repeated (version 1.1.9)

gausscop: Multivariate Gaussian Copula with Arbitrary Marginals

Description

gausscop fits multivariate repeated measurements models based on the Gaussian copula with a choice of marginal distributions. Dependence among responses is provided by the correlation matrix containing random effects and/or autoregression.

Usage

gausscop(
  response = NULL,
  distribution = "gamma",
  mu = NULL,
  shape = NULL,
  autocorr = "exponential",
  pmu = NULL,
  pshape = NULL,
  par = NULL,
  pre = NULL,
  delta = NULL,
  shfn = FALSE,
  common = FALSE,
  envir = parent.frame(),
  print.level = 0,
  ndigit = 10,
  gradtol = 1e-05,
  steptol = 1e-05,
  iterlim = 100,
  fscale = 1,
  stepmax = 10 * sqrt(theta %*% theta),
  typsize = abs(c(theta))
)

Value

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

Arguments

response

A list of two or three column matrices with response values, times, and possibly nesting categories, for each individual, one matrix or dataframe of response values, 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

The marginal distribution: exponential, gamma, Weibull, Pareto, inverse Gauss, logistic, Cauchy, Laplace, or Levy.

mu

The linear or nonlinear regression model to be fitted for the location parameter. For marginal distributions requiring positive response values, a log link is used. This model can be a function of the parameters or 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 that describes the location, returning a vector the same length as the number of observations.

shape

The linear or nonlinear regression model to be fitted for the log shape parameter. This can be a function of the parameters or 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 that describes the location. If it contains unknown parameters, the keyword mu may be used to specify a function of the location parameter.

autocorr

The form of the autocorrelation function: exponential is the usual \(\rho^{|t_i-t_j|}\); gaussian is \(\rho^{(t_i-t_j)^2}\); cauchy is \(1/(1+\rho(t_i-t_j)^2)\); spherical is \(((|t_i-t_j|\rho)^3-3|t_i-t_j|\rho+2)/2\) for \(|t_i-t_j|\leq1/\rho\) and zero otherwise.

pmu

Initial parameter estimates for the location regression model.

pshape

Initial parameter estimate for the shape regression model.

par

If supplied, an initial estimate for the autocorrelation parameter.

pre

Zero, one or two parameter estimates for the variance components, depending on the number of levels of nesting.

delta

Scalar or vector giving the unit of measurement for each response value, set to unity by default. For example, if a response is measured to two decimals, delta=0.01. Ignored if response has class, response or repeated.

shfn

If TRUE, the supplied shape function depends on the location function. The name of this location function must be the last argument of the shape function.

common

If TRUE, mu and shape must both be 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; 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 response. If response has class repeated, it is used as the environment.

print.level

Arguments for nlm.

ndigit

Arguments for nlm.

gradtol

Arguments for nlm.

steptol

Arguments for nlm.

iterlim

Arguments for nlm.

fscale

Arguments for nlm.

stepmax

Arguments for nlm.

typsize

Arguments for nlm.

Author

J.K. Lindsey

Details

With two levels of nesting, the first is the individual and the second will consist of clusters within individuals.

For clustered (non-longitudinal) data, where only random effects will be fitted, times are not necessary.

This function is designed to fit linear and nonlinear models with time-varying covariates observed at arbitrary time points. A continuous-time AR(1) and zero, one, or two levels of nesting can be handled.

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.)

References

Song, P.X.K. (2000) Multivariate dispersion models generated from Gaussian copula. Scandinavian Journal of Statistics 27, 305-320.

