Cochran–Mantel–Haenszel statistics

The Cochran–Mantel–Haenszel statistics (CMH) is a type of statistical analysis used primarily in the field of epidemiology and biostatistics to control for confounding variables in the assessment of the association between two binary variables (e.g., exposure and outcome) across different strata or groups. This method is particularly useful in the analysis of categorical data from multicenter trials or studies that have been stratified based on certain characteristics like age, location, or other demographic factors.

Overview

The Cochran–Mantel–Haenszel statistics are named after William G. Cochran, Nathan Mantel, and William Haenszel, who developed the method to provide a statistical test for the association between two categorical variables while controlling for at least one confounding variable. The CMH method is an extension of the Chi-squared test that is applied across different layers or strata of data.

Methodology

The CMH statistics involve calculating a weighted average of the odds ratios from each stratum. Unlike a simple pooling of data, which might introduce bias if the effect of the exposure differs across strata, the CMH method maintains the stratification of the data, thus preserving the validity of the statistical inference.

Formula

The CMH statistic is calculated using the formula: \[ \chi^2 = \frac{(N \sum_{i=1}^k O_i - T^2)^2}{N \sum_{i=1}^k V_i} \] where:

• \( O_i \) is the observed count of cases in the ith stratum,
• \( N \) is the total number of observations,
• \( T \) is the total sum of the products of row and column totals for each stratum,
• \( V_i \) is the variance in the ith stratum.

Applications

The CMH statistics are widely used in the analysis of clinical trial data where it is important to adjust for potential confounders that could affect the relationship between the treatment and the outcome. It is also used in observational studies where randomization is not possible.

Advantages

• **Control for Confounding**: CMH allows for the control of confounding variables across different strata, which can lead to more accurate estimates of the effect size.
• **Flexibility**: It can be used with matched or unmatched data and with any number of strata.
• **Efficiency**: Provides a more efficient use of the data as compared to stratified analysis where each stratum is analyzed separately.

Limitations

• **Assumption of Homogeneity**: The CMH test assumes that the odds ratios are homogeneous across strata. If this assumption is violated, the test may yield misleading results.
• **Binary Data**: The method is limited to binary outcome data, which may not be suitable for all types of research questions.

See also

• Biostatistics
• Epidemiology
• Odds ratio
• Chi-squared test
• Stratified sampling