# glmregNB: fit a negative binomial model with lasso (or elastic net),... In mpath: Regularized Linear Models

## Description

Fit a negative binomial linear model via penalized maximum likelihood. The regularization path is computed for the lasso (or elastic net penalty), snet and mnet penalty, at a grid of values for the regularization parameter lambda.

## Usage

 1 2 3 4 5 6 7 glmregNB(formula, data, weights, nlambda = 100, lambda=NULL, lambda.min.ratio = ifelse(nobs

## Arguments

 formula formula used to describe a model. data argument controlling formula processing via model.frame. weights observation weights. Default is 1 for each observation nlambda The number of lambda values - default is 100. lambda A user supplied lambda sequence lambda.min.ratio Smallest value for lambda, as a fraction of lambda.max, the (data derived) entry value (i.e. the smallest value for which all coefficients are zero). The default depends on the sample size nobs relative to the number of variables nvars. If nobs > nvars, the default is 0.001, close to zero. If nobs < nvars, the default is 0.05. alpha The L2 penalty mixing parameter, with 0≤α≤ 1. alpha=1 is lasso (mcp, scad) penalty; and alpha=0 the ridge penalty. gamma The tuning parameter of the snet or mnet penalty. rescale logical value, if TRUE, adaptive rescaling of the penalty parameter for penalty="mnet" or penalty="snet" with family other than "gaussian". See reference standardize Logical flag for x variable standardization, prior to fitting the model sequence. The coefficients are always returned on the original scale. Default is standardize=TRUE. If variables are in the same units already, you might not wish to standardize. penalty.factor This is a number that multiplies lambda to allow differential shrinkage of coefficients. Can be 0 for some variables, which implies no shrinkage, and that variable is always included in the model. Default is same shrinkage for all variables. thresh Convergence threshold for coordinate descent. Defaults value is 1e-6. maxit.theta Maximum number of iterations for estimating theta scaling parameter maxit Maximum number of coordinate descent iterations for each lambda value; default is 1000. eps If a number is less than eps in magnitude, then this number is considered as 0 trace If TRUE, fitting progress is reported start, etastart, mustart, ... arguments for the link{glmreg} function init.theta initial scaling parameter theta theta.est Estimate scale parameter theta? Default is TRUE. Note, the algorithm may become slow. In this case, one may use glmreg function with family="negbin", and a fixed theta

.

 theta0 initial scale parameter vector theta, with length nlambda if theta.est=FALSE. Default is NULL convex Calculate index for which objective function ceases to be locally convex? Default is FALSE and only useful if penalty="mnet" or "snet". link link function, default is log penalty Type of regularization method estimation method model, x.keep, y.keep logicals. If TRUE the corresponding components of the fit (model frame, response, model matrix) are returned. contrasts the contrasts corresponding to levels from the respective models

## Details

The sequence of models implied by lambda is fit by coordinate descent. This is a lasso (mcp, scad) or elastic net (mnet, snet) regularization path for fitting the negative binomial linear regression paths, by maximizing the penalized log-likelihood. Note that the objective function is

-∑ (weights * loglik) + λ*penalty

if standardize=FALSE and

-\frac{weights}{∑(weights)} * loglik + λ*penalty

if standardize=TRUE.

## Value

An object with S3 class "glmreg", "glmregNB" for the various types of models.

 call the call that produced the model fit b0 Intercept sequence of length length(lambda) beta A nvars x length(lambda) matrix of coefficients. lambda The actual sequence of lambda values used dev The computed deviance. The deviance calculations incorporate weights if present in the model. The deviance is defined to be 2*(loglike_sat - loglike), where loglike_sat is the log-likelihood for the saturated model (a model with a free parameter per observation). nulldev Null deviance (per observation). This is defined to be 2*(loglike_sat -loglike(Null)); The NULL model refers to the intercept model. nobs number of observations

## Author(s)

Zhu Wang <zwang@connecticutchildrens.org>

## References

Breheny, P. and Huang, J. (2011) Coordinate descent algorithms for nonconvex penalized regression, with applications to biological feature selection. Ann. Appl. Statist., 5: 232-253.

Zhu Wang, Shuangge Ma, Michael Zappitelli, Chirag Parikh, Ching-Yun Wang and Prasad Devarajan (2014) Penalized Count Data Regression with Application to Hospital Stay after Pediatric Cardiac Surgery, Statistical Methods in Medical Research. 2014 Apr 17. [Epub ahead of print]

print, predict, coef and plot methods, and the cv.glmregNB function.
 1 2 3 4 5 6 7 8 9 ## Not run: data("bioChemists", package = "pscl") fm_nb <- glmregNB(art ~ ., data = bioChemists) coef(fm_nb) ### ridge regression fm <- glmregNB(art ~ ., alpha=0, data = bioChemists, lambda=seq(0.001, 1, by=0.01)) fm <- cv.glmregNB(art ~ ., alpha=0, data = bioChemists, lambda=seq(0.001, 1, by=0.01)) ## End(Not run) `