A Meta-Analysis for Simultaneously Estimating Individual Means with Shrinkage, Isotonic Regression and Pretests

Meta-analyses combine the estimators of individual means to estimate the common mean of a population. However, the common mean could be undefined or uninformative in some scenarios where individual means are “ordered” or “sparse”. Hence, assessments of individual means become relevant, rather than the common mean. In this article, we propose simultaneous estimation of individual means using the James–Stein shrinkage estimators, which improve upon individual studies’ estimators. We also propose isotonic regression estimators for ordered means, and pretest estimators for sparse means. We provide theoretical explanations and simulation results demonstrating the superiority of the proposed estimators over the individual studies’ estimators. The proposed methods are illustrated by two datasets: one comes from gastric cancer patients and the other from COVID-19 patients.

Maximizing Heritability of a Linear Combination of Traits

In 1971 Jones proposed an approximate procedure for finding that linear combination of scores which has maximum heritability in a twin sample. I give an exact small-sample procedure. I point out two problems: such procedures can over-optimize the heritability by capitalizing on chance, and confidence intervals and significance tests are needed. I give an approach using James-Stein shrinkage estimation and bootstrapped standard errors to address these problems. It appears that confidence intervals may be quite broad. To reduce the width of the confidence intervals, one can accept some small-sample bias in exchange for smaller sampling errors. The James-Stein approach to estimating coefficients is used to achieve reduced confidence interval width. I illustrate with a computational example using personality data.