gausscop {repeated} R Documentation

## 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.

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`.)

### 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(),
iterlim=100, fscale=1, stepmax=10*sqrt(theta%*%theta),
typsiz=abs(c(theta)))
fitted.gausscop(z)
residuals.gausscop(z)
```

### 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|<=1/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. `others` Arguments controlling `nlm`. `z` An object of class, `gausscop`.

### Value

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

J.K. Lindsey

### References

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

`carma`, `elliptic`, `finterp`, `gar`, `gettvc`, `glmm`, `gnlmm`, `gnlr`, `iprofile`, `kalseries`, `mprofile`, `potthoff`, `read.list`, `restovec`, `rmna`, `tcctomat`, `tvctomat`.

### Examples

```library(gnlm)
# 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
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)
# 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)
```

[Package repeated version 1.0 Index]