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One of the ANCOVA assumptions, as I read it here, says that:

Independent variables orthogonal to covariates. In traditional ANCOVA, the independents area assumed to be orthogonal to the factors. If the covariate is influenced by the categorical independents, then the control adjustment ANCOVA makes on the dependent variable prior to assessing the effects of the categorical independents will be biased since some indirect effects of the independents will be removed from the dependent.

See also the third assumption here.

Could you tell me how should I verify that this assumption is met, in a given model containing three factors and one covariate?

You can refer to SPSS or R ways of doing this.

Thank you very much

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One approach is to see if the covariate is correlated with the predictor variables. That is, if the ANCOVA is given by:

predicted ~ covariate + predictor1*predictor2*predictor3

Then first assess whether the covariate and the various predictor effects/interactions are correlated:

covariate ~ predictor1*predictor2*predictor3

If you find that the covariate is correlated with any of the predictor variables or their interaction, then you're violating the assumption you cite. If the covariate is categorical with more than 2 levels, you'll have to assess the correlations via multinomial regression.

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  • $\begingroup$ That's just plain wrong. One of the points of doing ANCOVA is to adjust for covariate effects that might be confounded with the predictor. $\endgroup$ – Aniko Dec 29 '10 at 16:34
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    $\begingroup$ No. ANCOVA is meant as a way to enhance power when you have a covariate related to the predicted variable but unrelated to the predictor variables. It is lamentably common that folks fail to understand this and instead attempt to use ANCOVA to "control for"/"regress out" effects of the covariate when they are correlated with the predictors. See Miller 2001: ncbi.nlm.nih.gov/pubmed/11261398 $\endgroup$ – Mike Lawrence Dec 29 '10 at 16:38
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    $\begingroup$ I stand corrected. $\endgroup$ – Aniko Dec 29 '10 at 17:10
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    $\begingroup$ I should note that I myself was taught and employed (ncbi.nlm.nih.gov/pubmed/19154749) the inappropriate practice of using ANCOVA to "control for" a covariate with a known relationship with other predictor variables. It's only been through later self-directed study that I discovered the Miller & Chapman paper and the errors of my ways! $\endgroup$ – Mike Lawrence Dec 29 '10 at 17:24
  • $\begingroup$ Do you mean that I'll have to perform a 3-way anova and look for significant effects? $\endgroup$ – George Dontas Dec 30 '10 at 12:24
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EDIT As pointed out in the comments below, this is an answer to a different question, one about ensuring that the effect of the covariate is the same in all groups.

You have to check for interactions of the covariate and the factors. So if the ANCOVA model was

lm1 <- lm(outcome ~ covariate + factor1*factor2*factor3)

then the you can add all the interactions and look at

lm2 <- lm(outcome ~ covariate*factor1*factor2*factor3)

followed by an F-test:

anova(lm1, lm2)

It might also make sense to add fewer interactions (especially if you don't have a lot of data), so that the large number of high order interactions don't eat up the power.

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    $\begingroup$ I thought that interaction between the covariates and the factors indicates violation of the homogeneity of regressions assumption (another assumption also mentioned in the first link I gave) $\endgroup$ – George Dontas Dec 29 '10 at 16:53
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    $\begingroup$ This approach tests whether the effect of the predictor variables (on the predicted variable) is affected by the covariate (or vice versa). In other words, this tests assumption 2 on the wikiversity entry linked in the question. This routine cannot speak to assumption 3, the focus of the poster's question, which is about the relationship between the values of the covariate and the values predictor variables. My answer speaks to this assumption. $\endgroup$ – Mike Lawrence Dec 29 '10 at 16:56

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