When we attempt to infer on mixed effects models, we need to deal with the fact that the variance components are unknown. Classically, variance components tend to be estimated separately using iterative estimation schemes employing Ordinary Least Squares (OLS), Expectation Maximisation (EM) or Restricted Maximum Likelihood (ReML), see (22) for details. As an example of a non-Bayesian approach, (Worsley) estimates variance components at each split-level of the model separately. At higher than first levels, they propose EM for estimation of the random effects variance contribution, in order to reduce bias in the variance estimation - a potential problem in higher-level analyses if simple OLS were used. Positivity of the random-effects variance, avoiding what is known as the `negative variance problem' (where mixed-effects variance estimates are lower than fixed-effects variances implying negative random-effects variance (19)), is partially addressed but not strictly enforced.
However, only in certain special cases (not including the model presented here) is it possible to derive analytical forms for the null distributions required by frequentist statistics. In the absence of analytical forms, frequentist solutions rely on null distributions derived from the data using such techniques as permutation tests (20). However, these lose the statistical power gained from educated assumptions about, for example, the distribution of the noise, and limit inference to the number of available points in the empirical null distribution. Bayesian statistics gives us a tool for inferring on any model we choose, and guarantees that uncertainty will be handled correctly.
(11) have proposed a parametric empirical-Bayesian (PEB) approach for estimation of the all-in-one multi-level model. Unlike (Worsley) they relate the parameters of interest to the full set of original data, i.e. they do not utilise the `summary statistics' approach. Conditional posterior point estimates are generated using EM which give rise to posterior probability maps.
Working in a fully Bayesian reference analysis framework we have the capacity to infer either using the summary statistic split-level (Worsley) approach, or the all-in-one (11) approach. However, all-in-one inference is not part of this paper and is an area of future work. The difference between an all-in-one inference based on the work described in this paper, and the work PEB of (11), is that they assume a multivariate Gaussian marginal posterior for the regression parameters (and then heuristically convert it to a t-statistic), whereas we work in a fully Bayesian framework using reference priors which we can validate as giving a multivariate t-distribution with certain degrees as freedom using MCMC. Without reference priors, (11) have nothing principled to drive the important choice of prior at the top-level and as a result assume flat priors.
Importantly, one of the results demonstrated in this paper is that the inference we would obtain at the top level will be approximately the same regardless of whether we infer using the summary statistic split-level (Worsley) or the all-in-one approaches (11) (assuming that first-level temporal autocorrelations are effectively known). However, it is very important to realise that there will be a difference if we look to infer at intermediate levels in the model. This is because in the all-in-one approach, the regression parameters at these intermediate levels will be regularised by the levels above in the hierarchy, whereas in the split-level approach they will not. Whether or not an experimenter would like to infer on, for example, a subject in isolation, or on a subject in the context of the group of which it is a member, is a choice for the experimenter to make.