R for SAS users: Lesson 3

What is this workshop?

• For SAS users who want to learn R
• SAS code is provided for comparison purposes
• Lesson 3 in a series

IMPORTANT NOTE: In this lesson, the numerical results of the R and SAS code may no longer be identical. This is because of details in model specifications and differences in the fitting algorithms between R and SAS.

Conceptual learning objectives

During this workshop, you will …

• Learn what generalized linear mixed models (GLMMs) are
• Learn more about different predictions you can make from models
• Learn about how to deal with more complex covariance structures

Practical skills

We will …

• Import the data from a CSV file
• Clean and reshape the data
• Calculate some summary statistics and make some exploratory plots
• Fit a generalized linear mixed-effects model with repeated measures error structure
• Make plots and tables of results

How to follow along with this workshop

• Slides and text version of lessons are online
  • usda-ree-ars.github.io/SEAStats
• Fill in R code in the worksheet (replace ... with code)
• You can always copy and paste code from text version of lesson if you fall behind

Load R packages

• glmmTMB is a sophisticated GLMM fitting package that is more flexible than lme4
• DHARMa lets us make residual diagnostic plots and tests for certain kinds of GLMM

library(tidyverse)
library(lme4)
library(easystats)
library(emmeans)
library(multcomp)
library(glmmTMB)
library(DHARMa)

theme_set(theme_bw())

CBPP dataset

Import data

SAS

filename csvFile url "https://usda-ree-ars.github.io/SEAStats/R_for_SAS_users/datasets/cbpp.csv";
proc import datafile = csvFile out = cbpp replace dbms = csv; run;

data cbpp; set cbpp;
    rate = incidence/size;
run;

R

data(cbpp)

Examine the data

glimpse(cbpp)

Exploratory plots

• Plot the proportion incidence/size for each herd at each time period

SAS

proc sgplot data = cbpp noautolegend;
    styleattrs datacontrastcolors = (black) datalinepatterns = (solid);
    series x = period y = rate / group = herd;
run;

R

ggplot(cbpp, aes(x = period, y = incidence/size, group = herd)) +
  geom_line() +
  theme_bw()

Fit model

• Measurements from different time periods from the same herd are not independent
• We must account for this with a linear mixed model
  • Response variable is the proportion of cattle suffering from CBPP, incidence/size,
  • Fixed effect is the discrete time period, period,
  • Random intercept for each herd, (1 | herd)

lmm_cbpp <- lmer(incidence/size ~ period + (1 | herd), data = cbpp)

• Model diagnostic plots look reasonably good for a small dataset

check_model(lmm_cbpp)

• Modeled estimates of the mean and 95% confidence interval of proportion CBPP for each period.

emmeans(lmm_cbpp, ~ period)

• What is wrong here?

• We cannot assume normally distributed error around mean proportions
• We have to use a generalized linear mixed model (GLMM)

• GLMM uses a link function to convert response variable to a scale where its error is roughly normal
• Once we’ve transformed the response to that scale, we can treat it just like a linear mixed model
• Afterward, we “undo” the transformation to interpret model means and confidence intervals on the original scale

GLMM for binary response variable

• Here the response is binary (0 = animal is healthy, 1 = animal has CBPP)
• We want to predict the probability a randomly chosen individual animal from a given herd at a given time will have CBPP
• Probability ranges from 0 to 1, so it can’t be normal (bell curve has no upper and lower bound on the \(x\) axis)

Formula for predicting the probability of disease for individual \(i\) at time period \(j\) in herd \(k\)

\[\text{logit}~\hat{y}_{i,j,k} = \beta_0 + P_j + u_k\]

• \(\hat{y}_{i,j,k}\): predicted probability of disease for individual \(i\) at time period \(j\) in herd \(k\)
• \(\beta_0\): intercept
• \(P_j\): fixed effect of time period \(j\)
• \(u_k\): random intercept for each herd

What is “logit?”

