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Fitting Linear Mixed-Effects Models Using lme4

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Maximum likelihood or restricted maximum likelihood (REML) estimates of the parameters in linear mixed-effects models can be determined using the lmer function in the lme4 package for R. As for most model-fitting functions in R, the model is described in an lmer call by a formula, in this case including both fixed- and random-effects terms. The formula and data together determine a numerical representation of the model from which the profiled deviance or the profiled REML criterion can be evaluated as a function of some of the model parameters. The appropriate criterion is optimized, using one of the constrained optimization functions in R, to provide the parameter estimates. We describe the structure of the model, the steps in evaluating the profiled deviance or REML criterion, and the structure of classes or types that represents such a model. Sufficient detail is included to allow specialization of these structures by users who wish to write functions to fit specialized linear mixed models, such as models incorporating pedigrees or smoothing splines, that are not easily expressible in the formula language used by lmer.
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JSS Journal of Statistical Software
October 2015, Volume 67, Issue 1. doi: 10.18637/jss.v067.i01
Fitting Linear Mixed-Effects Models Using lme4
Douglas Bates
University of Wisconsin-Madison
Martin Mächler
ETH Zurich
Benjamin M. Bolker
McMaster University
Steven C. Walker
McMaster University
Abstract
Maximum likelihood or restricted maximum likelihood (REML) estimates of the pa-
rameters in linear mixed-effects models can be determined using the lmer function in the
lme4 package for R. As for most model-fitting functions in R, the model is described in
an lmer call by a formula, in this case including both fixed- and random-effects terms.
The formula and data together determine a numerical representation of the model from
which the profiled deviance or the profiled REML criterion can be evaluated as a function
of some of the model parameters. The appropriate criterion is optimized, using one of
the constrained optimization functions in R, to provide the parameter estimates. We de-
scribe the structure of the model, the steps in evaluating the profiled deviance or REML
criterion, and the structure of classes or types that represents such a model. Sufficient
detail is included to allow specialization of these structures by users who wish to write
functions to fit specialized linear mixed models, such as models incorporating pedigrees or
smoothing splines, that are not easily expressible in the formula language used by lmer.
Keywords: sparse matrix methods, linear mixed models, penalized least squares, Cholesky
decomposition.
1. Introduction
The lme4 package (Bates, Maechler, Bolker, and Walker 2015) for R(RCore Team 2015)
provides functions to fit and analyze linear mixed models, generalized linear mixed models
and nonlinear mixed models. In each of these names, the term “mixed” or, more fully, “mixed
effects”, denotes a model that incorporates both fixed- and random-effects terms in a linear
predictor expression from which the conditional mean of the response can be evaluated. In this
paper we describe the formulation and representation of linear mixed models. The techniques
2Linear Mixed Models with lme4
used for generalized linear and nonlinear mixed models will be described separately, in a
future paper.
At present, the main alternative to lme4 for mixed modeling in Ris the nlme package (Pin-
heiro, Bates, DebRoy, Sarkar, and RCore Team 2015). The main features distinguishing
lme4 from nlme are (1) more efficient linear algebra tools, giving improved performance on
large problems; (2) simpler syntax and more efficient implementation for fitting models with
crossed random effects; (3) the implementation of profile likelihood confidence intervals on
random-effects parameters; and (4) the ability to fit generalized linear mixed models (al-
though in this paper we restrict ourselves to linear mixed models). The main advantage of
nlme relative to lme4 is a user interface for fitting models with structure in the residuals (var-
ious forms of heteroscedasticity and autocorrelation) and in the random-effects covariance
matrices (e.g., compound symmetric models). With some extra effort, the computational
machinery of lme4 can be used to fit structured models that the basic lmer function cannot
handle (see Appendix A).
The development of general software for fitting mixed models remains an active area of re-
search with many open problems. Consequently, the lme4 package has evolved since it was
first released, and continues to improve as we learn more about mixed models. However,
we recognize the need to maintain stability and backward compatibility of lme4 so that it
continues to be broadly useful. In order to maintain stability while continuing to advance
mixed-model computation, we have developed several additional frameworks that draw on
the basic ideas of lme4 but modify its structure or implementation in various ways. These
descendants include the MixedModels package (Bates 2015) in Julia (Bezanson, Karpinski,
Shah, and Edelman 2012), the lme4pureR package (Bates and Walker 2013) in R, and the
flexLambda development branch of lme4. The current article is largely restricted to describing
the current stable version of the lme4 package (1.1-10), with Appendix Adescribing hooks
into the computational machinery that are designed for extension development. The gamm4
(Wood and Scheipl 2014) and blme (Dorie 2015;Chung, Rabe-Hesketh, Dorie, Gelman, and
Liu 2013) packages currently make use of these hooks.
Another goal of this article is to contrast the approach used by lme4 with previous formu-
lations of mixed models. The expressions for the profiled log-likelihood and profiled REML
(restricted maximum likelihood) criteria derived in Section 3.4 are similar to those presented
in Bates and DebRoy (2004) and, indeed, are closely related to “Henderson’s mixed-model
equations” (Henderson Jr. 1982). Nonetheless there are subtle but important changes in
the formulation of the model and in the structure of the resulting penalized least squares
(PLS) problem to be solved (Section 3.6). We derive the current version of the PLS problem
(Section 3.2) and contrast this result with earlier formulations (Section 3.5).
This article is organized into four main sections (Sections 2,3,4and 5), each of which
corresponds to one of the four largely separate modules that comprise lme4. Before describing
the details of each module, we describe the general form of the linear mixed model underlying
lme4 (Section 1.1); introduce the sleepstudy data that will be used as an example throughout
(Section 1.2); and broadly outline lme4’s modular structure (Section 1.3).
1.1. Linear mixed models
Just as a linear model is described by the distribution of a vector-valued random response
variable, Y, whose observed value is yobs, a linear mixed model is described by the distribution
Journal of Statistical Software 3
of two vector-valued random variables: Y, the response, and B, the vector of random effects.
In a linear model the distribution of Yis multivariate normal,
Y N (Xβ +o, σ2W1),(1)
where nis the dimension of the response vector, Wis a diagonal matrix of known prior
weights, βis a p-dimensional coefficient vector, Xis an n×pmodel matrix, and ois a vector
of known prior offset terms. The parameters of the model are the coefficients βand the scale
parameter σ.
In a linear mixed model it is the conditional distribution of Ygiven B=bthat has such a
form,
(Y|B =b) N (Xβ +Zb +o, σ2W1),(2)
where Zis the n×qmodel matrix for the q-dimensional vector-valued random-effects variable,
B, whose value we are fixing at b. The unconditional distribution of Bis also multivariate
normal with mean zero and a parameterized q×qvariance-covariance matrix, Σ,
B N (0,Σ).(3)
As a variance-covariance matrix, Σmust be positive semidefinite. It is convenient to express
the model in terms of a relative covariance factor,Λθ, which is a q×qmatrix, depending on
the variance-component parameter,θ, and generating the symmetric q×qvariance-covariance
matrix, Σ, according to
Σθ=σ2ΛθΛ>
θ,(4)
where σis the same scale factor as in the conditional distribution (2).
Although Equations 2,3, and 4fully describe the class of linear mixed models that lme4 can
fit, this terse description hides many important details. Before moving on to these details,
we make a few observations:
This formulation of linear mixed models allows for a relatively compact expression for
the profiled log-likelihood of θ(Section 3.4, Equation 34).
The matrices associated with random effects, Zand Λθ, typically have a sparse structure
with a sparsity pattern that encodes various model assumptions. Sections 2.3 and 3.7
provide details on these structures, and how to represent them efficiently.
The interface provided by lme4’s lmer function is slightly less general than the model
described by Equations 2,3, and 4. To take advantage of the entire range of possibili-
ties, one may use the modular functions (Sections 1.3 and Appendix A) or explore the
experimental flexLambda branch of lme4 on Github.
1.2. Example
Throughout our discussion of lme4, we will work with a data set on the average reaction time
per day for subjects in a sleep deprivation study (Belenky et al. 2003). On day 0 the subjects
had their normal amount of sleep. Starting that night they were restricted to 3 hours of sleep
per night. The response variable, Reaction, represents average reaction times in milliseconds
(ms) on a series of tests given each Day to each Subject (Figure 1),
4Linear Mixed Models with lme4
Days of sleep deprivation
Average reaction time (ms)
200
250
300
350
400
450
335
02468
309
330
02468
331
310
02468
351
333
02468
371
332
02468
372
369
02468
334
349
02468
352
370
02468
337
350
02468
200
250
300
350
400
450
308
Figure 1: Average reaction time versus days of sleep deprivation by subject. Subjects ordered
(from left to right starting on the top row) by increasing slope of subject-specific linear
regressions.
R> str(sleepstudy)
'data.frame': 180 obs. of 3 variables:
$ Reaction: num 250 259 251 321 357 ...
$Days :num 0123456789...
$ Subject : Factor w/ 18 levels "308","309","310",..: 1 1 1 1 1 1 1..
Each subject’s reaction time increases approximately linearly with the number of sleep-
deprived days. However, subjects also appear to vary in the slopes and intercepts of these
relationships, which suggests a model with random slopes and intercepts. As we shall see,
such a model may be fitted by minimizing the REML criterion (Equation 39) using
R> fm1 <- lmer(Reaction ~ Days + (Days | Subject), sleepstudy)
The estimates of the standard deviations of the random effects for the intercept and the slope
are 24.74 ms and 5.92 ms/day. The fixed-effects coefficients, β, are 251.4 ms and 10.47 ms/day
for the intercept and slope. In this model, one interpretation of these fixed effects is that they
are the estimated population mean values of the random intercept and slope (Section 2.2).
We have chosen the sleepstudy example because it is a relatively small and simple example
to illustrate the theory and practice underlying lmer. However, lmer is capable of fitting
more complex mixed models to larger data sets. For example, we direct the interested reader
to RShowDoc("lmerperf", package = "lme4") for examples that more thoroughly exercise
the performance capabilities of lmer.
Journal of Statistical Software 5
Module Rfunction Description
Formula
module
(Section 2)lFormula Accepts a mixed-model formula, data, and
other user inputs, and returns a list of objects
required to fit a linear mixed model.
Objective
function
module
(Section 3)mkLmerDevfun Accepts the results of lFormula and returns a
function to calculate the deviance (or
restricted deviance) as a function of the
covariance parameters, θ.
Optimization
module
(Section 4)optimizeLmer Accepts a deviance function returned by
mkLmerDevfun and returns the results of the
optimization of that deviance function.
Output
module
(Section 5)mkMerMod Accepts an optimized deviance function and
packages the results into a useful object.
Table 1: The high-level modular structure of lmer.
1.3. High-level modular structure
The lmer function is composed of four largely independent modules. In the first module, a
mixed-model formula is parsed and converted into the inputs required to specify a linear mixed
model (Section 2). The second module uses these inputs to construct an Rfunction which
takes the covariance parameters, θ, as arguments and returns negative twice the log profiled
likelihood or the REML criterion (Section 3). The third module optimizes this objective
function to produce maximum likelihood (ML) or REML estimates of θ(Section 4). Finally,
the fourth module provides utilities for interpreting the optimized model (Section 5).
