# Prediction interval for lmer() mixed effects model in R

I want to get a prediction interval around a prediction from a lmer() model. I have found some discussion about this:

http://rstudio-pubs-static.s3.amazonaws.com/24365_2803ab8299934e888a60e7b16113f619.html

http://glmm.wikidot.com/faq

but they seem to not take the uncertainty of the random effects into account.

Here's a specific example. I am racing gold fish. I have data on the past 100 races. I want to predict the 101st, taking into account uncertainty of my RE estimates, and FE estimates. I am including a random intercept for fish (there are 10 different fish), and fixed effect for weight (less heavy fish are quicker).

library("lme4")

fish <- as.factor(rep(letters[1:10], each=100))
race <- as.factor(rep(900:999, 10))
oz <- round(1 + rnorm(1000)/10, 3)
sec <- 9 + rep(1:10, rep(100,10))/10 + oz + rnorm(1000)/10

fishDat <- data.frame(fishID = fish,
raceID = race, fishWt = oz, time = sec)
plot(fishDat$fishID, fishDat$time)

lme1 <- lmer(time ~ fishWt + (1 | fishID), data=fishDat)
summary(lme1)


Now, to predict the 101st race. The fish have been weighed and are ready to go:

newDat <- data.frame(fishID = letters[1:10],
raceID = rep(1000, 10),
fishWt = 1 + round(rnorm(10)/10, 3))
newDat$pred <- predict(lme1, newDat) newDat fishID raceID fishWt pred 1 a 1000 1.073 10.15348 2 b 1000 1.001 10.20107 3 c 1000 0.945 10.25978 4 d 1000 1.110 10.51753 5 e 1000 0.910 10.41511 6 f 1000 0.848 10.44547 7 g 1000 0.991 10.68678 8 h 1000 0.737 10.56929 9 i 1000 0.993 10.89564 10 j 1000 0.649 10.65480  Fish D has really let himself go (1.11 oz) and is actually predicted to lose to Fish E and Fish F, both of whom he has been better than in the past. However, now I want to be able to say, "Fish E (weighing 0.91oz) will beat Fish D (weighing 1.11oz) with probability p." Is there a way to make such a statement using lme4? I want my probability p to take into account my uncertainty in both the fixed effect, and the random effect. Thanks! PS looking at the predict.merMod documentation, it suggests "There is no option for computing standard errors of predictions because it is difficult to define an efficient method that incorporates uncertainty in the variance parameters; we recommend bootMer for this task," but by golly, I cannot see how to use bootMer to do this. It seems bootMer would be used to get bootstrapped confidence intervals for parameter estimates, but I could be wrong. ## UPDATED Q: OK, I think I was asking the wrong question. I want to be able to say, "Fish A, weighing w oz, will have a race time that is (lcl, ucl) 90% of the time." In the example I have laid out, Fish A, weighing 1.0 oz, will have a race time of 9 + 0.1 + 1 = 10.1 sec on average, with a standard deviation of 0.1. Thus, his observed race time will be between x <- rnorm(mean = 10.1, sd = 0.1, n=10000) quantile(x, c(0.05,0.50,0.95)) 5% 50% 95% 9.938541 10.100032 10.261243  90% of the time. I want a prediction function that attempts to give me that answer. Setting all fishWt = 1.0 in newDat, re-running the sim, and using (as suggested by Ben Bolker below) predFun <- function(fit) { predict(fit,newDat) } bb <- bootMer(lme1,nsim=1000,FUN=predFun, use.u = FALSE) predMat <- bb$t


gives

> quantile(predMat[,1], c(0.05,0.50,0.95))
5%      50%      95%
10.01362 10.55646 11.05462


This seems to actually be centered around the population average? As if it's not taking the FishID effect into account? I thought maybe it was a sample size issue, but when I bumped the number of observed races from 100 to 10000, I still get similar results.

I'll note bootMer uses use.u=FALSE by default. On the flip side, using

bb <- bootMer(lme1,nsim=1000,FUN=predFun, use.u = TRUE)


gives

> quantile(predMat[,1], c(0.05,0.50,0.95))
5%      50%      95%
10.09970 10.10128 10.10270


That interval is too narrow, and would seem to be a confidence interval for Fish A's mean time. I want a confidence interval for Fish A's observed race time, not his average race time. How can I get that?

