nnpois {addreg}

EM Algorithm for Identity-link Poisson GLM

Description

Finds the maximum likelihood estimate of an identity-link Poisson GLM using an EM algorithm, where each of the coefficients is restricted to be non-negative.

Usage

nnpois(y, x, standard, offset, start, control = addreg.control(),
       accelerate = c("em", "squarem", "pem", "qn"),
       control.method = list())

Arguments

y

non-negative integer response vector.

x

non-negative covariate matrix.

standard

standardising vector, where each element is a positive constant that (multiplicatively) standardises the fitted value of the corresponding element of the response vector. The default is a vector of ones.

offset

non-negative additive offset vector. The default is a vector of zeros.

start

starting values for the parameter estimates. Each element must be greater than control$bound.tol.

control

an addreg.control object, which controls the fitting process.

accelerate

a character string that determines the acceleration algorithm to be used, (partially) matching one of "em" (no acceleration – the default), "squarem", "pem" or "qn". See turboem for further details. Note that "decme" is not permitted.

control.method

a list of control parameters for the acceleration algorithm. See turboem for details of the parameters that apply to each algorithm. If not specified, the defaults are used.

Details

This is a workhorse function for addreg, and runs the EM algorithm to find the constrained non-negative MLE associated with an identity-link Poisson GLM. See Marschner (2010) for full details.

Value

A list containing the following components

coefficients

the constrained non-negative maximum likelihood estimate of the parameters.

residuals

the residuals at the MLE, that is y - fitted.values

fitted.values

the fitted mean values.

rank

the number of parameters in the model (named “rank" for compatibility — we assume that models have full rank)

family

included for compatibility — will always be poisson(identity).

linear.predictors

included for compatibility — same as fitted.values (as this is an identity-link model).

deviance

up to a constant, minus twice the maximised log-likelihood.

aic

a version of Akaike's An Information Criterion, minus twice the maximised log-likelihood plus twice the number of parameters.

aic.c

a small-sample corrected version of Akaike's An Information Criterion (Hurvich, Simonoff and Tsai, 1998).

null.deviance

the deviance for the null model, comparable with deviance. The null model will include the offset and an intercept.

iter

the number of iterations of the EM algorithm used.

weights

included for compatibility — a vector of ones.

prior.weights

included for compatibility — a vector of ones.

standard

the standard vector passed to this function.

df.residual

the residual degrees of freedom.

df.null

the residual degrees of freedom for the null model.

y

the y vector used.

converged

logical. Did the EM algorithm converge (according to conv.test)?

boundary

logical. Is the MLE on the boundary of the parameter space — i.e. are any of the coefficients < control$bound.tol?

loglik

the maximised log-likelihood.

nn.design

the non-negative x matrix used.

Author(s)

Mark W. Donoghoe [e-mail].

This function is based on code from Marschner, Gillett and O'Connell (2012) written by Alexandra Gillett.

References

Hurvich, C. M., J. S. Simonoff and C.-L. Tsai (1998). Smoothing parameter selection in nonparametric regression using an improved Akaike information criterion. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 60(2): 271–293.

Marschner, I. C. (2010). Stable computation of maximum likelihood estimates in identity link Poisson regression. Journal of Computational and Graphical Statistics 19(3): 666–683.

Marschner, I. C., A. C. Gillett and R. L. O'Connell (2012). Stratified additive Poisson models: Computational methods and applications in clinical epidemiology. Computational Statistics and Data Analysis 56(5): 1115–1130.