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Introduction to Mixed Models in R
Galin Jones
School of StatisticsUniversity of Minnesota
http://www.stat.umn.edu/∼galin
March 2011
Second in a Series
• Sponsored by Quantitative Methods Collaborative.
• Previous workshop “An Introduction to R” was given on 3/7by Professor Sanford Weisberg.
• Upcoming talk on 4/11 by Chris Winship of HarvardUniversity on causal inference.
Goals
• Brief review of first workshop.
• Definition of mixed models and why they may be useful.
• Using mixed models in R through two simple case studies.
Review
• R is “free” software and can be downloaded athttp://cran.r-project.org
• A package is a collection of functions designed for specifictasks. For example the package lme4 fits many mixed models.
• When you install R on your computer you get the “base”distribution. This includes many useful packages but others,such as lme4, you need to install separately.
• The R search engine is extremely usefulhttp://www.rseek.org
• Getting data into R (or any other software package) can bechallenging.
• Many R packages include built-in data sets and we will usetwo of these today.
Mixed Models
Mixed models are a large and complex topic, we will only justbarely get started with them today.
There are many varieties of mixed models:
• Linear mixed models (LMM)
• Nonlinear mixed models (NLM)
• Generalized linear mixed models (GLMM)
Our focus will be on linear mixed models. Much more discussion ofthis material can be found in the following books.
Extending the Linear Model with R by Julian Faraway
Mixed-Effects Models in S and S-PLUS by Jose Pinheiro andDouglas Bates
Factors, Effects and Treatments
Suppose apple slices are treated with five preservative compounds(A, B, C, D, E) with the goal of extending shelf life.
Response: Shelf life
Factor: Preservative
Treatments = Levels of a factor: A, B, C, D, E
Effect: Impact of compound on shelf life
µ=population mean shelf lifeµA= population mean shelf life with treatment A
effect of A = µA − µ
Fixed or Random?
Factor effects are either fixed or random.
• Fixed: The levels in the study represent all levels of interest
• Random: The levels in the study represent only a sample ofthe levels of interest.
Mixed models have both fixed and random effects.
In our example, preservative is fixed since A, B, C, D, E are theonly levels of interest.
Blocking
Block:
• A group of units formed so that units within the group are ashomogeneous as possible.
• Reduces the effects of variation among experimental units.
• Treatmets are randomly assigned to blocks.
Lets add to the example: Suppose 10 individual fruit are randomlychosen from a population of fruit and the 5 preservatives arerandomly assigned to 5 portions of fruit.
• Preservative is a fixed effect.
• Fruit is a (random) block effect.
Mixed models
Mixed models contain both fixed and random effects This hasseveral ramifications:
• Using random effects broadens the scope of inference. That is,inferences can be made on a statistical basis to the populationfrom which the levels of the random factor have been drawn.
• Naturally incorporates dependence in the model. Observationsthat share the same level of the random effects are beingmodeled as correlated.
• Using random factors often gives more accurate estimates.
• Sophisticated estimation and fitting methods must be used.
Rail Data
The “Rail” data is a built-in R data set. It can be found in theMEMSS and the NLME packages.
The commands
> library(nlme)
> data(Rail)
> ?Rail
will make the data available to us and produce a description.
Rail Data
Evaluation of Stress in Railway Rails
Description
The Rail data frame has 18 rows and 2 columns.
Format
This data frame contains the following columns:
Rail
an ordered factor identifying the rail on which
the measurement was made.
travel
a numeric vector giving the travel time for ultrasonic
head-waves in the rail (nanoseconds). The value given
is the original travel time minus 36,100 nanoseconds.
Rail Data
Summary
• Six rails were chosen from a group of rails.
• Each rail was tested 3 times.
• Measured the time it takes an ultrasonic wave to travel thelength of a rail.
• Rail is a factor (fixed or random?) and travel is the response.
Rail Data
> Rail
Grouped Data: travel ~ 1 | Rail
Rail travel
1 1 55
2 1 53
3 1 54
4 2 26
5 2 37
.
.
.
