# rlm

##### Robust Fitting of Linear Models

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

##### Usage

`rlm(x, ...)`## S3 method for class 'formula':
rlm(formula, data, weights, \dots, 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)

## S3 method for class 'default':
rlm(x, y, weights, \dots, 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 = 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)

##### Arguments

- formula
- a formula of the form
`y ~ x1 + x2 + ...`

. - data
- data frame from which variables specified in
`formula`

are preferentially to be taken. - 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
`NA`

s are found. Thefactory-fresh default action in Ris`na.omit`

, and can be changed by - 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 MM-estimation or (for the
`formula`

method only) find the model frame. MM-estimation is M-estimation with Tukey's biweight initialized by a specific S-estimator. See theDetails se - 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: 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 - 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 pa - scale.est
- method of scale estimation: re-scaled MAD of the residuals (default)
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 evaluation points.
- k, a, b, c
- tuning constants.
- deriv
`0`

or`1`

: compute values of the psi function or of its first derivative.

##### Details

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 $n \gg p$)
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. 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 `"inv.var"`

weights only.

##### 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)
```

*Documentation reproduced from package MASS, version 7.3-19, License: GPL-2 | GPL-3*