Fixed-effects Models
These models assume that the studies included in the meta-analysis have no differences in underlying study populations, patient selection criteria, patient's response to treatment, and the methods of treatment. The apparent differences in study results are assumed to be purely due to chance during sampling. Indeed, this method assumes that individual study effects sizes are random draws from a single frequency distribution of an effect size and that the only source of variation between individual study effect sizes is with-in study heterogeneity. In other words, patients who were enrolled in different studies but were assigned the same treatment are taken to be exchangeable.
If we denote individual study effects with Yi then this model can be expressed as below
A test for homogeneity can be performed as given above to confirm the presence of homogeneity between the study effect sizes and results are compared with chi-square distribution.
Various methods for performing a fixed-effect model meta-analysis for a binary outcome have been proposed and include Mantel-Haenszel's method, Woolf's method, Peto's method, and logistic regression. Although fixed effects model is easier to develop, it has several limitations. The most important limitation is the assumption of homogeneity which is quite unrealistic and counter-intuitive. There are hardly any two studies that have similar study designs, enroll same kind of patients, and give treatment in the same manner. Even if there are such trials, it is unlikely that they have been performed at the same time, and thus may suffer from the bias due to the improvements in health care. Furthermore, above and other statistical test for homogeneity have low power and may not detect heterogeneity between the study results.
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