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It looks like a common consensus that likelihood ratio (LR) test is preferred over log rank and Wald in Cox model when sample size is small. I did some research and couldn't find any clear answer

My superficial understanding is that the uncertainty of variance approximation as well as the normal distribution (instead of t-distribution) approximation in Wald test makes it less accurate when sample size is small. Log rank is non-parametric which makes it less accurate for small sample sizes (I have a feeling that my understanding is not quite correct...).

Anyway, Can anyone share some insight regarding why LR test is considered more accurate than the other two?

kjetil b halvorsen
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unicorn
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1 Answers1

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The logrank test is the score test from a Cox proportional hazards model, so it makes the same assumptions as the Cox model. The LR test, among the three commonly used tests (the other two being Wald and score) is the gold standard. It is typically more accurate for all sample sizes. One way to see this is to note that even with complete separation, which occurs more often with logistic models than with Cox, the LR test is fully accurate whereas standard errors used in Wald tests blow up rendering Wald tests useless when complete separation (infinite regression coefficient estimates) is in play.

The LR test also provides more accurate confidence intervals using confidence profiles. If you don't want to have to make approximations, Bayesian inference is exact.

Frank Harrell
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  • I read multiple times that LR test estimates both 'restricted' and 'unrestricted' models whereas Wald and Score test only estimates one. This seems to be the reason why LR test is more accurate. How should I understand these 'restricted' and 'unrestricted' models here? – unicorn May 27 '19 at 02:24
  • That's not why one is more accurate. And the score test requires two models. All tests are testing the same thing and in linear models are virtually identical. For non-linear models, the LR test is more accurate because it's transformation-invariant and doesn't involve standard errors, which can be problematic. – Frank Harrell May 27 '19 at 03:14