Interpret Generalized linear mixed model output

Question

I've built a generalized linear mixed model due to non-normal data (no transformation will make it normal). I'm new to mixed models and I'm unsure how to report the output in a paper.

Here is the model, insignificant interactions have been removed. Alcohol = quantity of alcohol someone drinks per day over a 10 day period. Student is binary yes (1) and no (0).

glmer1 <- glmer(Alcohol ~ + Student + (1|Rep), data = df1, family = "Gamma")

Here is the output

Generalized linear mixed model fit by maximum likelihood (Laplace Approximation) ['glmerMod']
 Family: Gamma  ( inverse )
Formula: Alcohol ~ +Student + (1 | Rep)
   Data: data

     AIC      BIC   logLik deviance df.resid 
 -1016.2  -1008.8    522.6  -1020.2      146 

Scaled residuals: 
    Min      1Q  Median      3Q     Max 
-2.2007 -0.5077  0.2630  0.6525  3.0007 

Random effects:
 Groups   Name        Variance Std.Dev.
 Rep      (Intercept) 20.5009  4.3278  
 Residual              0.1652  0.3999  
Number of obs: 153, groups:  Rep, 4

Fixed effects:
            Estimate Std. Error t value Pr(>|z|)    
(Intercept)   51.200      5.811   7.204 2.00e-09 ***
Student           -2.841      4.408  -2.127     0.02    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Correlation of Fixed Effects:
    (Intr)
Student -0.299
> anova(glmer1)
Analysis of Variance Table
    npar Sum Sq Mean Sq F value
Student    1 0.2927  0.3227  1.7608

Sorry for dumping the entire code. How would I report this in a paper? Thanks in advance.

Answer 1

I've built a generalized linear mixed model due to non-normal data (no transformation will make it normal). I'm new to mixed models and I'm unsure how to report the output in a paper.

One of the first things to understand about any regression model is that it is not a requirement for the raw data to follow any particular distribution. The assumption of normality is only required in some situations and refers to the conditional distribution of the response (conditioned on the covariates) - that is, the model errors. To investigate this we would assess normality of the residuals.

Here is the model, insignificant interactions have been removed

Why did you remove them ? This is often a mistake if you have sound theoretical / clinical reasons for expecting an interaction to be present. Did you conduct a power analysis prior to collecting the data ? Was the study sufficiently powered to detect the effect size(s) of the interaction(s) (and other estimates) ?

Note for your model:

glmer(Alcohol ~ + Student + (1|Rep), data = df1, family = "Gamma")

you are specifying Rep as a grouping factor for random intercepts, however you do not have a sufficient number of these do to so. You are asking the software to estimate a variance for a variable from only 4 observations of it. You could go ahead with this model, but I would recommend that you also fit the model with Rep as a fixed effect and ascertain whether the two models are consistent.

As for reporting the results, a lot depends on your research question(s) but presumably it will centre on the fixed effect for Student which in the gamma model above shows 1 negative association between Student and the outcome (however this will be on the reciprocal scale since the canonical link function for a gamma model is $\frac{1}{\mu}$.

Given the above, I would start with the model:

lm(Alcohol ~ Student + Rep, data = df1)