rlmer: Robust Scoring Equations Estimator for Linear Mixed Models

In robustlmm: Robust Linear Mixed Effects Models

Robust Scoring Equations Estimator for Linear Mixed Models

Description

Robust estimation of linear mixed effects models, for hierarchical nested and non-nested, e.g., crossed, datasets.

Usage

rlmer(
  formula,
  data,
  ...,
  method = c("DAStau", "DASvar"),
  setting,
  rho.e,
  rho.b,
  rho.sigma.e,
  rho.sigma.b,
  rel.tol = 1e-08,
  max.iter = 40 * (r + 1)^2,
  verbose = 0,
  doFit = TRUE,
  init,
  size_obr = FALSE,
  design.weights = NULL
)

lmerNoFit(formula, data = NULL, ..., initTheta)

Arguments

formula	a two-sided linear formula object describing the fixed-effects part of the model, with the response on the left of a ~ operator and the terms, separated by + operators, on the right. The vertical bar character "|" separates an expression for a model matrix and a grouping factor.
data	an optional data frame containing the variables named in formula. By default the variables are taken from the environment from which lmer is called.
...	Additional parameters passed to lmer to find the initial estimates. See lmer.
method	method to be used for estimation of theta and sigma, see Details.
setting	a string specifying suggested choices for the arguments rho.e, rho.sigma.e, rho.b and rho.sigma.b. Use "RSEn" (the default) or "RSEa". Both use smoothPsi for all the “rho” arguments. For rho.sigma.e, squared robustness weights are used (see psi2propII). "RSEn" uses the same tuning parameter as for rho.e, which leads to higher robustness but lower efficiency. "RSEa" adjusts the tuning parameter for higher asymptotic efficiency which results in lower robustness (k = 2.28 for default rho.e). For diagonal random effects covariance matrices, rho.sigma.b is treated exactly as rho.sigma.e. For block diagonal random effects covariance matrices (with correlation terms), regular robustness weights are used for rho.sigma.b, not squared ones, as they're not needed. But the tuning parameters are adjusted for both rho.b and rho.sigma.b according to the dimensions of the blocks (for both "RSEn" or "RSEa"). For a block of dimension 2 (e.g., correlated random intercept and slope) k = 5.14 is used.
rho.e	object of class psi_func, specifying the functions to use for the huberization of the residuals.
rho.b	object of class psi_func or list of such objects (see Details), specifying the functions to use for the huberization of the random effects.
rho.sigma.e	object of class psi_func, specifying the weight functions to use for the huberization of the residuals when estimating the variance components, use the psi2propII function to specify squared weights and custom tuning parameters.
rho.sigma.b	(optional) object of class psi_func or list of such objects, specifying the weight functions to use for the huberization of the random effects when estimating the variance components (see Details). Use psi2propII to specify squared weights and custom tuning parameters or chgDefaults for regular weights for variance components including correlation parameters.
rel.tol	relative tolerance used as criteria in the fitting process.
max.iter	maximum number of iterations allowed.
verbose	verbosity of output. Ranges from 0 (none) to 3 (a lot of output)
doFit	logical scalar. When doFit = FALSE the model is not fit but instead a structure with the model matrices for the random-effects terms is returned (used to speed up tests). When doFit = TRUE, the default, the model is fit immediately.
init	optional lmerMod- or rlmerMod-object to use for starting values, a list with elements ‘fixef’, ‘u’, ‘sigma’, ‘theta’, the string "ransac", or a function producing an lmerMod object. An lmerMod or rlmerMod passed here supplies the model, not only the starting values: the working object is built from it, so formula and data no longer have any effect. Pass such an object only when it was fitted on the same data; a warning is issued if its number of observations does not match nrow(data). To start from values obtained elsewhere while keeping the model given by formula and data, use the list form. Its components are optional — whatever is omitted keeps the value from the internal lmer initialisation, which is the right choice for ‘u’ when the values come from a fit on different observations. When init = "ransac", the high-breakdown RANSAC start is obtained by calling ransac_lme4(formula, data) with its default K = 200 subsamples and sub_frac = 0.5. Its purpose is to raise the breakdown point of the starting value, which is what makes redescending psi-functions usable: their estimating equations are non-convex, so a low-breakdown start can be drawn to a bad local solution. rlmer takes ‘fixef’, ‘sigma’ and ‘theta’ from that fit and fits the model on the full data. The string form is random (not reproducible without an outer set.seed); for fine control of K, sub_frac, or seed, use rlmer_ransac.
size_obr	logical scalar; if TRUE (default FALSE), the size-controlling weight w_delta in the block-diagonal variance-components scoring equation is replaced by the Hampel-OBR (Stahel 1987 / Hampel et al. 1986) form w_\tau(d) = \min(1, b_\tau / |d - s\kappa - a|), with b_\tau taken from the tuning constant of rho.sigma.b and a determined by Fisher consistency under \chi^2_s. The default finite-difference form w_\delta(d) = (\psi(d) - \psi(d-s\kappa))/s is not Hampel-OBR-optimal at the central model; this option recovers asymptotic Hampel-OBR efficiency for the variance-component magnitudes (a typically modest 1-2 percentage-point gain at matched gross-error sensitivity). Has no effect for diagonal V_b (block size 1).
design.weights	Mallows-type design weights for robustness to high-leverage design points: NULL (default, exact current behaviour), a numeric vector of deterministic weights \eta_i \in (0, 1] of length n, or "mcd" to compute \eta_i = \min(1, \chi^2_{p^*}(0.975) / d_i^2)^{1/2} from robust squared Mahalanobis distances d_i^2 of the non-constant fixed-effects covariates (covMcd); the "mcd" tuning (severity \gamma = 1, cutoff c = 0.975) is the simulation-backed default (see vignette("rlmer")). The weights multiply the e-side score contribution of each observation throughout the estimator – the \beta, u, \sigma and \theta equations, the model vcov, the Satterthwaite degrees of freedom and the influence diagnostics – bounding the influence of high-leverage design points. For designs without continuous covariates "mcd" reduces to \eta \equiv 1 (the unmodified estimator). Active design weights are supported for a single grouping factor only; rlmer stops with an error if the model has more than one. Leverage robustness costs a little efficiency at the clean model (about 1% at the default tuning; see vignette("rlmer")).
initTheta	parameter to initialize theta with (optional)