To illustrate this modularity, we recreate the fm1 object by a series of four modular steps;
the formula module,
R> parsedFormula <- lFormula(formula = Reaction ~ Days + (Days | Subject),
+ data = sleepstudy)
the objective function module,
R> devianceFunction <- do.call(mkLmerDevfun, parsedFormula)
the optimization module,
R> optimizerOutput <- optimizeLmer(devianceFunction)
and the output module,
R> mkMerMod(rho = environment(devianceFunction), opt = optimizerOutput,
+ reTrms = parsedFormula$reTrms, fr = parsedFormula$fr)
6Linear Mixed Models with lme4
2. Formula module
2.1. Mixed-model formulas
Like most model-fitting functions in R,lmer takes as its first two arguments a formula spec-
ifying the model and the data with which to evaluate the formula. This second argument,
data, is optional but recommended and is usually the name of an Rdata frame. In the R
lm function for fitting linear models, formulas take the form resp ~ expr, where resp deter-
mines the response variable and expr is an expression that specifies the columns of the model
matrix. Formulas for the lmer function contain special random-effects terms,
R> resp ~ FEexpr + (REexpr1 | factor1) + (REexpr2 | factor2) + ...
where FEexpr is an expression determining the columns of the fixed-effects model matrix, X,
and the random-effects terms, (REexpr1 | factor1) and (REexpr2 | factor2), determine
both the random-effects model matrix, Z(Section 2.3), and the structure of the relative co-
variance factor, Λθ(Section 2.3). In principle, a mixed-model formula may contain arbitrarily
many random-effects terms, but in practice the number of such terms is typically low.
2.2. Understanding mixed-model formulas
Before describing the details of how lme4 parses mixed-model formulas (Section 2.3), we
provide an informal explanation and then some examples. Our discussion assumes familiarity
with the standard Rmodeling paradigm (Chambers 1993).
Each random-effects term is of the form (expr | factor). The expression expr is evaluated
as a linear model formula, producing a model matrix following the same rules used in standard
Rmodeling functions (e.g., lm or glm). The expression factor is evaluated as an Rfactor.
One way to think about the vertical bar operator is as a special kind of interaction between
the model matrix and the grouping factor. This interaction ensures that the columns of the
model matrix have different effects for each level of the grouping factor. What makes this a
special kind of interaction is that these effects are modeled as unobserved random variables,
rather than unknown fixed parameters. Much has been written about important practical
and philosophical differences between these two types of interactions (e.g., Henderson Jr.
1982;Gelman 2005). For example, the random-effects implementation of such interactions
can be used to obtain shrinkage estimates of regression coefficients (e.g., Efron and Morris
1977), or account for lack of independence in the residuals due to block structure or repeated
measurements (e.g., Laird and Ware 1982).
Table 2provides several examples of the right-hand-sides of mixed-model formulas. The
first example, (1 | g), is the simplest possible mixed-model formula, where each level of
the grouping factor, g, has its own random intercept. The mean and standard deviation of
these intercepts are parameters to be estimated. Our description of this model incorporates
any nonzero mean of the random effects as fixed-effects parameters. If one wishes to specify
that a random intercept has a priori known means, one may use the offset function as in
the second model in Table 2. This model contains no fixed effects, or more accurately the
fixed-effects model matrix, X, has zero columns and βhas length zero.
We may also construct models with multiple grouping factors. For example, if the observations
are grouped by g2, which is nested within g1, then the third formula in Table 2can be used
Journal of Statistical Software 7
Formula Alternative Meaning
(1 | g) 1 + (1 | g) Random intercept with
fixed mean.
0 + offset(o) + (1 | g) -1 + offset(o) + (1 | g) Random intercept with
a priori means.
(1 | g1/g2) (1 | g1) + (1 | g1:g2) Intercept varying among
g1 and g2 within g1.
(1 | g1) + (1 | g2) 1 + (1 | g1) + (1 | g2) Intercept varying among
g1 and g2.
x+(x|g) 1+x+(1+x|g) Correlated random
intercept and slope.
x+(x||g) 1+x+(1|g)+(0+x|g) Uncorrelated random
intercept and slope.
Table 2: Examples of the right-hand-sides of mixed-effects model formulas. The names of
grouping factors are denoted g,g1, and g2, and covariates and a priori known offsets as x
and o.
to model variation in the intercept. A common objective in mixed modeling is to account
for such nested (or hierarchical) structure. However, one of the most useful aspects of lme4
is that it can be used to fit random effects associated with non-nested grouping factors. For
example, suppose the data are grouped by fully crossing two factors, g1 and g2, then the
fourth formula in Table 2may be used. Such models are common in item response theory,
where subject and item factors are fully crossed (Doran, Bates, Bliese, and Dowling 2007).
In addition to varying intercepts, we may also have varying slopes (e.g., the sleepstudy data,
Section 1.2). The fifth example in Table 2gives a model where both the intercept and slope
vary among the levels of the grouping factor.
Specifying uncorrelated random effects
By default, lme4 assumes that all coefficients associated with the same random-effects term
are correlated. To specify an uncorrelated slope and intercept (for example), one may either
use double-bar notation, (x || g), or equivalently use multiple random-effects terms, x + (1
|g)+(0+x|g), as in the final example of Table 2. For example, if one examined the
results of model fm1 of the sleepstudy data (Section 1.2) using summary(fm1), one would see
that the estimated correlation between the slope for Days and the intercept is fairly low (0.066)
(See Section 5.2 below for more on how to extract the random-effects covariance matrix.) We
may use double-bar notation to fit a model that excludes a correlation parameter:
R> fm2 <- lmer(Reaction ~ Days + (Days || Subject), sleepstudy)
Although mixed models where the random slopes and intercepts are assumed independent
are commonly used to reduce the complexity of random-slopes models, they do have one
subtle drawback. Models in which the slopes and intercepts are allowed to have a nonzero
correlation (e.g., fm1) are invariant to additive shifts of the continuous predictor (Days in
this case). This invariance breaks down when the correlation is constrained to zero; any shift
in the predictor will necessarily lead to a change in the estimated correlation, and in the
likelihood and predictions of the model. For example, we can eliminate the correlation in
8Linear Mixed Models with lme4
Symbol Size
nLength of the response vector, Y.
pNumber of columns of the fixed-effects model matrix, X.
q=Pk
iqiNumber of columns of the random-effects model matrix, Z.
piNumber of columns of the raw model matrix, Xi.
`iNumber of levels of the grouping factor indices, ii.
qi=pi`iNumber of columns of the term-wise model matrix, Zi.
kNumber of random-effects terms.
mi=pi+1
2Number of covariance parameters for term i.
m=Pk
imiTotal number of covariance parameters.
Table 3: Dimensions of linear mixed models. The subscript i= 1, . . . , k denotes a specific
random-effects term.
Symbol Size Description
Xin×piRaw random-effects model matrix.
Jin×`iIndicator matrix of grouping factor indices.
Xij pi×1Column vector containing jth row of Xi.
Jij `i×1Column vector containing jth row of Ji.
iinVector of grouping factor indices.
Zin×qiTerm-wise random-effects model matrix.
θmCovariance parameters.
Tipi×piLower triangular template matrix.
Λiqi×qiTerm-wise relative covariance factor.
Table 4: Symbols used to describe the structure of the random-effects model matrix and the
relative covariance factor. The subscript i= 1, . . . , k denotes a specific random-effects term.
fm1 simply by adding an amount equal to the ratio of the estimated among-subject standard
deviations multiplied by the estimated correlation (i.e., σslopeintercept ·ρslope:intercept) to
the Days variable. The use of models such as fm2 should ideally be restricted to cases where
the predictor is measured on a ratio scale (i.e., the zero point on the scale is meaningful, not
just a location defined by convenience or convention), as is the case here.
2.3. Algebraic and computational account of mixed-model formulas
The fixed-effects terms of a mixed-model formula are parsed to produce the fixed-effects model
matrix, X, in the same way that the Rlm function generates model matrices. However, a
mixed-model formula incorporates k1random-effects terms of the form (r | f) as well.
These kterms are used to produce the random-effects model matrix, Z(Equation 2), and
the structure of the relative covariance factor, Λθ(Equation 4), which are matrices that
typically have a sparse structure. We now describe how one might construct these matrices
from the random-effects terms, considering first a single term, (r | f), and then generalizing
to multiple terms. Tables 3and 4summarize the matrices and vectors that determine the
structure of Zand Λθ.
The expression, r, is a linear model formula that evaluates to an Rmodel matrix, Xi, of
size n×pi, called the raw random-effects model matrix for term i. A term is said to be a
Journal of Statistical Software 9
scalar random-effects term when pi= 1, otherwise it is vector-valued. For a simple, scalar
random-effects term of the form (1 | f),Xiis the n×1matrix of ones, which implies a
random intercept model.
The expression fevaluates to an Rfactor, called the grouping factor, for the term. For the
ith term, we represent this factor mathematically with a vector iiof factor indices, which is
an n-vector of values from 1, . . . , `i.1Let Jibe the n×`imatrix of indicator columns for ii.
Using the Matrix package (Bates and Maechler 2015) in R, we may construct the transpose
of Jifrom a factor vector, f, by coercing fto a ‘sparseMatrix’ object. For example,
R> (f <- gl(3, 2))
[1]112233
Levels: 1 2 3
R> (Ji <- t(as(f, Class = "sparseMatrix")))
6 x 3 sparse Matrix of class "dgCMatrix"
123
[1,] 1 . .
[2,] 1 . .
[3,] . 1 .
[4,] . 1 .
[5,] . . 1
[6,] . . 1
When k > 1we order the random-effects terms so that `1`2≥ · · · ≥ `k; in general, this
ordering reduces “fill-in” (i.e., the proportion of elements that are zero in the lower triangle
of Λ>
θZ>W ZΛθ+Ibut not in the lower triangle of its left Cholesky factor, Lθ, described
below in Equation 18). This reduction in fill-in provides more efficient matrix operations
within the penalized least squares algorithm (Section 3.2).
Constructing the random-effects model matrix
The ith random-effects term contributes qi=`ipicolumns to the model matrix Z. We group
these columns into a matrix, Zi, which we refer to as the term-wise model matrix for the ith
term. Thus q, the number of columns in Zand the dimension of the random variable, B, is
q=
k
X
i=1
qi=
k
X
i=1
`ipi.(5)
Creating the matrix Zifrom Xiand Jiis a straightforward concept that is, nonetheless,
somewhat awkward to describe. Consider Zias being further decomposed into `iblocks of
picolumns. The rows in the first block are the rows of Ximultiplied by the 0/1 values in
1In practice, fixed-effects model matrices and random-effects terms are evaluated with respect to a model
frame, ensuring that any expressions for grouping factors have been coerced to factors and any unused levels
of these factors have been dropped. That is, `i, the number of levels in the grouping factor for the ith
random-effects term, is well-defined.
