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 details, 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. See the section Risks and Risk Differences for details. By default, the procedure uses column 1 risks to compute the Gail-Simon test. If you specify the GAILSIMON(COLUMN=2) option, the procedure uses column 2 risks.

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

$\displaystyle Q-$	$\displaystyle =$	$\displaystyle \sum _ h ~ (d_ h / s_ h)^2 ~ I(d_ h > 0 )$
$\displaystyle Q+$	$\displaystyle =$	$\displaystyle \sum _ h ~ (d_ h / s_ h)^2 ~ I( d_ h < 0 )$
$\displaystyle Q$	$\displaystyle =$	$\displaystyle \min ~ (Q-, ~ Q+)$

where is the risk difference in table h, is the standard error of the risk difference, and equals 1 if and 0 otherwise. Similarly, equals 1 if and 0 otherwise. The q $2 \times 2$ tables (strata) are indexed by $h = 1, 2, \ldots , q$ .

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

$\displaystyle p(Q-)$	$\displaystyle =$	$\displaystyle \sum _ h ~ (1 - F_ h(Q-)) ~ B(h; n=q, p=0.5)$
$\displaystyle p(Q+)$	$\displaystyle =$	$\displaystyle \sum _ h ~ (1 - F_ h(Q+)) ~ B(h; n=q, p=0.5)$
$\displaystyle p(Q)$	$\displaystyle =$	$\displaystyle \sum _{h=1}^{q-1} ~ (1-F_ h(Q)) ~ B(h; n=(q-1), p=0.5)$

where $F_ h(\cdot )$ is the cumulative chi-square distribution function with h degrees of freedom and is the binomial probability function with parameters n and p. The statistic Q tests the null hypothesis of no qualitative interaction. The statistic tests the null hypothesis of positive risk differences. A small p-value for indicates negative differences; similarly, a small p-value for indicates positive risk differences.