BrainVoyager v23.0

# Random Effects (RFX) Group Analysis

In order to test whether results obtained for individual subjects are valid at the population level, the statistical procedure must assess the variability of observed effects across subjects. In such a random effects (RFX) analysis, individual subjects of a study are considered to be a representative sample of a population. If group effects are significant at the random effects level, the findings from the sample of subjects can be generalized to the population from which the subjects have been drawn. In a fixed effects (FFX) analysis, obtained group results can not be generalized to the population level since the data of all subjects is concatenated and analyzed as if it stems from a single subject. The error variance in a FFX analysis is, thus, estimated by the variability across individual measurement time points while in a RFX analysis, the error variance is estimated by the variability of subject-specific effects across subjects.

In order to explicitly estimate the variability of effects across subjects, a RFX analysis usually proceeds in two (or more) levels (e.g. Kirby, 1993). In a first level, the data for each subject is "collapsed" resulting in mean effect estimates per condition (level 1). The estimated first-level mean effects enter the second level as the new dependent variable (instead of the raw data) and are analyzed across subjects (group analysis). Since the analysis at the second level explicitly models the variability of the estimated effects across subjects, the obtained results can be generalized to the population from which the subjects (sample) were drawn. The first level is performed in BrainVoyager by running a General Linear Model estimating condition effects (beta values) separately for each subject. Instead of one set of beta values in fixed effects analysis, this step provides a separate set of beta values for each subject. At the second level, BrainVoyager offers two approaches to analyze the data, the summary statistics approach and the ANCOVA approach.

## The Summary Statistics Approach

In the summary statistics approach, the same contrast is specified across the beta values of each subject and the mean value of this contrast is tested against zero using a t-test. In a similar way, two groups are compared by computing the mean of the summary statistic (contrast) for each group followed by a t-test comparing the two group means. In case that a multi-subject GLM has been computed or loaded, these basic random-effects analysis options are available directly in BrainVoyager's Overlay GLM Options dialog. In order to handle experimental designs with more than two groups or with multiple factors, the ANCOVA dialog can be used. This dialog can be invoked directly from the Analysis menu but can also be invoked directly from the Overlay GLM Contrasts dialog by clicking the RFX ANCOVA button.

## The ANCOVA Approach

The mean condition effects (beta values per subject or contrasts) can be analyzed at the second level using a standard analysis of variance (ANOVA) approach allowing to model one or more within subjects (repeated measures) factors. If the study represents data from multiple groups of subjects, up to two between subjects factors for group comparisons can be added. Using the ANOVA approach, The statistical analysis at the second level does not differ from the usual statistical approach in other human (e.g. psychological or medical) studies. The only major difference to standard statistics is that the analysis is performed separately for each voxel (or vertex) requiring appropriate corrections for a massive multiple comparisons problem. In addition to the estimated subject-specific effects of the fMRI design (beta values or contrasts of first level analysis), additional external variables (e.g. an IQ value for each subject) may be incorporated as covariates at the second level extending the ANOVA approach to a ANCOVA (analysis of covariance) approach.

## RFX Analysis Steps

Random effects analysis in BrainVoyager QX operates in three major stages:

- Running a random-effects GLM (RFX-GLM)
- Specifying and running multi-factorial ANOVA designs
- Testing RFX main and interaction effects as well as contrasts

Copyright © 2023 Rainer Goebel. All rights reserved.