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Getting Started with Multilevel Modeling in R
Jared E. Knowles
Introduction
Analysts dealing with grouped data and complex hierarchical structures in their
data ranging from measurements nested within participants, to counties nested within
states or students nested within classrooms often find themselves in need of modeling
tools to reflect this structure of their data. In R there are two predominant ways
to fit multilevel models that account for such structure in the data. These tutorials
will show the user how to use both the lme4 package in R to fit linear and nonlinear
mixed effect models, and to use rstan to fit fully Bayesian multilevel models. The
focus here will be on how to fit the models in R and not the theory behind the models.
For background on multilevel modeling, see the references. [1]
This tutorial will cover getting set up and running a few basic models using lme4
in R.Future tutorials will cover:
- constructing varying intercept, varying slope, and varying slope and intercept models in R
- generating predictions and interpreting parameters from mixed-effect models
- generalized and non-linear multilevel models
- fully Bayesian multilevel models fit with
rstanor other MCMC methods
Setting up your enviRonment
Getting started with multilevel modeling in R is simple. lme4 is the canonical
package for implementing multilevel models in R, though there are a number of packages
that depend on and enhance its feature set, including Bayesian extensions. lme4
has been recently rewritten to improve speed and to incorporate a C++ codebase, and
as such the features of the package are somewhat in flux. Be sure to update the package
frequently.
To install lme4, we just run:
# Main version
install.packages("lme4")
# Or to install the dev version
library(devtools)
install_github("lme4", user = "lme4")
Read in the data
Multilevel models are appropriate for a particular kind of data structure where
units are nested within groups (generally 5+ groups) and where we want to model
the group structure of the data. For our introductory example we will start with
a simple example from the lme4 documentation and explain what the model is doing.
We will use data from Jon Starkweather at the University of North Texas. Visit the excellent tutorial available here for more.
library(lme4) # load library
library(arm) # convenience functions for regression in R
lmm.data <- read.table("http://www.unt.edu/rss/class/Jon/R_SC/Module9/lmm.data.txt",
header = TRUE, sep = ",", na.strings = "NA", dec = ".", strip.white = TRUE)
# summary(lmm.data)
head(lmm.data)
## id extro open agree social class school
## 1 1 63.69 43.43 38.03 75.06 d IV
## 2 2 69.48 46.87 31.49 98.13 a VI
## 3 3 79.74 32.27 40.21 116.34 d VI
## 4 4 62.97 44.41 30.51 90.47 c IV
## 5 5 64.25 36.86 37.44 98.52 d IV
## 6 6 50.97 46.26 38.83 75.22 d I
Here we have data on the extroversion of subjects nested within classes and within schools.
Fit the Non-Multilevel Models
Let's start by fitting a simple OLS regression of measures of openness, agreeableness,
and socialability on extroversion. Here we use the display function in the
excellent arm package for abbreviated output. Other options include stargazer
for LaTeX typeset tables, xtable, or the ascii package for more flexible
plain text output options.
OLSexamp <- lm(extro ~ open + agree + social, data = lmm.data) display(OLSexamp) ## lm(formula = extro ~ open + agree + social, data = lmm.data) ## coef.est coef.se ## (Intercept) 57.84 3.15 ## open 0.02 0.05 ## agree 0.03 0.05 ## social 0.01 0.02 ## --- ## n = 1200, k = 4 ## residual sd = 9.34, R-Squared = 0.00
So far this model does not fit very well at all. The R model interface is quite a
simple one with the dependent variable being specified first, followed by the
~ symbol. The righ hand side, predictor variables, are each named. Addition
signs indicate that these are modeled as additive effects. Finally, we specify
that datframe on which to calculate the model. Here we use the lm function to
perform OLS regression, but there are many other options in R.
If we want to extract measures such as the AIC, we may prefer to fit a generalized
linear model with glm which produces a model fit through maximum likelihood
estimation. Note that the model formula specification is the same.
MLexamp <- glm(extro ~ open + agree + social, data = lmm.data) display(MLexamp) ## glm(formula = extro ~ open + agree + social, data = lmm.data) ## coef.est coef.se ## (Intercept) 57.84 3.15 ## open 0.02 0.05 ## agree 0.03 0.05 ## social 0.01 0.02 ## --- ## n = 1200, k = 4 ## residual deviance = 104378.2, null deviance = 104432.7 (difference = 54.5) ## overdispersion parameter = 87.3 ## residual sd is sqrt(overdispersion) = 9.34 AIC(MLexamp) ## [1] 8774
This results in a poor model fit. Let's look at a simple varying intercept model now.
