¥In a mixed
effects model, ANOVA cannot deal with unequal variances in the random factor between different levels of a fixed factor
HExample:
2-way layout, factor A=stimulus type (fixed effect), factor B=subject (random effect)
å As seen earlier, ANOVA can detect
differences in means between levels in A (different stimuli)
åBut if
the measurements from different stimuli also have significantly
different variances (e.g., more attentional wandering in
one task vs. another), then the ANOVA model for the signal is wrong
åIn
general, this ÒheteroscedasticityÓ problem is a difficult one, even in a 2-sample t-test; there is no exact F- or t-statistic
to test when the means and the variances might differ simultaneously
¥Although ANOVA
does allow somewhat for intra-subject correlations in measurements, it is not fully general
HExample:
2-way layout as above, 3 stimulus types in factor A; general correlation matrix between the 3 different types of
responses is
but ANOVA only properly deals with the case r12=r13=r23
(recall we are assuming subject effects are random;
this is the
correlation matrix for the intra-subject random
responses).
¥Possible
solution: general linear-quadratic minimum variance mixed effects modeling
HA
statistical theory not yet much applied to FMRI data (but it will be,
someday)
HQuestions
of sample size (number of subjects needed) will surely arise