growth (version 1.1.1)

elliptic: Nonlinear Multivariate Elliptically-contoured Repeated Measurements Models with AR(1) and Two Levels of Variance Components

Description

elliptic fits special cases of the multivariate elliptically-contoured distribution, the multivariate normal, Student t, and power exponential distributions. The latter includes the multivariate normal (power=1), a multivariate Laplace (power=0.5), and the multivariate uniform (power -> infinity) distributions as special cases. As well, another form of multivariate skew Laplace distribution is also available.

Usage

elliptic(response = NULL, model = "linear", distribution = "normal",
  times = NULL, dose = NULL, ccov = NULL, tvcov = NULL,
  nest = NULL, torder = 0, interaction = NULL,
  transform = "identity", link = "identity",
  autocorr = "exponential", pell = NULL, preg = NULL, covfn = NULL,
  pvar = var(y), varfn = NULL, par = NULL, pre = NULL,
  delta = NULL, shfn = FALSE, common = FALSE, twins = 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)))
# S3 method for elliptic
deviance(object, ...)
# S3 method for elliptic
fitted(object, recursive = FALSE, ...)
# S3 method for elliptic
residuals(object, recursive = FALSE, ...)
# S3 method for elliptic
print(x, digits = max(3, .Options$digits - 3),
  correlation = TRUE, ...)

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.

model

The model to be fitted for the location. Builtin choices are (1) linear for linear models with time-varying covariate; if torder > 0, a polynomial in time is automatically fitted; (2) logistic for a four-parameter logistic growth curve; (3) pkpd for a first-order one-compartment pharmacokinetic model. Otherwise, set this to 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, in which case ccov and tvcov cannot be used.

distribution

Multivariate normal, power exponential, Student t, or skew Laplace distribution. The latter is not an elliptical distribution. Note that the latter has a different parametrization of the skew (family) parameter than the univariate skew Laplace distribution in dskewlaplace: \(skew=\frac{\sigma(1-\nu^2)}{\sqrt{2}\nu}\). Here, zero skew yields a symmetric distribution.

times

When response is a matrix, a vector of possibly unequally spaced times when they are the same for all individuals or a matrix of times. Not necessary if equally spaced. Ignored if response has class, response or repeated.

dose

A vector of dose levels for the pkpd model, one per individual.

ccov

A vector or matrix containing time-constant baseline covariates with one line per individual, a model formula using vectors of the same size, or an object of class, tccov (created by tcctomat). If response has class, repeated, with a linear, logistic, or pkpd model, the covariates must be specified as a Wilkinson and Rogers formula unless none are to be used. For the pkpd and logistic models, all variables must be binary (or factor variables) as different values of all parameters are calculated for all combinations of these variables (except for the logistic model when a time-varying covariate is present). It cannot be used when model is a function.

tvcov

A list of vectors or matrices with time-varying covariates for each individual (one column per variable), a matrix or dataframe of such covariate values (if only one covariate), or an object of class, tvcov (created by tvctomat). If times are not the same as for responses, the list can be created with gettvc. If response has class, repeated, with a linear, logistic, or pkpd model, the covariates must be specified as a Wilkinson and Rogers formula unless none are to be used. Only one time-varying covariate is allowed except for the linear model; if more are required, set model equal to the appropriate mean function. This argument cannot be used when model is a function.

nest

When response is a matrix, a vector of length equal to the number of responses per individual indicating which responses belong to which nesting category. Categoriess must be consecutive increasing integers. This option should always be specified if nesting is present. Ignored if response has class, repeated.

torder

When the linear model is chosen, order of the polynomial in time to be fitted.

interaction

Vector of length equal to the number of time-constant covariates, giving the levels of interactions between them and the polynomial in time in the linear model.

transform

Transformation of the response variable: identity, exp, square, sqrt, or log.

link

Link function for the location: identity, exp, square, sqrt, or log. For the linear model, if not the identity, initial estimates of the regression parameters must be supplied (intercept, polynomial in time, time-constant covariates, time-varying covariates, in that order).

