    $\begingroup$ +1 Please note, though, that a covariate which is completely orthogonal to all other independent variables can still be uncorrelated with the dependent variable yet be significant in the ANCOVA. This is explained and illustrated at stats.stackexchange.com/a/28493. $\endgroup$ – whuber Jun 26 '14 at 16:31
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The answer is the latter: Might as well do regular ANOVA. If the DV and the covariate are not correlated then there isn't much point to doing an analysis of covariance: the covariate will not usefully extract variance from the DV. After all, the analysis is based on the assumption that the covariate is introducing variability into to DV in such a way that those who score low on the covariate score low on the DV; moderate on the covariate = moderate on the DV; high on the covariate = high on the DV (or the inverse if the DV and covariate are significantly negatively correlated). That's what makes something a "covariate". So one reason the various treatment groups may differ on the DV is the effect of the covariate. The analysis of covariance partials this effect of the covariate on the DV out of the effect(s) that the IVs have on the DV, giving you a "purer" look at the IVs' unique effect(s).

In addition to checking whether the DV and covariate are correlated at all it is important to look at the DV/covariate bivariate correlation to make sure they are LINEARLY correlated. If they are curvilinearly related then the covariate will not remove variance from the DV in a consistent way across the treatment groups. If this is the case, choose a different covariate, one that is strongly and linearly correlated with the DV.

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  • $\begingroup$ For the covariate to be useful it is not necessary for the DV and the covariate to be correlated. Instead, the covariate will improve the model provided its residuals with respect to all other independent variables (the ANOVA factors and any other covariates) are correlated with the residuals of the DV with respect to those same IVs. Provided we understand all your references to the DV and the covariate to really mean these residuals, then your answer would seem to be correct and contain useful advice. $\endgroup$ – whuber Jun 26 '14 at 15:55

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