Breslow-Day Test

TL;DR

• Checks whether odds ratios from multiple studies are sufficiently similar to justify pooling.
• Computes standardized residuals for each study, sums their squares to form the Breslow–Day statistic, and compares it to a chi-squared critical value with degrees of freedom = number of studies − 1.
• If the statistic exceeds the critical value, the homogeneity assumption is rejected and the pooled odds ratio should not be used.

Definition

The Breslow–Day test is a non-parametric statistical method used to assess the homogeneity of odds ratios in a meta-analysis by comparing observed odds ratios from different studies with the expected odds ratio (the average of all observed odds ratios).

Explanation

• The test assumes all included studies share the same true odds ratio. Under this assumption, observed odds ratios should differ only by random sampling error.

• Steps to conduct the Breslow–Day test:

  1. Calculate the observed odds ratio for each study.
  2. Calculate the expected odds ratio, which is the average of all the observed odds ratios.
  3. Calculate the standardized residual for each study: the difference between the observed odds ratio and the expected odds ratio, divided by the standard error of the observed odds ratio.
  4. Calculate the Breslow–Day statistic as the sum of the squared standardized residuals.
  5. 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.
  6. If the Breslow–Day statistic is greater than the critical value, the homogeneity assumption is rejected and the pooled odds ratio should not be used.

• The test is described in the context of meta-analysis to identify whether the pooled odds ratio is reliable; a significant result suggests the relationship may vary across the included studies.

Examples

Smoking and lung cancer meta-analysis

A meta-analysis includes four studies with the following observed odds ratios:

Study	Observed odds ratio
Study 1	1.5
Study 2	2.0
Study 3	1.2
Study 4	1.8

The expected odds ratio (the average) is:

$$\frac{1.5 + 2.0 + 1.2 + 1.8}{4} = 1.65$$

Standardized residuals are calculated as (observed − expected) / standard error:

Study	Observed OR	Standard error	Standardized residual
Study 1	1.5	0.1	(1.5 - 1.65) / 0.1 = -0.5
Study 2	2.0	0.2	(2.0 - 1.65) / 0.2 = 1.25
Study 3	1.2	0.3	(1.2 - 1.65) / 0.3 = -1.67
Study 4	1.8	0.4	(1.8 - 1.65) / 0.4 = 0.38

The Breslow–Day statistic is the sum of the squared standardized residuals:

$$\text{Breslow–Day statistic} = (-0.5)^2 + (1.25)^2 + (-1.67)^2 + (0.38)^2 = 2.32$$

Degrees of freedom = number of studies − 1 = 3. Comparing to a chi-squared critical value for 3 degrees of freedom:

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.

Use cases

• Assessing whether odds ratios from multiple studies in a meta-analysis are homogeneous enough to justify pooling.
• Determining whether the pooled odds ratio is reliable or whether study-specific variation invalidates pooling.

Notes or pitfalls

• The Breslow–Day test assumes that all studies share the same true odds ratio; significant deviation among observed odds ratios suggests this assumption is violated.
• The test is non-parametric and relies on comparison to a chi-squared distribution with degrees of freedom equal to the number of studies minus one.
• If the homogeneity assumption is rejected, the pooled odds ratio may not be reliable.

Related terms

• Odds ratio
• Meta-analysis
• Chi-squared distribution