15 5 50
16 6 80
17 6 85
18 6 83
Rail Data
> summary(Rail)
Rail travel
2:3 Min. : 26.00
5:3 1st Qu.: 50.25
1:3 Median : 66.50
6:3 Mean : 66.50
3:3 3rd Qu.: 85.00
4:3 Max. :100.00
> with(Rail, tapply(travel, Rail, mean))
2 5 1 6 3 4
31.66667 50.00000 54.00000 82.66667 84.66667 96.00000
> pdf(file="RailPlot1.pdf")
> with(Rail, plot(travel, Rail, xlab="Travel time"))
> dev.off()
Rail Data
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40 60 80 100
12
34
56
Travel time
Rail
Rail Data
• Between-rail variability is greater than within-rail variability.
• Within-rail variability is not constant.
• Mean travel time appears different for some rails2 5 1 6 3 4
We clearly need to account for the classification factor (Rail) in theanalysis.
Rail DataRail as a fixed effect
yij = βi + eij i = 1, . . . , 6 j = 1, 2, 3
where
• yij is the observed travel time for observation j on rail i .
• βi is the population mean travel time of rail i
• eij are independent and identically normally distributed withmean 0 and variance σ2 or
eijiid∼ N(0, σ2) .
This asumption means yijind∼ N(βi , σ
2)
> #Rail as a fixed effect
> r1.lm<-lm(travel ~ Rail - 1, data=Rail)
Rail Data
> r1.lm
Call:
lm(formula = travel ~ Rail - 1, data = Rail)
Coefficients:
Rail2 Rail5 Rail1 Rail6 Rail3 Rail4
31.67 50.00 54.00 82.67 84.67 96.00
This means that the oredered estimates of the βi are
β2 = 31.67 , . . . , β4 = 96.00
Rail Data
> summary(r1.lm)
Coefficients:
Estimate Std. Error t value Pr(>|t|)
Rail2 31.667 2.321 13.64 1.15e-08 ***
.
.
Rail4 96.000 2.321 41.35 2.59e-14 ***
Residual standard error: 4.021 on 12 degrees of freedom
Multiple R-squared: 0.9978,Adjusted R-squared: 0.9967
F-statistic: 916.6 on 6 and 12 DF, p-value: 2.971e-15
Note that the F-statistic and p-value are testing for any differencesbetween the rail effects. Also, the estimate of σ2 is 4.021 on 12degrees of freedom.
Rail Data
> with(Rail, bwplot(Rail ~ residuals(r1.lm)))
residuals(r1.lm)
2
5
1
6
3
4
−5 0 5
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Rail Data
The fixed effects model gives a good summary of the data but themain interest is in the population of rails. Also, we don’t believethe model is accurate since the 3 observations on each rail areclearly not independent.
Rail as a random effect
yij = β + bi + eij
where
• yij is the observed travel time for observation j on rail i .
• β is the population mean travel time
• bi is the deviation from β for the ith rail
Rail Data
• eij is the deviation for observation j on rail i from the meantravel time for rail i
biiid∼ N(0, σ2b) eij
iid∼ N(0, σ2) .
• Our assumptions imply that observations on the same rail arecorrelated, in fact,
corr =σ2b
σ2b + σ2.
Rail Data
> #Rail as random effect
> r2.lme<-lmer(travel ~ 1 + (1 | Rail),
REML=FALSE, data=Rail)
This notation takes some getting used to. Specifically,
(1 | Rail)
means that there is a single random factor which is constant withineach level and its levels are given by the grouping variable Rail.
Rail Data
> summary(r2.lme)
Linear mixed model fit by maximum likelihood
Formula: travel ~ 1 + (1 | Rail)
Random effects:
Groups Name Variance Std.Dev.
Rail (Intercept) 511.861 22.6243
Residual 16.167 4.0208
Number of obs: 18, groups: Rail, 6
Our best guess at σ2 and σ2b are
σ2b = 511.861 σ2 = 16.167
and the estimated correlation between observations on a rail is
corr =511.861
511.861 + 16.167= 0.969 .
