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In wikipedia, In wikipedia, the interpretation of high likelihood ratio test statistic is:

High values of the statistic mean that the observed outcome was nearly as likely to occur under the null hypothesis as the alternative, and so the null hypothesis cannot be rejected.

I can't understand why. Doesn't it mean that the observed data is more likely to occur under H0? Could anyone please explain it in an easy-to-understand way?

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The entirety of the Wikipedia paragraph you quoted is (bolding added by me):

The numerator corresponds to the likelihood of an observed outcome under the null hypothesis. The denominator corresponds to the maximum likelihood of an observed outcome, varying parameters over the whole parameter space. The numerator of this ratio is less than the denominator; so, the likelihood ratio is between 0 and 1. Low values of the likelihood ratio mean that the observed result was much less likely to occur under the null hypothesis as compared to the alternative. High values of the statistic mean that the observed outcome was nearly as likely to occur under the null hypothesis as the alternative, and so the null hypothesis cannot be rejected.

The point to understand about likelihood ratio tests is the statement in bold. The null hypothesis will always have a lower likelihood than the alternative. The likelihood ratio will always be less than (or equal to) 1, and the smaller it is the better the alternative is at fitting the data.

The reason the null model gives smaller likelihood is that it is a restricted model. It typically sets some parameters to zero. The alternative model is free to vary the restricted parameters in order to increase the likelihood of the data under the model, so the alternative model will always have higher (or at least as high) likelihood than the null hypothesis. The greater the likelihood increase under the alternative, the smaller the likelihood ratio.

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  • $\begingroup$ Thank you! But I think the denominator is not the likelihood under alternative hypothesis, it should be the likelihood under the entire parameter space(as it shows in Wiki). And this is also the reason why I can't understand"High values of the statistic mean that the observed outcome was nearly as likely to occur under the null hypothesis as the alternative" $\endgroup$ Nov 7 '20 at 13:31
  • $\begingroup$ Ok, then it sounds like your misunderstanding is matter of terminology. In this context the alternative hypothesis is the entire parameter space. In a single parameter test, for example, the null hypothesis might be that the parameter is zero and the alternative be that the parameter is any other number. In that case the parameter space is the Real Numbers while the null hypothesis restricts the parameter to a subset of that space (i.e., zero). So, yes, the denominator is the likelihood under the alternative hypothesis. $\endgroup$ Nov 7 '20 at 13:54
  • $\begingroup$ OMG, So in which context the alternative hypothesis means the opposite of null hypothesis? $\endgroup$ Nov 7 '20 at 14:23
  • $\begingroup$ Yeah, that's right. I see how it could be confusing because in other contexts the likelihood ratio compares one specific parameter value with one other. That situation has nice theoretical properties but in practice it's almost always used theta equals zero vs. (the composite alternative) theta is nonzero. $\endgroup$ Nov 7 '20 at 14:54

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