Chapter 13 : Analysis of Variance

Abstract

The mainstay of many scientific experiments is the factorial design. These comprise a number of experimental factors which are each expressed over a number of levels. Data are collected for each factor/level combination and then analysed using Analysis of Variance (ANOVA). The ANOVA uses F-tests to examine a pre-specified set of standard effects, eg. ‘main effects’ and ‘interactions’, as described in [Winer et al. 1991]. ANOVAs are commonly used in the analysis of PET, EEG, MEG and fMRI data. For PET, this analysis usually takes place at the ‘first’ level. This involves direct modelling of PET scans. For EEG, MEG and fMRI, ANOVAs are usually implemented at the ‘second level’. As described in the previous chapter, first level models are used to create contrast images for each subject. These are then used as data for a second level or ‘random effects’ analysis. Some different types of ANOVA are tabulated below. A two-way ANOVA, for example, is an ANOVA with 2 factors; a K1-by-K2 ANOVA is a two-way ANOVA with K1 levels of one factor and K2 levels of the other. A repeated measures ANOVA is one in which the levels of one or more factors are measured from the same unit (e.g, subjects). Repeated measures ANOVAs are also sometimes called within-subject ANOVAs, whereas designs in which each level is measured from a different group of subjects are called between-subject ANOVAs. Designs in which some factors are within-subject, and others between-subject, are sometimes called mixed designs. This terminology arises because in a between-subject design the difference between levels of a factor is given by the difference between subject responses eg. the difference between levels 1 and 2 is given by the difference between those subjects assigned to level 1 and those assigned to level 2. In a within-subject design, the levels of a factor are expressed within each subject eg. the difference between levels 1 and 2 is given by the average difference of subject responses to levels 1 and 2. This is like the difference between two-sample t-tests and paired t-tests. The benefit of repeated measures is that we can match the measurements better. However, we must allow for the possibility that the measurements are correlated (so-called ‘non-sphericity’ see below). The level of a factor is also sometimes referred to as a ‘treatment’ or a ‘group’ and each factor/level combination is referred to as a ‘cell’ or ‘condition’. For each type of ANOVA, we describe the relevant statistical models and show how they can be implemented in a GLM. We also give examples of how main effects and interactions can be tested for using F-contrasts.

Cite this paper

@inproceedings{Penny2006Chapter1, title={Chapter 13 : Analysis of Variance}, author={W . D . Penny}, year={2006} }