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<table width="100%" summary="page for lqs {MASS}"><tr><td>lqs {MASS}</td><td align="right">R Documentation</td></tr></table>
<h2>Resistant Regression</h2>
<h3>Description</h3>
<p>
Fit a regression to the <EM>good</EM> points in the dataset, thereby
achieving a regression estimator with a high breakdown point.
<code>lmsreg</code> and <code>ltsreg</code> are compatibility wrappers.
</p>
<h3>Usage</h3>
<pre>
lqs(x, ...)
## S3 method for class 'formula':
lqs(formula, data, ...,
method = c("lts", "lqs", "lms", "S", "model.frame"),
subset, na.action, model = TRUE,
x.ret = FALSE, y.ret = FALSE, contrasts = NULL)
## Default S3 method:
lqs(x, y, intercept = TRUE, method = c("lts", "lqs", "lms", "S"),
quantile, control = lqs.control(...), k0 = 1.548, seed, ...)
lmsreg(...)
ltsreg(...)
</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>subset</code></td>
<td>
an index vector specifying the cases to be used in
fitting. (NOTE: If given, this argument must be named exactly.)</td></tr>
<tr valign="top"><td><code>na.action</code></td>
<td>
function to specify the action to be taken if
<code>NA</code>s are found. The default action is for the procedure to
fail. Alternatives include <code><a href="../../stats/html/na.fail.html">na.omit</a></code> and
<code><a href="../../stats/html/na.fail.html">na.exclude</a></code>, which lead to omission of
cases with missing values on any required variable. (NOTE: If
given, this argument must be named exactly.)
</td></tr>
<tr valign="top"><td><code>model, x.ret, y.ret</code></td>
<td>
logical. If <code>TRUE</code> the model frame,
the model matrix and the response are returned, respectively.</td></tr>
<tr valign="top"><td><code>contrasts</code></td>
<td>
an optional list. See the <code>contrasts.arg</code>
of <code><a href="../../stats/html/model.matrix.html">model.matrix.default</a></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>intercept</code></td>
<td>
should the model include an intercept?</td></tr>
<tr valign="top"><td><code>method</code></td>
<td>
the method to be used. <code>model.frame</code> returns the model frame: for the
others see the <code>Details</code> section. Using <code>lmsreg</code> or
<code>ltsreg</code> forces <code>"lms"</code> and <code>"lts"</code> respectively.
</td></tr>
<tr valign="top"><td><code>quantile</code></td>
<td>
the quantile to be used: see <code>Details</code>. This is over-ridden if
<code>method = "lms"</code>.
</td></tr>
<tr valign="top"><td><code>control</code></td>
<td>
additional control items: see <code>Details</code>.</td></tr>
<tr valign="top"><td><code>k0</code></td>
<td>
the cutoff / tuning constant used for <i>chi()</i>
and <i>psi()</i> functions when <code>method = "S"</code>, currently
corresponding to Tukey's ‘biweight’.</td></tr>
<tr valign="top"><td><code>seed</code></td>
<td>
the seed to be used for random sampling: see <code>.Random.seed</code>. The
current value of <code>.Random.seed</code> will be preserved if it is set..
</td></tr>
<tr valign="top"><td><code>...</code></td>
<td>
arguments to be passed to <code>lqs.default</code> or
<code>lqs.control</code>, see <code>control</code> above and <code>Details</code>.</td></tr>
</table>
<h3>Details</h3>
<p>
Suppose there are <code>n</code> data points and <code>p</code> regressors,
including any intercept.
</p>
<p>
The first three methods minimize some function of the sorted squared
residuals. For methods <code>"lqs"</code> and <code>"lms"</code> is the
<code>quantile</code> squared residual, and for <code>"lts"</code> it is the sum
of the <code>quantile</code> smallest squared residuals. <code>"lqs"</code> and
<code>"lms"</code> differ in the defaults for <code>quantile</code>, which are
<code>floor((n+p+1)/2)</code> and <code>floor((n+1)/2)</code> respectively.
For <code>"lts"</code> the default is <code>floor(n/2) + floor((p+1)/2)</code>.
</p>
<p>
The <code>"S"</code> estimation method solves for the scale <code>s</code>
such that the average of a function chi of the residuals divided
by <code>s</code> is equal to a given constant.
