convergence: Assessing Convergence for Fitted Models

Description

[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.

Arguments

Details

Convergence controls

the controls for the nloptwrap optimizer (the default for lmer) are

ftol_abs

(default 1e-6) stop on small change in deviance

ftol_rel

(default 0) stop on small relative change in deviance

xtol_abs

(default 1e-6) stop on small change of parameter values

xtol_rel

(default 0) stop on small relative change of parameter values

maxeval

(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

rhobeg

(default 2e-3) initial radius of the trust region

rhoend

(default 2e-7) final radius of the trust region

maxfun

(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

FtolAbs

(default 1e-5) stop on small change in deviance

FtolRel

(default 1e-15) stop on small relative change in deviance

XtolRel

(default 1e-7) stop on small change of parameter values

maxfun

(default 10000) maximum number of function evaluations

Theoretical issues

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 \(10^{-4}\)). 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
1. 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).
2. 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 \(10^{5}\)); (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.

Examples

Run this code

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

## compute reciprocal condition number of Hessian
H <- fm1@optinfo$derivs$Hessian
Matrix::rcond(H)

## 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

}