[R-sig-ME] Shrinkage of ORs in a glmm

[lorenz.gygax at art.admin.ch]

Dear Mixed-modelers,

in a recently submitted paper, we used a glmm to estimate the risk of claw injuries in dairy cows on a set of 36 farms that differed in the type of flooring and were visited three times.

So far, we have worked with glmmPQL but may now switch to glmer because we were asked to calculate LR-tests by one of the reviewers. In addition, we are asked that we shrink our estimated odds-ratios at the population level. So far we have calculated these as e to the power of the estimated parameters.

I have thought that shrinkage happens automatically in a mixed-effects model but this does not seem to be the case depending on the numerical implementation (see comment of the reviewer below). Is this additional shrinkage indeed necessary and is there an easy implementation on how this shrinkage can be done in e.g. lme4?

Many thanks for your advice, Lorenz
-
Lorenz Gygax
Federal Veterinary Office FVO
Centre for proper housing of ruminants and pigs

>From the generalized linear mixed model, two types of odds ratio OR can be derived: subject specific or marginal odds ratios. In the former case, we have the OR for an individual farm, as if it was changing from one floor type to another. In the latter case, we have the OR in terms of population proportions. The latter involves shrinkage of estimated effects on the logit scale. This is a feature mentioned in number of papers about generalized linear mixed models, e.g. Liang & Zeeger (1986), Zeeger and Liang (1986),  Engel et al. (1995). With the GEE method, shrinkage is automatic, but with the other methods mentioned [PQL], it is not.

Without shrinkage, the OR is not in terms of an expected proportion for a random farm, but in terms of median proportions over farms. No mention is made of any shrinkage, so I take it that the first type of OR was calculated. The difference between the two types of OR has to do with the averaging (mathematical integration) over the random farm effects: either on the logit scale or on the original scale of proportions. The distinction between the types of OR becomes important when there are sizeable farm effects (a sizeable component of variance for random farm effects). The former OR, which averages at the logit scale, includes odds ratio's that involve either relatively small or large proportions (close to 0, or close to 1). These are changes that are of less practical interest (between proportions close to 0 or between proportions close to 1). The latter OR, that involves averaging at the original scale, will produce OR values closer to 1, and actually represents the OR
at the population level.

Engel, B., Buist, W. & Visscher, A. (1995). Inference for threshold models with
    variance components from the generalized linear mixed model perspective. Gen.
    Sel. Evol. 27, 15-32.
Liang, K.Y. & Zeeger, S.L. (1986). Longitudinal data analysis using generalized
    linear models. Biometrika 73, 13-22.
Zeeger, S.L. & Liang, K.Y. (1986). Longitudinal data analysis for discrete and
    continuous outcomes. Biometrics 42, 121-130.