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I am new to statistics and I have been doing some reading on ANCOVA lately. There is something that is slightly confusing me:

When motivating the use of the ANCOVA model, many of the resources that I have found on the web describe some experiment, where a one-way ANOVA yields that the categorical independent variable indeed influences the continuous dependent variable. One example is the introductory part of this.

What I have read so far makes it seems like one needs to perform an ANCOVA to control for the effects of the covariates only if statistically significant differences between the population means were found in a one-way ANOVA beforehand. Is this correct or did I get something wrong?

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I think this is an incorrect notion; it's quite possible that failing to include a covariate will make the ANOVA insignificant, for two reasons - each of them directly related to reasons why the covariate might have been included in the first place

1) The first is that including important covariates reduces error variance, which leads to an increase in power; failing to include them (and thereby facing reduced power) means you're less likely to reject the null for the ANOVA.

http://en.wikipedia.org/wiki/Analysis_of_covariance#Increase_Power

2) the second is that failing to include an important covariate can shift the coefficient estimates, perhaps making the ANOVA insignificant when there's actually a strong effect once the covariate is adjusted for (but in any case leaving you with potentially biased estimates).

http://en.wikipedia.org/wiki/Analysis_of_covariance#Adjusting_Preexisting_Differences

See also Simpson's Paradox. The illustration is particularly relevant to this question.

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  • $\begingroup$ Agreed. I think statistics textbooks probably present them in that order because the ANOVA is simpler, and the ANCOVA can be seen as an extension of it. If you believe that your additional predictors will explain substantial variance in your model, there's no reason to tie your hands by insisting on finding significance without them. The whole point of covariates is to increase your power to find effects. (unless you're interested in the covariates as predictors in their own right, in which case it makes even less sense to leave them out.) $\endgroup$
    – octern
    Feb 16 '14 at 16:52

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