Random-effects model for meta-analysis of clinical trials: An update

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Abstract

The random-effects model is often used for meta-analysis of clinical studies. The method explicitly accounts for the heterogeneity of studies through a statistical parameter representing the inter-study variation. We discuss several iterative and non-iterative alternative methods for estimating the inter-study variance and hence the overall population treatment effect. We show that the leading methods for estimating the inter-study variance are special cases of a general method-of-moments estimate of the inter-study variance. The general method suggests two new two-step methods. The iterative estimate is statistically optimal and it can be easily calculated on a spreadsheet program, such as Microsoft Excel, available on the desktop of most researchers. The two-step methods approximate the optimal iterative method better than the earlier one-step non-iterative methods.

Introduction

Meta-analysis is a statistical technique for combining estimated treatment effects from independent comparable clinical trials (studies). Such analyses have become increasingly popular in medical research where information about treatment efficacy is available from a number of clinical trials with inconclusive or inconsistent results.

A major difficulty in integrating the findings from various studies stems from the sometimes diverse nature of the studies being combined. The studies may differ, for example, in terms of patient characteristics or methods employed. To account for such inter-study differences, DerSimonian and Laird [1] proposed a simple random effects model which allows for treatment effects to vary across studies and uses a simple non-iterative method to estimate the inter-study treatment effect variance. Because it incorporates inter-study differences into the analysis of overall treatment efficacy, and because of its simplicity, the method [1] continues to be widely used. Nevertheless, indiscriminate or inappropriate use of any approach to meta-analysis of clinical trials can lead to misleading inferences about treatment effects [2], and the need for careful consideration of methods in drawing statistical inferences from comparable but heterogeneous studies remains critical.

In this paper, we first review the random-effects model for meta-analysis of clinical trials and introduce a general method-of-moments estimate for the inter-study variance which includes several existing estimates as special cases. In addition to the non-iterative method proposed by DerSimonian and Laird [1], an iterative estimate of the inter-study variance based on a random effects model was proposed by Paule and Mandel [3] for inter-laboratory studies. This estimate was subsequently shown to be statistically optimal [4] and can be easily calculated on a spreadsheet program. Another non-iterative estimate of the inter-study variance component based on a random-effects model was proposed by Cochran [5]. In contrast to the non-iterative DerSimonian and Laird as well as the iterative Paule and Mandel estimates, the estimate based on Cochran's ANOVA assumes that each study provides equal information and is of equal sample size.

We show that the inter-study variance estimates based on the methods of Cochran, DerSimonian and Laird, and Paule and Mandel are all special cases of a general method-of-moments estimate for the inter-study variance with slightly different weights assigned to the studies. The general method-of-moments estimate suggests two-step alternatives to the one-step non-iterative procedures based on Cochran's ANOVA and the DerSimonian and Laird methods. We illustrate and compare the estimates from the five methods in several examples, and based on the empirical evidence, suggest improvements to the commonly used one-step non-iterative random-effects model estimates.

Section snippets

Methods

We consider the problem of combining estimated treatment effects from a series of k comparative clinical studies, where the data from each study consist of the number of patients in treatment and control groups, nTi and nCi, and the proportion of patients with some event in each of the two groups, rTi and rCi. A random effects model for meta-analysis stipulates that the observed treatment effect, yi, from the i-th clinical study is made up of two additive components: the true treatment effect

Database

We compare the methods discussed in the previous section using data from six reviews published from 1981 to 2003. These reviews include articles from the New England Journal of Medicine [10], the British Medical Journal [11], Lancet [12], the Journal of the American Medical Association [13], [14] and Hepatology [15]. We briefly describe the six reviews identifying each by its first author or the study name:

Baum [10]: This is a survey of a number of studies that evaluate the efficacy of

Comparison

To compare the five statistical estimates for τ2 and μ discussed in the Methods section, we use the reported data from the six reviews discussed in the previous section. We recognize that this set of reviews may not be representative of all meta-analyses published in the medical literature. In each review, the data from the i-th study consist of the total number of subjects in the treatment and control groups (nTi and nCi) and the proportion of subjects with some event of interest in each of

Summary

We discuss two non-iterative methods and one iterative method for estimating the inter-study component of variance in a random-effects model for meta-analysis of clinical studies. We show that all three methods are special cases of a general method-of-moments estimate for the inter-study variance with slightly different weights assigned to the studies. In Cochran's ANOVA method, each study is assigned an equal weight while the weights in the DerSimonian and Laird method are inversely

Acknowledgments

We are grateful to Katherine Halvorsen and Fern Hunt for their critical readings of the various drafts of this article and for many insightful comments.

Certain software is identified in this paper for illustration. Such identification is not intended to imply recommendation or endorsement by the National Institute of Standards and Technology or the National Institutes of Health, nor is it intended to imply that the software is necessarily the best available for the purpose.

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