Multiple outputation: Inference for complex clustered data by averaging analyses from independent data

This article applies a simple method for settings where one has clustered data, but statistical methods are only available for independent data. We assume the statistical method provides us with a normally distributed estimate, I and an estimate of its variance (σ) over cap (2). We randomly select a data point from each cluster and apply our statistical method to this independent data. We repeat this multiple times, and use the average of the associated (θ) over cap 's as our estimate. An estimate of the variance is given by the average of the (σ) over cap (2)'s minus the sample variance of the (θ) over cap 's. We call this procedure multiple outputation, as all excess data within each cluster is thrown out multiple times. Hoffman, Sen, and Weinberg (2001, Biometrika 88, 1121-1134) introduced this approach for generalized linear models when the cluster size is related to outcome. In this article, we demonstrate the broad applicability of the approach. Applications to angular data, p-values, vector parameters, Bayesian inference, genetics data, and random cluster sizes are discussed. In addition, asymptotic normality of estimates based on all possible outputations, as well as a finite number of outputations, is proven given weak conditions. Multiple outputation provides a simple and broadly applicable method for analyzing clustered data. It is especially suited to settings where methods for clustered data are impractical, but can also be applied generally as a quick and simple tool.