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library(varTestnlme)

The varTesnlme package is very easy to use. Below are small examples on how to run it for linear, generalized linear and nonlinear mixed-effect models. A more detailed description is available in the paper doi:10.18637/jss.v107.i06.

Mixed-effect models can be run using nlme or lme4, but also using saemix. varTestnlme can be used to compare two nested models using likelihood ratio tests, where the variance of at least one random effect is tested equal to 0. Fixed effects can also be tested simultaneously, as well as covariances.

# Load the packages
library(nlme)
library(lme4)
library(saemix)
library(EnvStats)

Linear models

Here we focus on models run using lme4 and nlme, but saemix can also be used.

Case 1 : testing the variance of one random effect

We illustrate the results on the Orthodont dataset, which is part of the nlme package. We are interested in modeling the distance between the pituitary and the pterygomaxillary fissure (in mm) as a function of age, in 27 children. We will fit a random slope and random intercept model, and test whether the slope is random or not.

We first need to fit the two nested models: the full model corresponding to \(H_1\) and the null model corresponding to \(H_0\), where there is no random effect associated to age.

data("Orthodont")

# using nlme, with correlated slope and intercept
m1.nlme <- lme(distance ~ 1 + Sex + age + age*Sex, random = pdSymm(Subject ~ 1 + age), data = Orthodont, method = "ML")
m0.nlme <- lme(distance ~ 1 + Sex + age + age*Sex, random = ~ 1 | Subject, data = Orthodont, method = "ML")

# using lme4, with correlated slope and intercept
m1.lme4 <- lmer(distance ~ 1 + Sex + age + age*Sex + (1 + age | Subject), data = Orthodont, REML = FALSE)
#> Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl = control$checkConv, :
#> Model failed to converge with max|grad| = 0.0464695 (tol = 0.002, component 1)
m0.lme4 <- lmer(distance ~ 1 + Sex + age + age*Sex + (1 | Subject), data = Orthodont, REML = FALSE)

Now we can run the likelihood ratio test using the varTestnlme package. The function returns an object from S3 class htest.

vt1.nlme <- varCompTest(m1.nlme,m0.nlme)
#> Variance components testing in mixed effects models
#> Testing that the variance of the random effect associated to age is equal to 0
#> Likelihood ratio test statistic:
#>  LRT = 0.8331072
#> 
#> p-value from exact weights: 0.5103454
#> 
vt1.lme4 <- varCompTest(m1.lme4,m0.lme4)
#> Variance components testing in mixed effects models
#> Testing that the variance of the random effect associated to age is equal to 0
#> Likelihood ratio test statistic:
#>  LRT = 0.8326426
#> 
#> p-value from exact weights: 0.5104889
#>

Using the EnvStats package, nice printing options are available for objects of type htest:

print(vt1.nlme)
#> Variance components testing in mixed effects models
#> Testing that:
#>  variance of the random effect associated to age is equal to 0
#> against the alternative that:
#>  variance of the random effect associated to age > 0 
#> 
#>  Likelihood ratio test statistic:
#>  LRT =  0.8331072
#> 
#>  exact p-value: 0.5103454

It is also possible to access the components of the object using $ or [[:

vt1.nlme$statistic
#>       LRT 
#> 0.8331072
vt1.nlme$p.value
#>    pvalue.weights     pvalue.sample pvalue.lowerbound pvalue.upperbound 
#>         0.5103454                NA         0.5103454         0.5103454

For the p-value, four different values are provided:

  1. the p-value computed using the chi-bar-square expression as a mixture of chi-square distributions. This requires that the weights of the chi-bar-square are available, either because their exact expression is known (only in some simple cases), or because the user ask for their approximation via the option pval.comp = "both" or pval.comp = "approx".
  2. the p-value computed using a sample from the chi-bar-square distribution. This sample is used to approximate the chi-bar-square weights, and thus the corresponding p-value is only available when pval.comp = "both" or pval.comp = "approx".
  3. the lower bound on the p-value: this is always available
  4. the upper bound on the p-value: this is always available

Case 2 : testing the variance of one effect with uncorrelated random effects

# using nlme, with uncorrelated slope and intercept
m1diag.nlme <- lme(distance ~ 1 + Sex + age + age*Sex, random = pdDiag(Subject ~ 1 + age), data = Orthodont, method = "ML")

# using lme4, with uncorrelated slope and intercept
m1diag.lme4 <- lmer(distance ~ 1 + Sex + age + age*Sex + (1 + age || Subject), data = Orthodont, REML = FALSE)

vt1diag.nlme <- varCompTest(m1diag.nlme,m0.nlme)
#> Variance components testing in mixed effects models
#> Testing that the variance of the random effect associated to age is equal to 0
#> Likelihood ratio test statistic:
#>  LRT = 0.5304105
#> 
#> p-value from exact weights: 0.2332172
#> 
vt1diag.lme4 <- varCompTest(m1diag.lme4,m0.lme4)
#> Variance components testing in mixed effects models
#> Testing that the variance of the random effect associated to age is equal to 0
#> Likelihood ratio test statistic:
#>  LRT = 0.5304106
#> 
#> p-value from exact weights: 0.2332171
#>

