# 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? Jun 19, 2019 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, 2019 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"' Jun 19, 2019 at 21:10

I will just provide a counter-point to Robert's answer by building on the comment by User11852.

In fitting generalized linear models, the normality of residuals is not necessarily assumed. Often, in large samples, the residuals of a GLM will trend to normally distributed; however, standard residual analyses can produce false positive rejections of a correctly specified model. There are alternative kinds of residuals for GLMs, like deviance residuals and Anscombe residuals, that do tend to be normally distributed. Standard Pearson residuals, however, are often not normal. There are several StackExchange questions talking about this: example 1, example 2, example 3, and example 4.

In short, it is not appropriate to use the assumption of non-normal residuals as a basis for rejecting a GLM since it's not an assumption of the GLMs. You can check the assumptions for a gamma regression like you've run here.

The much more important aspect of model building is that you select a model that is appropriate to the data generation process you're modeling. The gamma distribution, for example, is appropriate when your data are continuous, restricted to only positive values, and when you expect the variance in your data to increase as the mean increases. Even if all the assumptions of a standard ordinary least squares regression are met, it doesn't mean that the model is appropriate to your data. For example, it looks like all of your data are positive, so a model that predicts observing negative values does not make sense because those predictions are meaningless and could (potentially) never be observed. Regardless of whether the assumptions are satisfied, the model is misspecified if it is predicting values that can't be observed. Many models are robust to assumption violations, and there are ways to adjust parameter estimates to address assumption violations (e.g., heteroskedastic-consistent standard errors, bootstrapping standard errors). Those kinds of corrections are important when you want to do inference on your model parameters, but a fundamental assumption of all linear models is that the model is correctly specified. You don't want to do inference on a model that is making non-sensical predictions.

Long story short, use whatever kind of (G)LM that is appropriate to your outcome data generation process and then tweak the model to make valid inferences (e.g., compensate for assumption violations).

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? Jun 19, 2019 at 17:19
• I needed to update lme4, now it works Jun 19, 2019 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? Jun 19, 2019 at 17:49
• To clarify: in the above instances of solving the singularity/residuals issue it worked only with the GLMM, not LMM.... Jun 19, 2019 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... Jun 19, 2019 at 22:07

You automatically assumed that random effects capture the correlation structure in the serial observations. The equal correlation pattern induced by random effects (compound symmetry) is unlikely to hold in serial data especially with a long time span. You would do well to consider serial correlation models not requiring random effects. A general form of such models is Markov semiparametric ordinal models which have the great advantages of allowing for very flexible correlation structure and not having to decide on how best to transform the response variable. Detailed case studies may be found here and in the longitudinal model chapter of RMS course notes.