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; IOU is the integrated Ornstein-Uhlenbeck process, \((2\rho \min(t_i,t_j)+\exp(-\rho t_i) \)\(+\exp(-\rho t_j)-1 -\exp(\rho|ti-t_j|))/2\rho^3\).

pell

Initial estimate of the power parameter of the multivariate power exponential distribution, of the degrees of freedom parameter of the multivariate Student t distribution, or of the asymmetry parameter of the multivariate Laplace distribution. If not supplied for the latter, asymmetry depends on the regression equation in model.

preg

Initial parameter estimates for the regression model. Only required for linear model if the link is not the identity or a variance (dispersion) function is fitted.

covfn

Either a function or a formula beginning with ~, specifying how the covariance depends on covariates: either a linear regression function in the Wilkinson and Rogers notation or a general function with named unknown parameters.

pvar

Initial parameter estimate for the variance or dispersion. If more than one value is provided, the log variance/dispersion depends on a polynomial in time. With the pkpd model, if four values are supplied, a nonlinear regression for the variance/dispersion is fitted.

varfn

The builtin variance (dispersion) function has the variance/dispersion proportional to a function of the location: pvar*v(mu) = identity or square. If pvar contains two initial values, an additive constant is included: pvar(1)+pvar(2)*v(mu). Otherwise, either a function or a formula beginning with ~, specifying either a linear regression function in the Wilkinson and Rogers notation or a general function with named unknown parameters for the log variance can be supplied. If it contains unknown parameters, the keyword mu may be used to specify a function of the location parameter.

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. If covfn is specified, this contains the initial estimates of the regression parameters.

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 variance (dispersion) function depends on the mean function. The name of this mean function must be the last argument of the variance/dispersion function.

common

If TRUE, mu and varfn 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 preg. If FALSE, parameters are distinct between the two functions and indexing starts at one in each function.

twins

Only possible when there are two observations per individual (e.g. twin data). If TRUE and covfn is supplied, allows the covariance to vary across pairs of twins with the diagonal "variance" of the covariance matrix remaining constant.

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.

object

An object of class, elliptic.

...

additional arguments.

recursive

If TRUE, recursive residuals or fitted values are given; otherwise, marginal ones. In all cases, raw residuals are returned, not standardized by the standard deviation (which may be changing with covariates or time).

x

An object of class, elliptic.

digits

number of digits to print.

correlation

logical; print correlations.

Value

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

Methods (by generic)

• deviance: Deviance method

• fitted: Fitted method

• residuals: Residuals method

• print: Print method

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. Recall that zero correlation (all zeros off-diagonal in the covariance matrix) only implies independence for the multivariate normal 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.)

Recursive fitted values and residuals are only available for the multivariate normal distribution with a linear model without a variance function and with either an AR(1) of exponential form and/or one level of random effect. In these cases, marginal and individual profiles can be plotted using mprofile and iprofile and residuals with plot.residuals.

References

Lindsey, J.K. (1999) Multivariate elliptically-contoured distributions for repeated measurements. Biometrics 55, 1277-1280.

Kotz, S., Kozubowski, T.J., and Podgorski, K. (2001) The Laplace Distribution and Generalizations. A Revisit with Applications to Communications, Economics, Engineering, and Finance. Basel: Birkhauser, Ch. 6.

See Also

carma, dpowexp, dskewlaplace, finterp, gar, gettvc, gnlmix, glmm, gnlmm, gnlr, iprofile, kalseries, mprofile, potthoff, read.list, restovec, rmna, tcctomat, tvctomat.

