Illustrating Model Effects
As the complexity of the model fit grows, it becomes harder and harder to interpret the substantive effect of parameters in the model.
Let’s start with a medium-sized example model using the InstEval data provided by the lme4 package. These data represent university lecture evaluations at ETH Zurich made by students. In this data, s is an individual student, d is an individual lecturer, studage is the semester the student is enrolled, lectage is how many semesters back the lecture with the rating took place, dept is the department of the lecture, and y is an integer 1:5 representing the ratings of the lecture from “poor” to “very good”:
library(lme4)
head(InstEval)
#> s d studage lectage service dept y
#> 1 1 1002 2 2 0 2 5
#> 2 1 1050 2 1 1 6 2
#> 3 1 1582 2 2 0 2 5
#> 4 1 2050 2 2 1 3 3
#> 5 2 115 2 1 0 5 2
#> 6 2 756 2 1 0 5 4
str(InstEval)
#> 'data.frame': 73421 obs. of 7 variables:
#> $ s : Factor w/ 2972 levels "1","2","3","4",..: 1 1 1 1 2 2 3 3 3 3 ...
#> $ d : Factor w/ 1128 levels "1","6","7","8",..: 525 560 832 1068 62 406 3 6 19 75 ...
#> $ studage: Ord.factor w/ 4 levels "2"<"4"<"6"<"8": 1 1 1 1 1 1 1 1 1 1 ...
#> $ lectage: Ord.factor w/ 6 levels "1"<"2"<"3"<"4"<..: 2 1 2 2 1 1 1 1 1 1 ...
#> $ service: Factor w/ 2 levels "0","1": 1 2 1 2 1 1 2 1 1 1 ...
#> $ dept : Factor w/ 14 levels "15","5","10",..: 14 5 14 12 2 2 13 3 3 3 ...
#> $ y : int 5 2 5 3 2 4 4 5 5 4 ...
Starting with a simple model:
After fitting the model we can make use of the first function provided by merTools, fastdisp which modifies the function arm:::display to more quickly display a summary of the model without calculating the model sigma:
We see some interesting effects. First, our decision to include student and lecturer effects seems justified as there is substantial variance within these groups. Second, there do appear to be some effects by age and for lectures given as a service by an outside lecturer. Let’s look at these in more detail. One way to do this would be to plot the coefficients together in a line to see which deviate from 0 and in what direction. To get a confidence interval for our fixed effect coefficients we have a number of options that represent a tradeoff between coverage and computation time – see confint.merMod for details.
An alternative is to simulate values of the fixed effects from the posterior using the function arm::sim. Our next tool, FEsim, is a convenience wrapper to do this and provide an informative data frame of the results.
We can present these results graphically, using ggplot2:
However, an easier option is:
Random Effects
Next, we might be interested in exploring the random effects. Again, we create a dataframe of the values of the simulation of these effects for the individual levels.
The result is a dataframe with estimates of the values of each of the random effects provided by the arm::sim() function. groupID represents the identfiable level for the variable for one random effect, term represents whether the simulated values are for an intercept or which slope, and groupFctr identifies which of the (1|x) terms the values represent. To make unique identifiers for each term, we need to use both the groupID and the groupFctr term in case these two variables use overlapping label names for their groups. In this case:
Most important is producing caterpillar or dotplots of these terms to explore their variation. This is easily accomplished with the dotplot function:
However, these graphics do not provide much control over the results. Instead, we can use the plotREsim function in merTools to gain more control over plotting of the random effect simulations.
The result is a ggplot2 object which can be modified however the user sees fit. Here, we’ve established that most student and professor effects are indistinguishable from zero, but there do exist extreme outliers with both high and low averages that need to be accounted for.
Subtantive Effects
A logical next line of questioning is to see how much of the variation in a rating can be caused by changing the student rater and how much is due to the fixed effects we identified above. This is a very difficult problem to solve, but using simulation we can examine the model behavior under a range of scenarios to understand how the model is reflecting changes in the data. To do this, we use another set of functions available in merTools.
The simplest option is to pick an observation at random and then modify its values deliberately to see how the prediction changes in response. merTools makes this task very simple:
The draw function takes a random observation from the data in the model and extracts it as a dataframe. We can now do a number of operations to this observation:
More interesting, let’s programatically modify this observation to see how the predicted value changes if we hold everything but one variable constant.
The function wiggle allows us to create a new dataframe with copies of the variable that modify just one value. Chaining together wiggle calls, we can see how the variable behaves under a number of different scenarios simultaneously.
The result allows us to graphically display the effect of each level of lectage on an observation that is otherwise identical. This is plotted here against a horizontal line representing the mean of the observed ratings, and two finer lines showing plus or minus one standard deviation of the mean.
This is nice, but selecting a random observation is not very satisfying as it may not be very meaningful. To address this, we can instead take the average observation:
Here, the average observation is identified based on either the modal observation for factors or on the mean for numeric variables. Then, the random effect terms are set to the level equivalent to the median effect – very close to 0.
example3 <- wiggle(example1, varlist = "service",
valueslist = list(c("0", "1")))
example3$yhat <- predict(m1, newdata = example3)
ggplot(example3, aes(x = service, y = yhat)) + geom_line(aes(group = 1)) +
theme_bw() + ylim(c(1, 5)) +
geom_hline(yintercept = mean(InstEval$y), linetype = 2) +
geom_hline(yintercept = mean(InstEval$y) + sd(InstEval$y), linetype = 3) +
geom_hline(yintercept = mean(InstEval$y) - sd(InstEval$y), linetype = 3)
Here we can see that for the average observation, whether the lecture is outside of the home department has a very slight negative effect on the overall rating. Might the individual professor or student have more of an impact on the overall rating? To answer this question we need to wiggle the same observation across a wide range of student or lecturer effects.
