# Miscellaneous stuff

- Unequal sample size of trials/events

The beta weights of those regressors are estimated as follows by using the method of least square estimation (LSE):

vector of beta's = (X'X)^(-1) * X'Y

where X is the design matrix, and Y is the column vector of FMRI time series at one voxel. The meaning of the above estimation is this

(1) X'Y sums over the information from all the events for each condition type. For example, if there are n1 events of condition type A and n2 events of type B, then X'Y collects the information into a vector for types A and B plus the baseline.

(2) The information for each condition type is corrected by the overlap correction matrix X'X, which calibers each type based on the potential overlaps in the time course (i.e., set by lags in 3dDeconvolve).

These estimates are unabiased, sufficient, efficient, and consistent. The relevent property of the estimates to your question here is the consistency: A sequence of estimators is said to be consistent if it converges in probability to the true value of the parameter. In the case of regression analysis of 3dDeconvolve, the convergence rate is proportional to 1/n, where n is number of events.

Back to your case, the convergence rates for condition types A and B are in the order of 1/n1 and 1/n2 respectively. The activation tests for types A (whether beta1 = 0) and B (whether beta2 = 0), contrast tests (such as whether beta1 = beta2) are done by the following t statistics:

t = beta1/s(beta1) (type A) t = beta2/s(beta2) (type B)

t =(beta1 - beta2)/[combination of s(beta1) and s(beta2)]

If n1 = 2*n2, the variance for estimate of beta1 is two times smaller than the one for that of beta2. If both n1 and n2 are small, this is a little concern. If both are big enough (such as 10 and 20), you should not worry about this since the imbalance is pretty marginal.

The issue of unequal sample size of trials comes up from time to time. It does raise some serious questions regarding both individual subject and group analyses.

At individual subject level, unequal sample size of trials leads to the difficulty of contrast testing. As I mentioned last year, the severety of the problem can be alleviated if the sampe sizes, although unequal, are big enough.

Another concern with unequal sample size arises at group level because equal variance is assumed among the regression coefficients (beta weights). The violation of the assumption, so-called heteroscedasticity, is obvious, but very difficult to handle. And variances from individual subject level are usually not carried over for group analysis for FMRI data analysis. I might be wrong, but it seems to me in the statistics world there is no systematic approach to testing and correcting the unequal variance problem.

Homogeneity of variance for one-way between-subjects ANOVA

Compound symmetry: (1) Homogeneity of variance; (2) Homogeneity of covariance

One-way within-subjects ANOVA is more sensitive to violations of its assumptions than is the one-way between-subjects ANOVA!

However, the problem might not be as severe as it sounds. Compared to cross-subject variability, unequal variances due to different sample size of trials are usually much less serious under most circumstances. There are definitely some exceptions and things may change in the future, but this is why currently there is no specific correction for this heteroscedasticity problem.

- Why interactions and simple effects?

pp. 254