# Should I use a linear mixed model or a generalized mixed model?

I have a test dataset with repeated measures, different individuals sampled at different time points, here measured in days. I want to know if I should use a GLMM or a LMM to see how well, if at all, a binary variable can predict a measurement: Measure ~ VarResult + (1|Sample) + (1|TimeDays)

I tested whether the response variable is normally distributed and found that it is more log-normally distributed:

library(fitdistrplus)
normal <- fitdist(testdata$$Measure, "norm") lognormal <- fitdist(testdata$$Measure, "lnorm")
gofstat(lognormal)
#AIC = -685.7581
gofstat(normal)
#AIC = -677.5334


I tested if the residuals of the models are normally distributed:

plot(resid(fitLMM))
plot(resid(fitGLMM))
#The plots show that they are randomly distributed


Lastly, I tested the models directly:

fitLMM = lmer(Measure ~ VarResult + (1|Sample) + (1|TimeDays),data=testdata)
fitGLMM = glmer(Measure ~VarResult + (1|Sample) + (1|TimeDays), data=testdata,family=Gamma(link = "log"))
anova(fitLMM,fitGLMM)
#Df     AIC     BIC logLik deviance Chisq Chi Df Pr(>Chisq)
#fitGLMM  5 -823.55 -810.58 416.78  -833.55
#fitLMM   6 -698.64 -683.07 355.32  -710.64     0      1          1


In summary: I initially assumed that since the data was not normally distributed I should use an GLMM, but I later found that it is moreso the distribution of residuals from the fit model. Just from the residuals, it seems like a LMM would suffice. However, looking at the AIC values from the models, it seems that the GLMM fits the data moreso. Which should I use? Is there a better set of methods to determine which one to use?

testdata = read.csv("Sample,Measure,TimeDays,VarResult
635,0.032378049,280,Neg
635,0.036529268,455,Neg
734,0.038922822,389,Pos
734,0.037950697,590,Neg
4,0.029629965,343,Neg
4,0.043117073,516,Pos
253,0.037353833,253,Neg
521,0.05366324,366,Neg
521,0.054729094,366,Neg
317,0.031040418,265.5,Neg
317,0.03427108,440,Neg
90,0.029745819,77,Pos
90,0.040464111,419,Pos
33,0.04897561,451,Neg
695,0.033675261,356.5,Neg
695,0.042414111,532,Neg
695,0.037702787,1460,Neg
559,0.027809582,98,Pos
56,0.035823868,259,Neg
811,0.044923519,84.5,Neg
811,0.040836063,287,Pos
196,0.037169686,282,Neg
196,0.053865157,4000,Neg
359,0.028349826,94.5,Neg
359,0.042155052,298,Neg
100,0.039143902,422,Neg
764,0.030491115,104.5,Pos
764,0.036705749,426,Pos
669,0.028559408,92,Pos
669,0.042163763,280,Pos
297,0.028658188,91.5,Pos
297,0.038996167,799,Pos
207,0.024137282,212.5,Pos
207,0.041345819,471,Pos
835,0.038783275,269.5,Neg
835,0.039457491,458,Neg
835,0.040020035,1825,Neg
472,0.025335366,98,Pos
472,0.058070209,289,Pos
274,0.030207143,206.5,Pos
274,0.04186777,403,Pos
274,0.025599652,206.5,Pos
274,0.043535366,403,Pos
22,0.027589547,80.5,Pos
22,0.039029965,255,Neg
22,0.04518223,2500,Neg
679,0.029500174,85.5,Pos
679,0.045858885,293,Neg
603,0.032273345,415.5,Pos
603,0.028848258,625,Pos
438,0.032180662,156,Pos
438,0.039858537,351,Neg
565,0.039438502,96.5,Pos
564,0.026607143,186,Pos
564,0.048023345,381,Neg
667,0.030010976,78,Pos
553,0.028255923,90.5,Neg
553,0.052350348,309,Neg
75,0.027937979,91.5,Neg
75,0.042420557,274,Neg
265,0.03024878,253,Pos
265,0.029622822,434,Neg
193,0.027783972,109,Pos
193,0.03874007,283,Pos
818,0.032143031,84.5,Pos
818,0.046759408,258,Neg
818,0.046601916,2500,Pos
427,0.027909233,101,Pos
427,0.039481882,290,Pos
767,0.039266202,84,Pos
767,0.041849652,265,Pos
84,0.029524913,87,Pos
84,0.03609878,283,Pos
84,0.039199129,1095,Neg
42,0.028929094,100,Pos
691,0.030785889,255,Neg
691,0.036512544,86.5,Pos
691,0.035471603,255,Neg
268,0.040618293,94,Neg
268,0.045518467,274,Neg
268,0.045215505,94,Neg
268,0.039156446,274,Neg
704,0.029968815,179,Pos
704,0.039189373,523,Pos
785,0.035352787,112,Pos
785,0.042238328,281,Pos
509,0.032170209,454,Pos
509,0.035958188,944,Pos
532,0.032875958,395.5,Pos
532,0.041398084,1206,Pos
182,0.063621951,340.5,Neg
155,0.039058014,396,Neg
231,0.049140592,125.5,Neg
797,0.028355226,329,Neg
797,0.043909582,811,Pos
73,0.040794425,483,Pos
73,0.041904007,713,Pos
530,0.031278049,103,Neg

