Check mixed models for boundary fits

Check mixed models for boundary fits.

Usage

check_singularity(x, tolerance = 1e-05, ...)

Arguments

x: A mixed model.
tolerance: Indicates up to which value the convergence result is accepted. The larger tolerance is, the stricter the test will be.
...: Currently not used.

Value

TRUE if the model fit is singular.

Details

If a model is "singular", this means that some dimensions of the variance-covariance matrix have been estimated as exactly zero. This often occurs for mixed models with complex random effects structures.

"While singular models are statistically well defined (it is theoretically sensible for the true maximum likelihood estimate to correspond to a singular fit), there are real concerns that (1) singular fits correspond to overfitted models that may have poor power; (2) chances of numerical problems and mis-convergence are higher for singular models (e.g. it may be computationally difficult to compute profile confidence intervals for such models); (3) standard inferential procedures such as Wald statistics and likelihood ratio tests may be inappropriate." (lme4 Reference Manual)

There is no gold-standard about how to deal with singularity and which random-effects specification to choose. Beside using fully Bayesian methods (with informative priors), proposals in a frequentist framework are:

avoid fitting overly complex models, such that the variance-covariance matrices can be estimated precisely enough (Matuschek et al. 2017)
use some form of model selection to choose a model that balances predictive accuracy and overfitting/type I error (Bates et al. 2015, Matuschek et al. 2017)
"keep it maximal", i.e. fit the most complex model consistent with the experimental design, removing only terms required to allow a non-singular fit (Barr et al. 2013)
since version 1.1.9, the glmmTMB package allows to use priors in a frequentist framework, too. One recommendation is to use a Gamma prior (Chung et al. 2013). The mean may vary from 1 to very large values (like 1e8), and the shape parameter should be set to a value of 2.5. You can then update() your model with the specified prior. In glmmTMB, the code would look like this:
```
# "model" is an object of class gmmmTMB
prior <- data.frame(
  prior = "gamma(1, 2.5)",  # mean can be 1, but even 1e8
  class = "ranef"           # for random effects
)
model_with_priors <- update(model, priors = prior)
```
Large values for the mean parameter of the Gamma prior have no large impact on the random effects variances in terms of a "bias". Thus, if 1 doesn't fix the singular fit, you can safely try larger values.

Note the different meaning between singularity and convergence: singularity indicates an issue with the "true" best estimate, i.e. whether the maximum likelihood estimation for the variance-covariance matrix of the random effects is positive definite or only semi-definite. Convergence is a question of whether we can assume that the numerical optimization has worked correctly or not.

References

Bates D, Kliegl R, Vasishth S, Baayen H. Parsimonious Mixed Models. arXiv:1506.04967, June 2015.
Barr DJ, Levy R, Scheepers C, Tily HJ. Random effects structure for confirmatory hypothesis testing: Keep it maximal. Journal of Memory and Language, 68(3):255-278, April 2013.
Chung Y, Rabe-Hesketh S, Dorie V, Gelman A, and Liu J. 2013. "A Nondegenerate Penalized Likelihood Estimator for Variance Parameters in Multilevel Models." Psychometrika 78 (4): 685–709. doi:10.1007/s11336-013-9328-2
Matuschek H, Kliegl R, Vasishth S, Baayen H, Bates D. Balancing type I error and power in linear mixed models. Journal of Memory and Language, 94:305-315, 2017.
lme4 Reference Manual, https://cran.r-project.org/package=lme4

Examples

data(sleepstudy, package = "lme4")
set.seed(123)
sleepstudy$mygrp <- sample(1:5, size = 180, replace = TRUE)
sleepstudy$mysubgrp <- NA
for (i in 1:5) {
  filter_group <- sleepstudy$mygrp == i
  sleepstudy$mysubgrp[filter_group] <-
    sample(1:30, size = sum(filter_group), replace = TRUE)
}

model <- lme4::lmer(
  Reaction ~ Days + (1 | mygrp / mysubgrp) + (1 | Subject),
  data = sleepstudy
)
#> boundary (singular) fit: see help('isSingular')
check_singularity(model)
#> [1] TRUE

# Fixing singularity issues using priors in glmmTMB
# Example taken from `vignette("priors", package = "glmmTMB")`
dat <- readRDS(system.file("vignette_data", "gophertortoise.rds",
               package = "glmmTMB"))
model <- glmmTMB::glmmTMB(
  shells ~ prev + offset(log(Area)) + factor(year) + (1 | Site),
  family = poisson,
  data = dat
)
# singular fit
check_singularity(model)
#> [1] TRUE

# impose Gamma prior on random effects parameters
prior <- data.frame(
  prior = "gamma(1, 2.5)", # mean can be 1, but even 1e8
  class = "ranef" # for random effects
)
model_with_priors <- update(model, priors = prior)
# no singular fit
check_singularity(model_with_priors)
#> [1] FALSE