## UPDATE 2, ALMOST:

I thought I found what I was looking for in Gelman and Hill (2007) , page 273. Need to utilize the arm package.

library("arm")


For Fish A:

x.tilde <- 1    #observed fishWt for new race
sigma.y.hat <- sigma.hat(lme1)$sigma$data        #get uncertainty estimate of our model
coef.hat <- as.matrix(coef(lme1)$fishID)[1,] #get intercept (random) and fishWt (fixed) parameter estimates y.tilde <- rnorm(1000, coef.hat %*% c(1, x.tilde), sigma.y.hat) #simulate quantile (y.tilde, c(.05, .5, .95)) 5% 50% 95% 9.930695 10.100209 10.263551  For all the fishes: x.tilde <- rep(1,10) #assume all fish weight 1 oz #x.tilde <- 1 + rnorm(10)/10 #alternatively, draw random weights as in original example sigma.y.hat <- sigma.hat(lme1)$sigma$data coef.hat <- as.matrix(coef(lme1)$fishID)
y.tilde <- matrix(rnorm(1000, coef.hat %*% matrix(c(rep(1,10), x.tilde), nrow = 2 , byrow = TRUE), sigma.y.hat), ncol = 10, byrow = TRUE)
quantile (y.tilde[,1], c(.05, .5, .95))
5%       50%       95%
9.937138 10.102627 10.234616


Actually, this probably isn't exactly what I want. I'm only taking into account the overall model uncertainty. In a situation where I have, say, 5 observed races for Fish K and 1000 observed races for Fish L, I think the uncertainty associated with my prediction for Fish K should be much larger than the uncertainty associated with my prediction for Fish L.

Will look further into Gelman and Hill 2007. I feel I may end up having to switch to BUGS (or Stan).

## UPDATE THE 3rd:

Perhaps I am conceptualizing things poorly. Using the predictInterval() function given by Jared Knowles in an answer below gives intervals that aren't quite what I would expect...

library("lattice")
library("lme4")
library("ggplot2")

fish <- c(rep(letters[1:10], each = 100), rep("k", 995), rep("l", 5))
oz <- round(1 + rnorm(2000)/10, 3)
sec <- 9 + c(rep(1:10, each = 100)/10,rep(1.1, 995), rep(1.2, 5)) + oz + rnorm(2000)

fishDat <- data.frame(fishID = fish, fishWt = oz, time = sec)
dim(fishDat)
plot(fishDat$fishID, fishDat$time)

lme1 <- lmer(time ~ fishWt + (1 | fishID), data=fishDat)
summary(lme1)
dotplot(ranef(lme1, condVar = TRUE))


I have added two new fish. Fish K, for whom we have observed 995 races, and Fish L, for whom we have observed 5 races. We have observed 100 races for Fish A-J. I fit the same lmer() as before. Looking at the dotplot() from the lattice package:

By default, dotplot() reorders the random effects by their point estimate. The estimate for Fish L is on the top line, and has a very wide confidence interval. Fish K is on the third line, and has a very narrow confidence interval. This makes sense to me. We have lots of data on Fish K, but not a lot of data on Fish L, so we are more confident in our guesstimate about Fish K's true swimming speed. Now, I would think this would lead to a narrow prediction interval for Fish K, and a wide prediction interval for Fish L when using predictInterval(). Howeva:

newDat <- data.frame(fishID = letters[1:12],
fishWt = 1)

preds <- predictInterval(lme1, newdata = newDat, n.sims = 999)
preds
ggplot(aes(x=letters[1:12], y=fit, ymin=lwr, ymax=upr), data=preds) +
geom_point() +
geom_linerange() +
labs(x="Index", y="Prediction w/ 95% PI") + theme_bw()


All of those prediction intervals appear to be identical in width. Why isn't our prediction for Fish K narrower the others? Why isn't our prediction for Fish L wider than others?

• predictInterval includes the error/uncertainty for both the fixed and random effect terms. In dotplot you are only seeing the uncertainty due to the random portion of the prediction, essentially the uncertainty around the estimate of the fish specific intercepts. If your model has a lot of uncertainty in the fixed parameter fishWt and this parameter drives most of the predicted value, then the uncertainty around any specific fish intercept is trivial and you won't see a big difference in the width of the intervals. We should make this more clear in predictInterval results. – jknowles Jan 4 '16 at 17:23

This question and excellent exchange was the impetus for creating the predictInterval function in the merTools package. bootMer is the way to go, but for some problems it is not feasible computationally to generate bootstrapped refits of the whole model (in cases where the model is large).

