I am studing relationship between the competition facing a hospital and the death at 30 days within it. I performed mixed-effect model assuming that patient in same hospital should be more correlated. Hospital(finessGeoDP) and Trimester are in random effect. HHI_cat is the index or competition (with four level)

Here is below the script of the model and the output.

**Contents**hide

# MODEL

`MultModel<-glmer(dc30 ~HHI_cat+age_cat+Sexe+Urgence+neoadj+ denutrition+score_charlson_cat+Acte+ Nbre.sejour_cat+statutHop2+Fdep09_cat3+ (1|Trimestre)+(1|finessGeoDP), data =data_Final,family=binomial(link="logit"), control=glmerControl(optimizer="bobyqa", optCtrl=list(maxfun=2e5))) `

# OUTPUT

I calculated the odds ratio of fixed-effects using function `exp()`

I also calculated the confident interval of odds using the `standard error*1.96`

However, I am not used to interpreting the results of random effects.

How to interpret the variance for finessGeoDP (Hospital ID) and Trimester. Do I have to convert these coef with `exp()`

before interpreting them?

Coul I calculate the confident interval of the variance using the `SD*1.96?`

Is there an interest in determining the significance of random effects?

Could results of random effects influence the interpretation of fixed effects?

` AIC BIC logLik deviance df.resid 42319.9 42578.0 -21133.9 42267.9 151533 Scaled residuals: Min 1Q Median 3Q Max -1.0389 -0.2019 -0.1446 -0.1108 15.6751 Random effects: Groups Name Variance Std.Dev. finessGeoDP (Intercept) 0.12824 0.3581 Trimestre (Intercept) 0.03333 0.1826 Number of obs: 151559, groups: finessGeoDP, 711; Trimestre, 20 Fixed effects: Estimate Std. Error z value Pr(>|z|) (Intercept) -4.41959 0.11735 -37.663 < 2e-16 *** HHI_catUn.peu.compétif -0.01905 0.05663 -0.336 0.736554 HHI_catmoy.competif -0.02566 0.06121 -0.419 0.675128 HHI_catTrès.competitif -0.20815 0.06389 -3.258 0.001122 ** age_cat61-70 ans 0.31443 0.05653 5.562 2.67e-08 *** age_cat71-80 ans 0.62614 0.05461 11.466 < 2e-16 *** age_cat81-90 ans 1.29198 0.05346 24.169 < 2e-16 *** age_catPlus de 90 ans 1.86270 0.07069 26.349 < 2e-16 *** SexeHomme 0.30788 0.02935 10.489 < 2e-16 *** UrgenceOui 1.07916 0.03549 30.408 < 2e-16 *** neoadjOui 0.20516 0.04978 4.122 3.76e-05 *** denutritionOui 0.35383 0.03156 11.210 < 2e-16 *** score_charlson_cat3-4 0.26342 0.04129 6.379 1.78e-10 *** score_charlson_cat>4 0.88358 0.03925 22.512 < 2e-16 *** ActeAutres 0.43596 0.05404 8.068 7.15e-16 *** Actecolectomie_gauche -0.14714 0.03827 -3.844 0.000121 *** ActeResection rectale -0.39737 0.07856 -5.058 4.24e-07 *** Acteresection_multiple_CCR 0.08006 0.05210 1.537 0.124376 ActeRRS -0.17226 0.04293 -4.013 6.01e-05 *** Nbre.sejour_cat51-100 -0.17283 0.04731 -3.653 0.000259 *** Nbre.sejour_cat>100 -0.37517 0.07712 -4.865 1.15e-06 *** statutHop2Hpt.non.univ -0.10931 0.07480 -1.461 0.143940 Fdep09_cat3Niv.moy 0.00302 0.03668 0.082 0.934384 Fdep09_cat3Niv.sup. -0.04000 0.03960 -1.010 0.312553 `

#### Best Answer

How to interpret the variance for finessGeoDP (Hospital ID) and Trimester. Do I have to convert these coef with exp() before interpreting them?

No, this would simply be wrong. Typically models with random effects are either interpreted

in terms of

*variance components*— common e.g. in population genetics, and very much harder to do for generalized linear (rather than "ordinary" linear) mixed models, i.e. with a non-Gaussian response variable. In this case you would look at the*proportion*of variance explained by each term, i.e. you would say something like "variation among groups in`finessGeoDP`

explains about 80% (0.12/0.15) of the variance while`Trimestre`

explains the remaining 20% (0.03/0.15). In the mixed case this is tricky because the decomposition includes neither the variability explained by the fixed-effect parameters, nor by binomial variation. (If you want to do things this way you should probably look into the plethora of plausible pseudo-$R^2$ measures for GLMM.)in terms of standard deviations; I generally find this more useful because the standard deviations are on the same (log-odds) scale as the fixed-effect estimates; for example, you could say that a "typical" range encompassing 95% of the variation in

`finessGeoDP`

would be about 4$sigma$=1.44; this is of about the same magnitude as the largest fixed-effect parameters.

Could I calculate the confident interval of the variance using the SD*1.96?

No. The SD here is *not* a measure of the uncertainty of the random effect parameter, it's just the value on the standard-deviation scale (i.e. $sqrt{textrm{variance}}$). Furthermore, even if you did have the standard error of the SD (or variance) estimate, these intervals are based on a Gaussian sampling distribution, which is usually a poor approximation. `confint(fitted_model,parm="theta_")`

will give you more reliable *likelihood profile confidence intervals* (warning, this is computationally intensive).

Is there an interest in determining the significance of random effects?

I would say usually not, but it is interesting in some contexts/to some people. Since we know that variances are always >0, p-values of random effects don't have the same sensible interpretation of "can we reliably determine the *sign* of this effect?" that applies to fixed-effect parameters.

Could results of random effects influence the interpretation of fixed effects?

Sure. (Otherwise there would be a *lot* of analyses where we don't care about the random effects *per se* and could save ourselves a lot of trouble by running simpler GLMs.)

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