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<html><head><title>R: Robust Fitting of Linear Models</title>
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<table width="100%" summary="page for rlm {MASS}"><tr><td>rlm {MASS}</td><td align="right">R Documentation</td></tr></table>
<h2>Robust Fitting of Linear Models</h2>
<h3>Description</h3>
<p>
Fit a linear model by robust regression using an M estimator.
</p>
<h3>Usage</h3>
<pre>
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 = 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)
</pre>
<h3>Arguments</h3>
<table summary="R argblock">
<tr valign="top"><td><code>formula</code></td>
<td>
a formula of the form <code>y ~ x1 + x2 + ...</code>.
</td></tr>
<tr valign="top"><td><code>data</code></td>
<td>
data frame from which variables specified in <code>formula</code> are
preferentially to be taken.
</td></tr>
<tr valign="top"><td><code>weights</code></td>
<td>
a vector of prior weights for each case.
</td></tr>
<tr valign="top"><td><code>subset</code></td>
<td>
An index vector specifying the cases to be used in fitting.
</td></tr>
<tr valign="top"><td><code>na.action</code></td>
<td>
A function to specify the action to be taken if <code>NA</code>s are found.
The ‘factory-fresh’ default action in <font face="Courier New,Courier" color="#666666"><b>R</b></font> is
<code><a href="../../stats/html/na.fail.html">na.omit</a></code>, and can be changed by
<code><a href="../../base/html/options.html">options</a>(na.action=)</code>.
</td></tr>
<tr valign="top"><td><code>x</code></td>
<td>
a matrix or data frame containing the explanatory variables.
</td></tr>
<tr valign="top"><td><code>y</code></td>
<td>
the response: a vector of length the number of rows of <code>x</code>.
</td></tr>
<tr valign="top"><td><code>method</code></td>
<td>
currently either M-estimation or MM-estimation or (for the
<code>formula</code> 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.
</td></tr>
<tr valign="top"><td><code>wt.method</code></td>
<td>
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?
</td></tr>
<tr valign="top"><td><code>model</code></td>
<td>
should the model frame be returned in the object?
</td></tr>
<tr valign="top"><td><code>x.ret</code></td>
<td>
should the model matrix be returned in the object?
</td></tr>
<tr valign="top"><td><code>y.ret</code></td>
<td>
should the response be returned in the object?
</td></tr>
<tr valign="top"><td><code>contrasts</code></td>
<td>
optional contrast specifications: se <code><a href="../../VGAM/html/notdocumentedyet.html">lm</a></code>.
</td></tr>
<tr valign="top"><td><code>w</code></td>
<td>
(optional) initial down-weighting for each case.
</td></tr>
<tr valign="top"><td><code>init</code></td>
<td>
(optional) initial values for the coefficients OR a method to find
initial values OR the result of a fit with a <code>coef</code> component. Known
methods are <code>"ls"</code> (the default) for an initial least-squares fit
using weights <code>w*weights</code>, and <code>"lts"</code> for an unweighted
least-trimmed squares fit with 200 samples.
</td></tr>
<tr valign="top"><td><code>psi</code></td>
<td>
the psi function is specified by this argument. It must give
(possibly by name) a function <code>g(x, ..., deriv)</code> that for
<code>deriv=0</code> returns psi(x)/x and for <code>deriv=1</code> returns
psi'(x). Tuning constants will be passed in via <code>...</code>.
</td></tr>
<tr valign="top"><td><code>scale.est</code></td>
<td>
method of scale estimation: re-scaled MAD of the residuals (default)
or Huber's proposal 2 (which can be selected by either <code>"Huber"</code>
or <code>"proposal 2"</code>).
</td></tr>
<tr valign="top"><td><code>k2</code></td>
<td>
tuning constant used for Huber proposal 2 scale estimation.
</td></tr>
<tr valign="top"><td><code>maxit</code></td>
<td>
the limit on the number of IWLS iterations.
