How to determine if a change in F value when introducing a covariate, is significant? My question may be very basic but I hope that you can help me.
I have got two groups (A and B). I am measuring two continuous biological measures (X and Y) which are significantly positively correlated. Theoretically it is clear that X can influence Y but not vice versa.
Y has been measured two times (Y1 and Y2) and should be analyzed by repeated measurements analysis.
The effect of group on X is strong and significant (ANOVA).
The effect of group on Y is also significant (repeated measurements ANOVA) but when introducing X as a covariate (repeated measurements ANCOVA), it is very small and not significant anymore. To my opinion, this probably means that group is influencing Y only indirectly.
I would like to know if the difference in p values and/or effect sizes between the group effect on Y  with and without X as a covariate, is significant.
Is there a way to test this?
 A: Yes there is an appropriate F test comparing the hierarchical models lower order model without X compared with the higher order model with X included.  Alternatively you can test to see whether or not the coefficient for X is significantly different from 0.  This is done with a t test.  Normailty of residuals is assumed for this.
A: Let me preface my answer with saying that I am still a novice- please use this answer with a grain of salt and use it as a starting point to research this method to confirm it is right for you.  If anyone else thinks this is wrong, please correct me.
I assume that you have a response variable (ie the thing you are interested in learning about) that you have not mentioned, and that X, Y, and group are all your predictor variables.
It sounds like you could do stepwise elimination of variables to take care of this entire set of tests in a single test, to answer your question.
Your full model (with all interactions) would be:
Response variable=Group + X + Y + GroupX + GroupY + XY + GroupX*Y
(with Group, X, and Y being the effect of these single variables on your response variables, and the various interactions (like X*Y) being the effect that X has in the presence of Y, etc.
I would enter this full model, then run a stepwise regression and look at the AIC values to see which model gives you the smallest AIC.  I am sure you can do this in SPSS.
The procedure will remove the factors or interactions that do not significantly assist the fit of the model, leaving only the models or interaction that significantly help the fit of the model.  You'll have to find a method to deal with the fact that your covariates are correlated.
If you have the effects you are describing in the final model that is accepted, it might look like this:
Response= Group + X + Y +GroupX + GroupY
With the 3 way interaction dropping out, because Y does not help predict your response when X AND group are included in the model.  This is assuming that X, Y, and Group actually affect your response individually.
Once you get your response, you will have to interpret what any significant interactions (or lack thereof) mean.
I hope this is at least a little helpful and not too confusing.