</p>
<p>
The <code>control</code> argument is a list with components
<dl>
<dt><code>psamp</code>:</dt><dd>the size of each sample. Defaults to <code>p</code>.</dd>
<dt><code>nsamp</code>:</dt><dd>the number of samples or <code>"best"</code> (the
default) or <code>"exact"</code> or <code>"sample"</code>.
If <code>"sample"</code> the number chosen is <code>min(5*p, 3000)</code>,
taken from Rousseeuw and Hubert (1997).
If <code>"best"</code> exhaustive enumeration is done up to 5000 samples;
if <code>"exact"</code> exhaustive enumeration will be attempted however
many samples are needed.</dd>
<dt><code>adjust</code>:</dt><dd>should the intercept be optimized for each
sample? Defaults to <code>TRUE</code>.</dd>
</dl>
<h3>Value</h3>
<p>
An object of class <code>"lqs"</code>. This is a list with components
</p>
<table summary="R argblock">
<tr valign="top"><td><code>crit</code></td>
<td>
the value of the criterion for the best solution found, in
the case of <code>method == "S"</code> before IWLS refinement.</td></tr>
<tr valign="top"><td><code>sing</code></td>
<td>
character. A message about the number of samples which
resulted in singular fits.</td></tr>
<tr valign="top"><td><code>coefficients</code></td>
<td>
of the fitted linear model</td></tr>
<tr valign="top"><td><code>bestone</code></td>
<td>
the indices of those points fitted by the best sample
found (prior to adjustment of the intercept, if requested).</td></tr>
<tr valign="top"><td><code>fitted.values</code></td>
<td>
the fitted values.</td></tr>
<tr valign="top"><td><code>residuals</code></td>
<td>
the residuals.</td></tr>
<tr valign="top"><td><code>scale</code></td>
<td>
estimate(s) of the scale of the error. The first is based
on the fit criterion. The second (not present for <code>method ==
"S"</code>) is based on the variance of those residuals whose absolute
value is less than 2.5 times the initial estimate.</td></tr>
</table>
<h3>Note</h3>
<p>
There seems no reason other than historical to use the <code>lms</code> and
<code>lqs</code> options. LMS estimation is of low efficiency (converging
at rate <i>n^{-1/3}</i>) whereas LTS has the same asymptotic efficiency
as an M estimator with trimming at the quartiles (Marazzi, 1993, p.201).
LQS and LTS have the same maximal breakdown value of
<code>(floor((n-p)/2) + 1)/n</code> attained if
<code>floor((n+p)/2) <= quantile <= floor((n+p+1)/2)</code>.
The only drawback mentioned of LTS is greater computation, as a sort
was thought to be required (Marazzi, 1993, p.201) but this is not
true as a partial sort can be used (and is used in this implementation).
</p>
<p>
Adjusting the intercept for each trial fit does need the residuals to
be sorted, and may be significant extra computation if <code>n</code> is large
and <code>p</code> small.
</p>
<p>
Opinions differ over the choice of <code>psamp</code>. Rousseeuw and Hubert
(1997) only consider p; Marazzi (1993) recommends p+1 and suggests
that more samples are better than adjustment for a given computational
limit.
</p>
<p>
The computations are exact for a model with just an intercept and
adjustment, and for LQS for a model with an intercept plus one
regressor and exhaustive search with adjustment. For all other cases
the minimization is only known to be approximate.
</p>
<h3>Author(s)</h3>
<p>
B. D. Ripley
</p>
<h3>References</h3>
<p>
P. J. Rousseeuw and A. M. Leroy (1987)
<EM>Robust Regression and Outlier Detection.</EM>
Wiley.
</p>
<p>
A. Marazzi (1993)
<EM>Algorithms, Routines and S Functions for Robust Statistics.</EM>
Wadsworth and Brooks/Cole.
</p>
<p>
P. Rousseeuw and M. Hubert (1997) Recent developments in PROGRESS. In
<EM>L1-Statistical Procedures and Related Topics</EM>,
ed Y. Dodge, IMS Lecture Notes volume <B>31</B>, pp. 201–214.
</p>
<h3>See Also</h3>
<p>
<code><a href="predict.lqs.html">predict.lqs</a></code>
</p>
<h3>Examples</h3>
<pre>
set.seed(123) # make reproducible
lqs(stack.loss ~ ., data = stackloss)
lqs(stack.loss ~ ., data = stackloss, method = "S", nsamp = "exact")
</pre>
<hr><div align="center">[Package <em>MASS</em> version 7.2-44 <a href="00Index.html">Index]</a></div>
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