rlm {MASS}R Documentation

Robust Fitting of Linear Models


Fit a linear model by robust regression using an M estimator.


rlm(x, ...)

## S3 method for class 'formula':
rlm(formula, data, weights, ..., subset, na.action,
    method = c("M", "MM", "model.frame"),
    wt.method = c("inv.var", "case"),
    model = TRUE, x.ret = TRUE, y.ret = FALSE, contrasts = NULL)

## Default S3 method:
rlm(x, y, weights, ..., w = rep(1, nrow(x)),
    init, psi = psi.huber, scale.est, k2 = 1.345,
    method = c("M", "MM"), wt.method = c("inv.var", "case"),
    maxit = 20, acc = 1e-4, test.vec = "resid", lqs.control = NULL)

psi.huber(u, k = 1.345, deriv = 0)
psi.hampel(u, a = 2, b = 4, c = 8, deriv = 0)
psi.bisquare(u, c = 4.685, deriv = 0)


formula a formula of the form y ~ x1 + x2 + ....
data data frame from which variables specified in formula are preferentially to be taken.
weights prior weights for each case.
subset An index vector specifying the cases to be used in fitting.
na.action A function to specify the action to be taken if NAs are found. The default action is for the procedure to fail. An alternative is na.omit, which leads to omission of cases with missing values on any required variable.
x a matrix or data frame containing the explanatory variables.
y the response: a vector of length the number of rows of x.
method currently either M-estimation or find the model frame. MM estimation is M-estimation with Tukey's biweight initialized by a specific S-estimator. See the details section.
wt.method are the weights case weights (giving the relative importance of case, so a weight of 2 means there are two of these) or the inverse of the variances, so a weight of two means this error is twice as variable?
model should the model frame be returned in the object?
x.ret should the model matrix be returned in the object?
y.ret should the response be returned in the object?
contrasts optional contrast specifications: se lm.
w (optional) initial down-weighting for each case.
init (optional) initial values for the coefficients OR a method to find initial values OR the result of a fit with a coef component. Known methods are "ls" (the default) for an initial least-squares fit using weights w*weights, and "lts" for an unweighted least-trimmed squares fit with 200 samples.
psi the psi function is specified by this argument. It must give (possibly by name) a function g(x, ..., deriv) that for deriv=0 returns psi(x)/x and for deriv=1 returns psi'(x). Tuning constants will be passed in via ....
scale.est method of scale estimation: re-scaled MAD of the residuals or Huber's proposal 2 (which can be selected by either "Huber" or "proposal 2").
k2 tuning constant used for Huber proposal 2 scale estimation.
maxit the limit on the number of IWLS iterations.
acc the accuracy for the stopping criterion.
test.vec the stopping criterion is based on changes in this vector.
... additional arguments to be passed to rlm.default or to the psi function.
lqs.control An optional list of control values for lqs.
u numeric vector of evalation points.
k,a,b,c tuning constants
deriv 0 or 1: compute values of the psi function or of its first derivative.


Fitting is done by iterated re-weighted least squares (IWLS).

Psi functions are supplied for the Huber, Hampel and Tukey bisquare proposals as psi.huber, psi.hampel and psi.bisquare. Huber's corresponds to a convex optimization problem and gives a unique solution (up to collinearity). The other two will have multiple local minima, and a good starting point is desirable.

Selecting method = "MM" selects a specific set of options which ensures that the estimator has a high breakdown point. The initial set of coefficients and the final scale are selected by an S-estimator with k0 = 1.548; this gives (for n >> p) breakdown point 0.5. The final estimator is an M-estimator with Tukey's biweight and fixed scale that will inherit this breakdown point provided c > k0; this is true for the default value of c that corresponds to 95% relative efficiency at the normal.


An object of class "rlm" inheriting from "lm". The additional components not in an lm object are

s the robust scale estimate used
w the weights used in the IWLS process
psi the psi function with parameters substituted
conv the convergence criteria at each iteration
converged did the IWLS converge?


P. J. Huber (1981) Robust Statistics. Wiley.

F. R. Hampel, E. M. Ronchetti, P. J. Rousseeuw and W. A. Stahel (1986) Robust Statistics: The Approach based on Influence Functions. Wiley.

A. Marazzi (1993) Algorithms, Routines and S Functions for Robust Statistics. Wadsworth & Brooks/Cole.

Venables, W. N. and Ripley, B. D. (2002) Modern Applied Statistics with S. Fourth edition. Springer.

See Also

lm, lqs.


summary(rlm(stack.loss ~ ., stackloss))
rlm(stack.loss ~ ., stackloss, psi = psi.hampel, init = "lts")
rlm(stack.loss ~ ., stackloss, psi = psi.bisquare)

[Package MASS version 7.2-23 Index]