Estimation of P-value of MAX test with double triangle diagram for 2 x 3 SNP case-control tables.

Single nucle otide polymorphisms (SNPs) are the most popular markers in genetic epidemiology. Multiple tests have been applied to evaluate genetic effect of SNPs, such as Pearson's test with two degrees of freedom, three tests with one degree of freedom (chi(2) tests for dominant and recessive modes and Cockran-Armitage trend test for additive mode) as well as MAX3 test and MAX test, which are combination of four tests mentioned earlier. Because MAX test is a combination of Pearson's test of two degrees of freedom and two tests of one degree of freedom, the probability density function (pdf) of MAX statistics does not match pdf of chi(2) distribution of either one or two degrees of freedom. In order to calculate P-value of MAX test, we introduced a new diagram, Double Triangle Diagram, which was an extension of de Finetti diagram in population genetics which characterized all of the tests for 2 x 3 tables. In the diagram the contour lines of MAX statistics were consisted of elliptic curves and two tangent lines to the ellipses in the space. We normalized the ellipses into regular circles and expressed P-value of MAX test in an integral form. Although a part of the integral was not analytically solvable, it was calculable with arbitrary accuracy by dividing the area under pdf into finite rectangles. We confirmed that P-values from our method took uniform distribution from 0 to 1 in three example marginal count sets and concluded that our method was appropriate to give P-value of MAX test for 2 x 3 tables.