Case 3 : testing all the variances

In the previous section, the weights of the chi-bar-square distribution where available explicitly. However, it is not always the case. By default, since the computation of these weights can be time consuming, the function is computing bounds on the p-value. In many cases this can be enough to decide whether to reject or not the null hypothesis. If more precision is wanted or needed, it is possible to specify it via the option pval.comp, which then needs to be set to either pval.comp="approx" or to pval.comp="both". In both cases, the control argument can be used to control the computation process. It is a list which contains three slots: M (default to 5000), the size of the Monte Carlo computation, parallel (default to FALSE) to specify whether computation should be parallelized, and nbcores (default to 1) to set the number of cores to be used in case of parallel computing.

m0noRE <- lm(distance ~ 1 + Sex + age + age*Sex, data = Orthodont)

vt <- varCompTest(m1diag.nlme,m0noRE,pval.comp = "both")
#> Variance components testing in mixed effects models
#> Testing that the covariance matrix of Intercept and age is equal to 0
#> Likelihood ratio test statistic:
#>  LRT = 50.13311
#> 
#> p-value from estimated weights: 2.319657e-12
#> bounds on p-value: lower 7.18311e-13 upper 7.215163e-12
#> 
vt2 <- varCompTest(m1diag.lme4,m0noRE)
#> Variance components testing in mixed effects models
#> Testing that the covariance matrix of (Intercept) and age is equal to 0
#> Likelihood ratio test statistic:
#>  LRT = 50.13311
#> bounds on p-value: lower 7.18311e-13 upper 7.215163e-12
#>

By default, the FIM is extracted from the packages, but it is also possible to compute it via parametric bootstrap. In this case, simply use the option fim="compute". The default bootstrap sampling size is B=1000 but it can be changed. To get the exact p-value one can use

varCompTest(m1diag.nlme, m0noRE, fim = "compute", pval.comp = "both", control = list(B=100))
varCompTest(m1diag.lme4, m0noRE, fim = "compute", pval.comp = "both", control = list(B=100))

Generalized linear model

m1 <- glmer(cbind(incidence, size - incidence) ~ period + (1 | herd),
             family = binomial, data = cbpp)
m0 <- glm(cbind(incidence, size - incidence) ~ period,
             family = binomial, data = cbpp)
varCompTest(m1,m0)
#> Variance components testing in mixed effects models
#> Testing that the variance of the random effect associated to (Intercept) is equal to 0
#> Likelihood ratio test statistic:
#>  LRT = 14.00527
#> 
#> p-value from exact weights: 9.114967e-05
#>

Nonlinear model

Testing that one variance is equal to 0 in a model with two correlated random effects, using the Theophylline dataset and the nlme package.

# with nlme
fm1Theo.nlme <- nlme(conc ~ SSfol(Dose, Time, lKe, lKa, lCl),
                     Theoph,
                     fixed = lKe + lKa + lCl ~ 1,
                     start=c( -2.4, 0.45, -3.2),
                     random = pdSymm(lKa + lCl ~ 1))
fm2Theo.nlme <- nlme(conc ~ SSfol(Dose, Time, lKe, lKa, lCl),
                     Theoph,
                     fixed = lKe + lKa + lCl ~ 1,
                     start=c( -2.4, 0.45, -3.2),
                     random = pdDiag(lCl ~ 1))
varCompTest(fm1Theo.nlme,fm2Theo.nlme)
#> Variance components testing in mixed effects models
#> Testing that the variance of the random effect associated to lKa is equal to 0
#> Likelihood ratio test statistic:
#>  LRT = 79.02084
#> 
#> p-value from exact weights: 3.773164e-18
#>

Testing that one variance is null in a model with 3 correlated random effects, using the Theophylline dataset and the lme4 package.

# with lme4
Th.start  <- c(lKe = -2.4, lKa = 0.45, lCl = -3.2)
nm1  <- nlmer(conc ~ SSfol(Dose , Time ,lKe , lKa , lCl) ~
                0+lKe+lKa+lCl +(lKe+lKa+lCl|Subject),
              nAGQ=0,
              Theoph,
              start = Th.start)
nm0  <- nlmer(conc ~ SSfol(Dose , Time ,lKe , lKa , lCl) ~
                0+lKe+lKa+lCl +(lKa+lCl|Subject),
              nAGQ=0,
              Theoph,
              start = Th.start)
varCompTest(nm1,nm0)
#> Variance components testing in mixed effects models
#> Testing that the variance of the random effect associated to lKe is equal to 0
#> Likelihood ratio test statistic:
#>  LRT = 2.043925
#> 
#> p-value from exact weights: 0.4616141
#>

Testing for the presence of randomness in the model, using the nlme package.

fm1 <- nlme(conc ~ SSfol(Dose, Time, lKe, lKa, lCl),
                     Theoph,
                     fixed = lKe + lKa + lCl ~ 1,
                     start=c( -2.4, 0.45, -3.2),
                     random = pdDiag(lKe + lKa + lCl ~ 1))
fm0 <- nls(conc ~ SSfol(Dose, Time, lKe, lKa, lCl),
                     Theoph,
                     start=list(lKe=-2.4,lKa=0.45,lCl=-3.2))
varCompTest(fm1,fm0)
#> Variance components testing in mixed effects models
#> Testing that the covariance matrix of lKe and lKa and lCl is equal to 0
#> Likelihood ratio test statistic:
#>  LRT = 117.172
#> bounds on p-value: lower 1.31613e-27 upper 1.748295e-25
#>

We can see that there is no need to approximate the weights of the chi-bar-square distribution since the bounds on the p-value are sufficient to reject the null hypothesis at any classical level.

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