Fit a varying intercept model
Depending on disciplinary norms, our next step might be to fit a varying intercept
model using a grouping variable such as school or classes. Using the glm function
and the familiar formula interface, such a fit is easy:
MLexamp.2 <- glm(extro ~ open + agree + social + class, data = lmm.data) display(MLexamp.2) ## glm(formula = extro ~ open + agree + social + class, data = lmm.data) ## coef.est coef.se ## (Intercept) 56.05 3.09 ## open 0.03 0.05 ## agree -0.01 0.05 ## social 0.01 0.02 ## classb 2.06 0.75 ## classc 3.70 0.75 ## classd 5.67 0.75 ## --- ## n = 1200, k = 7 ## residual deviance = 99187.7, null deviance = 104432.7 (difference = 5245.0) ## overdispersion parameter = 83.1 ## residual sd is sqrt(overdispersion) = 9.12 AIC(MLexamp.2) ## [1] 8719 anova(MLexamp, MLexamp.2, test = "F") ## Analysis of Deviance Table ## ## Model 1: extro ~ open + agree + social ## Model 2: extro ~ open + agree + social + class ## Resid. Df Resid. Dev Df Deviance F Pr(>F) ## 1 1196 104378 ## 2 1193 99188 3 5190 20.8 3.8e-13 *** ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
This is called a fixed-effects specification often. This is simply the case of fitting a separate dummy variable as a predictor for each class. We can see this does not provide much additional model fit. Let's see if school performs any better.
MLexamp.3 <- glm(extro ~ open + agree + social + school, data = lmm.data) display(MLexamp.3) ## glm(formula = extro ~ open + agree + social + school, data = lmm.data) ## coef.est coef.se ## (Intercept) 45.02 0.92 ## open 0.01 0.01 ## agree 0.03 0.02 ## social 0.00 0.00 ## schoolII 7.91 0.27 ## schoolIII 12.12 0.27 ## schoolIV 16.06 0.27 ## schoolV 20.43 0.27 ## schoolVI 28.05 0.27 ## --- ## n = 1200, k = 9 ## residual deviance = 8496.2, null deviance = 104432.7 (difference = 95936.5) ## overdispersion parameter = 7.1 ## residual sd is sqrt(overdispersion) = 2.67 AIC(MLexamp.3) ## [1] 5774 anova(MLexamp, MLexamp.3, test = "F") ## Analysis of Deviance Table ## ## Model 1: extro ~ open + agree + social ## Model 2: extro ~ open + agree + social + school ## Resid. Df Resid. Dev Df Deviance F Pr(>F) ## 1 1196 104378 ## 2 1191 8496 5 95882 2688 <2e-16 *** ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
The school effect greatly improves our model fit. However, how do we interpret these effects?
table(lmm.data$school, lmm.data$class) ## ## a b c d ## I 50 50 50 50 ## II 50 50 50 50 ## III 50 50 50 50 ## IV 50 50 50 50 ## V 50 50 50 50 ## VI 50 50 50 50
Here we can see we have a perfectly balanced design with fifty observations in each combination of class and school (if only data were always so nice!).
Let's try to model each of these unique cells. To do this, we fit a model and use
the : operator to specify the interaction between school and class.
MLexamp.4 <- glm(extro ~ open + agree + social + school:class, data = lmm.data) display(MLexamp.4) ## glm(formula = extro ~ open + agree + social + school:class, data = lmm.data) ## coef.est coef.se ## (Intercept) 80.36 0.37 ## open 0.01 0.00 ## agree -0.01 0.01 ## social 0.00 0.00 ## schoolI:classa -40.39 0.20 ## schoolII:classa -28.15 0.20 ## schoolIII:classa -23.58 0.20 ## schoolIV:classa -19.76 0.20 ## schoolV:classa -15.50 0.20 ## schoolVI:classa -10.46 0.20 ## schoolI:classb -34.60 0.20 ## schoolII:classb -26.76 0.20 ## schoolIII:classb -22.59 0.20 ## schoolIV:classb -18.71 0.20 ## schoolV:classb -14.31 0.20 ## schoolVI:classb -8.54 0.20 ## schoolI:classc -31.86 0.20 ## schoolII:classc -25.64 0.20 ## schoolIII:classc -21.58 0.20 ## schoolIV:classc -17.58 0.20 ## schoolV:classc -13.38 0.20 ## schoolVI:classc -5.58 0.20 ## schoolI:classd -30.00 0.20 ## schoolII:classd -24.57 0.20 ## schoolIII:classd -20.64 0.20 ## schoolIV:classd -16.60 0.20 ## schoolV:classd -12.04 0.20 ## --- ## n = 1200, k = 27 ## residual deviance = 1135.9, null deviance = 104432.7 (difference = 103296.8) ## overdispersion parameter = 1.0 ## residual sd is sqrt(overdispersion) = 0.98 AIC(MLexamp.4) ## [1] 3396
This is very useful, but what if we want to understand both the effect of the
school and the effect of the class, as well as the effect of the schools and classes?
