Definition

Consider a sequence of $2 \times 2$ contingency tables

	Dead	Alive
Treatment 1	$a_{i}$	$b_{i}$	$n_{i 1}$
Treatment 2	$c_{i}$	$d_{i}$	$n_{i 2}$
	$m_{i 1}$	$m_{i 2}$	$n_{i}$
where $i = 1, 2, \dots, k$ is the indicator of hospital

Under the null hypothesis $H_{0} : p_{11} = p_{12}, p_{21} = p_{22}, \dots, p_{k 1} = p_{k 2}$ , where $p_{i 1} = \frac{a _{i}}{n _{i 1}}$ and $p_{i 2} = \frac{c _{i}}{n _{i 2}}$ , the test statistic for Mantel-Haenszel test is defined as $M H = \frac{i = 1 \sum k ( a _{i} - E ( a _{i} ))}{i = 1 \sum k Var ( a _{i} )} \to D N (0, 1)$ where $E (a_{i}) = \frac{m _{i} n _{i}}{n _{i}}$ and $Var (a_{i}) = \frac{n _{i 1} n _{i 2} m _{i 1} m _{i 2}}{n _{i}^{2} ( n _{i} - 1 )}$ .

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Mantel-Haenszel Test

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