Cochran-Mantel-Haenszel Test of Conditional Independence

Data

Observations from $K$ strata. Within stratum $k$, a $2\times 2$ contingency table with cell counts $n_{ijk}$ where $i,j\in 1,2$ index the row and column:

\begin{array}{c|cc|c} & \text{Col 1} & \text{Col 2} & \\ \hline \text{Row 1} & n_{11k} & n_{12k} & n_{1+k} \\ \text{Row 2} & n_{21k} & n_{22k} & n_{2+k} \\ \hline & n_{+1k} & n_{+2k} & n_{++k} \end{array}$$ ## Assumptions 1. Independence across strata 2. Column (response) marginal totals ${n_{+1k}, n_{+2k}}$ are fixed in each stratum 3. Row totals ${n_{1+k}, n_{2+k}}$ and overall total $n_{++k}$ are fixed 4. Common odds ratio across strata: $\theta_{XY(k)} = \theta$ for all $k$ ## Hypotheses - $H_0$: Conditional independence (i.e., $\theta = 1$) - $H_{1}: \theta\neq1$ ## Test statistic $$\text{CMH} = \frac{[\sum_k (n_{11k} - \mu_{11k})]^2}{\sum_k \text{var}(n_{11k})}$$ where under $H_0$, treating $n_{11k}$ as hypergeometric:

\begin{align} \mu_{11k}& = E(n_{11k}) = n_{1+k} n_{+1k} / n_{++k} \ \

\text{var}(n_{11k})& = \frac{n_{1+k} n_{2+k} n_{+1k} n_{+2k}}{n_{++k}^2 (n_{++k} - 1)} \end{align}

## Null distribution Under $H_0$, asymptotically: $$\text{CMH} \sim \chi^2_1$$