The basic step for a fixed-effects model involves the calculation of a weighted average of the treatment effect across all of the eligible studies.
For a continuous outcome variable, the measured effect is expressed as the difference between sample treatment and control means. The weight is expressed as the inverse of the variance of the difference between the sample means. Therefore, if the variance is large the study will be given a lower weight. If the variance is smaller, the weight of the study is larger.
For a binary outcome variable, the measured effect usually is expressed as the logarithm of the estimated odds ratio. The weight is expressed as the inverse of the variance of the logarithm of the estimated odds ratio. Basically, the weighting takes the same approach using this value.
Suppose that there are K studies.
The estimated treatment effect (e.g., difference between the sample treatment and control means) in the \(k^{th}\) study, \(k = 1, 2, \dots , K\), is \(Y_{k}\) .
The estimated variance of \(Y_k\) in the \(k^{th}\) study is \(S_k^2\).
The weight for the estimated treatment effect in the \(k^{th}\) study is \(w_k= \dfrac{1}{S_k^2}\).
The overall weighted treatment effect is:
\(Y=\dfrac{\left(\sum_{k=1}^{K}w_kY_k \right)}{ \left(\sum_{k=1}^{K}w_k \right)} \)
The estimated variance of Y, the weighted treatment effect, is:
\(S^2 = \dfrac{1}{\left(\sum_{k=1}^{K}w_k \right) }\)
Testing the null hypothesis of no treatment effect (e.g., \(H_0 \colon \mu_1 - \mu_0 = 0\) for a continuous outcome) is performed from assuming that \(\dfrac{|Y|}{S}\) asymptotically follows a standard normal distribution.
The \(100(1 - \alpha )\%\) confidence interval for the overall weighted treatment effect is:
\(Y - \left(z_1 - \dfrac{\alpha}{2} \times S\right), Y + \left(z_1 - \dfrac{\alpha}{2} \times S\right)\)
The statistic for testing \(H_0 \colon\) {study homogeneity} is
\(Q=\sum_{k=1}^{K}w_k(Y_k-Y)^2 \)
Q has an asymptotic \(\chi^{2}\) distribution with K - 1 degrees of freedom.
Alternative: Mantel-Haenszel Test Section
An alternative, fixed-effects approach for a binary outcome is to apply the Mantel-Haenszel test for the pooled odds ratio. The Mantel-Haenszel test for the pooled odds ratio assumes that the odds ratio is equal across all studies. For the \(k^{th}\) study, \(k = 1, 2, \dots , K\), a 2 × 2 table is constructed:
Control | Treatment | |
Failure | \(n_{0k} - r_{0k}\) | \(n_{1k} - r_{1k}\) |
Success | \(r_{0k}\) | \(r_{1k}\) |
The disadvantage of the Mantel-Haenszel approach, however, is that it cannot adjust for covariates/regressors. Many researchers now use logistic regression analysis to estimate the odds ratio from a study while adjusting for covariates/regressors, so the weighted approach described previously is more applicable.