10 Linear Mixed Models with lme4
the first column of Jiand similarly for the subsequent blocks. With these definitions we
may define the term-wise random-effects model matrix, Zi, for the ith term as a transposed
Khatri-Rao product,
Zi= (J>
iX>
i)>=
J>
i1X>
i1
J>
i2X>
i2
.
.
.
J>
in X>
in
,(6)
where and are the Khatri-Rao2(Khatri and Rao 1968) and Kronecker products, and J>
ij
and X>
ij are row vectors of the jth rows of Jiand Xi. These rows correspond to the jth
sample in the response vector, Y, and thus jruns from 1, . . . , n. The Matrix package for R
contains a KhatriRao function, which can be used to form Zi. For example, if we begin with
a raw model matrix,
R> (Xi <- cbind(1, rep.int(c(-1, 1), 3L)))
[,1] [,2]
[1,] 1 -1
[2,] 1 1
[3,] 1 -1
[4,] 1 1
[5,] 1 -1
[6,] 1 1
then the term-wise random-effects model matrix is,
R> (Zi <- t(KhatriRao(t(Ji), t(Xi))))
6 x 6 sparse Matrix of class "dgCMatrix"
[1,] 1 -1 . . . .
[2,] 1 1 . . . .
[3,] . . 1 -1 . .
[4,] . . 1 1 . .
[5,] . . . . 1 -1
[6,] . . . . 1 1
In particular, for a simple, scalar term, Ziis exactly Ji, the matrix of indicator columns. For
other scalar terms, Ziis formed by element-wise multiplication of the single column of Xiby
each of the columns of Ji.
Because each Ziis generated from indicator columns, its cross-product, Z>
iZiis block-
diagonal consisting of `idiagonal blocks each of size pi.3Note that this means that when
2Note that the original definition of the Khatri-Rao product is more general than the definition used in the
Matrix package, which is the definition we use here.
3To see this, note that by the properties of Kronecker products we may write the cross-product matrix
Z>
iZias Pn
j=1 Jij J>
ij Xij X>
ij . Because Jij is a unit vector along a coordinate axis, the cross-product Jij J>
ij
is a pi×pimatrix of all zeros except for a single 1along the diagonal. Therefore, the cross-products, Xij X>
ij ,
will be added to one of the `iblocks of size pi×pialong the diagonal of Z>
iZi.
Journal of Statistical Software 11
k= 1 (i.e., there is only one random-effects term, and Zi=Z), Z>Zwill be block diag-
onal. These block-diagonal properties allow for more efficient sparse matrix computations
(Section 3.7).
The full random-effects model matrix, Z, is constructed from k1blocks,
Z=hZ1Z2. . . Zki.(7)
By transposing Equation 7and substituting in Equation 6, we may represent the structure
of the transposed random-effects model matrix as follows,
Z>=
sample 1 sample 2 . . . sample n
J11 X11 J12 X12 . . . J1nX1nterm 1
J21 X21 J22 X22 . . . J2nX2nterm 2
.
.
..
.
.....
.
..
.
.
.(8)
Note that the proportion of elements of Z>that are structural zeros is
Pk
i=1 pi(`i1)
Pk
i=1 pi
.(9)
Therefore, the sparsity of Z>increases with the number of grouping factor levels. As the
number of levels is often large in practice, it is essential for speed and efficiency to take
account of this sparsity, for example by using sparse matrix methods, when fitting mixed
models (Section 3.7).
Constructing the relative covariance factor
The q×qcovariance factor, Λθ, is a block diagonal matrix whose ith diagonal block, Λi, is of
size qi, i = 1, . . . , k. We refer to Λias the term-wise relative covariance factor. Furthermore,
Λiis a homogeneous block diagonal matrix with each of the `ilower-triangular blocks on
the diagonal being a copy of a pi×pilower-triangular template matrix,Ti. The covariance
parameter vector, θ, of length mi=pi+1
2, consists of the elements in the lower triangle of
Ti, i = 1, . . . , k. To provide a unique representation we require that the diagonal elements of
the Ti, i = 1, . . . , k be non-negative.
The template, Ti, can be constructed from the number pialone. In Rcode we denote pias
nc. For example, if we set nc <- 3, we could create the template for term ias
R> (rowIndices <- rep(1:nc, 1:nc))
[1]122333
R> (colIndices <- sequence(1:nc))
[1]112123
R> (template <- sparseMatrix(rowIndices, colIndices,
+ x = 1 * (rowIndices == colIndices)))
12 Linear Mixed Models with lme4
3 x 3 sparse Matrix of class "dgCMatrix"
[1,] 1 . .
[2,] 0 1 .
[3,] 0 0 1
Note that the rowIndices and colIndices fill the entire lower triangle, which contains the
initial values of the covariance parameter vector, θ,
R> (theta <- template@x)
[1]100101
(because the @x slot of the sparse matrix template is a numeric vector containing the nonzero
elements). This template contains three types of elements: structural zeros (denoted by .), off-
diagonal covariance parameters (initialized at 0), and diagonal variance parameters (initialized
at 1). The next step in the construction of the relative covariance factor is to repeat the
template once for each level of the grouping factor to construct a sparse block diagonal
matrix. For example, if we set the number of levels, `i, to two, nl <- 2, we could create the
transposed term-wise relative covariance factor, Λ>
i, using the .bdiag function in the Matrix
package,
R> (Lambdati <- .bdiag(rep(list(t(template)), nl)))
6 x 6 sparse Matrix of class "dgTMatrix"
[1,]100...
[2,].10...
[3,]..1...
[4,]...100
[5,]....10
[6,].....1
For a model with a single random-effects term, Λ>would be the initial transposed relative
covariance factor Λ>
θitself.
The transposed relative covariance factor, Λ>
θ, that arises from parsing the formula and data
is set at the initial value of the covariance parameters, θ. However, during model fitting, it
needs to be updated to a new θvalue at each iteration. This is achieved by constructing
a vector of indices, Lind, that identifies which elements of theta should be placed in which
elements of Lambdat,
R> LindTemplate <- rowIndices + nc * (colIndices - 1) - choose(colIndices, 2)
R> (Lind <- rep(LindTemplate, nl))
[1]124356124356
For example, if we randomly generate a new value for theta,
Journal of Statistical Software 13
R> thetanew <- round(runif(length(theta)), 1)
we may update Lambdat as follows,
R> Lambdati@x <- thetanew[Lind]
Section 3.6 describes the process of updating the relative covariance factor in more detail.
3. Objective function module
3.1. Model reformulation for improved computational stability
In our initial formulation of the linear mixed model (Equations 2,3, and 4), the covari-
ance parameter vector, θ, appears only in the marginal distribution of the random effects
(Equation 3). However, from the perspective of computational stability and efficiency, it is
advantageous to reformulate the model such that θappears only in the conditional distribu-
tion for the response vector given the random effects. Such a reformulation allows us to work
with singular covariance matrices, which regularly arise in practice (e.g., during intermediate
steps of the nonlinear optimizer, Section 4).
The reformulation is made by defining a spherical4random-effects variable, U, with distribu-
tion
U N (0, σ2Iq).(10)
If we set
B=ΛθU,(11)
then Bwill have the desired N(0,Σθ)distribution (Equation 3). Although it may seem more
natural to define Uin terms of Bwe must write the relationship as in Equation 11 to allow
for singular Λθ. The conditional distribution (Equation 2) of the response vector given the
random effects may now be reformulated as
(Y|U =u) N (µY|U=u, σ 2W1),(12)
where
µY|U =u=X β +ZΛθu+o(13)
is a vector of linear predictors, which can be interpreted as a conditional mean (or mode).
Similarly, we also define µU|Y=yobs as the conditional mean (or mode) of the spherical random
effects given the observed value of the response vector. Note also that we use the usymbol
throughout to represent a specific value of the random variable, U.
3.2. Penalized least squares
Our computational methods for maximum likelihood fitting of the linear mixed model involve
repeated applications of the PLS method. In particular, the PLS problem is to minimize the
penalized weighted residual sum-of-squares,
r2(θ,β,u) = ρ2(θ,β,u) + kuk2,(14)
4N(µ, σ2I)distributions are called “spherical” because contours of the probability density are spheres.
14 Linear Mixed Models with lme4
over "u
β#, where
ρ2(θ,β,u) =
W1/2hyobs µY|U =ui
2(15)
is the weighted residual sum-of-squares. This notation makes explicit the fact that r2and ρ2
both depend on θ,β, and u. The reason for the word “penalized” is that the term, kuk2, in
Equation 14 penalizes models with larger magnitude values of u.
In the so-called “pseudo-data” approach we write the penalized weighted residual sum-of-
squares as the squared length of a block matrix equation,
r2(θ,β,u) =
"W1/2(yobs o)
0#"W1/2ZΛθW1/2X
Iq0#"u
β#
2
.(16)
This pseudo-data approach shows that the PLS problem may also be thought of as a standard
least squares problem for an extended response vector, which implies that the minimizing
value, "µU|Y=yobs
b
βθ#, satisfies the normal equations,
"Λ>
θZ>W(yobs o)
X>W(yobs o)#="Λ>
θZ>W ZΛθ+IΛ>
θZ>W X
X>W ZΛθX>W X #"µU |Y =yobs
b
βθ#,(17)
where µU|Y=yobs is the conditional mean of Ugiven that Y=yobs. Note that this conditional
mean depends on θ, although we do not make this dependency explicit in order to reduce
notational clutter.
The cross-product matrix in Equation 17 can be Cholesky decomposed,
"Λ>
θZ>W ZΛθ+IΛ>
θZ>W X
X>W ZΛθX>W X #="Lθ0
R>
ZX R>
X#"L>
θRZX
0RX#.(18)
We may use this decomposition to rewrite the penalized weighted residual sum-of-squares as
r2(θ,β,u) = r2(θ) +
L>
θ(uµU|Y=yobs ) + RZ X (βb
βθ)
2+
RX(βb
βθ)
2,(19)
where we have simplified notation by writing r2(θ,b
βθ,µU|Y=yobs )as r2(θ). This is an im-
portant expression in the theory underlying lme4. It relates the penalized weighted residual
sum-of-squares, r2(θ,β,u), with its minimum value, r2(θ). This relationship is useful in
the next section where we integrate over the random effects, which is required for maximum
likelihood estimation.
