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<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"><html xmlns="http://www.w3.org/1999/xhtml"><head><title>R: Robust estimations for negative binomial GLM</title>
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<table width="100%" summary="page for glmrob.nb"><tr><td>glmrob.nb</td><td style="text-align: right;">R Documentation</td></tr></table>
<h2>Robust estimations for negative binomial GLM</h2>
<h3>Description</h3>
<p><code>glmrob.nb</code> is used to fit a negative binomial model by robust (bounded influence) methods, of the Mallows' type. Two approaches are implemented: The PR approach (for <code>bounding.func = "T/T"</code> and <code>bounding.func = "HA/HA"</code>) achieves robustness in the response by applying a bounded function on the Pearson residuals while the UD approach (for <code>bounding.func = "BY/T"</code>) achieves robustness in the response by applying a bounded function on the unscaled deviance components.
</p>
<h3>Usage</h3>
<pre>
glmrob.nb(y, X, bounding.func = "T/T", offset = rep(0,length(y)), weights.on.x = 'none',
options.wx = list(mve.nobs = floor(length(y)*0.8), p.chisq = 0.95),
c.tukey.beta = 4, c.tukey.sig = 4, c.by.beta = 4,
a.hampel.beta = 3, b.hampel.beta = 4, c.hampel.beta = 5,
a.hampel.sig = 3, b.hampel.sig = 4, c.hampel.sig = 5,
param.ini, minsig = 1e-2, maxsig = 50, minmu = 1e-10, maxmu = 1e120,
maxit = 30, maxit.sig = 50, sig.prec = 1e-8, tol = 1e-4)
</pre>
<h3>Arguments</h3>
<table summary="R argblock">
<tr valign="top"><td><code>y</code></td>
<td>
<p>The count response variable consisting of non-negative integers.</p>
</td></tr>
<tr valign="top"><td><code>X</code></td>
<td>
<p>The matrix of covariates. Number of rows must match the length of <code>y</code>. Missing values are not supported and the corresponding rows should be discarded beforehand. A single vector full of ones is supported for an intercept-only model.</p>
</td></tr>
<tr valign="top"><td><code>bounding.func</code></td>
<td>
<p>The bounded functions applied either on the Pearson residuals or on the deviance components; choice among <code>"T/T"</code>, <code>"HA/HA"</code> and <code>"BY/T"</code>. See Details below.</p>
</td></tr>
<tr valign="top"><td><code>offset</code></td>
<td>
<p>A vector of length <code>length(y)</code> to be directly added to the linear predictor, that is a covariate whose coefficient is set to 1.</p>
</td></tr>
<tr valign="top"><td><code>weights.on.x</code></td>
<td>
<p>Character string specifying the weights on the design. Choice among <code>"none"</code>, all equal to one, and <code>"hard"</code>, hard-rejection weights based on robust Mahalanobis distances.</p>
</td></tr>
<tr valign="top"><td><code>options.wx</code></td>
<td>
<p>A list of options for the weights in the design. <code>mve.nobs</code> is the minimum number of observations enclosed in the Minimum Volume Ellipsoid used for computing robust Mahalanobis distances; passed on to cov.rob. <code>p.chisq</code> is the probability determining the chi squared quantile used as a cut-off value for the hard rejection weights.</p>
</td></tr>
<tr valign="top"><td><code>c.tukey.beta, c.tukey.sig</code></td>
<td>
<p>Tuning constants for Tukey's biweight function, for the robust M-estimations of beta and sigma respectively. Both have an impact for <code>bounding.func = "T/T"</code> while <code>c.tukey.sig</code> has also an impact for <code>bounding.func = "BY/T"</code>.</p>
</td></tr>
<tr valign="top"><td><code>c.by.beta</code></td>
<td>
<p>Tuning constant used for Bianco and Yohai (1996)'s function. Has an impact only for <code>bounding.func = "BY/T"</code>.</p>
</td></tr>
<tr valign="top"><td><code>a.hampel.beta, b.hampel.beta, c.hampel.beta, a.hampel.sig, b.hampel.sig, c.hampel.sig</code></td>
<td>
<p>Tuning constants <code>a</code>, <code>b</code> and <code>c</code> for Hampel's three part function, for the robust M-estimations of beta and sigma respectively. Have an impact only for <code>bounding.func = "HA/HA"</code>.</p>
