In this vignette, we discuss emmeans's rudimentary capabilities for constructing prediction intervals.
Prediction is not the central purpose of the emmeans package. Even its name
refers to the idea of obtaining marginal averages of fitted values; and it is a
rare situation where one would want to make a prediction of the average of
several observations. We can certainly do that if it is truly desired,
but almost always, predictions should be based on the reference grid itself
(i.e., not the result of an emmeans() call),
inasmuch as a reference grid comprises combinations of model predictors.
A prediction interval requires an estimate of the error standard deviation,
because we need to account for both the uncertainty of our point predictions
and the uncertainty of outcomes centered on those estimates. By its current design,
we save the value (if any) returned by stats::sigma(object) when a reference grid is constructed for a model object. Not all models provide a sigma() method,
in which case an error is thrown if the error SD is not manually specified.
Also, in many cases, there may be a sigma() method, but it does not
return the appropriate value(s) in the context of the needed predictions.
(In an object returned by lme4::glmer(), for example,sigma() seems to always
returns 1.0.) Indeed, as will be seen in the example that follows,
one usually needs to construct a manual SD estimate when the model is a mixed-effects model.
So it is essentially always important to think very specifically about
whether we are using an appropriate value. You may check the value being
assumed by looking at the misc slot in the reference grid:
rg <- ref_grid(model)
rg@misc$sigma
Finally, sigma may be a vector, as long as it is conformable with the estimates
in the reference grid. This would be appropriate, for example, with a model
fitted by nlme::gls() with some kind of inhomogeneous error structure. It may take
some effort, as well as a clear understanding of the model and its structure, to
obtain suitable SD estimates. It was sugegsted to me that the function
easystats::get_variance() may be helpful – especially when working with
an unfamiliar model class. Personally, I prefer to make sure I understand the
structure of the model object and/or its summary to ensure I am not going astray.
To illustrate, consider the feedlot dataset provided with the package.
Here we have several herds of feeder cattle that are sent to feed lots and given
one of three diets. The weights of the cattle are measured at time of entry
(ewt) and at time of slaughter (swt). Different herds have possibly
different entry weights, based on breed and ranching practices, so we will
center each herd's ewt measurements, then use that as a covariate in a mixed
model:
feedlot = transform(feedlot, adj.ewt = ewt - predict(lm(ewt ~ herd)))
require(lme4)
feedlot.lmer <- lmer(swt ~ adj.ewt + diet + (1|herd), data = feedlot)
feedlot.rg <- ref_grid(feedlot.lmer, at = list(adj.ewt = 0))
summary(feedlot.rg) ## point predictions
## adj.ewt diet prediction SE df
## 0 Low 1029 25.5 12.0
## 0 Medium 998 26.4 13.7
## 0 High 1031 29.4 19.9
##
## Degrees-of-freedom method: kenward-roger
Now, as advised, let's look at the SDs involved in this model:
lme4::VarCorr(feedlot.lmer) ## for the model
## Groups Name Std.Dev.
## herd (Intercept) 77.087
## Residual 57.832
feedlot.rg@misc$sigma ## default in the ref. grid
## [1] 57.83221
So the residual SD will be assumed in our prediction intervals
if we don't specify something else. And we do want something else,
because in order to predict the slaughter weight of an arbitrary animal,
without regard to its herd, we need to account for the variation among
herds too, which is seen to be considerable. The two SDs reported by VarCorr()
are assumed to represent independent sources of variation, so they may be
combined into a total SD using the Pythagorean Theorem.
We will update the reference grid with the new value:
feedlot.rg <- update(feedlot.rg, sigma = sqrt(77.087^2 + 57.832^2))
We are now ready to form prediction intervals. To do so, simply call the
predict() function with an interval argument:
predict(feedlot.rg, interval = "prediction")
## adj.ewt diet prediction SE df lower.PL upper.PL
## 0 Low 1029 99.7 12.0 812 1247
## 0 Medium 998 99.9 13.7 783 1213
## 0 High 1031 100.7 19.9 821 1241
##
## Degrees-of-freedom method: kenward-roger
## Prediction intervals and SEs are based on an error SD of 96.369
## Confidence level used: 0.95
Note that the SEs for prediction are considerably greater than the SEs for
estimation in the original summary of feedlot.rg. Also, as a sanity check,
observe that these
prediction intervals cover about the same ground as the original data:
range(feedlot$swt)
## [1] 816 1248
By the way, we could have specified the desired sigma value as an additional
sigma argument in the predict() call, rather than updating the feedlot.rg
object.
Suppose, in our example, we want to predict swt for one or more particular herds.
Then the total SD we computed is not appropriate for that purpose, because that includes variation among herds.
But more to the point, if we are talking about particular herds, then we are really regarding herd as a fixed effect of interest; so the expedient thing to do is
to fit a different model where herd is a fixed effect:
feedlot.lm <- lm(swt ~ adj.ewt + diet + herd, data = feedlot)
So to predict slaughter weight for herds 9 and 19:
newrg <- ref_grid(feedlot.lm, at = list(adj.ewt = 0, herd = c("9", "19")))
predict(newrg, interval = "prediction", by = "herd")
## herd = 9:
## adj.ewt diet prediction SE df lower.PL upper.PL
## 0 Low 867 63.6 53 740 995
## 0 Medium 835 64.1 53 707 964
## 0 High 866 66.3 53 733 999
##
## herd = 19:
## adj.ewt diet prediction SE df lower.PL upper.PL
## 0 Low 1069 62.1 53 945 1194
## 0 Medium 1037 62.8 53 911 1163
## 0 High 1068 64.0 53 940 1197
##
## Prediction intervals and SEs are based on an error SD of 57.782
## Confidence level used: 0.95
This is an instance where the default sigma was already correct (being the only error SD we have available). The SD value is comparable to the residual SD in the previous model, and the prediction SEs are smaller than those for predicting over all herds.
For models fitted using Bayesian methods, these kinds of prediction intervals are
available only by forcing a frequentist analysis (frequentist = TRUE).
However, a better and more flexible approach with Bayesian models is to simulate
observations from the posterior predictive distribution. This is done via as.mcmc() and specifying a likelihood argument. An example is given in the “sophisticated models” vignette.