Clusters with random size: maximum likelihood versus weighted estimation

Abstract

The analysis of hierarchical data that take the form of clusters with random size has received considerable attention. The focus here is on samples that are very large in terms of number of clusters and/or members per cluster, on the one hand, as well as on very small samples (e.g., when studying rare diseases), on the other. Whereas maximum likelihood inference is straightforward in medium to large samples, in samples of sizes considered here it may become prohibitive. We propose sample-splitting (Molenberghs et al., 2011) as a way to replace iterative optimization of a likelihood that does not admit an analytical solution, with closed-form calculations. We use pseudo-likelihood (Molenberghs et al., 2014), consisting of computing weighted averages over solutions obtained for each cluster size occurring. As a result, the statistical properties of this approach need to be investigated. This is especially important because the minimal sufficient statistics involved are incomplete. The operational characteristics are studied using a first set of simulations. Simulations are also done to compare the proposed method to existing techniques developed to circumvent difficulties with unequal cluster sizes, such as multiple imputation. It follows that the proposed non-iterative methods have a strong beneficial impact on computation time; at the same time, the method is the most precise among its competitors considered. The findings are illustrated using data from a developmental toxicity study, where clusters are formed of fetuses within litters.

Fast two-stage estimator for clustered count data with overdispersion

“…In the LMM, it facilitates the estimation process, because with balanced clusters there are complete sufficient statistics and a closed-form MLE exists [13]. For more details on the split-sample methodology, we refer to [12].…”

“…However, V β has to be modified accordingly to estimate D. Furthermore, a specific covariance structure for D, e.g., compound-symmetry (CS) or autoregressive (AR), can be considered. Each case implies that a different system of equations needs to be solved to find an estimator of D. Findings by [12] and [11] for normally distributed hierarchical data with CS and AR structure can be extended to non-Gaussian outcomes, but arguable will be more complicated. In most cases, we may rely on approximations to estimate D. Therefore, a biased estimator is expected.…”

“…Depending on the model and the data size, the sub-samples can be independent or dependent. The method is not only fast, but it has also exhibited high efficiency with different clustering settings and types of outcomes [10][11][12].…”

A closed-form estimator for meta-analysis and surrogate markers evaluation

“…While useful for surrogacy evaluation, it is by no means restricted to that setting. Our estimator is based on so-called split-sample methodology and pseudo-likelihood (Molenberghs et al, 2011(Molenberghs et al, , 2014(Molenberghs et al, , 2018. Here, the sample is conveniently divided into subsamples, and the parameters are estimated in each one.…”

“…In our case, each subsample contains one single trial, leading to the so-called trial-by-trial or cluster-by-cluster estimator (Molenberghs et al, 2018). This method has been shown to exhibit good statistical performance and computational efficiency to analyze data with different clustering structures, such as autoregressive (AR) (Hermans et al, 2017) and compound-symmetry (Molenberghs et al, 2018). For the latter, the cluster-by-cluster estimator is consistent when the number of replicates per cluster increases more rapidly than the number of clusters.…”