• That’s the link function!
• Transforms the probability, which ranges from 0 to 1, to a scale that can take any value, positive or negative
• logit function transforms from the probability scale to the log-odds scale:

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

Fit binomial GLMM with logit link: SAS

proc glimmix data = cbpp;
    class herd period;
    model incidence/size = period / dist = binomial;
    random herd;
    weight size;
    lsmeans period / ilink cl diff oddsratio adjust = tukey;
run;

Fit binomial GLMM with logit link: R

• Use glmer() instead of lmer(), same formula as before
• Now we use family argument, for the “family” of response distributions we are using
• Logit link is the default but to be explicit we specify binomial(link = 'logit')
• weights = size accounts for the fact that the herds are different sizes; bigger herd = bigger contribution to the estimate

glmm_cbpp <- glmer(incidence/size ~ period + (1 | herd), weights = size, data = cbpp, family = binomial(link = 'logit'))

Model diagnostics

• The default linear mixed model diagnostics from check_model() are not relevant here but we still need to check for issues
• R package DHARMa simulates residuals and tests them
  • deviation from normality
  • overdispersion
  • outliers
  • homogeneity of variance

sim_resid_cbpp <- simulateResiduals(glmm_cbpp)
plot(sim_resid_cbpp)

Estimated marginal means of CBPP probability

• We use the argument type = 'response'
  • Uses the inverse link function to back-transform the means and 95% CI from the log-odds scale to the probability scale
  • In SAS the option is ilink (inverse link) in lsmeans and slice statements

emmeans(glmm_cbpp, pairwise ~ period, type = 'response', adjust = 'tukey')

Repeated measures

• Same plots were measured three times
• We cannot consider the three time points to be independent, we must model the covariance between them
• “Repeated measures” a.k.a. “split plot in time”

Import the data

SAS

filename csvFile2 url "https://usda-ree-ars.github.io/SEAStats/R_for_SAS_users/datasets/fungicide.csv";
proc import datafile = csvFile2 out = fungicide replace dbms = csv; run;

R

If you are using cloud server:

fungicide <- read_csv('data/fungicide.csv')

If you are running the code locally:

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

• Change some columns to factors, with control set as first reference level for the fungicide treatment column

fungicide <- fungicide %>%
  mutate(
    block = factor(block),
    plot = factor(plot),
    time = factor(time),
    fungicide = relevel(factor(fungicide), ref = 'None')
  )

Exploratory plots

• Plot raw data points and medians as semi-transparent horizontal bars
• Colorblind-friendly color palette (recommended)
• Change over time? Differences from control? Differences between treatments?

ggplot(fungicide, aes(x = time, color = fungicide, group = fungicide, y = n_healthy)) +
  geom_point(size = 2, position = position_dodge(width = .5)) +
  stat_summary(fun = 'median', geom = 'point', shape = 95, size = 10, position = position_dodge(width = .5), alpha = .5) +
  theme_bw() +
  scale_color_okabeito(order = c(9, 1, 2, 3))

• Plot time trends colored by block
• Use facet_wrap() to split treatments into different panels
• Differences between blocks?

ggplot(fungicide, aes(x = time, color = block, y = n_healthy)) +
  geom_line(aes(group = block)) +
  geom_point(size = 2) +
  facet_wrap(~ fungicide) +
  theme_bw() +
  scale_color_okabeito()

Accounting for repeated measures: 3 different covariance structures

• unstructured: separately estimated covariance parameter for each pair of time points
• compound symmetry a.k.a. CS: fixes the covariance parameter for each pair of time points at a single value
• first-order autoregressive a.k.a. AR(1): relates each time point to the one preceding it, assuming the same covariance between each pair of consecutive time points
• CS and AR(1) are less flexible but have fewer parameters; this can help avoid overfitting

Fitting the models

• Unstructured: us(0 + time | fungicide:block)
• Compound symmetry: cs(0 + time | fungicide:block)
• AR(1): ar1(0 + time | fungicide:block)

0 + time because we are only fitting the residual covariance for the time slope, not the intercept.