Unfortunately, this is not easily done with the standard glm.
MLexamp.5 <- glm(extro ~ open + agree + social + school * class - 1, data = lmm.data) display(MLexamp.5) ## glm(formula = extro ~ open + agree + social + school * class - ## 1, data = lmm.data) ## coef.est coef.se ## open 0.01 0.00 ## agree -0.01 0.01 ## social 0.00 0.00 ## schoolI 39.96 0.36 ## schoolII 52.21 0.36 ## schoolIII 56.78 0.36 ## schoolIV 60.60 0.36 ## schoolV 64.86 0.36 ## schoolVI 69.90 0.36 ## classb 5.79 0.20 ## classc 8.53 0.20 ## classd 10.39 0.20 ## schoolII:classb -4.40 0.28 ## schoolIII:classb -4.80 0.28 ## schoolIV:classb -4.74 0.28 ## schoolV:classb -4.60 0.28 ## schoolVI:classb -3.87 0.28 ## schoolII:classc -6.02 0.28 ## schoolIII:classc -6.54 0.28 ## schoolIV:classc -6.36 0.28 ## schoolV:classc -6.41 0.28 ## schoolVI:classc -3.65 0.28 ## schoolII:classd -6.81 0.28 ## schoolIII:classd -7.45 0.28 ## schoolIV:classd -7.24 0.28 ## schoolV:classd -6.93 0.28 ## schoolVI:classd 0.06 0.28 ## --- ## n = 1200, k = 27 ## residual deviance = 1135.9, null deviance = 4463029.9 (difference = 4461894.0) ## overdispersion parameter = 1.0 ## residual sd is sqrt(overdispersion) = 0.98 AIC(MLexamp.5) ## [1] 3396
Exploring Random Slopes
Another alternative is to fit a separate model for each of the school and class combinations. If we believe the relationsihp between our variables may be highly dependent on the school and class combination, we can simply fit a series of models and explore the parameter variation among them:
require(plyr)
modellist <- dlply(lmm.data, .(school, class), function(x) glm(extro ~ open +
agree + social, data = x))
display(modellist[[1]])
## glm(formula = extro ~ open + agree + social, data = x)
## coef.est coef.se
## (Intercept) 35.87 5.90
## open 0.05 0.09
## agree 0.02 0.10
## social 0.01 0.03
## ---
## n = 50, k = 4
## residual deviance = 500.2, null deviance = 506.2 (difference = 5.9)
## overdispersion parameter = 10.9
## residual sd is sqrt(overdispersion) = 3.30
display(modellist[[2]])
## glm(formula = extro ~ open + agree + social, data = x)
## coef.est coef.se
## (Intercept) 47.96 2.16
## open -0.01 0.03
## agree -0.03 0.03
## social -0.01 0.01
## ---
## n = 50, k = 4
## residual deviance = 47.9, null deviance = 49.1 (difference = 1.2)
## overdispersion parameter = 1.0
## residual sd is sqrt(overdispersion) = 1.02
We will discuss this strategy in more depth in future tutorials including how to performan inference on the list of models produced in this command.
Fit a varying intercept model with lmer
Enter lme4. While all of the above techniques are valid approaches to this problem,
they are not necessarily the best approach when we are interested explicitly in
variation among and by groups. This is where a mixed-effect modeling framework
is useful. Now we use the lmer function with the familiar formula interface,
but now group level variables are specified using a special syntax: (1|school)
tells lmer to fit a linear model with a varying-intercept group effect using
the variable school.