3.3. Probability densities
The residual sums-of-squares discussed in the previous section can be used to express various
probability densities, which are required for maximum likelihood estimation of the linear
Journal of Statistical Software 15
mixed model5,
fY|U (yobs |u) = |W|1/2
(2πσ2)n/2exp "ρ2(θ,β,u)
2σ2#,(20)
fU(u) = 1
(2πσ2)q/2exp "− kuk2
2σ2#,(21)
fY,U(yobs,u) = |W|1/2
(2πσ2)(n+q)/2exp "r2(θ,β,u)
2σ2#,(22)
fU |Y (u|yobs) = fY,U(yobs,u)
fY(yobs),(23)
where
fY(yobs) = ZfY,U(yobs ,u)du.(24)
The log-likelihood to be maximized can therefore be expressed as
L(θ,β, σ2|yobs) = log fY(yobs).(25)
The integral in Equation 24 may be more explicitly written as
fY(yobs) = |W|1/2
(2πσ2)(n+q)/2exp
r2(θ)
RX(βb
βθ)
2
2σ2
Zexp
L>
θ(uµU|Y=yobs ) + RZ X (βb
βθ)
2
2σ2
du,
(26)
which can be evaluated with the change of variables,
v=L>
θ(uµU|Y=yobs ) + RZ X (βb
βθ).(27)
The Jacobian determinant of the transformation from uto vis |Lθ|. Therefore we are able
to write the integral as
fY(yobs) = |W|1/2
(2πσ2)(n+q)/2exp
r2(θ)
RX(βb
βθ)
2
2σ2
Zexp "− kvk2
2σ2#|Lθ|1dv,
(28)
which by the properties of exponential integrands becomes,
exp L(θ,β, σ2|yobs) = fY(yobs) = |W|1/2|Lθ|1
(2πσ2)n/2exp
r2(θ)
RX(βb
βθ)
2
2σ2
.(29)
5These expressions only technically hold at the observed value, yobs , of the response vector, Y.
16 Linear Mixed Models with lme4
3.4. Evaluating and profiling the deviance and the REML criterion
We are now in a position to understand why the formulation in Equations 2and 3is particu-
larly useful. We are able to explicitly profile βand σout of the log-likelihood (Equation 25), to
find a compact expression for the profiled deviance (negative twice the profiled log-likelihood)
and the profiled REML criterion as a function of the relative covariance parameters, θ, only.
Furthermore these criteria can be evaluated quickly and accurately.
To estimate the parameters, θ,β, and σ2, we minimize negative twice the log-likelihood,
which can be written as
2L(θ,β, σ2|yobs) = log |Lθ|2
|W|+nlog(2πσ2) + r2(θ)
σ2+
RX(βb
βθ)
2
σ2.(30)
It is very easy to profile out β, because it enters into the ML criterion only through the
final term, which is zero if β=b
βθ, where b
βθis found by solving the penalized least-squares
problem in Equation 16. Therefore we can write a partially profiled ML criterion as
2L(θ, σ2|yobs) = log |Lθ|2
|W|+nlog(2πσ2) + r2(θ)
σ2.(31)
This criterion is only partially profiled because it still depends on σ2. Differentiating this
criterion with respect to σ2and setting the result equal to zero yields,
0 = n
b
σ2
θ
r2(θ)
b
σ4
θ
,(32)
which leads to a maximum profiled likelihood estimate,
b
σ2
θ=r2(θ)
n.(33)
This estimate can be substituted into the partially profiled criterion to yield the fully profiled
ML criterion,
2L(θ|yobs) = log |Lθ|2
|W|+n"1 + log 2πr2(θ)
n!#.(34)
This expression for the profiled deviance depends only on θ. Although q, the number of
columns in Zand the size of Σθ, can be very large indeed, the dimension of θis small,
frequently less than 10. The lme4 package uses generic nonlinear optimizers (Section 4) to
optimize this expression over θto find its maximum likelihood estimate.
The REML criterion
The REML criterion can be obtained by integrating the marginal density for Ywith respect
to the fixed effects (Laird and Ware 1982),
ZfY(yobs)dβ=|W|1/2|Lθ|1
(2πσ2)n/2exp "r2(θ)
2σ2#Zexp
RX(βb
βθ)
2
2σ2
dβ,(35)
Journal of Statistical Software 17
which can be evaluated with the change of variables,
v=RX(βb
βθ).(36)
The Jacobian determinant of the transformation from βto vis |RX|. Therefore we are able
to write the integral as
ZfY(yobs)dβ=|W|1/2|Lθ|1
(2πσ2)n/2exp "r2(θ)
2σ2#Zexp "− kvk2
2σ2#|RX|1dv,(37)
which simplifies to,
ZfY(yobs)dβ=|W|1/2|Lθ|1|RX|1
(2πσ2)(np)/2exp "r2(θ)
2σ2#.(38)
Minus twice the log of this integral is the (unprofiled) REML criterion,
2LR(θ, σ2|yobs) = log |Lθ|2|RX|2
|W|+ (np) log(2πσ2) + r2(θ)
σ2.(39)
Note that because βgets integrated out, the REML criterion cannot be used to find a point
estimate of β. However, we follow others in using the maximum likelihood estimate, b
βb
θ, at
the optimum value of θ=b
θ. The REML estimate of σ2is,
b
σ2
θ=r2(θ)
np,(40)
which leads to the profiled REML criterion,
2LR(θ|yobs) = log |Lθ|2|RX|2
|W|+ (np)"1 + log 2πr2(θ)
np!#.(41)
3.5. Changes relative to previous formulations
We compare the PLS problem as formulated in Section 3.2 with earlier versions and describe
why we use this version. What have become known as “Henderson’s mixed-model equations”
are given as Equation 6 of Henderson Jr. (1982) and would be expressed as
"X>X2X>Z2
Z>X2Z>Z2+Σ1#" b
βθ
µB|Y=yobs #="X>yobs2
Z>yobs2#,(42)
in our notation (ignoring weights and offsets, without loss of generality). The matrix written
as Rin Henderson Jr. (1982) is σ2Inin our formulation of the model.
Bates and DebRoy (2004) modified the PLS equations to
"Z>Z+Z>X
X>Z X>X#"µB|Y=yobs
b
βθ#="X>yobs
Z>yobs #,(43)
18 Linear Mixed Models with lme4
where θ=Λ>
θΛθ1=σ2Σ1is the relative precision matrix for a given value of θ. They
also showed that the profiled log-likelihood can be expressed (on the deviance scale) as
2L(θ) = log |Z>Z+|
||!+n"1 + log 2πr2(θ)
n!#.(44)
The primary difference between Equation 42 and Equation 43 is the order of the blocks in the
system matrix. The PLS problem can be solved using a Cholesky factor of the system matrix
with the blocks in either order. The advantage of using the arrangement in Equation 43
is to allow for evaluation of the profiled log-likelihood. To evaluate |Z>Z+|from the
Cholesky factor that block must be in the upper-left corner, not the lower right. Also, Zis
sparse whereas Xis usually dense. It is straightforward to exploit the sparsity of Z>Zin the
Cholesky factorization when the block containing this matrix is the first block to be factored.
If X>Xis the first block to be factored, it is much more difficult to preserve sparsity.
The main change from the formulation in Bates and DebRoy (2004) to the current formulation
is the use of a relative covariance factor, Λθ, instead of a relative precision matrix, θ, and
solving for the mean of U|Y =yobs instead of the mean of B|Y =yobs. This change improves
stability, because the solution to the PLS problem in Section 3.2 is well-defined when Λθis
singular whereas the formulation in Equation 43 cannot be used in these cases because θ
does not exist.
It is important to allow for Λθto be singular because situations where the parameter esti-
mates, b
θ, produce a singular Λb
θdo occur in practice. And even if the parameter estimates
do not correspond to a singular Λθ, it may be desirable to evaluate the estimation criterion
at such values during the course of the numerical optimization of the criterion.
Bates and DebRoy (2004) also provided expressions for the gradient of the profiled log-
likelihood expressed as Equation 44. These expressions can be translated into the current
formulation. From Equation 34 we can see that (again ignoring weights),
(2L(θ)) = log(|Lθ|2) + nlog(r2(θ))
=log(|Λ>
θZ>ZΛθ+I|) + nr2(θ)/r2(θ)
=log(|Λ>
θZ>ZΛθ+I|) + r2(θ)/(c
σ2).
(45)
The first gradient is easy to express but difficult to evaluate for the general model. The
individual elements of this gradient are
log(|Λ>
θZ>ZΛθ+I|)
∂θi
= tr
Λ>
θZ>ZΛθ
∂θiΛ>
θZ>ZΛθ+I1
= tr "LθL>
θ1 Λ>
θZ>ZΛθ
∂θi
+Λ>
θ
∂θi
Z>ZΛθ!#.
(46)
The second gradient term can be expressed as a linear function of the residual, with individual
elements of the form
∂r2(θ)
∂θi
=2u>Λ>
θ
∂θi
Z>yZΛθuXb
βθ,(47)
Journal of Statistical Software 19
Name/description Pseudocode Math Type
Mapping from covariance
parameters to relative
covariance factor
mapping function
Response vector yyobs (Section 1.1) double vector
Fixed-effects model matrix XX(Equation 2) double denseamatrix
Transposed random-effects
model matrix
Zt Z>(Equation 2) double sparse matrix
Square-root weights matrix sqrtW W1/2(Equation 2) double diagonal matrix
Offset offset o(Equation 2) double vector
aIn previous versions of lme4 a sparse Xmatrix, useful for models with categorical fixed-effect predictors
with many levels, could be specified; this feature is not currently available.
Table 5: Inputs into a linear mixed model.
using the results of Golub and Pereyra (1973). Although we do not use these results in lme4,
they are used for certain model types in the MixedModels package for Julia and do provide
improved performance.
3.6. Penalized least squares algorithm
For efficiency, in lme4 itself, PLS is implemented in compiled C++ code using the Eigen
(Guennebaud, Jacob, and and others 2015) templated C++ package for numerical linear
algebra. Here however, in order to improve readability we describe a version in pure R.
Section 3.7 provides greater detail on the techniques and concepts for computational efficiency,
which is important in cases where the nonlinear optimizer (Section 4) requires many iterations.
The PLS algorithm takes a vector of covariance parameters, θ, as inputs and returns the
profiled deviance (Equation 34) or the REML criterion (Equation 41). This PLS algorithm
consists of four main steps:
1. Update the relative covariance factor.
2. Solve the normal equations.
3. Update the linear predictor and residuals.
4. Compute and return the profiled deviance.
PLS also requires the objects described in Table 5, which define the structure of the model.
These objects do not get updated during the PLS iterations, and so it is useful to store various
matrix products involving them (Table 6). Table 7lists the objects that do get updated over
the PLS iterations. The symbols in this table correspond to a version of lme4 that is imple-
mented entirely in R(i.e., no compiled code as in lme4 itself). This implementation is called
lme4pureR and is currently available on Github (https://github.com/lme4/lme4pureR/).
PLS step I: Update relative covariance factor
The first step of PLS is to update the relative covariance factor, Λθ, from the current value
of the covariance parameter vector, θ. The updated Λθis then used to update the random-
20 Linear Mixed Models with lme4
Pseudocode Math
ZtW Z>W1/2
ZtWy Z>W yobs
ZtWX Z>W X
XtWX X>W X
XtWy X>W yobs
Table 6: Constant symbols involved in penalized least squares.
