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Subject:

Re: Ancova in SPM8

From:

Stephen J. Fromm

Date:

Fri, 13 May 2011 20:38:20 +0100

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 ```Hmm...it's hard for me to tell about centering just by reading the SPM options for it. I tried to load your job into SPM but it doesn't generate the design matrix, because obviously I don't have the data files. That said, I think the centering is probably correct just based on the design matrix schematic that you'd posted previously. First, an aside: when I loaded the job, it showed you had set ANCOVA to "yes" under the "factor" part of the design spec. That's a "different" ANCOVA; it's for PET. So change that to "no". (That said, I don't think it affected the way your design was set up.) So, getting back to the main question: I think the design is fine. It's just a matter of knowing how to interpret the results, ie what the betas are. So you can make appropriate contrasts. In rough, verbal terms, the betas for the first four columns are the main effect of condition, and the other columns are the condition-by-covariate interactions. If you want to more precisely delineate what's going on, write the model down symbolically: Condition: 1 <= i <= 4 Subject: 1 <= j <= 13 Covariate: 1 <= k <= 2     y_ij = beta_i + C_jk * gamma_ik + error (say). It's Einstein summation notation (sum over repeated indices), so it means     y_ij = beta_i + C_j1 * gamma_i1 + C_j2 * gamma_i2 + error * y is the data; it's indexed by condition and subject, but not by covariate * beta_i is the "beta" for condition i * C_jk {k=1,2} are the mean-centered covariates. They don't depend on condition, i, but only on subject, j. * gamma_ik is the "beta" for the condition-by-covariate interaction If we average over subject but not over condition, we get     y^hat_i. = beta_i Why? The C_j1 and C_j2 are mean-centered so go away. The error goes away since we're looking at predicted values in this type of formulation. So, the parameter estimates for the first four columns are the effect of condition. (Perhaps technically it's the differences of these...I'm not sure what conditions you're using, though you said in a previous post that they're already differences.) And the remaining parameters, what I've called the gammas, pretty clearly give the appropriate interactions. You can think of them as similar to the textbook ANCOVA examples, where the horizontal axis is the value of the covariate. Only difference with your case here is that you have two covariates not one. There's a subtlety to ANCOVA that always comes up. Here, e.g., beta_3 is the mean for condition three. But the meaning of that number is clouded because it's for a "typical" subject in your set of subjects, and for the average such subject the covariate is zero (precisely because it's been mean-centered). But that's not necessarily indicative of a truly generic subject. Though this issue is typically more at hand in cases where ANCOVA is used with a group factor, not a within-subject factor. ```