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If I replace the partial likelihood of the Cox model

https://en.wikipedia.org/wiki/Proportional_hazards_model#The_Cox_model

instead of the likelihood in the likelihood ratio test

https://en.wikipedia.org/wiki/Likelihood-ratio_test

What is the distribution of this test statistic? Is it still $\chi^2_q$?

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The partial likelihood ratio test is asymptotically $\chi^2_q$ as well, under conditions that guarantee consistency and asymptotic normality of the partial maximum likelihood estimators. This is a standard result in several textbooks.

See for instance this lecture notes:

https://www4.stat.ncsu.edu/~dzhang2/st745/chap6.pdf

or the book

Statistical Models Based on Counting Processes Authors: ANDERSEN, P.K., Borgan, O., Gill, R.D., Keiding, N.

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