# Solved – Poisson GLMM vs GLM in R (lme4)

I have been trying to sharpen my GLMM knowledge by working through some problems in Foundations of Linear and Generalized Linear Models. I am stuck on problem 9.36 which gives some homicide data then states "fit a Poisson GLMM. Interpret estimates. Show that the deviance decreases by 116.6 compared with the Poisson GLM, and intercept"

This is what I did

``library(lme4) homi <- read.table("http://www.stat.ufl.edu/~aa/glm/data/Homicides.dat",                    header = TRUE)  fit1 <- glm(count~race, family=poisson(link = log),data=homi)  fit2 <- glmer(count~1+(1|race), family=poisson(link = log),data=homi) summary(fit1) summary(fit2) ``

For `fit1`

``Call: glm(formula = count ~ race, family = poisson(link = log), data = homi)  Deviance Residuals:      Min       1Q   Median       3Q      Max   -1.0218  -0.4295  -0.4295  -0.4295   6.1874    Coefficients:             Estimate Std. Error z value Pr(>|z|)     (Intercept) -2.38321    0.09713  -24.54   <2e-16 *** race         1.73314    0.14657   11.82   <2e-16 *** --- Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1  (Dispersion parameter for poisson family taken to be 1)      Null deviance: 962.80  on 1307  degrees of freedom Residual deviance: 844.71  on 1306  degrees of freedom AIC: 1122  Number of Fisher Scoring iterations: 6 ``

and for `fit2`

``Generalized linear mixed model fit by maximum likelihood (Laplace   Approximation) [glmerMod]  Family: poisson  ( log ) Formula: count ~ 1 + (1 | race)    Data: homi       AIC      BIC   logLik deviance df.resid    1132.5   1142.8   -564.2   1128.5     1306   Scaled residuals:      Min      1Q  Median      3Q     Max  -0.7174 -0.3054 -0.3054 -0.3054 19.3426   Random effects:  Groups Name        Variance Std.Dev.  race   (Intercept) 0.739    0.8596   Number of obs: 1308, groups:  race, 2  Fixed effects:             Estimate Std. Error z value Pr(>|z|)   (Intercept)  -1.5235     0.6122  -2.489   0.0128 * ``

It seems like the deviance for the GLMM is `1128.5` while the deviance for the GLM is `844.71`. This is shows the GLM is fitting better than the GLMM which I think is the exact opposite solution from what question implied. I am not sure if I am looking at the correct output or if I setup the problem wrong.

Contents

There is probably something wrong with the question. The deviance for `fit1` can be computed with

``deviance(fit1)  # same as  sum(resid(fit1)^2) ``

But for the GLMM, the `lme4` packages uses another method documented in `?llikAIC`, which gives 1128 – higher than for the GLM.

Perhaps the question actually wants to test for the decrease in deviance when adding the single fixed effect, because it comes out as 118, close to your 116.6:

``fit1 <- glm(count~race, family=poisson(link = log),data=homi) fit0 <- glm(count~1, family=poisson(link = log),data=homi)  > anova(fit0, fit1) Analysis of Deviance Table  Model 1: count ~ 1 Model 2: count ~ race Resid. Df Resid. Dev Df Deviance 1      1307     962.80             2      1306     844.71  1   118.09 ``

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