It is not appropriate to classify participants as ‘responders’ or ‘non-responders’ unless this is how the fundamental data are generated, e.g., the outcome is all-or-nothing. As Stephen Senn has written about so extensively, it is usually impossible to really get a grip on ‘response’ for individual patients.
Changing methods doesn’t help with small sample size. In some cases, uses methods that cross-classifies data results in even smaller effective sample size because of intrinsic allowance for more interactions than we would usually have in a regression model.
Likelihood ratios are mainly useful with a binary outcome and a binary test. There they are useful for better understanding the test, and have major advantages over sensitivity and specificity. For binary X and binary Y, the odds ratio in a binary logistic model is the produce of the likelihood ratio positive and the likelihood ratio negative. But for the multivariable situation, statistical models are generally preferred over likelihood ratios.
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