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I have correlated data and am using a logistic regression mixed effects model to estimate the individual level (conditional) effect for a predictor of interest. I know that for standard marginal models, inference on model parameters using the Wald test is consistent for the likelihood ratio and score tests. They are usually approximately the same. Because the Wald is easy to compute and available in R output, I use that 99% of the time.

However, with a mixed effects model, I was intrigued to see a huge difference between the Wald test for the fixed effects, as they're reported in the model output in R, and a "by hand" likelihood ratio test--which involves actually fitting the reduced model. Intuitively, I can see why this might make a huge difference, because in the reduced model, the variance of the random effect is re-estimated and can substantially affect the likelihood.

Can someone explain

  1. How are the Wald test statistics computed in R for fixed effects?
  2. What is the information matrix for the estimated model parameters in a mixed effects model? (and is the same mx from which the Wald test statistics are calculated?)
  3. What are the differences in interpretation between the results from the two tests in the cases I described? which ones are generally motivated and used in the literature for inference?
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    $\begingroup$ wonder if this partly answers your question. $\endgroup$
    – qoheleth
    Jun 4, 2014 at 1:02

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The traditional Wald statistic to test the hypothesis H0 Lt = l for given L, r x p, and l, r x 1, is given by W = (Lt - l)' [L (X'H-1 X)-1 L' ]-1 (Lt - l) and asymptotically, this statistic has a chi-square distribution on r degrees of freedom. These are marginal tests, so that there is an adjustment for all other terms in the fixed part of the model. R is open source

  1. Do you have the source?
  2. What is your model exactly? Mixed effects is a fairly broad category, in as far as it resonates into the Fisher information matrix.
  3. You mean a likelihood ratio and a score test?
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