Principles of ANCOVA Modelling
Analysis of variance (ANOVA) models are linear models used to analyze most designs of scientific studies by relating a dependent (response) variable to one or more independent (explanatory) variables, called factors. Each factor of a ANOVA model represents a qualitative variable consisting of at least two levels (often also called treatments). Since any ANOVA model can be expressed with a General Linear Model (GLM), the explanatory variables are also called predictors. While a repeated measures GLM (multiple regression) with quantitative predictors is well suited for modeling the first-level (preprocessed) fMRI measurement points of single runs, the ANOVA framework is well-suited to describe the design at the second multi-subject level. The second-level analysis uses the estimated condition values (beta values or contrasts of betas) from the first level analysis as input (i.e. as dependent variable) for the second-level analysis. In all supported ANOVA models described below, the effects of subjects are viewed as random (random-effects (RFX) analysis). In addition to estimated beta or t values from a first-level GLM analysis, any multi-subject VMP (e.g Granger causality maps) or SMP (e.g. cortical thickness maps) can be used as input for the ANCOVA module.
In a single-factor fMRI study, an experimental factor could be defined as "auditory stimulation" and the levels of this factor could be specified as "human sounds" and "animal sounds". In multi-factor studies, two or more factors are investigated simultaneously to obtain information about their combined effects on the fMRI signal at each voxel or in selected brain regions. An example of a two-factorial fMRI study would be the simultaneous investigation of responses to auditory (factor A) and visual (factor B) stimuli, e.g. with the levels "sounds off" and "sounds on" for factor "auditory stimulation" and the levels "images off" and "images on" for factor B "visual stimulation". Such a study would lead to 2 x 2 = 4 factor-level combinations. When all combinations of the levels of all factors are included in a study, the factors are crossed. For two factors, crossing can be visualized in a two-way table:
|Visual \ Auditory||Sounds Off||Sounds On|
When the design of a study has factorial structure, it is often of interest to determine whether or not there are interaction effects among the individual factors. Interaction effects are differences between factor-level combinations, which can not be explained by additive main effects. In the example above, a brain area could respond to sound stimuli differently with respect to the presence or absence of a visual stimulus. An interaction effect, thus, asks whether a difference between levels in one factor (e.g., ["sounds on" - "sounds off"]) differs significantly with respect to the levels of another factor (e.g., "images off" vs "images on") and vice versa. If the differences between levels of one factor are not significantly different for all levels of the other factor, no interaction effect is present between the two factors, i.e. the changes in the dependent variable can be explained by additive main effects. Another important benefit of multi-factorial designs is increased sensitivity to detect differences between cell means of one factor since other included factor(s) may reduce the variability of the error term.
In most fMRI studies, each subject is tested under all experimental conditions, i.e. receives all "treatments" (factor-level combinations). This constitutes a repeated measures design. A factor with repeated measures is also called a within-subjects factor. Repeated measures designs have the important advantage that they provide good precision for comparing condition effects (treatments) because all sources of variability between subjects are excluded from the experimental error. One may view the subjects as serving as their own controls. In light of the nature of the fMRI signal (no absolute zero point), repeated measures designs should be used whenever possible. These designs have, however, also potential disadvantages known as interference effects. One type of interference effect is the order effect, which refers to the potential problem that a condition produces different effects depending on its position within the sequence of conditions. Another type of interference effect is the carryover effect. Repetition of the same condition and randomization of the order of different conditions independently for each subject should be used to minimize interference effects. For each subject, a random permutation should be used to define the condition order, and independent permutations should be selected for different subjects. Note that at the random-effects level (second level analysis) described here, these issues are assumed to be solved since the data is collapsed for each condition at the first level (estimated beta values); the mean effects of each condition (factor-level combinations) are therefore represented in the same order for each subject (e.g. in ANOVA ROI tables) and not in the order encountered by the subject.
While repeated measures designs should be used for fMRI studies whenever possible, many research questions require comparisons between subjects from different populations, e.g. a comparison of male vs female subjects, or healthy subjects vs subjects with a psychiatric disorder. Such grouping factors are called between-subjects factors. In a standard single factor (one-way) or two-factor ANOVA only one dependent variable is used. Since subjects are tested usually under several experimental conditions within an experiment (within-subject conditions), a specific condition or a contrast value need to be specified as the dependent variable from the subject-specific condition estimates of a GLM. Alternatively to this summary statistic approach, group comparisons can be expressed with designs containing both within-subjects and between-subjects factors. For non-fMRI data with a single dependent variable (e.g. cortical thickness measures), the single factor ANOVA is an appropriate model to compare different groups.
ANCOVA = ANOVA + Covariates
Analysis of covariance models combine analysis of variance with techniques from regression analysis. With respect to the design, ANCOVA models explain the dependent variable by combining categorical (qualitative) independent variables with continuous (quantitative) variables. There are special extensions to classical ANOVA calculations to estimate parameters for both categorical and continuous variables. ANCOVA models can, however, also be calculated using multiple regression analysis using a design matrix with a mix of dummy-coded qualitative and quantitative variables. In the latter approach, ANCOVA is considered as a special case of the General Linear Model (GLM) framework.
In BrainVoyager, the ANCOVA dialog and the Overlay RFX ANCOVA Tests dialog are used to specify and test ANCOVA designs with within-subjects factors, between-subjects factors and covariates. These dialogs currently support specification and testing of the following models covering the majority of designs used for neuroimaging studies:
- Single-factor repeated measures ANOVA
- Two-factors repeated measures ANOVA
- Three-factors repeated measures ANOVA
- Single-factor (one-way) ANOVA
- Two-factor (two-way) ANOVA supporting unbalanced designs
- ANOVA with one within-subjects and one between-subjects factor
- ANOVA with two within-subjects and one between-subjects factor
- Single-factor ANCOVA (analysis of covariance)
- Correlation analysis of subject-specific measures with dependent variable
Current Limitations and Future Developments
While subjects are treated as a random factor in all supported ANOVA models, the experimental factors of a multi-factorial design may be either fixed or random. If the factor levels are considered fixed, one is interested in the effects of the specific factors chosen (fixed factor). When the factor levels are a sample from a larger population of potential factor levels, one wants to draw inferences about the populations of factor levels (random factor). Analysis of variance models in which the factor levels are considered fixed are classified as ANOVA model I. Models for studies in which all factors are random are classified as ANOVA models II. Models in which some factors are fixed and some are random are classified as ANOVA models III (mixed effects model). At present BrainVoyager supports only ANOVA model I, i.e. all factors of the design are considered fixed. This model captures most fMRI studies since researchers are typically interested in the effects of each level of a factor. Models II and III are planned to be supported in a future release.
The supported models currently assume equal sample sizes for the groups of between factors (balanced studies). While tolerating slightly dfferent numbers of subjects in different groups, inferences for unbalanced data with fixed and random factors requires more complex procedures (e.g. maximum likelihood approach), which will be added in a future release. The two-factor ANOVA model (introduced with BrainVoyager 20.4) supports unbalanced designs (groups with unequal number of subjects).
If levels of one or more of the factors are unique to a particular level of another factor, the factors are called nested. At present, only fully-crossed factorial designs are supported; nested designs might be available in a future release.
Copyright © 2020 Rainer Goebel. All rights reserved.