• Most importantly, which of the two models fits your understanding of the problem? – usεr11852 Jun 19 '19 at 12:20
• From your description, GLMM is probably the better solution. But, what exactly is your outcome measure? Proportion? Percentage? Assuming that the outcome is bound between 0 and 1, and say is a proportion, consider a glmer with weights or a beta regression. Refer to these two questions and answers: stats.stackexchange.com/questions/87956/… stats.stackexchange.com/questions/189115/… – user139190 Jun 19 '19 at 13:31
• Thanks Michael. The response variable is a measure of drug efficacy, but it is weighted. According to the 2nd link you suggested, it was illustrated that the weighted model gives identical results to the adjusted model model accounting for the count number (insect count). If it gives identical results though, what is the point? In my case, since the unweighted measure in the response variable is not an integer, it throws an error with the 'family = "binomial"' – user250071 Jun 19 '19 at 21:10

If you actually test for normality in residuals, it is rejected for GLMM:

library(fBasics)
jarqueberaTest(resid(fitLMM))
jarqueberaTest(resid(fitGLMM))

shapiroTest(resid(fitLMM))
shapiroTest(resid(fitGLMM))


But before of that you need to evaluate the singularity problem in fitLMM. I got

> isSingular(fitLMM)
[1] TRUE


Apparently it is originated by sample effects that have zero deviation in your testdata.

• Interesting, thanks Robert. As per the singularity test, is this from the dataMaid package? I just ran the command you posted and it does not return true, but rather, an error. Care to elaborate on that one? – user250071 Jun 19 '19 at 17:19
• I needed to update lme4, now it works – user250071 Jun 19 '19 at 17:43
• Ok, so to avoid the singularity issue, I removed all instances of samples that had single values (I thought LMM could handle this?). This solved that issue. It also solved the normality of residuals issue as the p value for the jarqueberaTest is now ~0.6. To answer the actual question then: does all of this suggest I should use a GLMM? Was this the best way to determine this or is there an easier way? – user250071 Jun 19 '19 at 17:49
• To clarify: in the above instances of solving the singularity/residuals issue it worked only with the GLMM, not LMM.... – user250071 Jun 19 '19 at 21:29
• Clarification #2 (!!): I have never used the jarqueberaTest. I misinterpreted the null hypothesis. A p val < 0.05 rejects normality which means that the GLMM is not normally distributed. GLMMs are built for non-normal data, suggesting GLMMs are my method of choice I guess... – user250071 Jun 19 '19 at 22:07