The Breslow-Day test is a statistical method used to assess the homogeneity of odds ratios in a meta-analysis. It is a non-parametric test that compares the observed odds ratios of different studies with the expected odds ratio, which is the average of all the observed odds ratios.

To understand the Breslow-Day test, it is important to first understand what an odds ratio is and why it is used in meta-analysis. An odds ratio is a measure of the relationship between two variables, where one variable is the outcome of interest (e.g. disease status) and the other is a predictor or risk factor (e.g. exposure to a certain environmental toxin). The odds ratio is calculated as the ratio of the odds of the outcome in the exposed group to the odds of the outcome in the unexposed group.

In a meta-analysis, odds ratios from multiple studies are pooled and analyzed together to get a more precise estimate of the overall effect of a predictor on the outcome of interest. However, for the pooled odds ratio to be reliable, the individual studies included in the meta-analysis must have similar odds ratios. This is where the Breslow-Day test comes in.

The Breslow-Day test assumes that all the studies included in the meta-analysis have the same true odds ratio. If this assumption is true, then the observed odds ratios from each study should be similar, with some random variation due to sampling error. If the observed odds ratios are significantly different from each other, then this suggests that the assumption of homogeneity is violated and the pooled odds ratio may not be reliable.

To conduct the Breslow-Day test, the following steps are followed:

Calculate the observed odds ratio for each study included in the meta-analysis.

Calculate the expected odds ratio, which is the average of all the observed odds ratios.

Calculate the standardized residual for each study, which is the difference between the observed odds ratio and the expected odds ratio, divided by the standard error of the observed odds ratio.

Calculate the Breslow-Day statistic, which is the sum of the squared standardized residuals.

Compare the Breslow-Day statistic to the critical value from a chi-squared distribution table with degrees of freedom equal to the number of studies minus one.

If the Breslow-Day statistic is greater than the critical value, then the homogeneity assumption is rejected and the pooled odds ratio should not be used.

To illustrate the Breslow-Day test, consider a meta-analysis of the relationship between smoking and lung cancer. The meta-analysis includes four studies, with the following observed odds ratios:

Study 1: 1.5

Study 2: 2.0

Study 3: 1.2

Study 4: 1.8

The expected odds ratio is the average of the observed odds ratios, which is (1.5 + 2.0 + 1.2 + 1.8) / 4 = 1.65.

The standardized residual for each study is calculated as follows:

Study 1: (1.5 – 1.65) / 0.1 = -0.5

Study 2: (2.0 – 1.65) / 0.2 = 1.25

Study 3: (1.2 – 1.65) / 0.3 = -1.67

Study 4: (1.8 – 1.65) / 0.4 = 0.38

The Breslow-Day statistic is the sum of the squared standardized residuals, which is (-0.5)^2 + (1.25)^2 + (-1.67)^2 + (0.38)^2 = 2.32.

To determine if the homogeneity assumption is violated, we compare the Breslow-Day statistic to the critical value from a chi-squared distribution table with three degrees of freedom (since there are four studies minus one). If the Breslow-Day statistic is greater than the critical value, then the homogeneity assumption is rejected and the pooled odds ratio should not be used.

In this example, the Breslow-Day statistic of 2.32 is greater than the critical value of 7.82, indicating that the homogeneity assumption is violated and the pooled odds ratio should not be used. This suggests that the relationship between smoking and lung cancer may vary across the different studies, and a more nuanced analysis is needed to accurately estimate the overall effect.

In conclusion, the Breslow-Day test is a useful tool for assessing the homogeneity of odds ratios in a meta-analysis. By comparing the observed odds ratios from each study with the expected odds ratio, the Breslow-Day test can identify if the assumption of homogeneity is violated and the pooled odds ratio may not be reliable.