1. ## Ancova

Hey stats gurus,

Up until today I thought I conceptually (at least basically) understood what ANCOVA was doing but now I don't.

I have this data set (simple one-way ANOVA) that when I include the covariate it actually decreases the significance of the main effect... ?!? How is that possible and what does that mean?

I realize this is probably more complicated then can be explained here but if anyone has a quick and dirty answer to help me understand that would be great. What would be even better though, is if anyone knows of an online resource I could read up on in order to understand what's going on here. I don't really know what I'm looking for so my google searches have been futile.

2. "I have this data set (simple one-way ANOVA) that when I include the covariate it actually decreases the significance of the main effect... ?!? How is that possible and what does that mean?"

I am not an expert on this topic, but what immediately jumped out at me was correlation between the factor levels of the main effect and the covariate. I am trying to think through the math to ponder what I would do next if I was in this position at the moment. I think it would depend on what my goals were though.

(ps think about the most pathological covariate with one main effect possible: the main effect is just coding for the covariate falling into a certain range. Then the main effect could be beautifully predictive by itself while having it's statistical significance mauled by the inclusion of the actual covariate. It was just coding for the covariate in the first place!)

3. ... ?!? How is that possible and what does that mean?

Most likely this is because there is a small with-in treatment correlation (rho) between the variate (Y) and the covariate (X). The reduction in experimental error (Sigma^2) is follows:

Sigma^2*=(Sigma^2)(1-rho^2)(dfw/(dfw-1))

where Sigma^2* is associated with the ANCOVA, Sigma^2 is error associated with the ANOVA, and dfw are the degrees of freedom (within) for the ANOVA.

Note that you're losing one df in the denominator of the F ratio by including the covariate.

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