Lesson 5 learning objectives

At the end of this lesson, students will …

• Know what a generalized linear mixed model is and why you might want to use one.
• Fit a generalized linear mixed model to binary outcome data.
• Fit a generalized linear mixed model to count outcome data.

GLMMs

• Taking the log transformation of the response variable won’t always work
• The next step is to enter the world of generalized linear mixed models (GLMMs)
• General linear mixed model and generalized linear mixed model begin with the same letters, so it’s a little bit confusing!

Load needed packages and dataset (simulated)

library(tidyverse)
library(lme4)
library(easystats)

disease <- read_csv('https://usda-ree-ars.github.io/SEAStats/mixed_models_in_R/datasets/disease.csv')

Summarize the data

• How many animals in each pen and treatment got the disease?
• Use group_by() and summarize() to make a table

disease %>%
  group_by(inoc, pen) %>%
  summarize(no_disease = sum(disease == 0), disease = sum(disease == 1))

• What if we tried to just use a linear mixed model as we have done before?
• We code “disease absent” as 0 and “disease present” as 1 to make it numeric

lmm_disease <- lmer(disease ~ inoc + (1|pen), data = disease)

• Seems reasonable at first glance
• The intercept is 0.78, so the baseline rate is 78% disease
• The coefficient on inoculation is -0.64, so inoculation treatment makes disease go down to 14%
• What is wrong with that?

• Let’s say we had a population where the baseline rate of disease was only 40%.
• Our model would predict -24% disease when inoculated
• It makes more sense to think about relative changes to odds.
• If the baseline rate is very low, it can’t be reduced much more
• If the baseline rate is high, it’s easier to reduce

Diagnostics of linear model

check_model(lmm_disease)

• Residuals of this model do not really meet assumptions
• Not normally distributed with homogeneous variance
• Predicted values of the model go well outside the 0 to 1 range
• Very poor match between the data (green line) and predictions (blue lines)

Solving the problem

• Model binary response with a generalized linear mode
• Use a link function to convert the predicted response value into a scale that can be normally distributed
• Predict the probability that an individual from each pen with or without the inoculation treatment will get the disease
• Probability ranges from 0 to 1, so it cannot really have normally distributed error

What is that “logit” thing???

\[\text{logit}~\hat{y}_{i,j} = \beta_0 + \beta_1 inoc_i + u_j\]

• Logit, or log odds, is the link function

\[\text{logit}~p = \log \frac {p}{1-p}\]

• Transforms the probability (ranges from 0 to 1) to log-odds scale (can take any value, + or -)
• R function is qlogis()
• Inverse is plogis() (converts log odds back to probability)

Graph of the logit function

• Maps the values from 0 to 1 to negative to positive infinity.
• More or less a straight line between about 0.25 and 0.75
• Starts to get steep as you get close to the boundaries

Fit the logit model

• Binomial (aka logistic) GLMM with a logit link function
• Binomial means response variable can have two values (0 and 1, or no and yes)
• Now we use glmer() instead of lmer()
• Same model formula as with lmer()
• New argument, family, refers to the “family” of error distributions
• binomial(link = 'logit') used here, other links are possible

glmm_disease <- glmer(disease ~ inoc + (1|pen), data = disease, family = binomial(link = 'logit'))

Diagnostics of GLMM

check_model(glmm_disease)

• Looks better, especially the predictive check
• Model only predicts values between 0 and 1
• Makes logical sense and matches the observations

Exploring model summary

summary(glmm_disease)

• Intercept is 1.39 and treatment coefficient is -3.37. What does that mean???
• The coefficients are on the log odds scale and need to be back-transformed to probability scale

Back-transforming intercept

• 1.39 is the fitted log odds of disease in uninoculated control, averaged across the random effect of pen
• Transform this log odds back to the probability scale ranging from 0 to 1
• Use the inverse logit function, plogis()

plogis(1.39)

[1] 0.8005922

• Population-level prediction: a random individual who isn’t inoculated has ~80% probability of disease

Back-transforming slope

• -3.37 is the change in the log odds between uninoculated and inoculated
• Add intercept and slope, then take the inverse logit

plogis(1.39 - 3.37)

[1] 0.1213188

• Population-level prediction: random uninoculated individual has ~12% probability of disease
• We will not have to do these predictions “by hand” – wait for next lesson!

Effects in GLMMs

• GLMM can have all the same kinds of effects as “plain” LMMs
  • Continuous and categorical fixed effect predictors
  • Interaction effects
  • Crossed and nested random effects
  • Random intercepts and random slopes

GLMM with count response variable

• Response variable can only be a non-negative integer
• Discrete values, instead of continuous like a normal distribution
• Model cannot predict negative counts
• We can use GLMM with Poisson distribution with a log link function
• Poisson distribution is bounded over non-negative integers
• Ideal for count data (as long as there are not too many 0 values)

• 15 experimental blocks (block column), each containing four plots
• Number of borers per plot was counted on two different dates
• We will only consider the latest count (count2 column)

Load the dataset

data('stirret.borers', package = 'agridat')

If you are running R locally and didn’t/can’t install the agridat package, I included the CSV in the example datasets

stirret.borers <- read_csv('https://usda-ree-ars.github.io/SEAStats/mixed_models_in_R/datasets/stirret.borers.csv')

Examine the data

Histogram of the data by treatment, with means as a line (don’t worry about the code for now)

borer_means <- stirret.borers %>%
  group_by(trt) %>%
  summarize(count2_mean = mean(count2))

ggplot(stirret.borers, aes(x = count2)) +
  geom_histogram(bins = 10) +
  geom_vline(aes(xintercept = count2_mean), data = borer_means, color = 'red') +
  facet_wrap(~ trt) +
  theme_bw()

• Quite a few counts of zero or close to it
• “Both” and “Late” treatments have the lowest mean counts, then “Early”, then “None”

• Rearrange the factor levels of the trt column so control level 'None' is first
• It will be the reference level in the model, for easier interpretation

stirret.borers <- stirret.borers %>%
  mutate(trt = factor(trt, levels = c('None', 'Early', 'Late', 'Both')))

Fit as general linear model

• For proof of concept

lmm_borers <- lmer(count2 ~ trt + (1|block), data = stirret.borers)

Fit as generalized linear model

• Use family = poisson(link = 'log')
• Log is the link function that works best with the Poisson distribution

glmm_borers <- glmer(count2 ~ trt + (1|block), data = stirret.borers, family = poisson(link = 'log'))

Diagnostic plots

Compare LMM and Poisson GLMM diagnostics

check_model(lmm_borers)
check_model(glmm_borers)

• Residual diagnostics actually look acceptable for both models
• The count data are really not that far off from normal
• But LMM produces a lot of negative predictions
• GLMM allows us to make more meaningful predictions

Coefficients of GLMM

summary(glmm_borers)

• Back-transform coefficient estimates from the link function scale back to the original scale of the data
• Coefficients on log scale so we use exponentiation (exp()) as the inverse

• Intercept (3.42) is the estimated mean for the “None” treatment (control group) on a log scale

exp(3.42)

[1] 30.56942

• For other treatment groups, add coefficients to intercept and then exponentiate
• For example, “Early” treatment coefficient was -0.24

exp(3.42 - 0.24)

[1] 24.04675