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

`rlm(x, …)`# S3 method for 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)

# S3 method for default
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 = 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

an optional data frame, list or environment 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

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 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 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 `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 (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.

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

the robust scale estimate used

the weights used in the IWLS process

the psi function with parameters substituted

the convergence criteria at each iteration

did the IWLS converge?

a working residual, weighted for `"inv.var"`

weights only.

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"`

.

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.

```
# NOT RUN {
summary(rlm(stack.loss ~ ., stackloss))
rlm(stack.loss ~ ., stackloss, psi = psi.hampel, init = "lts")
rlm(stack.loss ~ ., stackloss, psi = psi.bisquare)
# }
```

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