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I have a question regarding the comparison of the following two multilevel models:

  1. Null model: outcome.nullmodel <- lmer(outcome ~ 1 + (1 | ID), data=multileveldata)

  2. Random slopes model: outcome.rs <- lmer(outcome ~ 1+ predictor_L1+(1+predictor_L1|ID), data = multileveldata)

I read in some articles that a random slopes model of that kind was compared to a null model/intercept only via deviance/likelihood-ratio test. But is it appropriate to use the deviance test here? I am not sure about that because the models differ in the fixed AND the random effect. Therefore, I would assume they are non-nested and should be compared using information criteria such as AIC and/or BIC instead of likelihood ration test. Or am I wrong?

Thank you very much in advance for your support!

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The models are still nested: you get model 1 by setting the coefficient of predictor_L1 and the variance component for predictor_L1 and the covariance component to zero. That's just three parameters.

It's a slightly non-standard likelihood ratio test, because the null value of the variance component is on the boundary of the parameter space, but there isn't an issue with nesting.

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