Fit Linear Mixed-Effects Models

Fit a linear mixed-effects model (LMM) to data, via REML or maximum likelihood.

Usage

lmer(formula, data = NULL, REML = TRUE, control = lmerControl(),
     start = NULL, verbose = 0L, subset, weights, na.action,
     offset, contrasts = NULL, devFunOnly = FALSE)

Arguments

formula: a two-sided linear formula object describing both the fixed-effects and random-effects part of the model, with the response on the left of a ~ operator and the terms, separated by + operators, on the right. Random-effects terms are distinguished by vertical bars (|) separating expressions for design matrices from grouping factors. Two vertical bars (||) can be used to specify multiple uncorrelated random effects for the same grouping variable. (Because of the way it is implemented, the ||-syntax works only for design matrices containing numeric (continuous) predictors; to fit models with independent categorical effects, see dummy or the lmer_alt function from the afex package.)
data: an optional data frame containing the variables named in formula. By default the variables are taken from the environment from which lmer is called. While data is optional, the package authors strongly recommend its use, especially when later applying methods such as update and drop1 to the fitted model (such methods are not guaranteed to work properly if data is omitted). If data is omitted, variables will be taken from the environment of formula (if specified as a formula) or from the parent frame (if specified as a character vector).
REML: logical scalar - Should the estimates be chosen to optimize the REML criterion (as opposed to the log-likelihood)?
control: a list (of correct class, resulting from lmerControl() or glmerControl() respectively) containing control parameters, including the nonlinear optimizer to be used and parameters to be passed through to the nonlinear optimizer, see the *lmerControl documentation for details.
start: a named list of starting values for the parameters in the model. For lmer this can be a numeric vector or a list with one component named "theta".
verbose: integer scalar. If > 0 verbose output is generated during the optimization of the parameter estimates. If > 1 verbose output is generated during the individual penalized iteratively reweighted least squares (PIRLS) steps.
subset: an optional expression indicating the subset of the rows of data that should be used in the fit. This can be a logical vector, or a numeric vector indicating which observation numbers are to be included, or a character vector of the row names to be included. All observations are included by default.
weights: an optional vector of ‘prior weights’ to be used in the fitting process. Should be NULL or a numeric vector. Prior weights are not normalized or standardized in any way. In particular, the diagonal of the residual covariance matrix is the squared residual standard deviation parameter sigma times the vector of inverse weights. Therefore, if the weights have relatively large magnitudes, then in order to compensate, the sigma parameter will also need to have a relatively large magnitude.
na.action: a function that indicates what should happen when the data contain NAs. The default action (na.omit, inherited from the 'factory fresh' value of getOption("na.action")) strips any observations with any missing values in any variables.
offset: this can be used to specify an a priori known component to be included in the linear predictor during fitting. This should be NULL or a numeric vector of length equal to the number of cases. One or more offset terms can be included in the formula instead or as well, and if more than one is specified their sum is used. See model.offset.
contrasts: an optional list. See the contrasts.arg of model.matrix.default.
devFunOnly: logical - return only the deviance evaluation function. Note that because the deviance function operates on variables stored in its environment, it may not return exactly the same values on subsequent calls (but the results should always be within machine tolerance).

Value

An object of class merMod (more specifically, an object of subclass lmerMod), for which many methods are available (e.g. methods(class="merMod"))

Note

In earlier version of the lme4 package, a method argument was used. Its functionality has been replaced by the REML argument.

Also, lmer(.) allowed a family argument (to effectively switch to glmer(.)). This has been deprecated in summer 2013, and been disabled in spring 2019.

Details

If the formula argument is specified as a character vector, the function will attempt to coerce it to a formula. However, this is not recommended (users who want to construct formulas by pasting together components are advised to use as.formula or reformulate); model fits will work but subsequent methods such as drop1, update may fail.
When handling perfectly collinear predictor variables (i.e. design matrices of less than full rank), [gn]lmer is not quite as sophisticated as some simpler modeling frameworks such as lm and glm. While it does automatically drop collinear variables (with a message rather than a warning), it does not automatically fill in NA values for the dropped coefficients; these can be added via fixef(fitted.model,add.dropped=TRUE). This information can also be retrieved via attr(getME(fitted.model,"X"),"col.dropped").
the deviance function returned when devFunOnly is TRUE takes a single numeric vector argument, representing the theta vector. This vector defines the scaled variance-covariance matrices of the random effects, in the Cholesky parameterization. For models with only simple (intercept-only) random effects, theta is a vector of the standard deviations of the random effects. For more complex or multiple random effects, running getME(.,"theta") to retrieve the theta vector for a fitted model and examining the names of the vector is probably the easiest way to determine the correspondence between the elements of the theta vector and elements of the lower triangles of the Cholesky factors of the random effects.

