Predicting probabilities in R with mixed effects model - r

Question

I have defined a binary response mixed effects model using the R function glmer as follows:
fit <-glmer(binary_r ~ cat1 + (1 | SUBJECTIDf) + (1 | cat2) + (1 | cat1:cat2),
family = binomial("logit"), data = mydata))
where cat1 and cat2 are categorical variables and SUBJECTIDf denotes the factor variable tagging the individual subjects of the study. Also, SUBJECTIDf and cat2 are cross-classified factors.
I would like to do the following for the above model:
create a table which presents the probability of a positive response for each combination of categories pertaining to cat1 and cat2;
create a plot (possibly a caterpillar plot) which displays the probabilities defined under a);
and
create summary statistics for the probabilities defined under 1., including the minimum and maximum probabilities, across all combinations of categories pertaining to cat1 and cat2.
I am assuming that in order to achieve the above outcomes, it would be appropriate to assume that the individual probabilities defined under 1., above are formed as averages or medians across all subjects for a given combination of cat1 and cat2.
I would be most grateful for advice on how to proceed.

Answer 1

Since you didn't give a reproducible example I'm going to simulate one ... this part of the answer is only setting up an example data set.
## crossed cat2 and SUBJECTIDf
dd <- expand.grid(cat2=factor(letters[1:10]),
SUBJECTIDf=factor(1:10))
## each subject gets one value of cat1:
## for example, half get A while half get B
cat_tab <- data.frame(SUBJECTIDf=factor(1:10),cat1=rep(c("A","B"),5))
dd <- merge(dd,cat_tab)
I'm having a little trouble accommodating the design consideration that cat1 and cat2 are not cross-classified. I'll just knock out some categories:
dd <- with(dd,dd[!(cat1=="A" & cat2 %in% c("a","b","c")),])
Now we have the design set up, we simulate response values:
library(lme4)
form <- binary_r ~ cat1 + (1 | SUBJECTIDf) + (1 | cat2) + (1 | cat1:cat2)
dd$binary_r <- simulate(form[-2], ## RHS only
family=binomial,
newdata=dd,
newparams=list(beta=0:1,
theta=c(2,4,1)),
seed=101)[[1]]
At this point we come in with the model fit you've suggested above.
fit <- glmer(form, family = binomial, data=dd)
create a table which presents the probability of a positive response for each combination of categories pertaining to cat1 and cat2;
By setting re.form below to exclude subject ID, we are implicitly computing the value for the hypothetical median individual (i.e., random effect set to zero; mean and median predictions coincide on the logit scale, but not once we back-transform to the probability scale).
Get unique combinations of cat1 and cat2 found in the data:
newdd <- unique(dd[,c("cat1","cat2")])
newdd$SUBJECTIDf <- NA ## need to have SUBJECTIDf in the data frame ...
t1 <- predict(fit,newdata=newdd,type="response",
re.form=~(1|cat2)+(1|cat1:cat2))
newdd <- data.frame(newdd[,c("cat1","cat2")],pred=t1)
head(newdd)
## cat1 cat2 pred
## 4 A d 0.215336024
## 5 A e 0.944897414
## 6 A f 0.036751551
## 7 A g 0.003819873
## 8 A h 0.970115614
## 9 A i 0.003819873
We could also compute predictions for all individuals as follows:
## we happen to have a factorial design, but expand.grid() would
## e.g. fill in missing values
newdd2 <- unique(dd[,c("cat1","cat2","SUBJECTIDf")])
t2 <- predict(fit,newdata=newdd2,type="response",
re.form=NULL)
newdd2$pred <- t2
head(newdd2)
In order to create the plot (see below) we would have to summarize across individuals within category combinations.
create a plot (possibly a caterpillar plot) which displays the probabilities defined under a);
library(ggplot2); theme_set(theme_bw())
ggplot(newdd,aes(cat2,pred,colour=cat1))+
geom_point()+scale_colour_brewer(palette="Set1")
Or, aggregating the subject-specific predictions:
ggplot(newdd2,aes(cat2,pred,colour=cat1))+
stat_summary(fun.y=mean,geom="point")+
scale_colour_brewer(palette="Set1")
We could use reorder() on the cat2 categories to try to get a more sensible order, but since there is a cat1:cat2 interaction, that might not work too well. Caterpillar plots (i.e. getting uncertainties on predictions) are a little trickier, because of the difficulty of getting uncertainty on predictions that combine uncertainty in conditional modes (values of individual random effects) and fixed effects. Can be done by (1) assuming conditional modes and fixed effects are independent or (2) parametric bootstrapping (bootMer), but both are a little more trouble than I'm willing to take at the moment ...
create summary statistics for the probabilities defined under 1., including the minimum and maximum probabilities, across all combinations of categories pertaining to cat1 and cat2.
This doesn't really make sense to me unless we are going the disaggregated route. If we have predicted for each combination of cat1 and cat2, then we only have a single value for each combination (i.e., no "min/max" probabilities). Aggregating is easy in base R, e.g.
aggregate(pred~cat1:cat2,data=newdd2,
FUN=function(x) c(min=min(x),max=max(x)))
or in the tidyverse:
library(dplyr)
newdd2 %>% group_by(cat1,cat2) %>%
summarise(min=min(pred),max=max(pred))