fmr {gnlm}

Generalized Nonlinear Regression Models with Two or Three Point Mixtures

Description

fmr fits user specified nonlinear regression equations to the location parameter of the common one and two parameter distributions. (The log of the scale parameter is estimated to ensure positivity.)

Usage

fmr(y = NULL, distribution = "normal", mu = NULL, mix = NULL,
  linear = NULL, pmu = NULL, pmix = NULL, pshape = NULL,
  censor = "right", exact = FALSE, wt = 1, delta = 1,
  common = FALSE, envir = parent.frame(), print.level = 0,
  typsize = abs(p), ndigit = 10, gradtol = 1e-05, stepmax = 10 *
  sqrt(p %*% p), steptol = 1e-05, iterlim = 100, fscale = 1)

Arguments

y

A response vector for uncensored data, a two column matrix for binomial data or censored data, with the second column being the censoring indicator (1: uncensored, 0: right censored, -1: left censored), or an object of class, response (created by restovec) or repeated (created by rmna or lvna). If the repeated data object contains more than one response variable, give that object in envir and give the name of the response variable to be used here.

distribution

Either a character string containing the name of the distribution or a function giving the -log likelihood and calling the location and mixture functions. Distributions are binomial, beta binomial, double binomial, multiplicative binomial, Poisson, negative binomial, double Poisson, multiplicative Poisson, gamma count, Consul, geometric, normal, inverse Gauss, logistic, exponential, gamma, Weibull, extreme value, Pareto, Cauchy, Student t, Laplace, and Levy. (For definitions of distributions, see the corresponding [dpqr]distribution help.)

mu

A user-specified function of pmu, and possibly linear, giving the regression equation for the location. This may contain a linear part as the second argument to the function. It may also be a formula beginning with ~, specifying either a linear regression function for the location parameter in the Wilkinson and Rogers notation or a general function with named unknown parameters. If it contains unknown parameters, the keyword linear may be used to specify a linear part. If nothing is supplied, the location is taken to be constant unless the linear argument is given.

mix

A user-specified function of pmix, and possibly linear, giving the regression equation for the mixture parameter. This may contain a linear part as the second argument to the function. It may also be a formula beginning with ~, specifying either a linear regression function for the mixture parameter in the Wilkinson and Rogers notation or a general function with named unknown parameters. If it contains unknown parameters, the keyword linear may be used to specify a linear part. If nothing is supplied, this parameter is taken to be constant. This parameter is the logit of the mixture probability.

linear

A formula beginning with ~ in W&R notation, or list of two such expressions, specifying the linear part of the regression function for the location or location and mixture parameters.

pmu

Vector of initial estimates for the location parameters. If mu is a formula with unknown parameters, their estimates must be supplied either in their order of appearance in the expression or in a named list.

pmix

Vector of initial estimates for the mixture parameters. If mix is a formula with unknown parameters, their estimates must be supplied either in their order of appearance in the expression or in a named list.

pshape

An initial estimate for the shape parameter.

censor

right, left, or both indicating where the mixing distribution is placed. both is only possible for binomial data.

exact

If TRUE, fits the exact likelihood function for continuous data by integration over intervals of observation given in delta, i.e. interval censoring.

wt

Weight vector.

delta

Scalar or vector giving the unit of measurement (always one for discrete data) for each response value, set to unity by default - for example, if a response is measured to two decimals, delta=0.01. If the response is transformed, this must be multiplied by the Jacobian. The transformation cannot contain unknown parameters. For example, with a log transformation, delta=1/y.

common

If TRUE, mu and mix must both be either functions with, as argument, a vector of parameters having some or all elements in common between them so that indexing is in common between them or formulae with unknowns. All parameter estimates must be supplied in pmu. If FALSE, parameters are distinct between the two functions and indexing starts at one in each function.

envir

Environment in which model formulae are to be interpreted or a data object of class, repeated, tccov, or tvcov; the name of the response variable should be given in y. If y has class repeated, it is used as the environment.

print.level

Arguments controlling nlm.

typsize

Arguments controlling nlm.

ndigit

Arguments controlling nlm.

gradtol

Arguments controlling nlm.

stepmax

Arguments controlling nlm.

steptol

Arguments controlling nlm.

iterlim

Arguments controlling nlm.

fscale

Arguments controlling nlm.