Examples

## linear mixed models - reference values from older code
(fm1 <- lmer(Reaction ~ Days + (Days | Subject), sleepstudy))
#> Linear mixed model fit by REML ['lmerMod']
#> Formula: Reaction ~ Days + (Days | Subject)
#>    Data: sleepstudy
#> REML criterion at convergence: 1743.628
#> Random effects:
#>  Groups   Name        Std.Dev. Corr
#>  Subject  (Intercept) 24.741       
#>           Days         5.922   0.07
#>  Residual             25.592       
#> Number of obs: 180, groups:  Subject, 18
#> Fixed Effects:
#> (Intercept)         Days  
#>      251.41        10.47  
summary(fm1)# (with its own print method; see class?merMod % ./merMod-class.Rd
#> 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 
#> 
#> Random effects:
#>  Groups   Name        Variance Std.Dev. Corr
#>  Subject  (Intercept) 612.10   24.741       
#>           Days         35.07    5.922   0.07
#>  Residual             654.94   25.592       
#> Number of obs: 180, groups:  Subject, 18
#> 
#> Fixed effects:
#>             Estimate Std. Error t value
#> (Intercept)  251.405      6.825  36.838
#> Days          10.467      1.546   6.771
#> 
#> Correlation of Fixed Effects:
#>      (Intr)
#> Days -0.138

str(terms(fm1))
#> Classes 'terms', 'formula'  language Reaction ~ Days
#>   ..- attr(*, "variables")= language list(Reaction, Days)
#>   ..- attr(*, "factors")= int [1:2, 1] 0 1
#>   .. ..- attr(*, "dimnames")=List of 2
#>   .. .. ..$ : chr [1:2] "Reaction" "Days"
#>   .. .. ..$ : chr "Days"
#>   ..- attr(*, "term.labels")= chr "Days"
#>   ..- attr(*, "order")= int 1
#>   ..- attr(*, "intercept")= int 1
#>   ..- attr(*, "response")= int 1
#>   ..- attr(*, ".Environment")=<environment: 0x561f6ed52738> 
#>   ..- attr(*, "predvars")= language list(Reaction, Days)
stopifnot(identical(terms(fm1, fixed.only=FALSE),
                    terms(model.frame(fm1))))
attr(terms(fm1, FALSE), "dataClasses") # fixed.only=FALSE needed for dataCl.
#>  Reaction      Days   Subject 
#> "numeric" "numeric"  "factor" 

