Some remarks on likelihood factorization

Various types of likelihood factorization are reviewed and some general statistical consequences noted. In one broad class there is noniterative asymptotically efficient combination of information across factors via generalized least squares. This is used to discuss missing information in simple binary problems. It is shown that with observations on a 2 x 2 table supplemented by independent observations on each margin the maximum likelihood estimate of the odds ratio differs from that based on the complete table but is not asymptotically more efficient.