Details

For the Poisson and related distributions, the mixture involves the zero category. For the binomial and related distributions, it involves the two extreme categories. For all other distributions, it involves either left or right censored individuals. A user-specified -log likelihood can also be supplied for the distribution.

Nonlinear regression models can be supplied as formulae where parameters are unknowns in which case factor variables cannot be used and parameters must be scalars. (See finterp.)

The printed output includes the -log likelihood (not the deviance), the corresponding AIC, the maximum likelihood estimates, standard errors, and correlations.

Value

A list of class gnlm is returned that contains all of the relevant information calculated, including error codes.

Author(s)

J.K. Lindsey

See Also

finterp, glm, gnlr, gnlr3, lm.

Examples

sex <- c(rep(0,10),rep(1,10))
sexf <- gl(2,10)
age <- c(8,10,12,12,8,7,16,7,9,11,8,9,14,12,12,11,7,7,7,12)
y <- cbind(c(9.2, 7.3,13.0, 6.9, 3.9,14.9,17.8, 4.8, 6.4, 3.3,17.2,
	14.4,17.0, 5.0,17.3, 3.8,19.4, 5.0, 2.0,19.0),
	c(0,1,0,1,1,1,0,1,0,1,1,1,1,1,1,1,1,1,1,1))
# y <- cbind(rweibull(20,2,2+2*sex+age),rbinom(20,1,0.7))
# log linear regression with Weibull distribution with a point mass
#   for right censored individuals
mu <- function(p) exp(p[1]+p[2]*sex+p[3]*age)
fmr(y, dist="Weibull", mu=mu, pmu=c(4,0,0), pmix=0.5, pshape=1)
# or equivalently
fmr(y, dist="Weibull", mu=function(p,linear) exp(linear),
	linear=~sexf+age, pmu=c(4,0,0), pmix=0.5, pshape=1)
# or
fmr(y, dist="Weibull", mu=~exp(b0+b1*sex+b2*age), pmu=list(b0=4,b1=0,b2=0),
	pmix=0.5, pshape=1)
#
# include logistic regression for the mixture parameter
mix <- function(p) p[1]+p[2]*sex
fmr(y, dist="Weibull", mu=~exp(a+b*age), mix=mix, pmu=c(4,0),
	pmix=c(10,0), pshape=0.5)
# or equivalently
fmr(y, dist="Weibull", mu=function(p,linear) exp(linear),
	linear=list(~age,~sexf), pmu=c(4,0), pmix=c(10,0), pshape=0.5)
# or
fmr(y, dist="Weibull", mu=~exp(b0+b1*age), mix=~c0+c1*sex,
	pmu=list(b0=4,b1=0), pmix=list(c0=10,c1=0), pshape=0.5)
#
# generate zero-inflated negative binomial data
x1 <- rpois(50,4)
x2 <- rpois(50,4)
ind <- rbinom(50,1,1/(1+exp(-1-0.1*x1)))
y <- ifelse(ind,rnbinom(50,3,mu=exp(1+0.2*x2)),0)
# standard Poisson models
gnlr(y, dist="Poisson", mu=~exp(a), pmu=1)
gnlr(y, dist="Poisson", mu=~exp(linear), linear=~x2, pmu=c(1,0.2))
# zero-inflated Poisson ZIP
fmr(y, dist="Poisson", mu=~exp(a), pmu=1, pmix=0)
fmr(y, dist="Poisson", mu=~exp(linear), linear=~x2, pmu=c(1,0.2), pmix=0)
fmr(y, dist="Poisson", mu=~exp(a), mix=~x1, pmu=1, pmix=c(1,0))
fmr(y, dist="Poisson", mu=~exp(linear), linear=~x2, mix=~x1, pmu=c(1,0.2),
	pmix=c(1,0))
# zero-inflated negative binomial
fmr(y, dist="negative binomial", mu=~exp(a), pmu=1, pshape=0, pmix=0)
fmr(y, dist="negative binomial", mu=~exp(linear), linear=~x2, pmu=c(1,0.2),
	pshape=0, pmix=0)
fmr(y, dist="negative binomial", mu=~exp(a), mix=~x1, pmu=1, pshape=0,
       pmix=c(1,0))
fmr(y, dist="negative binomial", mu=~exp(linear), linear=~x2, mix=~x1,
	pmu=c(1,0.2), pshape=0, pmix=c(1,0))