Details

Overview:

This function implements the Robust Scoring Equations estimator for linear mixed effect models. It can be used much like the function lmer in the package lme4. The supported models are the same as for lmer (gaussian family only). The robust approach used is based on the robustification of the scoring equations and an application of the Design Adaptive Scale approach.

Example analyses and theoretical details on the method are available in the vignette (see vignette("rlmer")).

Models are specified using the formula argument, using the same syntax as for lmer. Additionally, one also needs to specify what robust scoring or weight functions are to be used (arguments starting with rho.). By default a smoothed version of the Huber function is used. Furthermore, the method argument can be used to speed up computations at the expense of accuracy of the results.

Computation methods:

Currently, there are two different methods available for fitting models. They only differ in how the consistency factors for the Design Adaptive Scale estimates are computed. Available fitting methods for theta and sigma.e:

• DAStau (default): For this method, the consistency factors are computed using numerical quadrature. This is slower but yields more accurate results. This is the direct analogue to the DAS-estimate in robust linear regression.

• DASvar: This method computes the consistency factors using a direct approximation which is faster but less accurate. For complex models with correlated random effects with more than one correlation term, this is the only method available.

DAStau supports blocks of random effects of dimension at most 2; for fits containing a larger block it falls back to DASvar with a warning. An experimental option, options(robustlmm.dastau.mc = TRUE), lifts this restriction via a Monte-Carlo calibration of the consistency factors; it is simulation-validated but not backed by a finite-sample theorem, see robustlmm-options.

Weight functions:

The tuning parameters of the weight functions “rho” can be used to adjust robustness and efficiency of the resulting estimates (arguments rho.e, rho.b, rho.sigma.e and rho.sigma.b). Better robustness will lead to a decrease of the efficiency. With the default setting, setting = "RSEn", the tuning parameters are set to yield estimates with approximately 95% efficiency for the fixed effects. The variance components are estimated with a lower efficiency but better robustness properties.

One has to use different weight functions and tuning parameters for simple variance components and for such including correlation parameters. By default, they are chosen appropriately to the model at hand. However, when using the rho.sigma.e and rho.sigma.b arguments, it is up to the user to specify the appropriate function. See asymptoticEfficiency for methods to find tuning parameters that yield a given asymptotic efficiency.

• For simple variance components and the residual error scale use the function psi2propII to change the tuning parameters. This is similar to Proposal 2 in the location-scale problem (i.e., using the squared robustness weights of the location estimate for the scale estimate; otherwise the scale estimate is not robust).

