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I are analyzing time to event using a Cox model. There is a continuous variable C, and I are conducting a likelihood ratio test comparing the following two models to see if the variable C should be added.

model_a ~ A + B

model_b ~ A + B + C

anova(model_a, model_b, test = "LRT")

In this case, can we call model_a a null model, or is the null model an intercept-only model. There is no intercept-only model in Cox model, so there is no null model?

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  • $\begingroup$ Whether "variable C should be added" to your model shouldn't just be based on this type of statistical significance test. See Frank Harrell's online notes. Particularly with Cox models it's a good idea to include all predictors that are expected to be associated with outcome, based on your understanding of the subject matter, provided that you aren't overfitting the data. That's even if some predictors don't pass a test of statistical significance in your particular data sample. $\endgroup$
    – EdM
    Jan 18, 2023 at 15:26
  • $\begingroup$ Thank you for your comment. $\endgroup$
    – Totti
    Jan 19, 2023 at 6:03

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A standard likelihood ratio test compares two nested models, i.e., one model is more complex than the other, and the simpler model is actually a submodel of the more complex model (e.g., by having one or more parameters equal to zero). In such a test the null hypothesis (null model) is always the simpler one of the two models. If the test rejects, the likelihood of the more complex model looks so much better than that of the simpler model that this would be very unlikely to happen if the simpler model actually were true.

In your case the simpler model and therefore the null model is indeed model_a.

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  • $\begingroup$ Thank you for your comment, I understand that model_a will be the null model in this case. Your other comments were also helpful. $\endgroup$
    – Totti
    Jan 19, 2023 at 6:03

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