Name/description Pseudocode Math Type
Relative covariance factor lambda Λθ(Equation 4) double sparse lower-
triangular matrix
Random-effects Cholesky
factor
LLθ(Equation 18) double sparse triangu-
lar matrix
Intermediate vector in the
solution of the normal
equations
cu cu(Equation 48) double vector
Block in the full Cholesky
factor
RZX RZX (Equation 18) double dense matrix
Cross-product of the
fixed-effects Cholesky factor
RXtRX R>
XRX(Equation 50) double dense matrix
Fixed-effects coefficients beta β(Equation 2) double vector
Spherical conditional modes uu(Section 3.1) double vector
Non-spherical conditional
modes
bb(Equation 2) double vector
Linear predictor mu µY|U =u(Equation 13) double vector
Weighted residuals wtres W1/2(yobs µ)double vector
Penalized weighted residual
sum-of-squares
pwrss r2(θ)(Equation 19) double
Twice the log determinant
random-effects Cholesky
factor
logDet log |Lθ|2double
Table 7: Quantities updated during an evaluation of the linear mixed model objective function.
effects Cholesky factor, Lθ(Equation 18). The mapping from the covariance parameters to
the relative covariance factor can take many forms, but in general involves a function that
takes θinto the values of the nonzero elements of Λθ.
If Λθis stored as an object of class ‘dgCMatrix’ from the Matrix package for R, then we may
update Λθfrom θby,
R> Lambdat@x[] <- mapping(theta)
where mapping is an Rfunction that both accepts and returns a numeric vector. The nonzero
elements of sparse matrix classes in Matrix are stored in a slot called x.
In the current version of lme4 (1.1-10) the mapping from θto Λθis represented as an R
integer vector, Lind, of indices, so that
Journal of Statistical Software 21
R> mapping <- function(theta) theta[Lind]
The index vector Lind is computed during the model setup and stored in the function’s
environment. Continuing the example from Section 2.3, consider a new value for theta,
R> thetanew <- c(1, -0.1, 2, 0.1, -0.2, 3)
To put these values in the appropriate elements in Lambdati, we use mapping,
R> Lambdati@x[] <- mapping(thetanew)
R> Lambdati
6 x 6 sparse Matrix of class "dgTMatrix"
[1,] 1 -0.1 2.0 . . .
[2,] . 0.1 -0.2 . . .
[3,] . . 3.0 . . .
[4,] . . . 1 -0.1 2.0
[5,] . . . . 0.1 -0.2
[6,] . . . . . 3.0
This Lind approach can be useful for extending the capabilities of lme4 by using the modular
approach to fitting mixed models. For example, Appendix A.1 shows how to use Lind to fit
a model where two random slopes are uncorrelated, but both slopes are correlated with an
intercept.
The mapping from the covariance parameters to the relative covariance factor is treated
differently in other implementations of the lme4 approach to linear mixed models. At the other
extreme, the flexLambda branch of lme4 and the lme4pureR package provide the capabilities
for a completely general mapping. This added flexibility has the advantage of allowing a
much wider variety of models (e.g., compound symmetry, auto-regression). However, the
disadvantage of this approach is that it becomes possible to fit a much wider variety of ill-
posed models. Finally, if one would like such added flexibility with the current stable version
of lme4, it is always possible to use the modular approach to wrap the Lind-based deviance
function in a general mapping function taking a parameter to be optimized, say φ, into θ.
However, this approach is likely to be inefficient in many cases.
The update method from the Matrix package efficiently updates the random-effects Cholesky
factor, Lθ, from a new value of θand the updated Λθ.
R> L <- update(L, Lambdat %*% ZtW, mult = 1)
The mult = 1 argument corresponds to the addition of the identity matrix to the upper-left
block on the left-hand-side of Equation 18.
PLS step II: Solve normal equations
With the new covariance parameters installed in Λθ, the next step is to solve the normal
equations (Equation 17) for the current estimate of the fixed-effects coefficients, b
βθ, and the
22 Linear Mixed Models with lme4
conditional mode, µU |Y=yobs . We solve these equations using a sparse Cholesky factorization
(Equation 18). In a complex model fit to a large data set, the dominant calculation in the
evaluation of the profiled deviance (Equation 34) or REML criterion (Equation 41) is this
sparse Cholesky factorization (Equation 18). The factorization is performed in two phases: a
symbolic phase and a numeric phase. The symbolic phase, in which the fill-reducing permu-
tation Pis determined along with the positions of the nonzeros in Lθ, does not depend on
the value of θ. It only depends on the positions of the nonzeros in ZΛθ. The numeric phase
uses θto determine the numeric values of the nonzeros in Lθ. Using this factorization, the
solution proceeds by the following steps,
Lθcu=PΛ>
θZ>W y (48)
LθRZX =PΛ>
θZ>W X (49)
R>
XRX=X>W X R>
ZX RZ X (50)
R>
XRXb
βθ=X>W y RZX cu(51)
L>
θP u =cuRZX b
βθ(52)
which can be solved using the Matrix package with,
R> cu[] <- as.vector(solve(L, solve(L, Lambdat %*% ZtWy, system = "P"),
+ system = "L"))
R> RZX[] <- as.vector(solve(L, solve(L, Lambdat %*% ZtWX, system = "P"),
+ system = "L"))
R> RXtRX <- as(XtWX - crossprod(RZX), "dpoMatrix")
R> beta[] <- as.vector(solve(RXtRX, XtWy - crossprod(RZX, cu)))
R> u[] <- as.vector(solve(L, solve(L, cu - RZX %*% beta,
+ system = "Lt"), system = "Pt"))
Notice the nested calls to solve. The inner calls of the first two assignments determine
and apply the permutation matrix (system = "P"), whereas the outer calls actually solve
the equation (system = "L"). In the assignment to u[], the nesting is reversed in order to
return to the original permutation.
PLS step III: Update linear predictor and residuals
The next step is to compute the linear predictor, µY|U (Equation 13), and the weighted resid-
uals with new values for b
βθand µB|Y=yobs . In lme4pureR these quantities can be computed
as
R> b[] <- as.vector(crossprod(Lambdat, u))
R> mu[] <- as.vector(crossprod(Zt, b) + X %*% beta + offset)
R> wtres <- sqrtW * (y - mu)
where brepresents the current estimate of µB|Y=yobs .
PLS step IV: Compute profiled deviance
Finally, the updated linear predictor and weighted residuals can be used to compute the
profiled deviance (or REML criterion), which in lme4pureR proceeds as
Journal of Statistical Software 23
R> pwrss <- sum(wtres^2) + sum(u^2)
R> logDet <- 2 * determinant(L, logarithm = TRUE)$modulus
R> if (REML) logDet <- logDet + determinant(RXtRX, logarithm = TRUE)$modulus
R> attributes(logDet) <- NULL
R> profDev <- logDet + degFree * (1 + log(2 * pi * pwrss) - log(degFree))
The profiled deviance consists of three components: (1) log-determinant(s) of Cholesky factor-
ization (logDet), (2) the degrees of freedom (degFree), and the penalized weighted residual
sum-of-squares (pwrss).
3.7. Sparse matrix methods
In fitting linear mixed models, an instance of the PLS problem (17) must be solved at each
evaluation of the objective function during the optimization (Section 4) with respect to θ.
Because this operation must be performed many times it is worthwhile considering how to
provide effective evaluation methods for objects and calculations involving the sparse matrices
associated with random-effects terms (Sections 2.3).
The CHOLMOD library of Cfunctions (Chen, Davis, Hager, and Rajamanickam 2008), on
which the sparse matrix capabilities of the Matrix package for Rand the sparse Cholesky
factorization in Julia are based, allows for separation of the symbolic and numeric phases.
Thus we perform the symbolic phase as part of establishing the structure representing the
model (Section 2). Furthermore, because CHOLMOD functions allow for updating Lθdirectly
from the matrix Λ>
θZ>without actually forming Λ>
θZ>ZΛθ+Iwe generate and store Z>
instead of Z(note that we have ignored the weights matrix, W, for simplicity). We can
update Λ>
θZ>directly from θwithout forming Λθand multiplying two sparse matrices.
Although such a direct approach is used in the MixedModels package for Julia, in lme4 we
first update Λ>
θthen form the sparse product Λ>
θZ>. A third alternative, which we employ
in lme4pureR, is to compute and save the cross-products of the model matrices, Xand Z,
and the response, y, before starting the iterations. To allow for case weights, we save the
products X>W X ,X>W y,Z>W X,Z>W y and Z>W Z (see Table 6).
We wish to use structures and algorithms that allow us to take a new value of θand evaluate
the Lθ(Equation 18) efficiently. The key to doing so is the special structure of Λ>
θZ>W1/2.
To understand why this matrix, and not its transpose, is of interest we describe the sparse
matrix structures used in Julia and in the Matrix package for R.
Dense matrices are stored in Rand in Julia as a one-dimensional array of contiguous storage
locations addressed in column-major order. This means that elements in the same column
are in adjacent storage locations whereas elements in the same row can be widely separated
in memory. For this reason, algorithms that work column-wise are preferred to those that
work row-wise.
Although a sparse matrix can be stored in a triplet format, where the row position, column
position and element value of the nonzeros are recorded as triplets, the preferred storage
forms for actual computation with sparse matrices are compressed sparse column (CSC) or
compressed sparse row (CSR, Davis 2006, Chapter 2). The CHOLMOD (and, more generally,
the SuiteSparse package of Clibraries; Davis et al. 2015) uses the CSC storage format. In
this format the nonzero elements in a column are in adjacent storage locations and access to
all the elements in a column is much easier than access to those in a row (similar to dense
24 Linear Mixed Models with lme4
matrices stored in column-major order).
The matrices Zand ZΛθhave the property that the number of nonzeros in each row, Pk
i=1 pi,
is constant. For CSC matrices we want consistency in the columns rather than the rows, which
is why we work with Z>and Λ>
θZ>rather than their transposes.
An arbitrary m×nsparse matrix in CSC format is expressed as two vectors of indices, the row
indices and the column pointers, and a numeric vector of the nonzero values. The elements
of the row indices and the nonzeros are aligned and are ordered first by increasing column
number then by increasing row number within column. The column pointers are a vector of
size n+ 1 giving the location of the start of each column in the vectors of row indices and
nonzeros.
4. Nonlinear optimization module
The objective function module produces a function which implements the penalized least
squares algorithm for a particular mixed model. The next step is to optimize this function
over the covariance parameters, θ, which is a nonlinear optimization problem. The lme4
package separates the efficient computational linear algebra required to compute profiled
likelihoods and deviances for a given value of θfrom the nonlinear optimization algorithms,
which use general-purpose nonlinear optimizers.
One benefit of this separation is that it allows for experimentation with different nonlinear
optimizers. Throughout the development of lme4, the default optimizers and control param-
eters have changed in response to feedback from users about both efficiency and convergence
properties. lme4 incorporates two built-in optimization choices, an implementation of the
Nelder-Mead simplex algorithm adapted from Steven Johnson’s NLopt Clibrary (Johnson
2014) and a wrapper for Powell’s BOBYQA algorithm, implemented in the minqa package
(Bates, Mullen, Nash, and Varadhan 2014) as a wrapper around Powell’s original Fortran
code (Powell 2009). More generally, lme4 allows any user-specified optimizer that (1) can
work with an objective function (i.e., does not require a gradient function to be specified),
(2) allows box constraints on the parameters, and (3) conforms to some simple rules about
argument names and structure of the output. An internal wrapper allows the use of the op-
timx package (Nash and Varadhan 2011), although the only optimizers provided via optimx
that satisfy the constraints above are the nlminb and L-BFGS-B algorithms that are them-
selves wrappers around the versions provided in base R. Several other algorithms from Steven
Johnson’s NLopt package are also available via the nloptr wrapper package (e.g., alternate
implementations of Nelder-Mead and BOBYQA, and Powell’s COBYLA algorithm).