• For multi-dimensional blocks of random effects modeled, e.g., a model with correlated random intercept and slope, (referred to as block diagonal case below), use the chgDefaults function to change the tuning parameters. The parameter estimation problem is multivariate, unlike the case without correlation where the problem was univariate. For the employed estimator, this amounts to switching from simple scale estimates to estimating correlation matrices. Therefore different weight functions have to be used. Squaring of the weights (using the function psi2propII) is no longer necessary. To yield estimates with the same efficiency, the tuning parameters for the block diagonal are larger than for the simple case. Tables of tuning parameters are given in Table 2 and 3 of the vignette (vignette("rlmer")).

Recommended tuning parameters:

For a more robust estimate, use setting = "RSEn" (the default). For higher efficiency, use setting = "RSEa". The settings described in the following paragraph are used when setting = "RSEa" is specified.

For the smoothed Huber function the tuning parameters to get approximately 95% efficiency are k=1.345 for rho.e and k=2.28 for rho.sigma.e (using the squared version). For simple variance components, the same can be used for rho.b and rho.sigma.b. For variance components including correlation parameters, use k=5.14 for both rho.b and rho.sigma.b. Tables of tuning parameter are given in Table 2 and 3 of the vignette (vignette("rlmer")).

Specifying (multiple) weight functions:

If custom weight functions are specified using the argument rho.b (rho.e) but the argument rho.sigma.b (rho.sigma.e) is missing, then the squared weights are used for simple variance components and the regular weights are used for variance components including correlation parameters. The same tuning parameters will be used when setting = "RSEn" is used. To get higher efficiency either use setting = "RSEa" (and only set arguments rho.e and rho.b). Or specify the tuning parameters by hand using the psi2propII and chgDefaults functions.

To specify separate weight functions rho.b and rho.sigma.b for different variance components, it is possible to pass a list instead of a psi_func object. The list entries correspond to the groups as shown by VarCorr(.) when applied to the model fitted with lmer. A set of correlated random effects count as just one group.

lmerNoFit:

The lmerNoFit function can be used to get trivial starting values. This is mainly used to verify the algorithms to reproduce the fit by lmer when starting from trivial initial values.

Value

object of class rlmerMod.

Prior weights

weights is passed through to lmer and has the usual meaning, \mathrm{Var}(e_i) = \sigma^2 / w_i. Weighted method = "DASvar" fits made with robustlmm 3.2-0 to 3.4-5 are wrong and should be refitted: the Design Adaptive Scale counted the prior weights twice, so \hat\sigma, the variance components and every standard error derived from them were biased by an amount that depended on the scale the weights happened to be supplied on. The default method = "DAStau" was affected only through its starting value, and in practice only at rho.e = rho.b = cPsi. Unweighted fits are unaffected in every version. See the 3.5.0 entry of news(package = "robustlmm") for the magnitudes and the measured exposure of each method.

Author(s)

Manuel Koller, with thanks to Vanda Lourenço for improvements.

See Also

lmer, vignette("rlmer")

Examples

## dropping of VC
system.time(print(rlmer(Yield ~ (1|Batch), Dyestuff2, method="DASvar")))

## Not run: 
  ## Default method "DAStau"
  system.time(rfm.DAStau <- rlmer(Yield ~ (1|Batch), Dyestuff))
  summary(rfm.DAStau)
  ## DASvar method (faster, less accurate)
  system.time(rfm.DASvar <- rlmer(Yield ~ (1|Batch), Dyestuff,
                                  method="DASvar"))
  ## compare the two
  compare(rfm.DAStau, rfm.DASvar)

  ## Fit variance components with higher efficiency
  ## psi2propII yields squared weights to get robust estimates
  ## this is the same as using rlmer's argument `setting = "RSEa"`
  rlmer(diameter ~ 1 + (1|plate) + (1|sample), Penicillin,
        rho.sigma.e = psi2propII(smoothPsi, k = 2.28),
        rho.sigma.b = psi2propII(smoothPsi, k = 2.28))

  ## use chgDefaults for variance components including
  ## correlation terms (regular, non squared weights suffice)
  ## this is the same as using rlmer's argument `setting = "RSEa"`
  rlmer(Reaction ~ Days + (Days|Subject), sleepstudy,
        rho.sigma.e = psi2propII(smoothPsi, k = 2.28),
        rho.b = chgDefaults(smoothPsi, k = 5.14, s=10),
        rho.sigma.b = chgDefaults(smoothPsi, k = 5.14, s=10))

## End(Not run)

## Not run: 
  ## start from lmer's initial estimate, not its fit
  rlmer(Yield ~ (1|Batch), Dyestuff, init = lmerNoFit)

## End(Not run)

robustlmm documentation built on July 30, 2026, 5:11 p.m.