[2 solutions for estimating odds ratios with zeros].

Abstract

Two solutions are proposed for the estimation of odds ratios (OR) when one or the two elements of the principal (A, D) or secondary (B, C) diagonals of a 2 x 2 matrix (A, B, C, D) are 0. The OR estimate is AD/BC. If A or D are 0, OR = 0; if B or C are 0, the OR is undefined. Analytical solution. This solution conserves the marginal totals. If B = 0 and C = 0, the OR cannot be less than AD/1 (the minimal acceptable value), then the equation (A-X) (D-X)/X2 = KAD/1 searches for that X which subtracted to A and B and added to B (0) and C (0) yields an OR K times AD; if B = 0 and C > 0 then (A-X) (D-X)/X (C+X) = AD/C; if B > 0 and C = 0, then B replaces C in the latter equation. If A and D are 0, X2/(B-X) (C-X) = 1/KBC; if A = 0 and D > 0, X (D+X)/(B-X) (C-X) = D/KBC; if A > 0 and D = 0, A replaces D in the latter equation. K can be taken at the maximum Chi squared value. Probabilistic solution. Zeros are replaced by ones and the elements of the diagonal without zeros are increased proportionally until the exact probability (Fisher) of this new matrix is equal or the nearest less than the exact probability of the original matrix. Since small numbers increase the estimation bias, the Haldane's correction should be always applied. This correction adds 0.5 to A, B C and D to estimate the OR (In OR) and adds 1 to these elements to estimate their variance.