Examples

# NOT RUN {
# linear models
y <- matrix(rnorm(40),ncol=5)
x1 <- gl(2,4)
x2 <- gl(2,1,8)
# independence with time trend
elliptic(y, ccov=~x1, torder=2)
# AR(1)
elliptic(y, ccov=~x1, torder=2, par=0.1)
elliptic(y, ccov=~x1, torder=3, interact=3, par=0.1)
# random intercept
elliptic(y, ccov=~x1+x2, interact=c(2,0), torder=3, pre=2)
#
# nonlinear models
time <- rep(1:20,2)
dose <- c(rep(2,20),rep(5,20))
mu <- function(p) exp(p[1]-p[3])*(dose/(exp(p[1])-exp(p[2]))*
	(exp(-exp(p[2])*time)-exp(-exp(p[1])*time)))
shape <- function(p) exp(p[1]-p[2])*time*dose*exp(-exp(p[1])*time)
conc <- matrix(rnorm(40,mu(log(c(1,0.3,0.2))),sqrt(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 <- ifelse(conc>0,conc,0.01)
# with builtin function
# independence
elliptic(conc, model="pkpd", preg=log(c(0.5,0.4,0.1)), dose=c(2,5))
# AR(1)
elliptic(conc, model="pkpd", preg=log(c(0.5,0.4,0.1)), dose=c(2,5),
	par=0.1)
# add variance function
elliptic(conc, model="pkpd", preg=log(c(0.5,0.4,0.1)), dose=c(2,5),
	par=0.1, varfn=shape, pvar=log(c(0.5,0.2)))
# multivariate power exponential distribution
elliptic(conc, model="pkpd", preg=log(c(0.5,0.4,0.1)), dose=c(2,5),
	par=0.1, varfn=shape, pvar=log(c(0.5,0.2)), pell=1,
	distribution="power exponential")
# multivariate Student t distribution
elliptic(conc, model="pkpd", preg=log(c(0.5,0.4,0.1)), dose=c(2,5),
	par=0.1, varfn=shape, pvar=log(c(0.5,0.2)), pell=5,
	distribution="Student t")
# multivariate Laplace distribution
elliptic(conc, model="pkpd", preg=log(c(0.5,0.4,0.1)), dose=c(2,5),
	par=0.1, varfn=shape, pvar=log(c(0.5,0.2)),
	distribution="Laplace")
# or equivalently with user-specified function
# independence
elliptic(conc, model=mu, preg=log(c(0.5,0.4,0.1)))
# AR(1)
elliptic(conc, model=mu, preg=log(c(0.5,0.4,0.1)), par=0.1)
# add variance function
elliptic(conc, model=mu, preg=log(c(0.5,0.4,0.1)), par=0.1,
	varfn=shape, pvar=log(c(0.5,0.2)))
# multivariate power exponential distribution
elliptic(conc, model=mu, preg=log(c(0.5,0.4,0.1)), par=0.1,
	varfn=shape, pvar=log(c(0.5,0.2)), pell=1,
	distribution="power exponential")
# multivariate Student t distribution
elliptic(conc, model=mu, preg=log(c(0.5,0.4,0.1)), par=0.1,
	varfn=shape, pvar=log(c(0.5,0.2)), pell=5,
	distribution="Student t")
# multivariate Laplace distribution
elliptic(conc, model=mu, preg=log(c(0.5,0.4,0.1)), par=0.1,
	varfn=shape, pvar=log(c(0.5,0.2)), pell=5,
	distribution="Laplace")
# or with user-specified formula
# independence
elliptic(conc, model=~exp(absorption-volume)*
	dose/(exp(absorption)-exp(elimination))*
	(exp(-exp(elimination)*time)-exp(-exp(absorption)*time)),
	preg=list(absorption=log(0.5),elimination=log(0.4),
	volume=log(0.1)))
# AR(1)
elliptic(conc, model=~exp(absorption-volume)*
	dose/(exp(absorption)-exp(elimination))*
	(exp(-exp(elimination)*time)-exp(-exp(absorption)*time)),
	preg=list(absorption=log(0.5),elimination=log(0.4),volume=log(0.1)),
	par=0.1)
# add variance function
elliptic(conc, model=~exp(absorption-volume)*
	dose/(exp(absorption)-exp(elimination))*
	(exp(-exp(elimination)*time)-exp(-exp(absorption)*time)),
	preg=list(absorption=log(0.5),elimination=log(0.4),volume=log(0.1)),
	varfn=~exp(b1-b2)*time*dose*exp(-exp(b1)*time),
	par=0.1, pvar=list(b1=log(0.5),b2=log(0.2)))
# variance as function of the mean
elliptic(conc, model=~exp(absorption-volume)*
	dose/(exp(absorption)-exp(elimination))*
	(exp(-exp(elimination)*time)-exp(-exp(absorption)*time)),
	preg=list(absorption=log(0.5),elimination=log(0.4),volume=log(0.1)),
	varfn=~d*log(mu),shfn=TRUE,par=0.1, pvar=list(d=1))
# multivariate power exponential distribution
elliptic(conc, model=~exp(absorption-volume)*
	dose/(exp(absorption)-exp(elimination))*