This flexibility assists with diagnosing convergence problems – it is easy to switch among
several algorithms to determine whether the problem lies in a failure of the nonlinear opti-
mization stage, as opposed to a case of model misspecification or unidentifiability or a problem
with the underlying PLS algorithm. To date we have only observed PLS failures, which arise
if X>W X R>
ZX RZ X becomes singular during an evaluation of the objective function, with
badly scaled problems (i.e., problems with continuous predictors that take a very large or
very small numerical range of values).
The requirement for optimizers that can handle box constraints stems from our decision to
parameterize the variance-covariance matrix in a constrained space, in order to allow for
singular fits. In contrast to the approach taken in the nlme package (Pinheiro et al. 2015),
Journal of Statistical Software 25
which goes to some lengths to use an unconstrained variance-covariance parameterization
(the log-Cholesky parameterization; Pinheiro and Bates 1996), we instead use the Cholesky
parameterization but require the elements of θcorresponding to the diagonal elements of the
Cholesky factor to be non-negative. With these constraints, the variance-covariance matrix is
singular if and only if any of the diagonal elements is exactly zero. Singular fits are common
in practical data-analysis situations, especially with small- to medium-sized data sets and
complex variance-covariance models, so being able to fit a singular model is an advantage:
when the best fitting model lies on the boundary of a constrained space, approaches that try
to remove the constraints by fitting parameters on a transformed scale will often give rise
to convergence warnings as the algorithm tries to find a maximum on an asymptotically flat
surface (Bolker et al. 2013).
In principle the likelihood surfaces we are trying to optimize over are smooth, but in practice
using gradient information in optimization may or may not be worth the effort. In special
cases, we have a closed-form solution for the gradients (Equations 4547), but in general
we will have to approximate them by finite differences, which is expensive and has limited
accuracy. (We have considered using automatic differentiation approaches to compute the
gradients more efficiently, but this strategy requires a great deal of additional machinery, and
would have drawbacks in terms of memory requirements for large problems.) This is the
primary reason for our switching to derivative-free optimizers such as BOBYQA and Nelder-
Mead in the current version of lme4, although as discussed above derivative-based optimizers
based on finite differencing are available as an alternative.
There is most likely further room for improvement in the nonlinear optimization module; for
example, some speed-up could be gained by using parallel implementations of derivative-free
optimizers that evaluated several trial points at once (Klein and Neira 2013). In practice
users most often have optimization difficulties with poorly scaled or centered data – we are
working to implement appropriate diagnostic tests and warnings to detect these situations.
5. Output module
Here we describe some of the methods in lme4 for exploring fitted linear mixed models (Ta-
ble 8), which are represented as objects of the S4 class ‘lmerMod. We begin by describing the
theory underlying these methods (Section 5.1) and then continue the sleepstudy example
introduced in Section 1.2 to illustrate these ideas in practice.
5.1. Theory underlying the output module
Covariance matrix of the fixed-effect coefficients
In the lm function, the variance-covariance matrix of the coefficients is the inverse of the
cross-product of the model matrix, times the residual variance (Chambers 1993). The inverse
cross-product matrix is computed by first inverting the upper triangular matrix resulting from
the QR decomposition of the model matrix, and then taking its cross-product,
Varθ "µU |Y=yobs
ˆ
β#!=σ2"L>
θRZX
0RX#1"Lθ0
R>
ZX R>
X#1
.(53)
26 Linear Mixed Models with lme4
Because of normality, the marginal covariance matrix of ˆ
βis just the lower-right p-by-pblock
of Varθ "µU |Y=yobs
ˆ
β#!.This lower-right block is
Varθ (ˆ
β) = σ2R1
X(R>
X)1,(54)
which follows from the Schur complement identity (Horn and Zhang 2005, Theorem 1.2).
This matrix can be extracted from a fitted ‘merMod’ object as
R> RX <- getME(fm1, "RX")
R> sigma2 <- sigma(fm1)^2
R> sigma2 * chol2inv(RX)
[,1] [,2]
[1,] 46.574573 -1.451097
[2,] -1.451097 2.389463
which could be computed with lme4 as vcov(fm1).
The square-root diagonal of this covariance matrix contains the estimates of the standard
errors of fixed-effects coefficients. These standard errors are used to construct Wald confidence
intervals with confint(., method = "Wald"). Such confidence intervals are approximate,
and depend on the assumption that the likelihood profile of the fixed effects is linear on the
ζscale.
Profiling
The theory of likelihood profiles is straightforward: the deviance (or likelihood) profile,
2Lp(), for a focal model parameter Pis the minimum value of the deviance conditioned
on a particular value of P. For each parameter of interest, our goal is to evaluate the de-
viance profile for many points – optimizing over all of the non-focal parameters each time –
over a wide enough range and with high enough resolution to evaluate the shape of the profile
(in particular, whether it is quadratic, which would allow use of Wald confidence intervals and
tests) and to find the values of Psuch that 2Lp(P) = 2L(b
P) + χ2(1 α), which represent
the profile confidence intervals. While profile confidence intervals rely on the asymptotic dis-
tribution of the minimum deviance, this is a much weaker assumption than the log-quadratic
likelihood surface required by Wald tests.
An additional challenge in profiling arises when we want to compute profiles for quantities
of interest that are not parameters of our PLS function. We have two problems in using the
deviance function defined above to profile linear mixed models. First, a parameterization of
the random-effects variance-covariance matrix in terms of standard deviations and correla-
tions, or variances and covariances, is much more familiar to users, and much more relevant
as output of a statistical model, than the parameters, θ, of the relative covariance factor –
users are interested in inferences on variances or standard deviations, not on θ. Second, the
fixed-effects coefficients and the residual standard deviation, both of which are also of interest
to users, are profiled out (in the sense used above) of the deviance function (Section 3.4), so
we have to find a strategy for estimating the deviance for values of βand σ2other than the
profiled values.
Journal of Statistical Software 27
To handle the first problem we create a new version of the deviance function that first takes
a vector of standard deviations (and correlations), and a value of the residual standard devi-
ation, maps them to a θvector, and then computes the PLS as before; it uses the specified
residual standard deviation rather than the PLS estimate of the standard deviation (Equa-
tion 33) in the deviance calculation. We compute a profile likelihood for the fixed-effects
parameters, which are profiled out of the deviance calculation, by adding an offset to the
linear predictor for the focal element of β. The resulting function is not useful for general
nonlinear optimization – one can easily wander into parameter regimes corresponding to in-
feasible (non-positive semidefinite) variance-covariance matrices – but it serves for likelihood
profiling, where one focal parameter is varied at a time and the optimization over the other
parameters is likely to start close to an optimum.
In practice, the profile method systematically varies the parameters in a model, assessing
the best possible fit that can be obtained with one parameter fixed at a specific value and
comparing this fit to the globally optimal fit, which is the original model fit that allowed all
the parameters to vary. The models are compared according to the change in the deviance,
which is the likelihood ratio test statistic. We apply a signed square root transformation
to this statistic and plot the resulting function, which we call the profile zeta function or
ζ, versus the parameter value. The signed aspect of this transformation means that ζis
positive where the deviation from the parameter estimate is positive and negative otherwise,
leading to a monotonically increasing function which is zero at the global optimum. A ζ
value can be compared to the quantiles of the standard normal distribution. For example,
a 95% profile deviance confidence interval on the parameter consists of the values for which
1.96 < ζ < 1.96. Because the process of profiling a fitted model can be computationally
intensive – it involves refitting the model many times – one should exercise caution with
complex models fit to large data sets.
The standard approach to this computational challenge is to compute ζat a limited number
of parameter values, and to fill in the gaps by fitting an interpolation spline. Often one is able
to invert the spline to obtain a function from ζto the focal parameter, which is necessary
in order to construct profile confidence intervals. However, even if the points themselves are
monotonic, it is possible to obtain non-monotonic interpolation curves. In such a case, lme4
falls back to linear interpolation, with a warning.
The last part of the technical specification for computing profiles is deciding on a strategy
for choosing points to sample. In one way or another one wants to start at the estimated
value for each parameter and work outward either until a constraint is reached (i.e., a value
of 0 for a standard deviation parameter or a value of ±1for a correlation parameter), or
until a sufficiently large deviation from the minimum deviance is attained. lme4’s profiler
chooses a cutoff φbased on the 1αmax critical value of the χ2distribution with a number
of degrees of freedom equal to the total number of parameters in the model, where αmax is
set to 0.05 by default. The profile computation initially adjusts the focal parameter piby
an amount = 1.01ˆpifrom its ML or REML estimate ˆpi(or by = 0.001 if ˆpiis zero, as
in the case of a singular variance-covariance model). The local slope of the likelihood profile
(ζ(ˆpi+)ζ(ˆpi))/ is used to pick the next point to evaluate, extrapolating the local slope
to find a new that would be expected to correspond to the desired step size ζ(equal to
φ/8by default, so that 16 points would be used to cover the profile in both directions if the
log-likelihood surface were exactly quadratic). Some fail-safe testing is done to ensure that
the step chosen is always positive, and less than a maximum; if a deviance is ever detected
28 Linear Mixed Models with lme4
that is lower than that of the ML deviance, which can occur if the initial fit was wrong due
to numerical problems, the profiler issues an error and stops.
Parametric bootstrapping
To avoid the asymptotic assumptions of the likelihood ratio test, at the cost of greatly in-
creased computation time, one can estimate confidence intervals by parametric bootstrapping
– that is, by simulating data from the fitted model, refitting the model, and extracting the
new estimated parameters (or any other quantities of interest). This task is quite straightfor-
ward, since there is already a simulate method, and a refit function which re-estimates the
(RE)ML parameters for new data, starting from the previous (RE)ML estimates and re-using
the previously computed model structures (including the fill-reducing permutation) for effi-
ciency. The bootMer function is thus a fairly thin wrapper around a simulate/refit loop,
with a small amount of additional logic for parallel computation and error-catching. (Some
of the ideas of bootMer are adapted from merBoot (Sánchez-Espigares and Ocaña 2009), a
more ambitious framework for bootstrapping lme4 model fits which unfortunately seems to
be unavailable at present.)
Conditional variances of random effects
It is useful to clarify that the conditional covariance concept in lme4 is based on a simplifi-
cation of the linear mixed model. In particular, we simplify the model by assuming that the
quantities, β,Λθ, and σ, are known (i.e., set at their estimated values). In fact, the only
way to define the conditional covariance is at fixed parameter values. Our approximation
here is using the estimated parameter values for the unknown “true” parameter values. In
this simplified case, Uis the only quantity in the statistical model that is both random and
unknown.
