lmrob: MM-type Estimators for Linear Regression

Description

Computes fast MM-type estimators for linear (regression) models.

Usage

lmrob(formula, data, subset, weights, na.action, method = "MM",
      model = TRUE, x = !control$compute.rd, y = FALSE,
      singular.ok = TRUE, contrasts = NULL, offset = NULL,
      control = NULL, init = NULL, ...)

Value

An object of class lmrob; a list including the following components:

coefficients: The estimate of the coefficient vector
scale: The scale as used in the M estimator.
residuals: Residuals associated with the estimator.

converged: TRUE if the IRWLS iterations have converged.
iter: number of IRWLS iterations
rweights: the “robustness weights” \(\psi(r_i/S) / (r_i/S)\).
fitted.values: Fitted values associated with the estimator.

init.S: The list returned by lmrob.S() or lmrob.M.S() (for MM-estimates, i.e., method="MM" or "SM" only)
init: A similar list that contains the results of intermediate estimates (not for MM-estimates).

rank: the numeric rank of the fitted linear model.
cov: The estimated covariance matrix of the regression coefficients
df.residual: the residual degrees of freedom.

weights: the specified weights (missing if none were used).
na.action: (where relevant) information returned by model.frame on the special handling of NAs.
offset: the offset used (missing if none were used).
contrasts: (only where relevant) the contrasts used.
xlevels: (only where relevant) a record of the levels of the factors used in fitting.
call: the matched call.
terms: the terms object used.

model: if requested (the default), the model frame used.
x: if requested, the model matrix used.
y: if requested, the response used.

In addition, non-null fits will have components assign, and qr relating to the linear fit, for use by extractor functions such as summary.

Arguments

formula: a symbolic description of the model to be fit. See lm and formula for more details.
data: an optional data frame, list or environment (or object coercible by as.data.frame to a data frame) containing the variables in the model. If not found in data, the variables are taken from environment(formula), typically the environment from which lmrob is called.
subset: an optional vector specifying a subset of observations to be used in the fitting process.
weights: an optional vector of weights to be used in the fitting process (in addition to the robustness weights computed in the fitting process).
na.action: a function which indicates what should happen when the data contain NAs. The default is set by the na.action setting of options, and is na.fail if that is unset. The “factory-fresh” default is na.omit. Another possible value is NULL, no action. Value na.exclude can be useful.
method: string specifying the estimator-chain. MM is interpreted as SM. See Details, notably the currently recommended setting = "KS2014".
model, x, y: logicals. If TRUE the corresponding components of the fit (the model frame, the model matrix, the response) are returned.
singular.ok: logical. If FALSE (the default in S but not in R) a singular fit is an error.
contrasts: an optional list. See the contrasts.arg of model.matrix.default.
offset: this can be used to specify an a priori known component to be included in the linear predictor during fitting. An offset term can be included in the formula instead or as well, and if both are specified their sum is used.
control: a list specifying control parameters; use the function lmrob.control(.) and see its help page.
init: an optional argument to specify or supply the initial estimate. See Details.
...: additional arguments can be used to specify control parameters directly instead of (but not in addition to!) via control.

Author

(mainly:) Matias Salibian-Barrera and Manuel Koller

Details

Overview:

This function computes an MM-type regression estimator as described in Yohai (1987) and Koller and Stahel (2011). By default it uses a bi-square redescending score function, and it returns a highly robust and highly efficient estimator (with 50% breakdown point and 95% asymptotic efficiency for normal errors). The computation is carried out by a call to lmrob.fit().

The argument setting of lmrob.control is provided to set alternative defaults as suggested in Koller and Stahel (2011) (setting="KS2011"; now do use its extension setting="KS2014"). For further details, see lmrob.control.

Initial Estimator init:

The initial estimator may be specified using the argument init. This can either be a string, a function or a list. A string can be used to specify built in internal estimators (currently S and M-S, see See also below). A function taking arguments x, y, control, mf (where mf stands for model.frame) and returning a list containing at least the initial coefficients as coefficients and the initial scale estimate scale. Or a list giving the initial coefficients and scale as coefficients and scale. See also Examples.

Note that if the init argument is a function or list, the method argument must not contain the initial estimator, e.g., use MDM instead of SMDM.

The default, equivalent to init = "S", uses as initial estimator an S-estimator (Rousseeuw and Yohai, 1984) which is computed using the Fast-S algorithm of Salibian-Barrera and Yohai (2006), calling lmrob.S(). That function, since March 2012, by default uses nonsingular subsampling which makes the Fast-S algorithm feasible for categorical data as well, see Koller (2012). Note that convergence problems may still show up as warnings, e.g.,


  S refinements did not converge (to refine.tol=1e-07) in 200 (= k.max) steps

and often can simply be remedied by increasing (i.e. weakening) refine.tol or increasing the allowed number of iterations k.max, see lmrob.control.

