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I am designing a simple study where I ask participants a problem. Then I code the answers as either correct or incorrect. I have a prediction from the literature that the percentage of correct answers should be approximately 20%.

My thought is to define two expected distributions (20% correct as H1 or alternative hypothesis vs 50% correct as H0 or null hypothesis), then calculate chi-square goodness of fit statistics for each of the expected distribution.

My question is: Can I calculate p value under the two hypotheses (20% correct vs 50% correct), then use their ratio as a likelihood ratio measure? By the definition of likelihood ratio (p(x|H0) / p(x|H1)), it seems to make sense but I haven't been able to find an example in which ratios of two p values to be used as a likelihood ratio.

Is this approach reasonable? Or can you refer me to an alternative way to analyze this question?

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    $\begingroup$ A p-value is not a likelihood, so I very much doubt that this makes sense. Why not calculate a proper likelihood ratio? $\endgroup$ Feb 6 at 1:54

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It has been proposed at least a few times to use a plot of p-value vs the value null hypothesis parameter as a pseudo-likelihood like object (see for example The p-value Function and Statistical Inference by DAS Fraser). If you really feel that you should work with p-values then consider plotting the full function rather than working with just two particular points, but I think that you would be better served by a more conventional likelihood function.

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