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I am curious as to why a likelihood ratio cannot give positive evidence for the null, since it is a model comparison.

Indeed, this is more confusing given the fact that Bayes Factors are similar comparisons (nearly equivalent in some cases) and yet can give evidence for the null.

Why can a likelihood ratio not give evidence for a null, but a Bayes factor can?

EDIT: Please see comment from @Silverfish explaining how this question is not answered by previous posts about NHST. Essentially my question asks: why do we treat evidence from likelihood ratios differently to evidence from Bayes factors, when both are model comparisons? This is not the same as ‘why can’t we accept the null in NNHST?’ Which is a question I understand.

Silverfish
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HereItIs
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  • Consider data that are inconsistent with both the null and any alternative. – whuber Jun 19 '23 at 17:13
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    The question of why NHST can only reject but not accept the null hypothesis has been asked many times before. – Durden Jun 19 '23 at 18:22
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    I think the point at the heart of this question is "why do we treat evidence from likelihood ratios differently to evidence from Bayes factors, when both are model comparisons?". That strikes me as sufficiently different from "why NHST can only reject but not accept the null" to not be a duplicate... even if someone could give a reasonable answer that isn't much more than "because we use likelihood ratios in the NHST framework" followed by a link to some of the previous questions. (I suspect it would be possible to write a little more detail than that, but even that's not a trivial duplicate) – Silverfish Jun 19 '23 at 22:20
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    I agree. This is more about the nature of the LRT more than NHST in and of itself. – Shawn Hemelstrand Jun 20 '23 at 00:40
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    Agreed these posts regarding NHST do not answer the specific question I’ve asked, or the one kindly paraphrased by @Silverfish. – HereItIs Jun 20 '23 at 05:46
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    I think it was a mistake closing the question. Dear moderators, which of the four answers to the 'duplicate' question do you think addresses the question asked here? – Graham Bornholt Jun 20 '23 at 07:15

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The likelihood ratio itself (not the test) can be used for model comparison and can provide positive evidence for one hypothesis over another (according to the Likelihood school). However, you mentioned positive evidence for the null hypothesis which raises the question of whether the likelihood ratio test (LRT) is what you are referring to.

The LRT does not compare models because (like all tests) tail probability calculations such as the p-value are only based on the null.

Testing can be regarded as addressing the question: Are the data (statistically) consistent with the null hypothesis? It is about the absolute plausibility of the null. In contrast, Bayes factors (and some non-testing uses of likelihood ratios) are concerned with the relative plausibility of two hypotheses.

Graham Bornholt
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The premise is false. A ratio of likelihoods can (does) quantify the evidence from the data in favour of the parameter value (or model) that has the higher likelihood against the parameter value (or model) that has the lower likelihood. If the null parameter value that is serving as your null gives the numerator likelihood in the ratio then that ratio does tell you how strongly the evidence favours (ratio > 1) or disfavours (ratio < 1) the null, but only relative to the parameter value that gives the denominator likelihood.

Even where the data are inconsistent with both parameter values (as in @whuber's comment) the ratio will tell you how much the evidence favours one of the values relative to the other (it can be a tie, but that is unusual). There would be a different parameter value that is favoured against both of those in the ratio, but that does not change the meaning of the likelihood ratio. (That circumstance should show that a full examination of the likelihood function is a better idea than a fixation on two specific parameter values.)

Michael Lew
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Why can a likelihood ratio not give evidence for the null since it is a model comparison?

This is a loaded question and it contains a false statement "a likelihood ratio can not give evidence for the null". It is not true that we can not give evidence for the null.

The situation is instead that we can not accept the null (and neither can we accept other hypotheses). The point of observations and experiments is to find a data driven answer to questions by excluding/eliminating what is (probably) not the answer (Popper's falsification), such as with two one-sided t-tests (TOST).

That idea of the 'not accepting a hypothesis' has been a topic several times before

Sextus Empiricus
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