lme4 (version 1.1-15)

convergence: Assessing Convergence for Fitted Models

Description

The lme4 package 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 reliably whether such algorithms have converged is difficult. For example, evaluating the Karush-Kuhn-Tucker conditions (convergence criteria which in the simplest case of non-constrained optimization reduce 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 when they have to be estimated by finite differences (as in the case of lme4; direct computation of derivatives based on analytic expressions may eventually be available for some special classes, but we have not yet implemented them) they may not be sufficiently accurate for reliable convergence testing.

• 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. don't scale random-effects (Cholesky) gradients, 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, or

  2. scale gradients by the inverse Cholesky factor of the Hessian, equivalent to scaling by the estimated Wald standard error of the estimated parameters. The latter approach is used in the current version of lme4; it has the disadvantage that it requires us to estimate the Hessian (although the Hessian is required for reliable estimation of the fixed-effect standard errors for GLMMs in any case).

• 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.

If you do see convergence warnings, and want to trouble-shoot/double-check the results, the following steps are recommended (examples are given below):

• double-check the model specification and the data for mistakes

• center and scale continuous predictor variables (e.g. with scale)

• check for singularity: if any of the diagonal elements of the Cholesky factor are zero or very small, the convergence testing methods may be inappropriate (see examples)

• double-check the Hessian calculation with the more expensive Richardson extrapolation method (see examples)

• restart the fit from the apparent optimum, or from a point perturbed slightly away from the optimum

• try all available optimizers (e.g. several different implementations of BOBYQA and Nelder-Mead, L-BFGS-B from optim, nlminb, …) via the allFit() function, see ‘5.’ in the examples. 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.

To quote Douglas Adams, we apologize for the inconvenience.

See Also

lmerControl

Examples

# NOT RUN {
fm1 <- lmer(Reaction ~ Days + (Days | Subject), sleepstudy)

## 1. center and scale predictors:
ss.CS <- transform(sleepstudy, Days=scale(Days))
fm1.CS <- update(fm1, data=ss.CS)

## 2. check singularity
diag.vals <- getME(fm1,"theta")[getME(fm1,"lower") == 0]
any(diag.vals < 1e-6) # FALSE

## 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)

## 5. try all available optimizers

  source(system.file("utils", "allFit.R", package="lme4"))
  fm1.all <- allFit(fm1)
  ss <- summary(fm1.all)
  ss$ fixef               ## extract fixed effects
  ss$ llik                ## log-likelihoods
  ss$ sdcor               ## SDs and correlations
  ss$ theta               ## Cholesky factors
  ss$ which.OK            ## which fits worked

# }