Variance Components

Researchers often refer to different ``group analyses'', the most common being ``fixed-effects'' and ``mixed-effects''. What these terms are actually referring to are different inter-session (or inter-subject) noise (variance) models. We now summarise what the terms and associated models mean.

We start with the equation for the $t$ statistic:

$$t = \frac{\mbox{mean effect}}{\sqrt{\mbox{variance(mean effect)}}},$$ (1)

i.e., we are asking how big the mean effect size is compared with the ``noise'' (the mean's standard deviation¹). The standard deviation is the square root of either the fixed-effects variance of the mean or the random-effects variance of the mean.

With fixed-effects modelling, we assume that we are only interested in the factors and levels present in the study, and therefore our higher-level fixed-effects variance $FV$ is derived from pooling² the first-level (within-session) variances (of first-level effect size mean) $FV_i$, according to:

$$FV = \frac{\sum(FV_i)}{n^2}, DoF_{FV} = \sum{DoF_{FVi}},$$ (2)

where $DoF$ is the degrees of freedom, which, in the case of FMRI time series, is usually large. This modelling therefore ignores the cross-session (or cross-subject) variance completely and the results cannot be generalised outside of the group of sessions/subjects involved in the study.

With simple mixed-effects³modelling, we derive the mixed-effects variance $MV$ directly from the variance of the first-level parameter estimates $PE_i$ (effect sizes) or contrasts of parameter estimates:

$$MV = \frac{var(PE_i)}{n}, DoF_{MV} = n-1,$$ (3)

with a (normally) much smaller DoF than with fixed-effects. Thus the modelling uses the cross-session (or cross-subject) variance, and the results (which are generally ``more conservative'' than with a fixed-effects analysis) are relevant to the whole population from which the group of sessions/subjects was taken.

The mixed-effects variance is the sum of the fixed-effects (within-session) variance and random-effects (pure inter-session) variance (though note that simple estimation methods calculate this directly, as above, and do not explicitly use the fixed-effects variance). Therefore the estimated mixed-effects variance should in theory and in practice be larger than the fixed-effects variance. We expect that when there is large inter-session variance there will be a large difference between fixed- and random-effects analyses.

There have recently been significant developments in group-level analysis. For example, it has been shown in [1] that there is value in carrying up lower-level variances to higher-level analyses of mixed-effects variance, and one implementation of this, using Bayesian modelling/estimation methodology has been reported in [3]. Whilst the dataset used in this paper may well prove useful in investigating these developments further, this is beyond the scope of this paper. Instead, this paper concentrates primarily on two other questions, namely the magnitude of session variability, and the effect that first-level analysis methodologies can have on its effect. Therefore, for mixed-effects analyses in this paper, we have only used ordinary least squares (OLS) estimators (see equation 3 and [12]).