Examples

# linear models
y <- matrix(rgamma(40,1,1),ncol=5)+rep(rgamma(8,0.5,1),5)
x1 <- c(rep(0,4),rep(1,4))
reps <- rmna(restovec(y),ccov=tcctomat(x1))
# independence with default gamma marginals
# compare with gnlm::gnlr(y, pmu=1, psh=0, dist="gamma", env=reps)
gausscop(y, pmu=1, pshape=0, env=reps)
gausscop(y, mu=~x1, pmu=c(1,0), pshape=0, env=reps)
# AR(1)
gausscop(y, pmu=1, pshape=0, par=0.1, env=reps)
if (FALSE) {
# random effect
gausscop(y, pmu=1, pshape=0, pre=0.1, env=reps)
# try other marginal distributions
gausscop(y, pmu=1, pshape=0, pre=0.1, env=reps, dist="Weibull")
gausscop(y, pmu=1, pshape=0, pre=0.1, env=reps, dist="inverse Gauss",
	stepmax=1)
gausscop(y, pmu=1, pshape=0, pre=0.1, env=reps, dist="Cauchy")
#
# first-order one-compartment model
# create data objects for formulae
dose <- c(2,5)
dd <- tcctomat(dose)
times <- matrix(rep(1:20,2), nrow=2, byrow=TRUE)
tt <- tvctomat(times)
# vector covariates for functions
dose <- c(rep(2,20),rep(5,20))
times <- rep(1:20,2)
# functions
mu <- function(p) exp(p[1]-p[3])*(dose/(exp(p[1])-exp(p[2]))*
	(exp(-exp(p[2])*times)-exp(-exp(p[1])*times)))
shape <- function(p) exp(p[1]-p[2])*times*dose*exp(-exp(p[1])*times)
lmu <- function(p) p[1]-p[3]+log(dose/(exp(p[1])-exp(p[2]))*
	(exp(-exp(p[2])*times)-exp(-exp(p[1])*times)))
lshape <- function(p) p[1]-p[2]+log(times*dose)-exp(p[1])*times
# response
#conc <- matrix(rgamma(40,shape(log(c(0.1,0.4))),
#	scale=mu(log(c(1,0.3,0.2))))/shape(log(c(0.1,0.4))),ncol=20,byrow=TRUE)
#conc[,2:20] <- conc[,2:20]+0.5*(conc[,1:19]-matrix(mu(log(c(1,0.3,0.2))),
#	ncol=20,byrow=TRUE)[,1:19])
#conc <- restovec(ifelse(conc>0,conc,0.01),name="conc")
conc <- matrix(c(3.65586845,0.01000000,0.01000000,0.01731192,1.68707608,
	0.01000000,4.67338974,4.79679942,1.86429851,1.82886732,1.54708795,
	0.57592054,0.08014232,0.09436425,0.26106139,0.11125534,0.22685364,
	0.22896015,0.04886441,0.01000000,33.59011263,16.89115866,19.99638316,
	16.94021361,9.95440037,7.10473948,2.97769676,1.53785279,2.13059515,
	0.72562344,1.27832563,1.33917155,0.99811111,0.23437424,0.42751355,
	0.65702300,0.41126684,0.15406463,0.03092312,0.14672610),
	ncol=20,byrow=TRUE)
conc <- restovec(conc)
reps <- rmna(conc, ccov=dd, tvcov=tt)
# constant shape parameter
gausscop(conc, mu=lmu, pmu=log(c(1,0.4,0.1)), par=0.5, pshape=0, envir=reps)
# or
gausscop(conc, mu=~absorption-volume+
	log(dose/(exp(absorption)-exp(elimination))*
	(exp(-exp(elimination)*times)-exp(-exp(absorption)*times))),
	pmu=list(absorption=0,elimination=log(0.4),volume=log(0.1)),
	par=0.5, pshape=0, envir=reps)
# compare to gar autoregression
gar(conc, dist="gamma", times=1:20, mu=mu,
	preg=log(c(1,0.4,0.1)), pdepend=0.5, pshape=1)
#
# time dependent shape parameter
gausscop(conc, mu=lmu, shape=lshape,
	pmu=log(c(1,0.4,0.1)), par=0.5, pshape=c(-0.1,-0.1))
# or
gausscop(conc, mu=~absorption-volume+
	log(dose/(exp(absorption)-exp(elimination))*
	(exp(-exp(elimination)*times)-exp(-exp(absorption)*times))),
	shape=~b1-b2+log(times*dose)-exp(b1)*times,
	pmu=list(absorption=0,elimination=log(0.4),volume=log(0.1)),
	par=0.5, pshape=list(b1=-0.1,b2=-0.1), envir=reps)
#
# shape depends on location
lshape <- function(p, mu) p[1]*log(abs(mu))
gausscop(conc, mu=lmu, shape=lshape, shfn=TRUE, pmu=log(c(1,0.4,0.1)),
	par=0.5, pshape=1)
# or
gausscop(conc, mu=~absorption-volume+
	log(dose/(exp(absorption)-exp(elimination))*
	(exp(-exp(elimination)*times)-exp(-exp(absorption)*times))),
	shape=~d*log(abs(mu)), shfn=TRUE,
	pmu=list(absorption=0,elimination=log(0.4),volume=log(0.1)),
	par=0.5, pshape=list(d=1), envir=reps)
}