</td></tr>
<tr valign="top"><td><code>acc</code></td>
<td>
the accuracy for the stopping criterion.
</td></tr>
<tr valign="top"><td><code>test.vec</code></td>
<td>
the stopping criterion is based on changes in this vector.
</td></tr>
<tr valign="top"><td><code>...</code></td>
<td>
additional arguments to be passed to <code>rlm.default</code> or to the <code>psi</code>
function.
</td></tr>
<tr valign="top"><td><code>lqs.control</code></td>
<td>
An optional list of control values for <code><a href="lqs.html">lqs</a></code>.
</td></tr>
<tr valign="top"><td><code>u</code></td>
<td>
numeric vector of evalation points.
</td></tr>
<tr valign="top"><td><code>k, a, b, c</code></td>
<td>
tuning constants.
</td></tr>
<tr valign="top"><td><code>deriv</code></td>
<td>
<code>0</code> or <code>1</code>: compute values of the psi function or of its
first derivative.
</td></tr>
</table>
<h3>Details</h3>
<p>
Fitting is done by iterated re-weighted least squares (IWLS).
</p>
<p>
Psi functions are supplied for the Huber, Hampel and Tukey bisquare
proposals as <code>psi.huber</code>, <code>psi.hampel</code> and
<code>psi.bisquare</code>. 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.
</p>
<p>
Selecting <code>method = "MM"</code> 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 <code>k0 = 1.548</code>; this gives (for <i>n >> p</i>)
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 <code>c > k0</code>;
this is true for the default value of <code>c</code> that corresponds to
95% relative efficiency at the normal. Case weights are not
supported for <code>method = "MM"</code>.
</p>
<h3>Value</h3>
<p>
An object of class <code>"rlm"</code> inheriting from <code>"lm"</code>.
The additional components not in an <code>lm</code> object are
</p>
<table summary="R argblock">
<tr valign="top"><td><code>s</code></td>
<td>
the robust scale estimate used
</td></tr>
<tr valign="top"><td><code>w</code></td>
<td>
the weights used in the IWLS process
</td></tr>
<tr valign="top"><td><code>psi</code></td>
<td>
the psi function with parameters substituted
</td></tr>
<tr valign="top"><td><code>conv</code></td>
<td>
the convergence criteria at each iteration
</td></tr>
<tr valign="top"><td><code>converged</code></td>
<td>
did the IWLS converge?
</td></tr>
<tr valign="top"><td><code>wresid</code></td>
<td>
a working residual, weighted for <code>"inv.var"</code> weights only.
</td></tr>
</table>
<h3>References</h3>
<p>
P. J. Huber (1981)
<EM>Robust Statistics</EM>.
Wiley.
</p>
<p>
F. R. Hampel, E. M. Ronchetti, P. J. Rousseeuw and W. A. Stahel (1986)
<EM>Robust Statistics: The Approach based on Influence Functions</EM>.
Wiley.
</p>
<p>
A. Marazzi (1993)
<EM>Algorithms, Routines and S Functions for Robust Statistics</EM>.
Wadsworth & Brooks/Cole.
</p>
<p>
Venables, W. N. and Ripley, B. D. (2002)
<EM>Modern Applied Statistics with S.</EM> Fourth edition. Springer.
</p>
<h3>See Also</h3>
<p>
<code><a href="../../VGAM/html/notdocumentedyet.html">lm</a></code>, <code><a href="lqs.html">lqs</a></code>.
</p>
<h3>Examples</h3>
<pre>
summary(rlm(stack.loss ~ ., stackloss))
rlm(stack.loss ~ ., stackloss, psi = psi.hampel, init = "lts")
rlm(stack.loss ~ ., stackloss, psi = psi.bisquare)
</pre>
<hr><div align="center">[Package <em>MASS</em> version 7.2-44 <a href="00Index.html">Index]</a></div>
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