Significance of lod scores (3)

Derivation of asymptotic distribution of MLOD, lod score maximised over theta and heterozygote penetrance value f1

LR = P(D, M; theta=t, f1=f) / P(D, M; theta=0.5, f1=f)

LR = P(M | D; theta=t, f1=f) P(D; f1=f) / P(M | D; theta=0.5, f1=f) P(D; f1=f)

Since P(M | D; theta=0.5, f1=f) = P(M):

LR = P(M | D; theta=t, f1=f) / P(M)

Also:

P(M) = P(M | D; theta=t, f1=Kp), i.e. f0=f1=f2, no effect at disease locus

So:

LR = P(M | D; theta=t, f1=f) / P(M | D; theta=t, f1=Kp)

This is likelihood ratio of two nested hypotheses. Maximising the lod score, to get MLOD, is equivalent to maximising the conditional likelihood of the marker data given the disease data.

We expect 2ln(10)MLOD to be chi-squared with one degree of freedom. Similar reasoning applies to MALOD, which incorporates admixture, though then the test is one-tailed with two degrees of freedom.

lodsig2.html