In those cases, predictInterval is designed to use the arm::sim functions to generate distributions of parameters in the model and then to use those distributions to generate simulated values of the response given the newdata provided by the user. It's simple to use -- all you would need to do is:

library(merTools)
preds <- predictInterval(lme1, newdata = newDat, n.sims = 999)


You can specify a whole host of other values to predictInterval including setting the interval for the prediction intervals, choosing whether to report the mean or median of the distribution, and choosing whether or not to include the residual variance from the model.

It's not a full prediction interval because the variability of the theta parameters in the lmer object are not included, but all of the other variation is captured through this method, giving a pretty decent approximation.

• This looks awesome! Reading through the vignette now. Thanks! – hossibley Aug 23 '15 at 19:00
• The prediction intervals aren't quite what I expected. See update 3 above. – hossibley Dec 8 '15 at 16:57
• As an update, the latest development version of merTools does allow nested effects. It will be pushed to CRAN shortly. – jknowles Feb 18 '16 at 18:16
• What about unconditional prediction intervals? I have a model that I would like to generate prediction intervals on new data about which the random effects are unknown. Say you had a new fish that was not part of the original trials, but you do know it's weight. How could you generate a prediction interval with predictInterval()? – emudrak Feb 19 '16 at 20:46
• The only option I see in that case is to base the prediction interval on the fixed effect and model variability. Since you do not know what the group effect would be on the prediction, nor how precise it is, you could assign it to an unobserved factor level and predictInterval should just set the random effect to 0. If you wanted to do more than that, you could assign the same observation to various groups from the random effect distribution and simulate what the range of possible effects might be, and merTools contains examples of how to do this. – jknowles Feb 19 '16 at 22:09

Do this by making bootMer generate a set of predictions for each parametric bootstrap replicate:

predFun <- function(fit) {
predict(fit,newDat)
}
bb <- bootMer(lme1,nsim=200,FUN=predFun,seed=101)


The output of bootMer is in a not-terribly-transparent "boot" object, but we can get the raw predictions out of the $t component. How much of the time does Fish E beat Fish D? predMat <- bb$t
dim(predMat) ## 200 rows (PB reps) x 10 (predictions)


Fish E's times are in column 5, fish D's times are in column 4, so we just need to know the proportion that column 5 is less than column 4:

mean(predMat[,5]<predMat[,4])  ## 0.57

• I'm getting some unexpected results. If I set fishWt = 1 for all fish in newDat, I would expect the mean/median time for Fish A to be ~ 10.1, Fish B ~ 10.2, ... , Fish J ~ 11.0 (since their time in the training data is defined as: sec <- 9 + rep(1:10, rep(100,10))/10 + oz + rnorm(1000)/10). When I use predict(), prediction times for Fish A, E, and J are 10.09, 10.49, and 10.99, as expected. However, median times for the bootMer method you descirbe are: 10.52, 10.59, and 10.50. I would have expected more agreement? – hossibley Apr 22 '15 at 23:32
• Using use.u=TRUE as in: bb <- bootMer(lme1,nsim=200,FUN=predFun,seed=101,use.u=TRUE) seems to give me what I want. Thanks! – hossibley Apr 23 '15 at 1:58
• OK, this gets a little tricky. You need to look at the use.u argument to bootMer. The question is, when you say "uncertainty in the fixed effect and the random effect", what do you mean by 'the random effect'? Do you mean uncertainty in the random-effects variance, or in the conditional modes (i.e. the fish-specific effects)? You can use use.u=TRUE, but I don't think it will necessarily do what you want ... – Ben Bolker Apr 23 '15 at 2:01
• If I use use.u=TRUE, then the "values of u [stay] fixed at their estimated values". I interpret this as meaning, whatever our random effect point estimate for Fish A is, it is taken as The God's Honest Truth, if you will. bootMer assumes there is no error in our RE point estimate. If I use use.u=FALSE, does bootMer take into account the RE point estimates at all? It seems bootMer results when using use.u=FALSE are equivalent (or, asymptotically equiv) to using re.form=NA in the predict() statement. Is that true? – hossibley Apr 23 '15 at 15:41
• I think it's not implemented ATM, but you can extract the conditional variances of the conditional modes/BLUPs via c(attr(ranef(lme1,condVar=TRUE)[[1]],"postVar")) (they're all identical in this example), and then sample those values. – Ben Bolker Apr 30 '15 at 1:18