Rail Data
Fixed effects:
Estimate Std. Error t value
(Intercept) 66.500 9.285 7.162
The estimate of β is β = 66.5. For a new rail drawn for thepopulation this is our best guess at travel time while the predictedvalues for a new observation on each of the same 6 rails are
> fixef(r2.lme)+ranef(r2.lme)$Rail
(Intercept)
2 32.02957
5 50.17190
1 54.13023
6 82.49824
3 84.47740
4 95.69266
Rail Data
> qqnorm(resid(r2.lme), main="")
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−2 −1 0 1 2
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−20
24
6
Theoretical Quantiles
Sam
ple Q
uant
iles
Rail Data
> plot(fitted(r2.lme), resid(r2.lme), xlab="Fitted",
ylab="Residuals")
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30 40 50 60 70 80 90
−6−4
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Fitted
Resid
uals
Multilevel Models
Multilevel models
• are special cases of mixed models,
• are useful for data with a hierarchical structure, and
• can be implemented in a variety of R packages.
Joint Schools Project
> library(faraway)
> data(jsp)
> ?jsp
Description
Example Dataset from "Practical Regression and Anova"
Format
See for yourself
Source
See Reference
References
Reference details may be found in "Practical Regression
and Anova" by Julian Faraway
Joint Schools Project> str(jsp)
’data.frame’: 3236 obs. of 9 variables:
$ school : Factor w/ 49 levels "1","2","3","4",..: 1 1
1 1 1 1 1 1 1 1 ...
$ class : Factor w/ 4 levels "1","2","3","4": 1 1 1 1
1 1 1 1 1 1 ...
$ gender : Factor w/ 2 levels "boy","girl": 2 2 2 1 1
1 1 1 1 1 ...
$ social : Factor w/ 9 levels "1","2","3","4",..: 9 9
9 2 2 2 2 2 9 9 ...
$ raven : num 23 23 23 15 15 22 22 22 14 14 ...
$ id : Factor w/ 1192 levels "1","2","3","4",..: 1
1 1 2 2 3 3 3 4 4 ...
$ english: num 72 80 39 7 17 88 89 83 12 25 ...
$ math : num 23 24 23 14 11 36 32 39 24 26 ...
$ year : num 0 1 2 0 1 0 1 2 0 1 ...
Joint Schools Project
> summary(jsp)
school class gender social raven
48 : 206 1:1949 boy :1551 4 :1225 Min. : 4.00
33 : 131 2: 987 girl:1685 9 : 484 1st Qu.:21.00
4 : 131 3: 169 2 : 424 Median :25.00
31 : 107 4: 131 5 : 288 Mean :25.13
47 : 102 3 : 270 3rd Qu.:29.00
50 : 101 6 : 221 Max. :36.00
(Other):2458 (Other): 324
id english math year
1 : 3 Min. : 0.00 Min. : 1.00 Min. :0.0000
3 : 3 1st Qu.:31.00 1st Qu.:22.00 1st Qu.:0.0000
4 : 3 Median :54.00 Median :28.00 Median :1.0000
6 : 3 Mean :52.49 Mean :26.66 Mean :0.9379
7 : 3 3rd Qu.:75.00 3rd Qu.:33.00 3rd Qu.:2.0000
8 : 3 Max. :98.00 Max. :40.00 Max. :2.0000
(Other):3218
Joint Schools Project
#Subset data to focus on Year=2
> jsp.year2<-jsp[jsp$year==2,]
> plot(jitter(math) ~ jitter(raven), xlab="Raven Score",
ylab="Math Score", data=jsp.year2)
> boxplot(math ~ social, xlab="Social Class",
ylab="Math Score", data=jsp.year2)
> boxplot(math ~ gender, xlab="Gender", ylab="Math Score",
data=jsp.year2)
Joint Schools Project
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5 10 15 20 25 30 35
510
1520
2530
3540
Raven Score
Mat
h Sc
ore
There is clearly correlation between math score and raven score.
Joint Schools Project
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1 2 3 4 5 6 7 8 9
510
1520
2530
3540
Social Class
Mat
h Sc
ore
There are differences in math scores between the levels of socialclass.
Joint Schools Project
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boy girl
510
1520
2530
3540
Gender
Mat
h Sc
ore
Math scores are not different between the levels of gender.
Joint Schools Project
Center raven since we would otherwise be comparing to zero.
