Cosma Shalizi, an associate professor in statistics at Carnegie Mellon University, gives an interpretation of the likelihood-ratio threshold in an LR test: It’s the shadow price of statistical power:

[…]

Suppose we know the probability density of the noise

pand that of the signal isq. The Neyman-Pearson lemma, as many though not all schoolchildren know, says that then, among all tests off a given sizes, the one with the smallest miss probability, or highest power, has the form “say ‘signal’ ifq(x)/p(x) >t(s), otherwise say ‘noise’,” and that the thresholdtvaries inversely withs. The quantityq(x)/p(x) is thelikelihood ratio; the Neyman-Pearson lemma says that to maximize power, we should say “signal” if its sufficientlymore likelythan noise.The likelihood ratio indicates how different the two distributions — the two

hypotheses— are atx, the data-point we observed. It makes sense that the outcome of the hypothesis test should depend on this sort of discrepancy between the hypotheses. But why theratio, rather than, say, the differenceq(x) –p(x), or a signed squared difference, etc.? Can we make this intuitive?Start with the fact that we have an optimization problem under a constraint. Call the region where we proclaim “signal”

R. We want to maximize its probability when we are seeing a signal,Q(R), while constraining the false-alarm probability,P(R) =s. Lagrange tells us that the way to do this is to minimizeQ(R) –t[P(R) –s] overRandtjointly. So far the usual story; the next turn is usually “as you remember from the calculus of variations…”Rather than actually doing math, let’s think like economists. Picking the set

Rgives us a certain benefit, in the form of the powerQ(R), and a cost,tP(R). (Thetsterm is the same for allR.) Economists, of course, tell us to equatemarginalcosts and benefits. What is the marginal benefit of expandingRto include a small neighborhood around the pointx? Just, by the definition of “probability density”,q(x). The marginal cost is likewisetp(x). We should includexinRifq(x) >tp(x), orq(x)/p(x) >t. The boundary ofRis where marginal benefit equals marginal cost, and that is why we need the likelihoodratioand not the likelihooddifference, or anything else. (Except for a monotone transformation of the ratio, e.g. the log ratio.) The likelihood ratio thresholdtis, in fact, the shadow price of statistical power.

It seems sensible to me.

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