Method method:

The following chain of estimates is customizable via the method argument. There are currently two types of estimates available,

"M":: corresponds to the standard M-regression estimate.

"D":

stands for the Design Adaptive Scale estimate as proposed in Koller and Stahel (2011).

The method argument takes a string that specifies the estimates to be calculated as a chain. Setting method='SMDM' will result in an intial S-estimate, followed by an M-estimate, a Design Adaptive Scale estimate and a final M-step. For methods involving a D-step, the default value of psi (see lmrob.control) is changed to "lqq".

By default, standard errors are computed using the formulas of Croux, Dhaene and Hoorelbeke (2003) (lmrob.control option cov=".vcov.avar1"). This method, however, works only for MM-estimates (i.e., method = "MM" or = "SM"). For other method arguments, the covariance matrix estimate used is based on the asymptotic normality of the estimated coefficients (cov=".vcov.w") as described in Koller and Stahel (2011). The var-cov computation can be skipped by cov = "none" and (re)done later by e.g., vcov(<obj>, cov = ".vcov.w").

As of robustbase version 0.91-0 (April 2014), the computation of robust standard errors for method="SMDM" has been changed. The old behaviour can be restored by setting the control parameter cov.corrfact = "tauold".

References

Croux, C., Dhaene, G. and Hoorelbeke, D. (2003) Robust standard errors for robust estimators, Discussion Papers Series 03.16, K.U. Leuven, CES.

Koller, M. (2012) Nonsingular subsampling for S-estimators with categorical predictors, ArXiv e-prints https://arxiv.org/abs/1208.5595; extended version published as Koller and Stahel (2017), see lmrob.control.

Koller, M. and Stahel, W.A. (2011) Sharpening Wald-type inference in robust regression for small samples. Computational Statistics & Data Analysis 55(8), 2504--2515.

Maronna, R. A., and Yohai, V. J. (2000) Robust regression with both continuous and categorical predictors. Journal of Statistical Planning and Inference 89, 197--214.

Rousseeuw, P.J. and Yohai, V.J. (1984) Robust regression by means of S-estimators, In Robust and Nonlinear Time Series, J. Franke, W. Härdle and R. D. Martin (eds.). Lectures Notes in Statistics 26, 256--272, Springer Verlag, New York.

Salibian-Barrera, M. and Yohai, V.J. (2006) A fast algorithm for S-regression estimates, Journal of Computational and Graphical Statistics 15(2), 414--427.

Yohai, V.J. (1987) High breakdown-point and high efficiency estimates for regression. The Annals of Statistics 15, 642--65.

Yohai, V., Stahel, W.~A. and Zamar, R. (1991) A procedure for robust estimation and inference in linear regression; in Stahel and Weisberg (eds), Directions in Robust Statistics and Diagnostics, Part II, Springer, New York, 365--374; tools:::Rd_expr_doi("10.1007/978-1-4612-4444-8_20").

Examples

Run this code

data(coleman)
set.seed(0)
## Default for a very long time:
summary( m1 <- lmrob(Y ~ ., data=coleman) )

## Nowadays **strongly recommended** for routine use:
summary(m2 <- lmrob(Y ~ ., data=coleman, setting = "KS2014") )
##                                       ------------------

plot(residuals(m2) ~ weights(m2, type="robustness")) ##-> weights.lmrob()
abline(h=0, lty=3)

data(starsCYG, package = "robustbase")
## Plot simple data and fitted lines
plot(starsCYG)
  lmST <-    lm(log.light ~ log.Te, data = starsCYG)
(RlmST <- lmrob(log.light ~ log.Te, data = starsCYG))
abline(lmST, col = "red")
abline(RlmST, col = "blue")
## --> Least Sq.:/ negative slope  \ robust: slope ~= 2.2 % checked in ../tests/lmrob-data.R
summary(RlmST) # -> 4 outliers; rest perfect
vcov(RlmST)
stopifnot(all.equal(fitted(RlmST),
                    predict(RlmST, newdata = starsCYG), tol = 1e-14))
## FIXME: setting = "KS2011"  or  setting = "KS2014"  **FAIL** here

##--- 'init' argument -----------------------------------
## 1)  string
set.seed(0)
m3 <- lmrob(Y ~ ., data=coleman, init = "S")
stopifnot(all.equal(m1[-18], m3[-18]))
## 2) function
initFun <- function(x, y, control, ...) { # no 'mf' needed
    init.S <- lmrob.S(x, y, control)
    list(coefficients=init.S$coef, scale = init.S$scale)
}
set.seed(0)
m4 <- lmrob(Y ~ ., data=coleman, method = "M", init = initFun)
## list
m5 <- lmrob(Y ~ ., data=coleman, method = "M",
            init = list(coefficients = m3$init$coef, scale = m3$scale))
stopifnot(all.equal(m4[-17], m5[-17]))

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