Given this simplified linear mixed model, a standard result in Bayesian linear regression mod-
eling (Gelman et al. 2013) implies that the conditional distribution of the spherical random
effects given the observed response vector is Gaussian,
(U|Y =yobs) N (µU|Y=yobs ,b
σ2V),(55)
where
V=Λ>
b
θZ>W ZΛb
θ+Iq1=L1
b
θ>L1
b
θ(56)
is the unscaled conditional variance and
µU|Y=yobs =VΛ>
b
θZ>Wyobs oXb
β(57)
is the conditional mean/mode. Note that in practice the inverse in Equation 56 does not get
computed directly, but rather an efficient method is used that exploits the sparse structures.
The random-effects coefficient vector, b, is often of greater interest. The conditional covari-
ance matrix of Bmay be expressed as
b
σ2Λb
θVΛ>
b
θ.(58)
The hat matrix
The hat matrix, H, is a useful object in linear model diagnostics (Cook and Weisberg 1982).
In general, Hrelates the observed and fitted response vectors, but we specifically define it as
Journal of Statistical Software 29
µY|U =uo=H(yobs o).(59)
To find Hwe note that
µY|U =uo=hZΛXi"µU |Y=yobs
b
βθ#.(60)
Next we get an expression for "µU|Y =yobs
b
βθ#by solving the normal equations (Equation 17),
"µU|Y=yobs
b
βθ#="L>
θRZX
0RX#1"Lθ0
R>
ZX R>
X#1"Λ>
θZ>
X>#W(yobs o).(61)
By the Schur complement identity (Horn and Zhang 2005),
"L>
θRZX
0RX#1
=
L>
θ1L>
θ1RZX R1
X
0R1
X
.(62)
Therefore, we may write the hat matrix as
H= (C>
LCL+C>
RCR),(63)
where
LθCL=Λ>
θZ>W1/2(64)
and
R>
XCR=X>W1/2R>
ZX CL.(65)
The trace of the hat matrix is often used as a measure of the effective degrees of freedom
(e.g., Vaida and Blanchard 2005). Using a relationship between the trace and vec operators,
the trace of Hmay be expressed as
tr(H) = kvec(CL)k2+kvec(CR)k2.(66)
5.2. Using the output module
The user interface for the output module largely consists of a set of methods (Table 8) for
objects of class ‘merMod’, which is the class of objects returned by lmer. In addition to these
methods, the getME function can be used to extract various objects from a fitted mixed model
in lme4. Here we illustrate the use of several of these methods.
Updating fitted mixed models
To illustrate the update method for ‘merMod’ objects we construct a random intercept only
model. This task could be done in several ways, but we choose to first remove the random-
effects term (Days | Subject) and add a new term with a random intercept,
R> fm3 <- update(fm1, . ~ . - (Days | Subject) + (1 | Subject))
R> formula(fm3)
30 Linear Mixed Models with lme4
Generic Brief description of return value
anova Decomposition of fixed-effects contributions
or model comparison.
as.function Function returning profiled deviance or REML criterion.
coef Sum of the random and fixed effects for each level.
confint Confidence intervals on linear mixed-model parameters.
deviance Minus twice maximum log-likelihood.
(Use REMLcrit for the REML criterion.)
df.residual Residual degrees of freedom.
drop1 Drop allowable single terms from the model.
extractAIC Generalized Akaike information criterion
fitted Fitted values given conditional modes (Equation 13).
fixef Estimates of the fixed-effects coefficients, b
β
formula Mixed-model formula of fitted model.
logLik Maximum log-likelihood.
model.frame Data required to fit the model.
model.matrix Fixed-effects model matrix, X.
ngrps Number of levels in each grouping factor.
nobs Number of observations.
plot Diagnostic plots for mixed-model fits.
predict Various types of predicted values.
print Basic printout of mixed-model objects.
profile Profiled likelihood over various model parameters.
ranef Conditional modes of the random effects.
refit A model (re)fitted to a new set of observations of the response variable.
refitML A model (re)fitted by maximum likelihood.
residuals Various types of residual values.
sigma Residual standard deviation.
simulate Simulated data from a fitted mixed model.
summary Summary of a mixed model.
terms Terms representation of a mixed model.
update An updated model using a revised formula or other arguments.
VarCorr Estimated random-effects variances, standard deviations, and correlations.
vcov Covariance matrix of the fixed-effects estimates.
weights Prior weights used in model fitting.
Table 8: List of currently available methods for objects of the class ‘merMod.
Reaction ~ Days + (1 | Subject)
Model summary and associated accessors
The summary method for ‘merMod’ objects provides information about the model fit. Here
we consider the output of summary(fm1) in four steps. The first few lines of output indicate
that the model was fitted by REML as well as the value of the REML criterion (Equation 39)
at convergence (which may also be extracted using REMLcrit(fm1)). The beginning of the
Journal of Statistical Software 31
summary also reproduces the model formula and the scaled Pearson residuals,
Linear mixed model fit by REML ['lmerMod']
Formula: Reaction ~ Days + (Days | Subject)
Data: sleepstudy
REML criterion at convergence: 1743.6
Scaled residuals:
Min 1Q Median 3Q Max
-3.9536 -0.4634 0.0231 0.4634 5.1793
This information may also be obtained using standard accessor functions,
R> formula(fm1)
R> REMLcrit(fm1)
R> quantile(residuals(fm1, "pearson", scaled = TRUE))
The second piece of the summary output provides information regarding the random effects
and residual variation,
Random effects:
Groups Name Variance Std.Dev. Corr
Subject (Intercept) 612.09 24.740
Days 35.07 5.922 0.07
Residual 654.94 25.592
Number of obs: 180, groups: Subject, 18
which can also be accessed using,
R> print(vc <- VarCorr(fm1), comp = c("Variance", "Std.Dev."))
R> nobs(fm1)
R> ngrps(fm1)
R> sigma(fm1)
The print method for objects returned by VarCorr hides the internal structure of these
VarCorr.merMod’ objects. The internal structure of an object of this class is a list of
matrices, one for each random-effects term. The standard deviations and correlation ma-
trices for each term are stored as attributes, stddev and correlation, respectively, of the
variance-covariance matrix, and the residual standard deviation is stored as attribute sc. For
programming use, these objects can be summarized differently using their as.data.frame
method,
R> as.data.frame(VarCorr(fm1))
grp var1 var2 vcov sdcor
1 Subject (Intercept) <NA> 612.089963 24.74045195
2 Subject Days <NA> 35.071661 5.92213312
3 Subject (Intercept) Days 9.604306 0.06555113
4 Residual <NA> <NA> 654.941028 25.59181564
32 Linear Mixed Models with lme4
which contains one row for each variance or covariance parameter. The grp column gives
the grouping factor associated with this parameter. The var1 and var2 columns give the
names of the variables associated with the parameter (var2 is <NA> unless it is a covariance
parameter). The vcov column gives the variances and covariances, and the sdcor column
gives these numbers on the standard deviation and correlation scales.
The next chunk of output gives the fixed-effects estimates,
Fixed effects:
Estimate Std. Error t value
(Intercept) 251.405 6.825 36.84
Days 10.467 1.546 6.77
Note that there are no pvalues. The fixed-effects point estimates may be obtained separately
via fixef(fm1). Conditional modes of the random-effects coefficients can be obtained with
ranef (see Section 5.1 for information on the theory). Finally, we have the correlations
between the fixed-effects estimates
Correlation of Fixed Effects:
(Intr)
Days -0.138
The full variance-covariance matrix of the fixed-effects estimates can be obtained in the usual
Rway with the vcov method (Section 5.1). Alternatively, coef(summary(fm1)) will return
the full fixed-effects parameter table as shown in the summary.
Diagnostic plots
lme4 provides tools for generating most of the standard graphical diagnostic plots (with the
exception of those incorporating influence measures, e.g., Cook’s distance and leverage plots),
in a way modeled on the diagnostic graphics of nlme (Pinheiro and Bates 2000). In particular,
the familiar plot method in base Rfor linear models (objects of class ‘lm’) can be used to
create the standard fitted vs. residual plots,
R> plot(fm1, type = c("p", "smooth"))
scale-location plots,
R> plot(fm1, sqrt(abs(resid(.))) ~ fitted(.), type = c("p", "smooth"))
and quantile-quantile plots (from lattice),
R> qqmath(fm1, id = 0.05)
In contrast to the plot method for ‘lm’ objects, these scale-location and Q-Q plots are based
on raw rather than standardized residuals.
In addition to these standard diagnostic plots, which examine the validity of various assump-
tions (linearity, homoscedasticity, normality) at the level of the residuals, one can also use
the dotplot and qqmath methods for the conditional modes (i.e., ‘ranef.mer’ objects gen-
erated by ranef(fit)) to check for interesting patterns and normality in the conditional
Journal of Statistical Software 33
modes. lme4 does not provide influence diagnostics, but these (and other useful diagnostic
procedures) are available in the dependent packages HLMdiag and influence.ME (Loy and
Hofmann 2014;Nieuwenhuis, Te Grotenhuis, and Pelzer 2012).
Finally, posterior predictive simulation (Gelman and Hill 2006) is a generally useful diagnostic
tool, adapted from Bayesian methods, for exploring model fit. Users pick some summary
statistic of interest. They then compute the summary statistic for an ensemble of simulations,
and see where the observed data falls within the simulated distribution; if the observed data is
extreme, we might conclude that the model is a poor representation of reality. For example,
using the sleep study fit and choosing the interquartile range of the reaction times as the
summary statistic:
R> iqrvec <- sapply(simulate(fm1, 1000), IQR)
R> obsval <- IQR(sleepstudy$Reaction)
R> post.pred.p <- mean(obsval >= c(obsval, iqrvec))
The (one-tailed) posterior predictive pvalue of 0.78 indicates that the model represents the
data adequately, at least for this summary statistic. In contrast to the full Bayesian case,
the procedure described here does not allow for the uncertainty in the estimated parameters.
However, it should be a reasonable approximation when the residual variation is large.
Sequential decomposition and model comparison
Objects of class ‘merMod’ have an anova method which returns Fstatistics corresponding to
the sequential decomposition of the contributions of fixed-effects terms. In order to illustrate
this sequential ANOVA decomposition, we fit a model with orthogonal linear and quadratic
Days terms,
R> fm4 <- lmer(Reaction ~ polyDays[ , 1] + polyDays[ , 2] +
+ (polyDays[ , 1] + polyDays[ , 2] | Subject),
+ within(sleepstudy, polyDays <- poly(Days, 2)))
R> anova(fm4)
Analysis of Variance Table
Df Sum Sq Mean Sq F value
polyDays[, 1] 1 23874.5 23874.5 46.0757
polyDays[, 2] 1 340.3 340.3 0.6567
The relative magnitudes of the two sums of squares indicate that the quadratic term explains
much less variation than the linear term. Furthermore, the magnitudes of the two Fstatistics
strongly suggest significance of the linear term, but not the quadratic term. Notice that this
is only an informal assessment of significance as there are no pvalues associated with these
Fstatistics, an issue discussed in more detail in the next subsection (“Computing pvalues”,
p. 29).
