rlm {MASS}  R Documentation 
Robust Fitting of Linear Models
Description
Fit a linear model by robust regression using an M estimator.
Usage
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 = "ls", psi = psi.huber,
scale.est = c("MAD", "Huber", "proposal 2"), k2 = 1.345,
method = c("M", "MM"), wt.method = c("inv.var", "case"),
maxit = 20, acc = 1e4, 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)
Arguments
formula 
a formula of the form 
data 
an optional data frame, list or environment from which variables
specified in 
weights 
a vector of 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 
x 
a matrix or data frame containing the explanatory variables. 
y 
the response: a vector of length the number of rows of 
method 
currently either Mestimation or MMestimation or (for the

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 half 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: see 
w 
(optional) initial downweighting 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 
psi 
the psi function is specified by this argument. It must give
(possibly by name) a function 
scale.est 
method of scale estimation: rescaled MAD of the residuals (default)
or Huber's proposal 2 (which can be selected by either 
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 
lqs.control 
An optional list of control values for 
u 
numeric vector of evaluation points. 
k , a , b , c 
tuning constants. 
deriv 

Details
Fitting is done by iterated reweighted 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 Sestimator
with k0 = 1.548
; this gives (for n \gg p
)
breakdown point 0.5.
The final estimator is an Mestimator 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. Case weights are not
supported for method = "MM"
.
Value
An object of class "rlm"
inheriting from "lm"
.
Note that the df.residual
component is deliberately set to
NA
to avoid inappropriate estimation of the residual scale from
the residual mean square by "lm"
methods.
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? 
wresid 
a working residual, weighted for 
Note
Prior to version 7.352
, offset terms in formula
were omitted from fitted and predicted values.
References
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
Examples
summary(rlm(stack.loss ~ ., stackloss))
rlm(stack.loss ~ ., stackloss, psi = psi.hampel, init = "lts")
rlm(stack.loss ~ ., stackloss, psi = psi.bisquare)