Statistics/LLR

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A generalized log-likelihood ratio test is a test statistic computed by taking the ratio of the maximum probability under the constraint of the null hypothesis to the maximum probability with that constraint relaxed. If this ratio is given by λ and the null hypothesis holds, then -2 log λ has a particularly handy asymptotic distribution. Many common test statistics such as the Z Test, the F Test and Pearson's χ2 test can be phrased as generalized log-likelihood ratios or approximations thereof.

Many of these approximations were quite useful when computers did not exists, but now that taking a log is really no more vexing than multiplying two numbers, other approximations may be more useful, especially in special cases were the approximations are suspect.

If we write the likelihood as computed by our model of an observed outcome X to be L(X|ω) where ω is set of parameters for our model, then the generalized likelihood ratio is

max L(X | w )
H0
l =
max L(X | w )
H

If H0 is a strict subset of H that is generated by setting some of the ω to zero and the true value of ω really is taken from H0, then -2 log λ will be asymptotically χ2 distributed with degrees of freedom equal to the difference in dimensionality of H and H0.

For instance, in the case of Pearson's test, we might try to compare two coins to determine whether they have the same probability of coming up heads. Our observation can be put into a contingency table with rows corresponding to the coin and columns corresponding to heads or tails. The elements of the contingency table will be the number of times the coin for that row came up heads or tails. The contents of this table are our observation X.

Heads Tails
Coin 1 k1H k1T
Coin 2 k2H k2T

Here ω consists of the parameters p1H, p1T, p2H, and p2T which are the probability that coin 1 (2) comes up heads (tails). The hypothesis space H is defined by the usual constraints on a distribution, pij ≥ 0, pij ≤ 1, and piH + piT = 1. The null hypothesis H0 is the sub-space where p1j = p2j. In all of these constraints, i = 1,2 and j = H,T.

The hypothesis and null hypothesis can be rewritten slightly so that they satisfy the constraints for the log-likelihood ratio to have the desired nice distribution. Since the constraint causes the two-dimensional H to be reduced to the one-dimensional H0, the asymptotic distribution for the test will be χ^2(1), the χ^2 distribution with one degree of freedom.

For the general contingency table, we can write the log-likelihood ratio test as

pij
-2 log l = S k ij log
i,j mij