Simple effect testing at group analysis level
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Assume a model of one-way within-subject (repeated-measures) ANOVA (3dANOVA2 -type 3)
Yij = μ + αi+ βj + εij
where,
Yij independent variable – regression coefficient (% signal change) from individual subject analysis;
μ constant – grand mean;
αi constants subject to Σαi = 0 – simple effect of factor A at level i, i = 1, 2, ..., a;
βj independent N(0, σp2) – random effect of subject j, j = 1, 2, ..., b;
εij independent N(0, σ2) – random error or within-subject variability or interaction between the factor of interest and subject.
Assumptions are:
E(Yij) = μ + αi, Var(Yij) = σp2 + σ2, Cov(Yij, Yi'j) = σp2 (i ‡ i'), Cov(Yij,,Yi'j') = 0 (j ‡ j');
Correlation between any two levels (αi and αj) of factor A: σp2/(σp2 + σ2).
For simple effect testing
H0: αi = 0
the t statistic for this null hypothesis was implemented in 3dANOVA2 -type 3 as
H0: αi = 0
the t statistic for this null hypothesis was implemented in 3dANOVA2 -type 3 as
t = Yi·/sqrt(MSAS/b) (as shown for option -amean in the current version of 3dANOVA2 manual)
where Yi· (with a bar at the top) is the sample mean of Yi1, Yi2, ..., Yib, and MSAS is the mean squares of sums for the interaction term between factor A and subject (S). The problem with the above statistic is the following: The variance of the numerator Yi· does not match up with the expected value of the term MSAS:
E(Yi·) = μ,
Var (Yi·) = (σp2 + σ2)/b, E(MSAS) = σ2
Var (Yi·) = (σp2 + σ2)/b, E(MSAS) = σ2
The consequence is that, if any two levels of factor A are positively correlated (e.g., σp2 ‡ 0) , Var (Yi·) ‡ E(MSAS) unless σp2 = 0, and the above t statistic would get inflated. Such misrepresentation could be very significant if cross-subject variability is large. Because of this inequality, the old t formula was not validly t-distributed.
Instead of testing the simple effect with the above statistic, it is more appropriate to do the following:
t = Yi· /sqrt(s2(Yi·)/b)
which is basically a one-sample t test with s2(Yi·) being standard error of samples of factor A at level i: Yi1, Yi2, ..., Yib. This approach is more aligned up with the general philosophy of testing with only the partial data involved in the null hypothesis.
The options -amean and -bmean in 3dANOVA3 have also been corrected based on the same principle.
The options -amean and -bmean in 3dANOVA3 have also been corrected based on the same principle.