</td></tr>
<tr valign="top"><td><code>param.ini</code></td>
<td>
<p>An optional vector of starting values for <code>c(sigma, beta)</code> (in that order). If missing, the MLE is computed as starting values for the robust estimates.</p>
</td></tr>
<tr valign="top"><td><code>minsig, maxsig</code></td>
<td>
<p>Lower and upper bounds passed on to uniroot, used for updating the robust estimate of sigma given beta. In case of multiple solutions (typically with low tuning constant values), increasing <code>minsig</code>, say to 0.05 or 0.1, may help avoiding spurious solutions arbitrarily close to 0.</p>
</td></tr>
<tr valign="top"><td><code>minmu, maxmu</code></td>
<td>
<p>Lower and upper bounds of fitted means, for numerical stability.</p>
</td></tr>
<tr valign="top"><td><code>maxit</code></td>
<td>
<p>Maximum number of iterations between robust updates of beta given sigma and sigma given beta.</p>
</td></tr>
<tr valign="top"><td><code>maxit.sig</code></td>
<td>
<p>Maximum number of iterations passed on to <code>uniroot</code> for the robust update of sigma.</p>
</td></tr>
<tr valign="top"><td><code>sig.prec</code></td>
<td>
<p>Numerical tolerance passed on to <code>uniroot</code> for the robust update of sigma.</p>
</td></tr>
<tr valign="top"><td><code>tol</code></td>
<td>
<p>Numerical tolerance for robust update of beta given sigma and for maximum likelihood starting values.</p>
</td></tr>
</table>
<h3>Details</h3>
<p>The function computes robust M-estimates of the regression parameter beta and the overdispersion parameter sigma of the negative binomial (generalized linear) model. The user can choose among three sets of estimators: <code>T/T</code> applies Tukey's biweight function on the Pearson residuals for estimating both beta and sigma; <code>HA/HA</code> uses Hampel's three part function on the Pearson residuals for both beta and sigma; <code>BY/T</code> applies Bianco and Yohai's function on the unscaled deviance components for estimating beta and uses Tukey's biweight for sigma.
</p>
<p>The parameterization used here is such that the conditional variance of the response given the covariates is <i>μ+σμ^2</i>, so that the Poisson distribution with mean <i>μ</i> is the limiting case when <i>σ</i> tends to 0. The sigma considered here is thus the inverse of the theta considered in glm.nb from the <span class="pkg">MASS</span> package.
</p>
<h3>Value</h3>
<p>The function returns a list with following components:
</p>
<table summary="R valueblock">
<tr valign="top"><td><code>coef</code></td>
<td>
<p>a vector whose first element is the robust estimate of sigma, followed by the intercept and by the covariates' coefficients robust estimates.</p>
</td></tr>
<tr valign="top"><td><code>stdev</code></td>
<td>
<p>a vector consisting of the (robust) standard errors of the estimates of beta.</p>
</td></tr>
<tr valign="top"><td><code>weights.y</code></td>
<td>
<p>a vector of length <code>length(y)</code> consisting of the resulting weights on the response.</p>
</td></tr>
<tr valign="top"><td><code>weights.x</code></td>
<td>
<p>a vector of length <code>length(y)</code> consisting of the resulting weights on the design.</p>
</td></tr>
</table>
<h3>Note</h3>
<p>This is version 0.4.</p>
<h3>Author(s)</h3>
<p>William H. Aeberhard, <a href="mailto:[email protected]">[email protected]</a>
</p>
<h3>References</h3>
<p>Aeberhard, W. H., Cantoni, E., Heritier, S. (2014) Robust Inference in the Negative Binomial Regression Model with an Application to Falls Data. <em>Biometrics</em> <b>70, </b> 920–931.
</p>
<p>Bianco, A. M., Yohai, V. J. (1996) Robust estimation in the logistic regression model. In <em>Robust Statistics, Data Analysis, and Computer Intensive Methods</em>, H. Rieder (ed), 17–34, New York: Springer-Verlag.
</p>
<p>Cantoni, E., Ronchetti, E. (2001) Robust Inference for Generalized Linear Models. <em>JASA</em> <b>96,</b> 1022–1030.
</p>
</body></html>