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

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

where is the risk difference in stratum h and

The column 1 risk difference in stratum h is computed as

where is the proportion of row 1 observations that are classified in column 1 and 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 (Sato 1989) as

where

The % confidence limits for the common risk difference are

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 , where is the midpoint of the score confidence interval and is the width of the confidence interval divided by . The summary score estimate of the common risk difference is computed as

where

The variance of is computed as

The % summary score confidence limits for the common risk difference are

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 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

The confidence levels of the stratum Wilson confidence limits are chosen so that the overall confidence coefficient (for the stratified Wilson confidence limits) is % (Yan and Su 2010).

Denote the lower and upper stratified Wilson score confidence limits for the common row 1 proportion as and , respectively, and denote the lower and upper stratified Wilson confidence limits for the common row 2 proportion as and , respectively. The % stratified Newcombe confidence limits for the common risk (proportion) difference are

where is the Mantel-Haenszel estimate of the common risk difference and

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 Confidence Limits for the Risk Difference. See also Kim and Won (2013).