Assessing Convergence for Fitted Models
[g]lmer fits may produce convergence warnings;
these do not necessarily mean the fit is incorrect (see
“Theoretical details” below). The following steps are recommended
assessing and resolving convergence warnings
(also see examples below):
double-check the model specification and the data
adjust stopping (convergence) tolerances for the nonlinear optimizer,
using the optCtrl argument to [g]lmerControl
(see “Convergence controls” below)
center and scale continuous predictor variables (e.g. with scale)
double-check the Hessian calculation with the more expensive Richardson extrapolation method (see examples)
restart the fit from the reported optimum, or from a point perturbed slightly away from the reported optimum
use allFit to try the fit with all available optimizers (e.g. several different implementations
of BOBYQA and Nelder-Mead, L-BFGS-B from optim, nlminb,
...). While this will of course be slow for large fits, we consider
it the gold standard; if all optimizers converge to values that
are practically equivalent, then we would consider the convergence
warnings to be false positives.
the controls for the nloptwrap optimizer (the default
for lmer) are
(default 1e-6) stop on small change in deviance
(default 0) stop on small relative change in deviance
(default 1e-6) stop on small change of parameter values
(default 0) stop on small relative change of parameter values
(default 1000) maximum number of function evaluations
Changing ftol_abs and xtol_abs to stricter values
(e.g. 1e-8) is a good first step for resolving convergence
problems, at the cost of slowing down model fits.
the controls for minqa::bobyqa (default for
glmer first-stage optimization) are
(default 2e-3) initial radius of the trust region
(default 2e-7) final radius of the trust region
(default 10000) maximum number of function evaluations
rhoend, which describes the scale of parameter uncertainty
on convergence, is approximately analogous to xtol_abs.
the controls for Nelder_Mead (default for
glmer second-stage optimization) are
(default 1e-5) stop on small change in deviance
(default 1e-15) stop on small relative change in deviance
(default 1e-7) stop on small change of parameter values
(default 10000) maximum number of function evaluations
lme4 uses general-purpose nonlinear optimizers (e.g. Nelder-Mead or Powell's BOBYQA method) to estimate the variance-covariance matrices of the random effects. Assessing the convergence of such algorithms reliably is difficult. For example, evaluating the Karush-Kuhn-Tucker conditions (convergence criteria which reduce in simple cases to showing that the gradient is zero and the Hessian is positive definite) is challenging because of the difficulty of evaluating the gradient and Hessian.
We (the lme4 authors and maintainers) are still in the process
of finding the best strategies for testing convergence. Some of the
relevant issues are
the gradient and Hessian are the basic ingredients of
KKT-style testing, but (at least for now) lme4 estimates
them by finite-difference approximations which are sometimes
unreliable.
The Hessian computation in particular represents a difficult tradeoff between computational expense and accuracy. At present the Hessian computations used for convergence checking (and for estimating standard errors of fixed-effect parameters for GLMMs) follow the ordinal package in using a naive but computationally cheap centered finite difference computation (with a fixed step size of 1e-4). A more reliable but more expensive approach is to use Richardson extrapolation, as implemented in the numDeriv package.
it is important to scale the estimated gradient at the estimate appropriately; two reasonable approaches are
scale gradients by the inverse Cholesky factor of the
Hessian, equivalent to scaling gradients by the
estimated Wald standard error
of the estimated parameters. lme4 uses this
approach; it requires the Hessian to be estimated (although the Hessian is
required for
reliable estimation of the fixed-effect standard errors for GLMMs
in any case).
use unscaled gradients on the random-effects parameters, since these are essentially already unitless (for LMMs they are scaled relative to the residual variance; for GLMMs they are scaled relative to the sampling variance of the conditional distribution); for GLMMs, scale fixed-effect gradients by the standard deviations of the corresponding input variable
Exploratory analyses suggest that (1) the naive estimation of the Hessian may fail for large data sets (number of observations greater than approximately 1e5); (2) the magnitude of the scaled gradient increases with sample size, so that warnings will occur even for apparently well-behaved fits with large data sets.
if (interactive()) {
fm1 <- lmer(Reaction ~ Days + (Days | Subject), sleepstudy)
## 1. decrease stopping tolerances
strict_tol <- lmerControl(optCtrl=list(xtol_abs=1e-8, ftol_abs=1e-8))
if (all(fm1@optinfo$optimizer=="nloptwrap")) {
fm1.tol <- update(fm1, control=strict_tol)
}
## 2. center and scale predictors:
ss.CS <- transform(sleepstudy, Days=scale(Days))
fm1.CS <- update(fm1, data=ss.CS)
## 3. recompute gradient and Hessian with Richardson extrapolation
devfun <- update(fm1, devFunOnly=TRUE)
if (isLMM(fm1)) {
pars <- getME(fm1,"theta")
} else {
## GLMM: requires both random and fixed parameters
pars <- getME(fm1, c("theta","fixef"))
}
if (require("numDeriv")) {
cat("hess:\n"); print(hess <- hessian(devfun, unlist(pars)))
cat("grad:\n"); print(grad <- grad(devfun, unlist(pars)))
cat("scaled gradient:\n")
print(scgrad <- solve(chol(hess), grad))
}
## compare with internal calculations:
fm1@optinfo$derivs
## 4. restart the fit from the original value (or
## a slightly perturbed value):
fm1.restart <- update(fm1, start=pars)
set.seed(101)
pars_x <- runif(length(pars),pars/1.01,pars*1.01)
fm1.restart2 <- update(fm1, start=pars_x,
control=strict_tol)
## 5. try all available optimizers
fm1.all <- allFit(fm1)
ss <- summary(fm1.all)
ss$ fixef ## fixed effects
ss$ llik ## log-likelihoods
ss$ sdcor ## SDs and correlations
ss$ theta ## Cholesky factors
ss$ which.OK ## which fits worked
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