How do we identify this range? merTools provides the REquantile function which helps to identify which levels of the grouping terms correspond to which quantile of the magnitude of the random effects:
Here we can see that group level 446 corresponds to the 25th percentile of the effect for the student groups, and level REquantile(m1, quantile = 0.25, groupFctr = "d") corresponds to the 25th percentile for the instructor group. Using this information we can reassign a specific observation to varying magnitudes of grouping term effects to see how much they might influence our final prediction.
example4 <- draw(m1, type = 'average')
example4 <- wiggle(example4, varlist = "s",
list(REquantile(m1, quantile = seq(0.1, 0.9, .1),
groupFctr = "s")))
example4$yhat <- predict(m1, newdata = example4)
ggplot(example4, aes(x = reorder(s, -yhat), y = yhat)) +
geom_line(aes(group = 1)) +
theme_bw() + ylim(c(1, 5)) +
geom_hline(yintercept = mean(InstEval$y), linetype = 2) +
geom_hline(yintercept = mean(InstEval$y) + sd(InstEval$y), linetype = 3) +
geom_hline(yintercept = mean(InstEval$y) - sd(InstEval$y), linetype = 3)
This figure is very interesting because it shows that moving across the range of student effects can have a larger impact on the score than the fixed effects we observed above. That is, getting a “generous” or a “stingy” rater can have a substantial impact on the final rating.
But, we can do even better. First, we can move beyond the average observation by taking advantage of the varList option to the function which allows us to specify a subset of the data to compute an average for.
Now we have the average observation with a student age of 2 and a lecture age of 4. We can then follow the same procedure as before to explore the effects on our subsamples. Before we do that, let’s fit a slightly more complex model that includes a random slope.
data(VerbAgg)
m2 <- glmer(r2 ~ Anger + Gender + btype + situ +
(1|id) + (1 + Gender|item), family = binomial,
data = VerbAgg)
example6 <- draw(m2, type = 'average', varList = list("id" = "149"))
example6$btype <- "scold"
example6$situ <- "self"
tempdf <- wiggle(example6, varlist = "Gender", list(c("M", "F")))
tempdf <- wiggle(tempdf, varlist = "item",
list(unique(VerbAgg$item)))
tempdf$yhat <- predict(m2, newdata = tempdf, type = "response",
allow.new.levels = TRUE)
ggplot(tempdf, aes(x = item, y = yhat, group = Gender)) +
geom_line(aes(color = Gender))+
theme_bw() + ylim(c(0, 1)) +
theme(axis.text.x = element_text(angle = 20, hjust=1),
legend.position = "bottom") + labs(x = "Item", y = "Probability")
Here we’ve shown that the effect of both the intercept and the gender slope on item simultaneously affect our predicted value. This results in the two lines for predicted values across the items not being parallel. While we can see this by looking at the results of the summary of the model object, using fastdisp in the merTools package for larger models, it is not intuitive what that effect looks like across different scenarios. merTools has given us the machinery to investigate this.
Uncertainty
The above examples make use of simulation to show the model behavior after changing some values in a dataset. However, until now, we’ve focused on using point estimates to represent these changes. The use of predicted point estimates without incorporating any uncertainty can lead to overconfidence in the precision of the model.
In the predictInterval function, discussed in more detail in another package vignette, we provide a way to incorporate three out of the four types of uncertainty inherent in a model. These are:
- Overall model uncertainty
- Uncertainty in fixed effect values
- Uncertainty in random effect values
- Uncertainty in the distribution of the random effects
1-3 are incorporated in the results of predictInterval, while capturing 4 would require making use of the bootMer function – options discussed in greater detail elsewhere. The main advantage of predictInterval is that it is fast. By leveraging the power of the arm::sim() function, we are able to generate prediction intervals for individual observations from very large models very quickly. And, it works a lot like predict:
exampPreds <- predictInterval(m2, newdata = tempdf,
type = "probability", level = 0.8)
tempdf <- cbind(tempdf, exampPreds)
ggplot(tempdf, aes(x = item, y = fit, ymin = lwr, ymax = upr,
group = Gender)) +
geom_ribbon(aes(fill = Gender), alpha = I(0.2), color = I("black"))+
theme_bw() + ylim(c(0, 1)) +
theme(axis.text.x = element_text(angle = 20),
legend.position = "bottom")+ labs(x = "Item", y = "Probability")
Here we can see there is barely any gender difference in terms of area of potential prediction intervals. However, by default, this approach includes the residual variance of the model. If we instead focus just on the uncertainty of the random and fixed effects, we get:
exampPreds <- predictInterval(m2, newdata = tempdf,
type = "probability",
include.resid.var = FALSE, level = 0.8)
tempdf <- cbind(tempdf[, 1:8], exampPreds)
ggplot(tempdf, aes(x = item, y = fit, ymin = lwr, ymax = upr,
group = Gender)) +
geom_ribbon(aes(fill = Gender), alpha = I(0.2), color = I("black"))+
geom_line(aes(color = Gender)) +
theme_bw() + ylim(c(0, 1)) +
theme(axis.text.x = element_text(angle = 20),
legend.position = "bottom") + labs(x = "Item", y = "Probability")
Here, more difference emerges, but we see that the differences are not very precise.
