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The Mantel-Haenszel Relative Risk

If all tables (created by individual stratas) are homogeneous (the Chi-square test of homogeneity for the RR), can check this condition), then, on the basis of these tables, the pooled relative risk with the confidence interval can be designated. Such relative risk is a weighted mean for a relative risk designated for the individual stratas. The usage of the weighted method, proposed by Mantel and Haenszel allows to include the contribution of the strata weights. Each strata of the input has an influence on the pooled relative risk construction (the greater size of the strata, the greater weight and the greater influence on the pooled relative risk).

Weights for individual stratas are designated according to the following formula:

$\begin{displaymath} g^{(s)}=\frac{O_{21}^{(s)}\left(O_{11}^{(s)}+O_{12}^{(s)}\right)}{n^{(s)}}, \end{displaymath}$

and the Mantel-Haenszel relative risk:

$\begin{displaymath} RR_{MH}=\frac{R}{S}, \end{displaymath}$

where:

$\displaystyle R=\sum_{s=1}^w\frac{O_{11}^{(s)}\left(O_{21}^{(s)}+O_{22}^{(s)}\right)}{n^{(s)}}$ ,

$\displaystyle S=\sum_{s=1}^wg^{(s)}$ .

The confidence interval for $log RR_{MH}$ is designated on the basis of the standard error calculated according to the following formula:

$\begin{displaymath} SE_{MH}=\sqrt{\frac{V}{RS}}, \end{displaymath}$

where:

$\displaystyle V=\sum_{s=1}^wV^{(s)}$ ,

$\displaystyle V^{(s)}=\frac{\left(O_{11}^{(s)}+O_{12}^{(s)}\right)\left(O_{21}^{(s)}+O_{22}^{(s)}\right)\left(O_{11}^{(s)}+O_{21}^{(s)}\right)-\left(O_{11}^{(s)}*O_{21}^{(s)}*n^{(s)}\right)}{\left(n^{(s)}\right)^2}$ .

The Manel-Hanszel Chi-square test for the $RR_{MH}$

The Mantel-Haenszel Chi-square test for the $RR_{MH}$ is used in the hypothesis verification about the significance of designated relative risk ( $RR_{MH}$ ). It should be calculated for large frequencies, in a contingency table.

Hypotheses:

$\begin{array}{cl} \mathcal{H}_0: & RR_{MH} = 1, \\ \mathcal{H}_1: & RR_{MH} \ne 1. \end{array}$

The test statistic is defined by:

$\begin{displaymath} \chi^2_{MH}=\frac{\left(\sum_{s=1}^wO_{11}^{(s)}-\sum_{s=1}^wE_{11}^{(s)}\right)^2}{V}, \end{displaymath}$

where:

$E_{11}^{(s)}=\frac{\left(O_{11}^{(s)}+O_{21}^{(s)}\right)\left(O_{11}^{(s)}+O_{12}^{(s)}\right)}{n^{(s)}}$ are the expected frequencies in the first contingency table cell, for individual stratas $s=1,2,...,w$ .

This statistic asymptotically (for large frequencies) has the Chi-square distribution with 1 degree of freedom.

The p-value, designated on the basis of the test statistic, is compared with the significance level $\alpha$ :

$\begin{array}{ccl} $ if $ p \le \alpha & \Longrightarrow & $ reject $ \mathcal{H}_0 $ and accept $ \mathcal{H}_1, \\ $ if $ p > \alpha & \Longrightarrow & $ there is no reason to reject $ \mathcal{H}_0. \\ \end{array}$

The Chi-square test of homogeneity for the $RR$

The Chi-square test of homogeneity for the $RR$ is used in the hypothesis verification that the variable creating stratas, is the modifying effect, i.e. it influences on the designated relative risk in the manner that, the relative risks are significant different for individual stratas.

Hypotheses:

$\begin{array}{cl} \mathcal{H}_0: & RR_{MH} = RR^{(s)}, $ for all the stratas $s=1,2,...,w$,$ \\ \mathcal{H}_1: & RR_{MH} \ne RR^{(s)}, $ for at least one strata.$ \end{array}$

The test statistic, using weighted least squares method, is defined by:

$\begin{displaymath} \chi^2=\sum_{s=1}^w v^{(s)}\left(\ln(RR^{(s)})-\ln(RR_{MH})\right)^2 \end{displaymath}$

where:

$v^{(s)}=\left(\frac{O_{12}^{(s)}}{O_{11}^{(s)}\left(O_{11}^{(s)}+O_{12}^{(s)}\right)}+\frac{O_{22}^{(s)}}{O_{21}^{(s)}\left(O_{21}^{(s)}+O_{22}^{(s)}\right)}\right)^{-1}$ .

This statistic asymptotically (for large frequencies) has the Chi-square distribution with the number of degrees of freedom calculated using the formula: $df=w-1$ .

The p-value, designated on the basis of the test statistic, is compared with the significance level $\alpha$ :

$\begin{array}{ccl} $ if $ p \le \alpha & \Longrightarrow & $ reject $ \mathcal{H}_0 $ and accept $ \mathcal{H}_1, \\ $ if $ p > \alpha & \Longrightarrow & $ there is no reason to reject $ \mathcal{H}_0. \\ \end{array}$

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The Mantel-Haenszel Relative Risk

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