# Solved – How to interpret random effect coefficients in glmer

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

# 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  ``

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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