To understand how these quantities are computed, let Ricontain the rows of RX(Equa-
tion 18) associated with the ith fixed-effects term. Then the sum of squares for term iis
SSi=b
β>R>
iRib
β.(67)
34 Linear Mixed Models with lme4
If DFiis the number of columns in Ri, then the Fstatistic for term iis
Fi=SSi
b
σ2DFi
.(68)
For multiple arguments, the anova method returns model comparison statistics.
R> anova(fm1, fm2, fm3)
refitting model(s) with ML (instead of REML)
Data: sleepstudy
Models:
fm3: Reaction ~ Days + (1 | Subject)
fm2: Reaction ~ Days + ((1 | Subject) + (0 + Days | Subject))
fm1: Reaction ~ Days + (Days | Subject)
Df AIC BIC logLik deviance Chisq Chi Df Pr(>Chisq)
fm3 4 1802.1 1814.8 -897.04 1794.1
fm2 5 1762.0 1778.0 -876.00 1752.0 42.0754 1 8.782e-11
fm1 6 1763.9 1783.1 -875.97 1751.9 0.0639 1 0.8004
The output shows χ2statistics representing the difference in deviance between successive
models, as well as pvalues based on likelihood ratio test comparisons. In this case, the
sequential comparison shows that adding a linear effect of time uncorrelated with the intercept
leads to an enormous and significant drop in deviance (deviance 42,p1010), while
the further addition of correlation between the slope and intercept leads to a trivial and non-
significant change in deviance (deviance 0.06). For objects of class ‘lmerMod’ the default
behavior is to refit the models with ML if fitted with REML = TRUE, which is necessary in
order to get sensible answers when comparing models that differ in their fixed effects; this
can be controlled via the refit argument.
Computing pvalues
One of the more controversial design decisions of lme4 has been to omit the output of pvalues
associated with sequential ANOVA decompositions of fixed effects. The absence of analytical
results for null distributions of parameter estimates in complex situations (e.g., unbalanced
or partially crossed designs) is a long-standing problem in mixed-model inference. While the
null distributions (and the sampling distributions of non-null estimates) are asymptotically
normal, these distributions are not tdistributed for finite size samples – nor are the corre-
sponding null distributions of differences in scaled deviances Fdistributed. Thus approximate
methods for computing the approximate degrees of freedom for tdistributions, or the denom-
inator degrees of freedom for Fstatistics (Satterthwaite 1946;Kenward and Roger 1997), are
at best ad hoc solutions.
However, computing finite-size-corrected pvalues is sometimes necessary. Therefore, although
the package does not provide them (except via parametric bootstrapping, Section 5.1), we
have provided a help page to guide users in finding appropriate methods:
R> help("pvalues")
Journal of Statistical Software 35
This help page provides pointers to other packages that provide machinery for calculating
pvalues associated with ‘merMod’ objects. It also suggests framing the inference problem as
a likelihood ratio test, achieved by supplying multiple ‘merMod’ objects to the anova method,
as well as alternatives to pvalues such as confidence intervals.
Previous versions of lme4 provided the mcmcsamp function, which generated a Markov chain
Monte Carlo sample from the posterior distribution of the parameters (assuming flat priors).
Due to difficulty in constructing a version of mcmcsamp that was reliable even in cases where
the estimated random-effects variances were near zero, mcmcsamp has been withdrawn.
Computing confidence intervals
As described above (Section 5.1), lme4 provides confidence intervals (using confint) via
Wald approximations (for fixed-effects parameters only), likelihood profiling, or parametric
bootstrapping (the boot.type argument selects the bootstrap confidence interval type).
As is typical for computationally intensive profile confidence intervals in R, the intervals can
be computed either directly from fitted ‘merMod’ objects (in which case profiling is done
as an interim step), or from a previously computed likelihood profile (of class ‘thpr’, for
“theta profile”). Parametric bootstrapping confidence intervals use a thin wrapper around
the bootMer function that passes the results to boot.ci from the boot package (Canty and
Ripley 2015;Davison and Hinkley 1997) for confidence interval calculation.
In the running sleep study example, the 95% confidence intervals estimated by all three
methods are quite similar. The largest change is a 26% difference in confidence interval
widths between profile and parametric bootstrap methods for the correlation between the
intercept and slope random effects ({0.54,0.98} vs. {0.48,0.68}).
General profile zeta and related plots
lme4 provides several functions (built on lattice graphics, Sarkar 2008) for plotting the profile
zeta functions (Section 5.1) and other related quantities.
The profile zeta plot (Figure 2;xyplot) is simply a plot of the profile zeta function
for each model parameter; linearity of this plot for a given parameter implies that the
likelihood profile is quadratic (and thus that Wald approximations would be reasonably
accurate).
The profile density plot (Figure 3;densityplot) displays an approximation of the
probability density function of the sampling distribution for each parameter. These
densities are derived by setting the cumulative distribution function (c.d.f) to be Φ(ζ)
where Φis the c.d.f. of the standard normal distribution. This is not quite the same as
evaluating the distribution of the estimator of the parameter, which for mixed models
can be very difficult, but it gives a reasonable approximation. If the profile zeta plot is
linear, then the profile density plot will be Gaussian.
The profile pairs plot (Figure 4;splom) gives an approximation of the two-dimensional
profiles of pairs of parameters, interpolated from the univariate profiles as described in
Bates and Watts (1988, Chapter 6). The profile pairs plot shows two-dimensional 50%,
80%, 90%, 95% and 99% marginal confidence regions based on the likelihood ratio, as
well as the profile traces, which indicate the conditional estimates of each parameter
36 Linear Mixed Models with lme4
ζ
−2
−1
0
1
2
20 30 40
σ1
−0.5 0.0 0.5
σ2
4 6 8 10
σ3
22 24 26 28 30
σ
240 250 260 270
(Intercept)
6 8 10 12 14
−2
−1
0
1
2
Days
Figure 2: Profile zeta plot: xyplot(prof.obj)
for fixed values of the other parameters. While panels above the diagonal show profiles
with respect to the original parameters (with random-effects parameters on the standard
deviation/correlation scale, as for all profile plots), the panels below the diagonal show
plots on the (ζi, ζj)scale. The below-diagonal panels allow us to see distortions from an
elliptical shape due to nonlinearity of the traces, separately from the one-dimensional
distortions caused by a poor choice of scale for the parameters. The ζscales provide, in
some sense, the best possible set of single-parameter transformations for assessing the
contours. On the ζscales the extent of a contour on the horizontal axis is exactly the
same as the extent on the vertical axis and both are centered about zero.
For users who want to build their own graphical displays of the profile, there is a method for
as.data.frame that converts profile (‘thpr’) objects to a more convenient format.
Computing fitted and predicted values; simulating
Because mixed models involve random coefficients, one must always clarify whether predic-
tions should be based on the marginal distribution of the response variable or on the distri-
bution that is conditional on the modes of the random effects (Equation 12). The fitted
method retrieves fitted values that are conditional on all of the modes of the random effects;
the predict method returns the same values by default, but also allows for predictions to be
made conditional on different sets of random effects. For example, if the re.form argument
is set to NULL (the default), then the predictions are conditional on all random effects in the
model; if re.form is ~ 0 or NA, then the predictions are made at the population level (all
random-effects values set to zero). In models with multiple random effects, the user can give
re.form as a formula that specifies which random effects are conditioned on.
lme4 also provides a simulate method, which allows similar flexibility in conditioning on
Journal of Statistical Software 37
density
0.00 0.02 0.04 0.06
10 20 30 40 50
σ1
0.0 0.2 0.4 0.6 0.8 1.0 1.2
−0.5 0.0 0.5 1.0
σ2
0.0 0.1 0.2 0.3
4 6 8 10
σ3
0.00 0.05 0.10 0.15 0.20 0.25
22 24 26 28 30
σ
0.000.010.020.03 0.040.050.06
230 240 250 260 270
(Intercept)
0.00 0.05 0.10 0.15 0.20 0.25
6 8 10 12 14 16
Days
Figure 3: Profile density plot: densityplot(prof.obj).
random effects; in addition it allows the user to choose (via the use.u argument) between
conditioning on the fitted conditional modes or choosing a new set of simulated condi-
tional modes (zero-mean Normal deviates chosen according to the estimated random-effects
variance-covariance matrices). Finally, the simulate method has a method for ‘formula
objects, which allows for de novo simulation in the absence of a fitted model. In this case,
the user must specify the random effects (θ), fixed effects (β), and residual standard devia-
tion (σ) parameters via the newparams argument. The standard simulation method (based
on a ‘merMod’ object) is the basis of parametric bootstrapping (Section 5.1) and posterior
predictive simulation (Section 5.2); de novo simulation based on a formula provides a flexible
framework for power analysis.
6. Conclusion
Mixed modeling is an extremely useful but computationally intensive technique. Computa-
tional limitations are especially important because mixed models are commonly applied to
moderately large data sets (104106observations). By developing an efficient, general, and
(now) well-documented platform for fitted mixed models in R, we hope to provide both a
practical tool for end users interested in analyzing data and a reusable, modular framework
for downstream developers interested in extending the class of models that can be easily and
efficiently analyzed in R.
We have learned much about linear mixed models in the process of developing lme4, both from
our own attempts to develop robust and reliable procedures and from the broad community
of lme4 users; we have attempted to describe many of these lessons here. In moving forward,
our main priorities are (1) to maintain the reference implementation of lme4 on the Compre-
38 Linear Mixed Models with lme4
Scatter Plot Matrix
.sig01
10 20 30 40 50
−3
−2
−1
0
.sig02
−0.5
0.0
0.5
1.0 0.5 1.0
0 1 2 3
.sig03
4
6
8
10 8 10
0 1 2 3
.sigma
22
24
26
28
30 26 28 30
0 1 2 3
(Intercept)
230
240
250
260
270 250260270
0 1 2 3
Days
6
8
10
12
14
16
0 1 2 3
Figure 4: Profile pairs plot: splom(prof.obj).
hensive RArchive Network (CRAN), developing relatively few new features; (2) to improve
the flexibility, efficiency and scalability of mixed-model analysis across multiple compatible
implementations, including both the MixedModels package for Julia and the experimental
flexLambda branch of lme4.
Acknowledgments
Rune Haubo Christensen, Henrik Singmann, Fabian Scheipl, Vincent Dorie, and Bin Dai
contributed ideas and code to the current version of lme4; the large community of lme4
users has exercised the software, discovered bugs, and made many useful suggestions. Søren
Højsgaard participated in useful discussions and Xi Xia and Christian Zingg exercised the code
and reported problems. We would like to thank the Banff International Research Station for
hosting a working group on lme4, and an NSERC Discovery grant and NSERC postdoctoral
fellowship for funding.
Journal of Statistical Software 39
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