## Maximum Likelihood (ML), and "monitor" iterations via 'verbose':
fm1_ML <- update(fm1, REML=FALSE, verbose = 1)
#> iteration: 1
#> 	f(x) = 1784.642296
#> iteration: 2
#> 	f(x) = 1790.125637
#> iteration: 3
#> 	f(x) = 1798.999624
#> iteration: 4
#> 	f(x) = 1803.853200
#> iteration: 5
#> 	f(x) = 1800.613981
#> iteration: 6
#> 	f(x) = 1798.604631
#> iteration: 7
#> 	f(x) = 1752.260737
#> iteration: 8
#> 	f(x) = 1797.587692
#> iteration: 9
#> 	f(x) = 1754.954110
#> iteration: 10
#> 	f(x) = 1753.695682
#> iteration: 11
#> 	f(x) = 1754.816999
#> iteration: 12
#> 	f(x) = 1753.106734
#> iteration: 13
#> 	f(x) = 1752.939377
#> iteration: 14
#> 	f(x) = 1752.256879
#> iteration: 15
#> 	f(x) = 1752.057448
#> iteration: 16
#> 	f(x) = 1752.022389
#> iteration: 17
#> 	f(x) = 1752.022728
#> iteration: 18
#> 	f(x) = 1751.971687
#> iteration: 19
#> 	f(x) = 1751.952603
#> iteration: 20
#> 	f(x) = 1751.948524
#> iteration: 21
#> 	f(x) = 1751.987176
#> iteration: 22
#> 	f(x) = 1751.983213
#> iteration: 23
#> 	f(x) = 1751.951971
#> iteration: 24
#> 	f(x) = 1751.946276
#> iteration: 25
#> 	f(x) = 1751.946698
#> iteration: 26
#> 	f(x) = 1751.947568
#> iteration: 27
#> 	f(x) = 1751.945312
#> iteration: 28
#> 	f(x) = 1751.944180
#> iteration: 29
#> 	f(x) = 1751.943533
#> iteration: 30
#> 	f(x) = 1751.942441
#> iteration: 31
#> 	f(x) = 1751.942170
#> iteration: 32
#> 	f(x) = 1751.942370
#> iteration: 33
#> 	f(x) = 1751.942278
#> iteration: 34
#> 	f(x) = 1751.942204
#> iteration: 35
#> 	f(x) = 1751.941309
#> iteration: 36
#> 	f(x) = 1751.940931
#> iteration: 37
#> 	f(x) = 1751.940567
#> iteration: 38
#> 	f(x) = 1751.940179
#> iteration: 39
#> 	f(x) = 1751.940082
#> iteration: 40
#> 	f(x) = 1751.940270
#> iteration: 41
#> 	f(x) = 1751.941501
#> iteration: 42
#> 	f(x) = 1751.939489
#> iteration: 43
#> 	f(x) = 1751.939392
#> iteration: 44
#> 	f(x) = 1751.939398
#> iteration: 45
#> 	f(x) = 1751.939425
#> iteration: 46
#> 	f(x) = 1751.939355
#> iteration: 47
#> 	f(x) = 1751.939490
#> iteration: 48
#> 	f(x) = 1751.939363
#> iteration: 49
#> 	f(x) = 1751.939345
#> iteration: 50
#> 	f(x) = 1751.939344
#> iteration: 51
#> 	f(x) = 1751.939345
#> iteration: 52
#> 	f(x) = 1751.939348
#> iteration: 53
#> 	f(x) = 1751.939344
(fm2 <- lmer(Reaction ~ Days + (Days || Subject), sleepstudy))
#> Linear mixed model fit by REML ['lmerMod']
#> Formula: Reaction ~ Days + ((1 | Subject) + (0 + Days | Subject))
#>    Data: sleepstudy
#> REML criterion at convergence: 1743.669
#> Random effects:
#>  Groups    Name        Std.Dev.
#>  Subject   (Intercept) 25.051  
#>  Subject.1 Days         5.988  
#>  Residual              25.565  
#> Number of obs: 180, groups:  Subject, 18
#> Fixed Effects:
#> (Intercept)         Days  
#>      251.41        10.47  
anova(fm1, fm2)
#> refitting model(s) with ML (instead of REML)
#> Data: sleepstudy
#> Models:
#> fm2: Reaction ~ Days + ((1 | Subject) + (0 + Days | Subject))
#> fm1: Reaction ~ Days + (Days | Subject)
#>     npar    AIC    BIC  logLik -2*log(L)  Chisq Df Pr(>Chisq)
#> fm2    5 1762.0 1778.0 -876.00    1752.0                     
#> fm1    6 1763.9 1783.1 -875.97    1751.9 0.0639  1     0.8004
sm2 <- summary(fm2)
print(fm2, digits=7, ranef.comp="Var") # the print.merMod()         method
#> Linear mixed model fit by REML ['lmerMod']
#> Formula: Reaction ~ Days + ((1 | Subject) + (0 + Days | Subject))
#>    Data: sleepstudy
#> REML criterion at convergence: 1743.669
#> Random effects:
#>  Groups    Name        Variance 
#>  Subject   (Intercept) 627.56905
#>  Subject.1 Days         35.85838
#>  Residual              653.58350
#> Number of obs: 180, groups:  Subject, 18
#> Fixed Effects:
#> (Intercept)         Days  
#>   251.40510     10.46729  
print(sm2, digits=3, corr=FALSE)       # the print.summary.merMod() method
#> Linear mixed model fit by REML ['lmerMod']
#> Formula: Reaction ~ Days + ((1 | Subject) + (0 + Days | Subject))
#>    Data: sleepstudy
#> 
#> REML criterion at convergence: 1743.7
#> 
#> Scaled residuals: 
#>    Min     1Q Median     3Q    Max 
#> -3.963 -0.463  0.020  0.465  5.186 
#> 
#> Random effects:
#>  Groups    Name        Variance Std.Dev.
#>  Subject   (Intercept) 627.6    25.05   
#>  Subject.1 Days         35.9     5.99   
#>  Residual              653.6    25.57   
#> Number of obs: 180, groups:  Subject, 18
#> 
#> Fixed effects:
#>             Estimate Std. Error t value
#> (Intercept)   251.41       6.89   36.51
#> Days           10.47       1.56    6.71

## Fit sex-specific variances by constructing numeric dummy variables
## for sex and sex:age; in this case the estimated variance differences
## between groups in both intercept and slope are zero ...
data(Orthodont,package="nlme")
Orthodont$nsex <- as.numeric(Orthodont$Sex=="Male")
Orthodont$nsexage <- with(Orthodont, nsex*age)
lmer(distance ~ age + (age|Subject) + (0+nsex|Subject) +
     (0 + nsexage|Subject), data=Orthodont)
#> boundary (singular) fit: see help('isSingular')
#> Linear mixed model fit by REML ['lmerMod']
#> Formula: distance ~ age + (age | Subject) + (0 + nsex | Subject) + (0 +  
#>     nsexage | Subject)
#>    Data: Orthodont
#> REML criterion at convergence: 442.6367
#> Random effects:
#>  Groups    Name        Std.Dev.  Corr 
#>  Subject   (Intercept) 2.3268096      
#>            age         0.2264158 -0.61
#>  Subject.1 nsex        0.0001559      
#>  Subject.2 nsexage     0.0000000      
#>  Residual              1.3100560      
#> Number of obs: 108, groups:  Subject, 27
#> Fixed Effects:
#> (Intercept)          age  
#>     16.7611       0.6602  
#> optimizer (nloptwrap) convergence code: 0 (OK) ; 0 optimizer warnings; 1 lme4 warnings