Conditional estimates for OR

When two independent samples from separate population are collected, a product binomial likelihood can be used. However, conditioning on margins, a different formulation is achieved, as follows (shai being population OR):

For shai = 1 under null hypothesis, the above expression is exactly the same as Fisher-Irwin exact test. More simplification reduces the likelihood to this Lc = (n1! n2! m1! m2!) / (N! a! b! c! d!).

Conditional estimates and confidence intervals for OR in R can be calculated as follows:

> m = matrix(c(a,b,c,d),2,2) > fisher.test(m, conf.level = 0.95)         Fisher's Exact Test for Count Data data:  m  p-value = 0.01075 alternative hypothesis: true odds ratio is not equal to 1  95 percent confidence interval:  1.225596 8.254335  sample estimates: odds ratio    3.024924  >  > # The same point estimate of conditional OR > # can be obtained by optimization: > cond.nonc.hyp = function(orhat,a, b,c,d) { +   n1 = a + b +   n2 = c + d +   m1 = a + c +   m2 = b + d +   N = n1 + n2 +   al = max(0, m1 - n2) +   au = min(m1, n1) +   i = al:au +   l = (choose(n1, a) * choose(N - n1, m1 - a) * orhat^a) / +   sum(choose(n1, i) * choose(N - n1, m1 - i) * orhat^i) +   return(l) + } > orhat = a*d/(b*c) # initial estimate > cmle  =  optim(orhat,  cond.nonc.hyp,  a = a,  b = b, c = c, d = d, +   method  =  "BFGS",  control  =  list(fnscale  =  -1), hessian = T) > cmle$par [1] 3.024922