# Simple effect testing at group analysis level

Assume a model of one-way within-subject (repeated-measures) ANOVA (3dANOVA2 -type 3)

Y

_{ij}= μ + α_{i}+ β_{j}+ ε_{ij}where,

Y

_{ij}independent variable – regression coefficient (% signal change) from individual subject analysis;

μ constant – grand mean;

α

_{i}constants subject to Σα_{i}= 0 – simple effect of factor A at level*i*,*i*= 1, 2, ...,*a*;β

_{j}independent*N*(0, σ_{p}^{2}) – random effect of subject*j*,*j*= 1, 2, ...,*b*;ε

_{ij}independent*N*(0, σ^{2}) – random error or within-subject variability or interaction between the factor of interest and subject.Assumptions are:

E(Y

_{ij}) = μ + α

_{i}, Var(Y

_{ij}) = σ

_{p}

^{2}+ σ

^{2}, Cov(Y

_{ij}, Y

_{i'j}) = σ

_{p}

^{2}(i ‡ i'), Cov(Y

_{ij},,Y

_{i'j'}) = 0 (j ‡ j');

Correlation between any two levels (α

_{i }and α_{j}) of factor A: σ_{p}^{2}/(σ_{p}^{2}+ σ^{2}).For simple effect testing

H

the

H

_{0}: α_{i}= 0the

*t*statistic for this null hypothesis was implemented in*3dANOVA2*-type 3 as

*t***=**

**Y**

_{i·}

**/sqrt(MSAS/**(as shown for option -amean in the current version of 3dANOVA2 manual)

*b*)

where

*Y*_{i·}(with a bar at the top) is the sample mean of Y_{i1}, Y_{i2}, ..., Y_{ib}, 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*Y*_{i· }does not match up with the expected value of the term*MSAS*: E(Y

Var (Y

_{i·}) = μ,Var (Y

_{i·}) = (σ_{p}^{2}+ σ^{2})/*b*, E(*MSAS*) = σ^{2}

The consequence is that, if any two levels of factor A are positively correlated (e.g., σ

_{p}^{2}‡ 0) , Var (Y_{i·}) ‡ E(*MSAS*) unless σ_{p}^{2}= 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-distributedInstead of testing the simple effect with the above statistic, it is more appropriate to do the following:

*t***=**

**Y**

_{i·}

**/sqrt(s**

^{2}

**(**Y

_{i}

_{·})/*b*)which is basically a one-sample

The options -amean and -bmean in 3dANOVA3 have also been corrected based on the same principle.

*t*test with*s*^{2}(Y_{i}_{·}) being standard error of samples of factor A at level*i*: Y_{i1}, Y_{i2}, ..., Y_{ib}. 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.