A two-stage hierarchical regression model for meta-analysis of epidemiologic nonlinear dose–response data

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Abstract

To estimate a summarized dose–response relation across different exposure levels from epidemiologic data, meta-analysis often needs to take into account heterogeneity across studies beyond the variation associated with fixed effects. We extended a generalized-least-squares method and a multivariate maximum likelihood method to estimate the summarized nonlinear dose–response relation taking into account random effects. These methods are readily suited to fitting and testing models with covariates and curvilinear dose–response relations.

Introduction

Many epidemiologic studies provide data on the relation between a quantitative exposure and an outcome as a series of dose-specific relative risks, with one category serving as the common reference. Examples include studies investigating dose–response relationship between alcohol intake and cardiovascular disease (Table 2) and those examining the relation of obesity and renal cell cancer (Table 6). It would be informative if the dose–response patterns among the studies can be summarized with a meta-analysis.

To summarize findings with an overall measure of association, meta-analysis has been an important method for quantitative synthesis of previous research. Heterogeneity over studies can be assessed by the existing tests such as DerSimonian and Laird’s Q test (DerSimonian and Laird, 1986), the likelihood ratio test (Stram and Lee, 1994), or τ2-bootstrap method (Takkouche et al., 1999). These tests however may have low statistical power especially when the number of studies to be included in the meta-analysis is small (Hardy and Thompson, 1998). Because tests for heterogeneity will often be underpowered, random-effects models are used routinely (Takkouche et al., 1999). The National Research Council report (National Research Council, 1992) recommends the use of random-effects approaches for meta-analysis and the exploration of sources of variation in study results.

Berlin et al. (1993) have discussed the application of random-effects regression in meta-analysis of epidemiologic dose–response data, where each study provides separate effect estimates for a series of doses (levels of exposure) with zero exposure as the reference. Their approach includes two stages that estimates a slope for each study at first stage, then derives an overall estimate of dose–response effect by weighted average of the individual slopes. They have introduced the method that can be readily applied when only linear dose–response relation is considered. For nonlinear dose–response relation, they mentioned that it would require using the covariance-adjusted multiple regression model which needs to develop an iterative fitting algorithm to estimate the between-study variance component for estimating and testing the overall effect of curvilinear dose–responses (Berlin et al., 1993). In the published meta-analysis papers that summarized a possible U- or J-shaped dose–response relationship (Bergström et al., 2001, Castelnuova et al., 2002, Hernán et al., 2002, Corrano et al., 2004), none have incorporated a model to estimate nonlinear dose–response relationships taking into account random effects. Castelnuova et al. (2002) did not consider random effects; Hernán et al. (2002) took into account random effects but did not consider the curvilinear terms; Bergström et al. (2001) considered a linear term with random effects but evaluated the quadratic term as a fixed effect; Corrano et al. (2004) estimated nonlinear trend using power transformation of the exposure variable to avoid the need to include a higher order (curvilinear) term in the analysis.

To estimate the summarized curvilinear dose–response relationship, the key issue is to find a method to estimate the between-study covariance components at the second stage in a two-stage procedure. In this report, to estimate the between-study covariance components, we have extended the generalized-least-squares (GLS) and multivariate maximum likelihood (MML) approaches, described by Berkey et al. (1998), for summarizing multiple outcomes with random effects to the pooling of study-specific slopes for nonlinear dose–response relation.

Section snippets

Effect estimates in epidemiologic dose–response relations

In a meta-analysis of epidemiologic dose–response relation, relative risks associated with different exposure categories from different studies are summarized to obtain an estimate of overall slope. For categorized exposure, levels of exposure are usually set at the midpoint of each category (Greenland and Longnecker, 1992, Berlin et al., 1993, Shi and Copas, 2004). The odds ratio is used as a measure of relative risk for case-control studies, and the relative risk estimates are natural

Example 1. Studies with zero exposure dose as reference

In the example we reviewed four published papers (Bianchi et al., 1993, Bobak et al., 2000, Malarcher et al., 2001, Vliegenthart et al., 2004) and summarized six sets of data (Table 2) investigating the dose–response relationship between alcohol intake and vascular disease. The six sets of data were used as if they were from six different studies to estimate the summarized dose–response relationship between alcohol intake and vascular disease. The alcohol consumption was estimated to be 0.05 mL

Discussion

Meta-analysis is a useful tool for summarizing and evaluating the existing literature on a dose–response relation. Berlin et al. (1993) have introduced a two-stage procedure for summarizing linear dose–response data taking into account the random effects. The first stage estimates the slope for each study, and the second stage derives the overall estimate of the linear dose–response effect with random effects by pooling the study-specific coefficient estimates from each study. Similarly, when

Acknowledgments

The authors thank Dr. Catherine S. Berkey of the Department of Medicine, Brigham and Women’s Hospital and Harvard Medical School for her advice on programming of the MML method in SAS. The authors also thank the reviewers and editor for valuable comments on the manuscript.

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