The FREQ Procedure

Common Risk Difference

PROC FREQ provides Mantel-Haenszel and summary score estimates for the common risk (proportion) difference for multiway $2 \times 2$ tables. PROC FREQ also provides stratified Newcombe confidence limits for the common risk difference.

Mantel-Haenszel Estimate of the Common Risk Difference

PROC FREQ computes the Mantel-Haenszel (Mantel and Haenszel, 1959) estimate of the common risk difference as

\[ \hat{d}_{\mi{MH}} = \left( \sum _ h \hat{d}_ h w_ h \right) ~ / ~ \left( \sum _ h w_ h \right) \]

where $\hat{d}_ h$ is the risk difference in stratum h and

\[ w_ h = n_{h1\cdot } n_{h2\cdot } / n_ h \]

The column 1 risk difference in stratum h is computed as

\[ \hat{d}_ h = \hat{p}_{h1} - \hat{p}_{h2} = (n_{h11} / n_{h1 \cdot }) - (n_{h21} / n_{h2 \cdot }) \]

where $\hat{p}_{h1}$ is the proportion of row 1 observations that are classified in column 1 and $\hat{p}_{h2}$ is the proportion or row 2 observations that are classified in column 1. The column 2 risk is computed in the same way. For more information, see Agresti (2013, p. 231).

PROC FREQ computes the variance of $\hat{d}_{\mi{MH}}$ (Sato, 1989) as

\[ \hat{\sigma }^2(\hat{d}_{\mi{MH}}) = \left( \hat{d}_{\mi{MH}} \sum _ h P_ h + \sum _ h Q_ h \right) / \left( \sum _ h w_ h \right)^2 \]

where

\[ P_ h = \left( n_{h1\cdot }^2 n_{h21} - n_{h2\cdot }^2 n_{h11} + n_{h1\cdot } n_{h2\cdot } (n_{h2\cdot }-n_{h1\cdot })/2 \right) / n_ h^2 \]
\[ Q_ h = \left( n_{h11} ( n_{h2\cdot } - n_{h21} ) + n_{h21} (n_{h1\cdot } - n_{h11} ) \right) / 2 n_ h \]

The $100(1-\alpha )$% confidence limits for the common risk difference are

\[ \hat{d}_{\mi{MH}} ~ \pm ~ \left( z_{\alpha /2} \times \hat{\sigma }(\hat{d}_{\mi{MH}}) \right) \]
Summary Score Estimate of the Common Risk Difference

PROC FREQ computes the summary score estimate of the common risk difference as described in Agresti (2013, p. 231). This estimate is computed from Miettinen-Nurminen (score) confidence limits for the stratum risk differences. For more information, see the section "Miettinen-Nurminen (Score) Confidence Limits." The score confidence interval for the risk difference in stratum h can be expressed as $\hat{d}_ h^\prime \pm z_{\alpha /2} s_ h^\prime $, where $\hat{d}_ h^\prime $ is the midpoint of the score confidence interval and $s_ h^\prime $ is the width of the confidence interval divided by $2z_{\alpha /2}$. The summary score estimate of the common risk difference is computed as

\[ \hat{d}_{\mi{S}} = \sum _ h \hat{d}_ h^\prime w_ h^\prime \]

where

\[ w_ h^\prime = ( 1 / {s_ h^\prime }^2 ) / \sum _ i (1/{s_ i^\prime }^2) \]

The variance of $\hat{d}_{\mi{S}}$ is computed as

\[ \hat{\sigma }^2(\hat{d}_{\mi{S}}) = 1 / \sum _ h ( 1 / {s_ h^\prime }^2 ) \]

The $100(1-\alpha )$% summary score confidence limits for the common risk difference are

\[ \hat{d}_{\mi{S}} ~ \pm ~ \left( z_{\alpha /2} \times \hat{\sigma }(\hat{d}_{\mi{S}}) \right) \]
Stratified Newcombe Confidence Limits for the Common Risk Difference

PROC FREQ computes stratified Newcombe confidence limits for the common risk (proportion) difference by using the method of Yan and Su (2010). The stratified Newcombe confidence limits are constructed from stratified Wilson confidence limits for the common (overall) row proportions.

PROC FREQ first computes individual Wilson confidence limits for the row proportions in each $2 \times 2$ table (stratum), as described in the section Wilson (Score) Confidence Limits. These stratum Wilson confidence limits are then combined to form stratified Wilson confidence limits for the overall row proportions by using Mantel-Haenszel weights, where the Mantel-Haenszel weight for stratum h is

\[ w_ h = n_{h1\cdot } n_{h2\cdot } / n_ h \]

The confidence levels of the stratum Wilson confidence limits are chosen so that the overall confidence coefficient (for the stratified Wilson confidence limits) is $100(1 - \alpha )$% (Yan and Su, 2010).

Denote the lower and upper stratified Wilson score confidence limits for the common row 1 proportion as $L_1$ and $U_1$, respectively, and denote the lower and upper stratified Wilson confidence limits for the common row 2 proportion as $L_2$ and $U_2$, respectively. The $100(1-\alpha )$% stratified Newcombe confidence limits for the common risk (proportion) difference are

\begin{eqnarray*} L & = & \hat{d}_{\mi{MH}} ~ - ~ z_{\alpha /2} \sqrt { \lambda _1 L_1 (1-L_1) + \lambda _2 U_2 (1-U_2) } \\[0.10in] U & = & \hat{d}_{\mi{MH}} ~ + ~ z_{\alpha /2} \sqrt { \lambda _2 L_2 (1-L_2) + \lambda _1 U_1 (1-U_1) } \end{eqnarray*}

where $\hat{d}_{\mi{MH}}$ is the Mantel-Haenszel estimate of the common risk difference and

\begin{eqnarray*} \lambda _1 & = & \sum _ h w_ h^2 / n_{h1\cdot } \\[0.05in] \lambda _2 & = & \sum _ h w_ h^2 / n_{h2\cdot } \end{eqnarray*}

When there is a single stratum, the stratified Newcombe confidence interval is equivalent to the (unstratified) Newcombe confidence interval. For more information, see the subsection "Newcombe Confidence Limits" in the section Risk Difference Confidence Limits. See also Kim and Won (2013).