Dear Xiuqin,
as it's probably a between-subject factor (no subjects were scanned at two or more centers), why not simply go with a factor "center" with Full factorial? This might result in a 2 x 2 x n ANOVA with group (patients, controls), sex, center (site 1, site 2, ... site n). You could also easily test whether there are any interactions with center. Probably not your main reseach focus but it might be the case that one center treats the late stages of the disease, so the difference between their controls and patients would be larger, or they use stricter criterions, then their patients might show less atrophy because they are diagnosed earlier.
Alternatively you could also add several covariates of course, one covariate with ones for site 2 and otherwise zeros, another with ones for site 3 and otherwise zeros, ... You might prefer this way if number of acquistions varies to a rather large extent between sites. But you could also set up contrast vectors within the 2 x 2 x n ANOVA model that take into account different number of subjects (weighting the larger site more strongly). But this should be something for our statisticians in the forum :-)
Best,
Helmut
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