The FREQ Procedure

Gail-Simon Test for Qualitative Interactions

The GAILSIMON option in the TABLES statement provides the Gail-Simon test for qualitative interaction for stratified 2 times 2 tables. For more information, see Gail and Simon (1985); Silvapulle (2001); Dmitrienko et al. (2005).

The Gail-Simon test is based on the risk differences in stratified 2 times 2 tables, where the risk difference is defined as the row 1 risk (proportion in column 1) minus the row 2 risk. For more information, see the section Risks and Risk Differences. By default, PROC FREQ uses column 1 risks to compute the Gail-Simon test. If you specify the GAILSIMON(COLUMN=2) option, PROC FREQ uses column 2 risks.

PROC FREQ computes the Gail-Simon test statistics as described in Gail and Simon (1985),

StartLayout 1st Row 1st Column upper Q minus 2nd Column equals 3rd Column sigma-summation Underscript h Endscripts left-parenthesis d Subscript h Baseline slash s Subscript h Baseline right-parenthesis squared upper I left-parenthesis d Subscript h Baseline greater-than 0 right-parenthesis 2nd Row 1st Column upper Q plus 2nd Column equals 3rd Column sigma-summation Underscript h Endscripts left-parenthesis d Subscript h Baseline slash s Subscript h Baseline right-parenthesis squared upper I left-parenthesis d Subscript h Baseline less-than 0 right-parenthesis 3rd Row 1st Column upper Q 2nd Column equals 3rd Column min left-parenthesis upper Q minus comma upper Q plus right-parenthesis EndLayout

where d Subscript h is the risk difference in table h, s Subscript h is the standard error of the risk difference, and upper I left-parenthesis d Subscript h Baseline greater-than 0 right-parenthesis equals 1 if d Subscript h Baseline greater-than 0 and 0 otherwise. Similarly, upper I left-parenthesis d Subscript h Baseline less-than 0 right-parenthesis equals 1 if d Subscript h Baseline less-than 0 and 0 otherwise. The q 2 times 2 tables (strata) are indexed by h equals 1 comma 2 comma ellipsis comma q.

The p-values for the Gail-Simon statistics are computed as

StartLayout 1st Row 1st Column upper P left-parenthesis upper Q minus right-parenthesis 2nd Column equals 3rd Column sigma-summation Underscript h Endscripts left-parenthesis 1 minus upper F Subscript h Baseline left-parenthesis upper Q minus right-parenthesis right-parenthesis upper B left-parenthesis h semicolon n equals q comma p equals 0.5 right-parenthesis 2nd Row 1st Column upper P left-parenthesis upper Q plus right-parenthesis 2nd Column equals 3rd Column sigma-summation Underscript h Endscripts left-parenthesis 1 minus upper F Subscript h Baseline left-parenthesis upper Q plus right-parenthesis right-parenthesis upper B left-parenthesis h semicolon n equals q comma p equals 0.5 right-parenthesis 3rd Row 1st Column upper P left-parenthesis upper Q right-parenthesis 2nd Column equals 3rd Column sigma-summation Underscript h equals 1 Overscript q minus 1 Endscripts left-parenthesis 1 minus upper F Subscript h Baseline left-parenthesis upper Q right-parenthesis right-parenthesis upper B left-parenthesis h semicolon n equals left-parenthesis q minus 1 right-parenthesis comma p equals 0.5 right-parenthesis EndLayout

where upper F Subscript h Baseline left-parenthesis dot right-parenthesis is the cumulative chi-square distribution function with h degrees of freedom and upper B left-parenthesis h semicolon n comma p right-parenthesis is the binomial probability function with parameters n and p. The statistic Q tests the null hypothesis of no qualitative interaction. The statistic upper Q minus tests the null hypothesis of positive risk differences. A small p-value for upper Q minus indicates negative differences; similarly, a small p-value for upper Q plus indicates positive risk differences.

Last updated: December 09, 2022