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Suppose I use R to fit a Generalized Linear Mixed Model from the binomial family and with a logit link. How do I obtain the prediction intervals (as opposed to the confidence intervals) for the fixed effects (as opposed to the random effects) in a way that incorporates the variability described by the random effects? Thank you.

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We figured out a good way to do this that is good enough for our purposes (which is utilizing the predictive distribution to draw samples of empirically estimated fixed effects to calibrate an agent-based model). Here's the step-by-step process:

  1. Simulate the $\beta_j$'s from the variance-covariance matrix output of the GLMM.

  2. Sample the standard error of the standard deviation of the random intercepts from a chi-square distribution with $n - 1$ degrees of freedom. See this R-sig-ME post for details.

  3. From the $\beta_j$'s estimated in step one, sample the $\beta_j$'s again from a normal distribution with mean equal to the $\beta_j$ and standard deviation equal to that sampled in step two.

  4. Repeat steps 1-3 many times, saving the results at each iteration, to obtain an empirical predictive distribution of the $\beta_j$'s.

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  • $\begingroup$ "Simulate the $\beta_j$'s from the variance-covariance matrix output of the GLMM." --- from what distribution? The $\beta$s are fixed quantities. Do you mean to simulate some approximately pivotal quantity, about which you can make an asymptotic distributional argument? $\endgroup$ – Glen_b Sep 5 '13 at 23:21
  • $\begingroup$ Brash: Why wouldn't you just use parametric bootstrap? What you describe seems like a somewhat contrived version of it anyway. Also you give a wikipedia link, not a R-sig-ME one. Step 3 confuses me; could you please give a reference where you saw this procudure? (Steps 1-3) @Glen_b: I think the OP wants to say that he takes account of the variability due to the VCV matrix of the random effects as well as the standard error - but I might understood wrong... $\endgroup$ – usεr11852 Sep 6 '13 at 1:26
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    $\begingroup$ To expand on my comment: I'm not objecting to the basic concept, which I think is sound enough (I've done something similar to obtain prediction intervals for a GLM, and I've seen Ripley suggest the same procedure as I have used for GLMs), but I think it needs more explicit detail about the simulation, and a clearer justification of why it should be reasonable. $\endgroup$ – Glen_b Sep 6 '13 at 1:32
  • $\begingroup$ First thing's first: I fixed the link. It is now a R-sig-ME post. $\endgroup$ – Brash Equilibrium Sep 6 '13 at 16:30

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