MLexamp.6 <- lmer(extro ~ open + agree + social + (1 | school), data = lmm.data) display(MLexamp.6) ## lmer(formula = extro ~ open + agree + social + (1 | school), ## data = lmm.data) ## coef.est coef.se ## (Intercept) 59.12 4.10 ## open 0.01 0.01 ## agree 0.03 0.02 ## social 0.00 0.00 ## ## Error terms: ## Groups Name Std.Dev. ## school (Intercept) 9.79 ## Residual 2.67 ## --- ## number of obs: 1200, groups: school, 6 ## AIC = 5836.1, DIC = 5789 ## deviance = 5806.5
We can fit multiple group effects with multiple group effect terms.
MLexamp.7 <- lmer(extro ~ open + agree + social + (1 | school) + (1 | class),
data = lmm.data)
display(MLexamp.7)
## lmer(formula = extro ~ open + agree + social + (1 | school) +
## (1 | class), data = lmm.data)
## coef.est coef.se
## (Intercept) 60.20 4.21
## open 0.01 0.01
## agree -0.01 0.01
## social 0.00 0.00
##
## Error terms:
## Groups Name Std.Dev.
## school (Intercept) 9.79
## class (Intercept) 2.41
## Residual 1.67
## ---
## number of obs: 1200, groups: school, 6; class, 4
## AIC = 4737.9, DIC = 4683
## deviance = 4703.6
And finally, we can fit nested group effect terms through the following syntax:
MLexamp.8 <- lmer(extro ~ open + agree + social + (1 | school/class), data = lmm.data) display(MLexamp.8) ## lmer(formula = extro ~ open + agree + social + (1 | school/class), ## data = lmm.data) ## coef.est coef.se ## (Intercept) 60.24 4.01 ## open 0.01 0.00 ## agree -0.01 0.01 ## social 0.00 0.00 ## ## Error terms: ## Groups Name Std.Dev. ## class:school (Intercept) 2.86 ## school (Intercept) 9.69 ## Residual 0.98 ## --- ## number of obs: 1200, groups: class:school, 24; school, 6 ## AIC = 3568.6, DIC = 3508 ## deviance = 3531.1
Here the (1|school/class) says that we want to fit a mixed effect term for varying
intercepts 1| by schools, and for classes that are nested within schools.
Fit a varying slope model with lmer
But, what if we want to explore the effect of different student level indicators
as they vary across classrooms. Instead of fitting unique models by school (or school/class)
we can fit a varying slope model. Here we modify our random effect term to include
variables before the grouping terms: (1 + open|school/class) tells R to fit
a varying slope and varying intercept model for schools and classes nested within
schools, and to allow the slope of the open variable to vary by school.
MLexamp.9 <- lmer(extro ~ open + agree + social + (1 + open | school/class),
data = lmm.data)
display(MLexamp.9)
## lmer(formula = extro ~ open + agree + social + (1 + open | school/class),
## data = lmm.data)
## coef.est coef.se
## (Intercept) 60.26 3.93
## open 0.01 0.01
## agree -0.01 0.01
## social 0.00 0.00
##
## Error terms:
## Groups Name Std.Dev. Corr
## class:school (Intercept) 2.62
## open 0.01 1.00
## school (Intercept) 9.51
## open 0.00 1.00
## Residual 0.98
## ---
## number of obs: 1200, groups: class:school, 24; school, 6
## AIC = 3574.7, DIC = 3506
## deviance = 3529.3
Conclusion
Fitting mixed effect models and exploring group level variation is very easy within the R language and ecosystem. In future tutorials we will explore comparing across models, doing inference with mixed-effect models, and creating graphical representations of mixed effect models to understand their effects.
Appendix
print(sessionInfo(), locale = FALSE) ## R version 3.0.1 (2013-05-16) ## Platform: x86_64-w64-mingw32/x64 (64-bit) ## ## attached base packages: ## [1] stats graphics grDevices utils datasets methods base ## ## other attached packages: ## [1] plyr_1.8 arm_1.6-10 MASS_7.3-29 lme4_1.0-5 ## [5] Matrix_1.1-0 lattice_0.20-24 knitr_1.5 ## ## loaded via a namespace (and not attached): ## [1] abind_1.4-0 coda_0.16-1 evaluate_0.5.1 formatR_0.10 ## [5] grid_3.0.1 minqa_1.2.1 nlme_3.1-113 splines_3.0.1 ## [9] stringr_0.6.2 tools_3.0.1
[1] Examples include Gelman and Hill, Gelman et al. 2013, etc.
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