Definition

For the $i$ -th sample, let $T_{i 1}, T_{i 2}, \dots, T_{i n_{i}}$ , where $i = 1, 2, \dots, K$ , be i.i.d. survival time with a distribution $F_{i}$ , and $C_{i 1}, C_{i 2}, \dots, C_{i n_{i}}$ be i.i.d. censoring time with a distribution $G_{i}$ . We can observe $(X_{ij}, δ_{ij}), j = 1, 2, \dots, n_{i}$ where $X_{ij} = min (T_{ij}, C_{ij})$ and $δ_{ij} = I (T_{ij} \leq C_{ij})$ is the censoring indicator.

The generalized Mantel-Haenszel test is an extension of Mantel-Haenszel Test used for more than two sample case.

Let $(z_{i}, ζ_{i}), i = 1, 2, \dots, N$ be the combined samples

For each uncensored time point, construct a $2 \times K$ table.

	$1$	$2$	$\dots$	$K$
Dead	$a_{i 1}$	$a_{i 2}$	$\dots$	$a_{i K}$	$m_{i 1}$
Alive			$\dots$		$m_{i 2}$
	$n_{i 1}$	$n_{i 2}$	$\dots$	$n_{i} K$	$N_{i}$

Under the null hypothesis $H_{0} : F_{1} = F_{2} = \dots = F_{K}$ , the test statistic of generalized Mantel-Haenszel test is defined as $M H = (a_{(- 1)} - E (a_{(- 1)}))^{t} Σ_{(1)} (a_{(- 1)} - E (a_{(- 1)})) \to D χ^{2} (K - 1)$ where $a_{(- 1)}$ , $Σ_{(- 1)}$ are the $K - 1$ -length vector and $(K - 1) \times (K - 1)$ matrix, calculated with the first population is deleted data (corner-point constraint).

The $a$ and $Σ$ is defined as $a - E (a) = i = 1 \sum K (a_{i} - E (a_{i}))$ where $E (a_{i}) = (\frac{m _{i 1} n _{i 1}}{N _{i}}, \dots, \frac{m _{i K} n _{i K}}{N _{i}})$ $Σ = i = 1 \sum K Cov (a_{i})$ where $Cov (a_{i}) = \frac{m _{i 1} m _{i 2}}{N _{i} - 1} \frac{n _{i 1}}{N _{i}} (1 - \frac{n _{i 1}}{N _{i}}) ⋱ - \frac{n _{ik} n _{i l}}{N _{i}^{2}} \frac{n _{i K}}{N _{i}} (1 - \frac{n _{i K}}{N _{i}})$

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Generalized Mantel-Haenszel Test

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