Quality Effects Model

Weighting by Inverse Variance

Because the results from different studies investigating different independent variables are measured on different scales, the dependent variable in a meta-analysis is some standardized measure of effect size. One of the commonly used effect measures in clinical trials is a relative risk (RR), when the outcome of the experiments is dichotomous (success vs. failure). The standard approach frequently used in meta-analysis in clinical research is termed the inverse variance method or fixed-effects model based on Woolf and first proposed by Gene Glass in 1976. The average effect size across all studies is computed, whereby the weights are equal to the inverse variance of each study's effect estimator. Larger studies and studies with less random variation are given greater weight than smaller studies. In the case of studies reporting a RR, the logRR has a standard error (se) given by

where PiT and PiC are the risks of the outcome in the treatment group and control groups, respectively, of the ith study and n is the number of patients in the respective groups. The weights (w) allocated to each of the studies are inversely proportional to the square of the standard error. Thus,

which gives greater weight to those studies with smaller standard errors. The combined effect size is computed by the weighted average as

where

Assuming these estimates are distributed normally, the 95% confidence limits are easily obtained by

The Conventional Approach to Heterogeneity

As can be seen above, the variability within studies is incorporated in the current approach to combining the effects by adjusting based on the variance of the estimates in each individual study. So, if a study reports a higher variance for its RR estimate, it would get lesser weight in the final combined estimate and vice versa. Although statistically very appealing, this approach does not take into account the innate variability that [Page 1164]exists between the studies arising from differences in the study protocols and how well they were executed and conducted. This major limitation has been well recognized and gave rise to the random-effects model approach. Selection of the random-effects model is often carried out using a simple test of homogeneity of the studies involved because of the demonstration that, in the presence of even slight between-study heterogeneity, the fixed-effects model results in inferences that substantially underestimate the variation in the data and in parameter estimates. Nevertheless, despite widespread use of this adjustment for heterogeneity, it is now thought that the choice of fixed-effects or random-effects meta-analysis should not be made on the basis of perceived heterogeneity but rather on the basis of purpose. The fixed-effects meta-analysis tests the null hypothesis that treatments were identical in all trials. When and if this is rejected, then the alternative hypothesis that may be asserted is, “There is at least one trial in which the treatments differed.” In other words, the random effects analysis works as a check on the robustness of conclusions from a fixed-effects model to failure in the assumption of homogeneity and cannot go beyond this causal “finding.”

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