	(exp(-exp(elimination)*time)-exp(-exp(absorption)*time)),
	preg=list(absorption=log(0.5),elimination=log(0.4),volume=log(0.1)),
	varfn=~exp(b1-b2)*time*dose*exp(-exp(b1)*time),
	par=0.1, pvar=list(b1=log(0.5),b2=log(0.2)), pell=1,
	distribution="power exponential")
# multivariate Student t distribution
elliptic(conc, model=~exp(absorption-volume)*
	dose/(exp(absorption)-exp(elimination))*
	(exp(-exp(elimination)*time)-exp(-exp(absorption)*time)),
	preg=list(absorption=log(0.5),elimination=log(0.4),volume=log(0.1)),
	varfn=~exp(b1-b2)*time*dose*exp(-exp(b1)*time),
	par=0.1, pvar=list(b1=log(0.5),b2=log(0.2)), pell=5,
	distribution="Student t")
# multivariate Laplace distribution
elliptic(conc, model=~exp(absorption-volume)*
	dose/(exp(absorption)-exp(elimination))*
	(exp(-exp(elimination)*time)-exp(-exp(absorption)*time)),
	preg=list(absorption=log(0.5),elimination=log(0.4),volume=log(0.1)),
	varfn=~exp(b1-b2)*time*dose*exp(-exp(b1)*time),
	par=0.1, pvar=list(b1=log(0.5),b2=log(0.2)), pell=5,
	distribution="Laplace")
#
# generalized logistic regression with square-root transformation
# and square  link
time <- rep(seq(10,200,by=10),2)
mu <- function(p) {
	yinf <- exp(p[2])
	yinf*(1+((yinf/exp(p[1]))^p[4]-1)*exp(-yinf^p[4]
		*exp(p[3])*time))^(-1/p[4])}
y <- matrix(rnorm(40,sqrt(mu(c(2,1.5,0.05,-2))),0.05)^2,ncol=20,byrow=TRUE)
y[,2:20] <- y[,2:20]+0.5*(y[,1:19]-matrix(mu(c(2,1.5,0.05,-2)),
	ncol=20,byrow=TRUE)[,1:19])
y <- ifelse(y>0,y,0.01)
# with builtin function
# independence
elliptic(y, model="logistic", preg=c(2,1,0.1,-1), trans="sqrt",
	link="square")
# the same model with AR(1)
elliptic(y, model="logistic", preg=c(2,1,0.1,-1), trans="sqrt",
	link="square", par=0.4)
# the same model with AR(1) and one component of variance
elliptic(y, model="logistic", preg=c(2,1,0.1,-1),
	trans="sqrt", link="square", pre=1, par=0.4)
# or equivalently with user-specified function
# independence
elliptic(y, model=mu, preg=c(2,1,0.1,-1), trans="sqrt",
	link="square")
# the same model with AR(1)
elliptic(y, model=mu, preg=c(2,1,0.1,-1), trans="sqrt",
	link="square", par=0.4)
# the same model with AR(1) and one component of variance
elliptic(y, model=mu, preg=c(2,1,0.1,-1),
	trans="sqrt", link="square", pre=1, par=0.4)
# or equivalently with user-specified formula
# independence
elliptic(y, model=~exp(yinf)*(1+((exp(yinf-y0))^b4-1)*
	exp(-exp(yinf*b4+b3)*time))^(-1/b4),
	preg=list(y0=2,yinf=1,b3=0.1,b4=-1), trans="sqrt", link="square")
# the same model with AR(1)
elliptic(y, model=~exp(yinf)*(1+((exp(yinf-y0))^b4-1)*
	exp(-exp(yinf*b4+b3)*time))^(-1/b4),
	preg=list(y0=2,yinf=1,b3=0.1,b4=-1), trans="sqrt",
	link="square", par=0.1)
# add one component of variance
elliptic(y, model=~exp(yinf)*(1+((exp(yinf-y0))^b4-1)*
	exp(-exp(yinf*b4+b3)*time))^(-1/b4),
	preg=list(y0=2,yinf=1,b3=0.1,b4=-1),
	trans="sqrt", link="square", pre=1, par=0.1)
#
# multivariate power exponential and Student t distributions for outliers
y <- matrix(rcauchy(40,mu(c(2,1.5,0.05,-2)),0.05),ncol=20,byrow=TRUE)
y[,2:20] <- y[,2:20]+0.5*(y[,1:19]-matrix(mu(c(2,1.5,0.05,-2)),
	ncol=20,byrow=TRUE)[,1:19])
y <- ifelse(y>0,y,0.01)
# first with normal distribution
elliptic(y, model="logistic", preg=c(1,1,0.1,-1))
elliptic(y, model="logistic", preg=c(1,1,0.1,-1), par=0.5)
# then power exponential
elliptic(y, model="logistic", preg=c(1,1,0.1,-1), pell=1,
	distribution="power exponential")
elliptic(y, model="logistic", preg=c(1,1,0.1,-1), par=0.5, pell=1,
	distribution="power exponential")
# finally Student t
elliptic(y, model="logistic", preg=c(1,1,0.1,-1), pell=1,
	distribution="Student t")
elliptic(y, model="logistic", preg=c(1,1,0.1,-1), par=0.5, pell=1,
	distribution="Student t")

# }