Quasi-Empirical Bayes methods for parameter estimation involving many small samples

Animal studies in pharmaceutical discovery and toxicology are not always statistically powered for estimation or hypothesis testing. Typically, only 3 to 5 animals are allocated per group, based on historical conventions or industry practice, particularly in early toxicology studies with several different types of controls and compounds at various concentrations. When we estimate means, variances, or other parameters under these conditions, often the confidence intervals generated will be of little practical use due to the small sample size. If, however, historical or even concurrent data with similar characteristics is available from comparable experiments, all data could be incorporated into the estimation by using an Empirical Bayesian approach. To implement this method, the existing data is used to determine prior distributions for the parameters of interest, which are then combined with the sample data of interest to produce posterior distributions. In our case study, we combined data from 30 different experiments to use as a basis for defining the prior distributions on the mean and standard deviation (SD). For practical reasons related to our application, we prefer to use the standard deviation instead of the variance or precision that are more commonly used in the Bayesian methodology. For the mean parameter, the prior distribution is approximated by a Normal distribution, covering the range of all samples. For SD, the prior distribution is approximated with a half-Normal, half-Cauchy, or Uniform with carefully chosen boundaries. An Empirical Bayes method is then applied, combining the selected prior distributions with observed data in each small experiment to obtain the posterior distribution for the mean and for the variance of that particular experiment. The strategy of using the combined data from multiple samples to develop a common prior distribution that borrows strength across all the available data reduces the variability of the estimates and improves the estimation of individual parameters. In effect, this method combines "borrowing strength" with "Empirical Bayes" in a way that suggests "Tukey meets Robbins"!