Fitting the models

fit_us <- glmmTMB(
  cbind(n_healthy, n_plants - n_healthy) ~ fungicide + time + fungicide:time + (1 | block) + us(0 + time | fungicide:block), 
  data = fungicide, family = binomial
)

fit_cs <- glmmTMB(
  cbind(n_healthy, n_plants - n_healthy) ~ fungicide + time + fungicide:time + (1 | block) + cs(0 + time | fungicide:block), 
  data = fungicide, family = binomial
)

fit_ar1 <- glmmTMB(
  cbind(n_healthy, n_plants - n_healthy) ~ fungicide + time + fungicide:time + (1 | block) + ar1(0 + time | fungicide:block), 
  data = fungicide, family = binomial
)

How did the models turn out?

• Unstructured covariance model does not even converge (too many parameters for small dataset)
• CS and AR1 converged
• Check diagnostics using simulated residuals

sim_resid_csmodel <- simulateResiduals(fit_cs)
plot(sim_resid_csmodel)

sim_resid_ar1model <- simulateResiduals(fit_ar1)
plot(sim_resid_ar1model)

AIC comparison of CS and AR1 models

• AIC measures how well the model fits the data with a penalty for having too many parameters (lower is better)
• Difference is very small; we will go with the AR1 model moving forward

AIC(fit_cs, fit_ar1)

Estimated marginal means

• Mean disease probability by fungicide treatment averaged across times, with pairwise contrasts

emm_fung <- emmeans(
  fit_ar1, pairwise ~ fungicide, 
  type = 'response', adjust = 'tukey'
)

• Mean disease probability by fungicide treatment within each time point, with pairwise contrasts

emm_fung_by_time <- emmeans(
  fit_ar1, pairwise ~ fungicide | time, 
  type = 'response', adjust = 'tukey'
)

Plots to visualize model predictions

• cld() summarizes multiple comparisons with letters
• Annotate means and 95% confidence interval error bars with letters
• Multiple comparisons are Tukey-adjusted, 95% CIs are Sidak-adjusted

cld_fung <- cld(emm_fung$emmeans, adjust = 'tukey', Letters = letters)
cld_fung_by_time <- cld(emm_fung_by_time$emmeans, adjust = 'tukey', Letters = letters)

plot_data <- bind_rows(as_tibble(cld_fung), as_tibble(cld_fung_by_time)) %>%
  mutate(time = fct_na_value_to_level(time, 'overall') %>%
           factor(labels = c(paste(c(12, 20, 42), 'days after planting'), 'overall')))

Fit models in SAS

• Similar but not identical models
• method = laplace, a maximum-likelihood method, specified to obtain AIC values

/* Unstructured covariance */
proc glimmix data = fungicide method = laplace;
    class block fungicide time;
    model n_healthy/n_plants = fungicide time fungicide*time;
    random intercept / subject = block;
    random time / subject = fungicide(block) type = un;
    lsmeans fungicide / ilink cl diff lines adjust = tukey;
    slice fungicide*time / sliceby = time ilink cl diff lines adjust = tukey;
run;

/* Compound symmetry */
proc glimmix data = fungicide method = laplace;
    class block fungicide time;
    model n_healthy/n_plants = fungicide time fungicide*time;
    random intercept / subject = block;
    random time / subject = fungicide(block) type = cs;
    lsmeans fungicide / ilink cl diff lines adjust = tukey;
    slice fungicide*time / sliceby = time ilink cl diff lines adjust = tukey;
run;

/* AR1 */
proc glimmix data = fungicide method = laplace;
    class block fungicide time;
    model n_healthy/n_plants = fungicide time fungicide*time;
    random intercept / subject = block;
    random time / subject = fungicide(block) type = ar(1);
    lsmeans fungicide / ilink cl diff lines adjust = tukey;
    slice fungicide*time / sliceby = time ilink cl diff lines adjust = tukey;
run;

Conclusion

What have you learned in this lesson?

• What GLMMs are and how link functions work
• How to fit GLMMs to repeated measures binomial data
• How to fit GLMMs with different error covariance structures
• How to compare means from GLMMs and plot them

Nice job! Pat yourselves on the back once again!