> jsp.y2$ctrraven<-jsp.y2$raven-mean(jsp.y2$raven)
> jsp1.lme<-lmer(math ~ ctrraven*social*gender +
(1 | school) + (1 | school:class), data=jsp.y2)
> qqnorm(resid(jsp1.lme),main="")
> plot(fitted(jsp1.lme) ~ resid(jsp1.lme), xlab="Fitted",
ylab="Residuals")
Joint Schools Project
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−3 −2 −1 0 1 2 3
−20
−15
−10
−50
510
15
Theoretical Quantiles
Sam
ple Q
uant
iles
There isn’t much of concern here.
Joint Schools Project
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−20 −15 −10 −5 0 5 10 15
1015
2025
3035
40
Fitted
Resid
uals
There is some evidence of non-constant variance. Transformation?
Joint Schools Project
> anova(jsp1.lme)
Analysis of Variance Table
Df Sum Sq Mean Sq F value
ctrraven 1 10218.0 10218.0 374.4037
social 8 615.7 77.0 2.8201
gender 1 21.6 21.6 0.7917
ctrraven:social 8 577.4 72.2 2.6445
ctrraven:gender 1 2.5 2.5 0.0905
social:gender 8 275.2 34.4 1.2605
ctrraven:social:gender 8 187.2 23.4 0.8573
Joint Schools Project
The p-values associated with the F -statistics are approximate andcan be too small when the number of cases is small. However, thisdata set is probably large enough to overcome this limitation. Theparametric bootstrap can be used if this is a concern.
> nrow(jsp.y2) - sum(anova(jsp1.lme)[,1]) - 1
[1] 917
> round(pf(anova(jsp1.lme)[,4], anova(jsp1.lme)[,1], 917,
lower.tail=FALSE),4)
[1] 0.0000 0.0043 0.3738 0.0072 0.7636 0.2607 0.5524
Joint Schools Project
> #Remove Gender
> jsp2.lme<-lmer(math ~ ctrraven*social + (1 | school) +
(1 | school:class), data=jsp.y2)
> qqnorm(resid(jsp2.lme),main="")
> plot(fitted(jsp2.lme) ~ resid(jsp2.lme), xlab="Fitted",
ylab="Residuals")
> qqnorm(ranef(jsp2.lme)$"school:class"[[1]],
main="School Effects")
> qqnorm(ranef(jsp2.lme)$"school:class"[[1]],
main="Class Effects")
Joint Schools Project
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−3 −2 −1 0 1 2 3
−20
−15
−10
−50
510
15
Theoretical Quantiles
Sam
ple Q
uant
iles
Still not much of concern.
Joint Schools Project
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−20 −15 −10 −5 0 5 10 15
1520
2530
3540
Fitted
Resid
uals
Constant variance is still a problem.
Joint Schools Project
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−2 −1 0 1 2
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School Effects
Theoretical Quantiles
Sam
ple Q
uant
iles
Joint Schools Project
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−2 −1 0 1 2
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Class Effects
Theoretical Quantiles
Sam
ple Q
uant
iles
Joint Schools Project
> anova(jsp2.lme)
Analysis of Variance Table
Df Sum Sq Mean Sq F value
ctrraven 1 10159.7 10159.7 374.3263
social 8 609.5 76.2 2.8073
ctrraven:social 8 564.6 70.6 2.6005
> nrow(jsp.y2) - sum(anova(jsp2.lme)[,1]) - 1
[1] 935
> round(pf(anova(jsp2.lme)[,4], anova(jsp2.lme)[,1], 935,
lower.tail=FALSE),4)
[1] 0.0000 0.0045 0.0082
Joint Schools Project
> sch.effects<-ranef(jsp2.lme)$school[[1]]
> summary(sch.effects)
Min. 1st Qu. Median Mean 3rd Qu. Max.
-2.45200 -0.69480 0.01107 0.00000 0.71840 2.44900
> raw.sch.effects<-coef(lm(math ~ school-1,jsp.y2))
> raw.sch.effects<-raw.sch.effects-mean(raw.sch.effects)
> plot(raw.sch.effects,sch.effects)
> sint<-c(9,14,29)
> text(raw.sch.effects[sint],sch.effects[sint]+0.2,
c("9","15","30"))
Joint Schools Project
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−6 −4 −2 0 2 4
−2−1
01
2
raw.sch.effects
sch.
effec
ts
9
15
30
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