Generalized autoregression

Usage

gar(response, distribution="normal", times=NULL, totals=NULL, censor=NULL,
	delta=NULL, mu=NULL, shape=NULL, shfn=F, common=F, preg=NULL,
	pdepend=NULL, pshape=NULL, transform="identity", link="identity",
	autocorr="exponential", order=1, envir=sys.frame(sys.parent()),
	print.level=0, ndigit=10, gradtol=0.00001,steptol=0.00001,
	fscale=1, iterlim=100, typsiz=abs(p), stepmax=10*sqrt(p%*%p))

Arguments

response A list of two or three column matrices with responses, corresponding times, and possibly a censor indicator, for each individual, one matrix or dataframe of responses, or an object of class, response (created by restovec) or repeated (created by rmna).
distribution The distribution to be fitted: Bernoulli, Poisson, exponential, negative binomial, mult Poisson, double Poisson, beta binomial, mult binomial, double binomial, normal, inverse Gauss, logistic, gamma, Weibull, Cauchy, Laplace, Levy, Pareto, gen(eralized) gamma, gen(eralized) logistic, Hjorth, Burr, gen(eralized) Weibull, gen(eralized) extreme value, gen(eralized) inverse Gauss, or power exponential.
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.
totals An appropriate scalar, vector, or matrix of binomial totals (only applicable for binomial, beta binomial, mult binomial, double binomial). Ignored if response has class, response or repeated.
censor If response is a matrix, a matrix of the same size containing the censor indicator: 1=uncensored, 0=right-censored, -1=left-censored. Ignored if response has class, response or repeated.
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. If the response has been pretransformed, this must be multiplied by the Jacobian. This transformation cannot contain unknown parameters. For example, with a log transformation, delta=1/y. (The delta values for the censored response are ignored.) The jacobian is calculated automatically for the transform option. Ignored if response has class, response or repeated.
mu A user-specified function of pmu giving the regression equation for the location. 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. It must yield a value for each observation on each individual.
shape An optional user-specified shape regression function; this may depend on the location (function) through its second argument, in which case, shfn must be set to TRUE. It may also be a formula beginning with ~, specifying either a linear regression function for the shape parameter in the Wilkinson and Rogers notation or a general function with named unknown parameters.
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 preg. If FALSE, parameters are distinct between the two functions and indexing starts at one in each function.
preg The initial parameter estimates for the location regression function. 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.
pdepend One or two estimates of the dependence parameters for the Kalman update. With one, it is Markovian and, with two, it is nonstationary. For the latter, the order must be one.
pshape Zero to two estimates for the shape parameters of the distribution if shape is not a function; otherwise, estimates for the parameters in this function, with one extra at the end for three-parameter distributions. If shape 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.
transform Transformation of the response variable: identity, exp, square, sqrt, or log.
link Link function for the mean: identity, exp, square, sqrt, log, logit, or cloglog (last two only for binary data).
autocorr The form of the (second if two) dependence 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; IOU is the integrated Ornstein-Uhlenbeck process, (2rho min(t_i,t_j)+exp(-rho t_i) +exp(-rho t_j)-1 -exp(rho|ti-t_j|))/2rho^3.
order First- or second-order stationary autoregression.
envir Environment in which model formulae are to be interpreted or a data object of class, repeated, tccov, or tvcov. If response has class repeated, it is used as the environment.
others Arguments controlling nlm.

Description

gar fits a first- or second-order generalized autoregression, possibly with Kalman update over time (first-order only).

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

Marginal and individual profiles can be plotted using profile and iprofile and residuals with plot.residuals.

Value

A list of classes gar and recursive is returned.

Author(s)

J.K. Lindsey

References

Lindsey, J.K. (1997) Applying Generalized Linear Models. Springer, pp. 93𤩕

Lambert, P. (1996) Statistics in Medicine 15, 1695-1708

See Also

carma, elliptic, finterp, gnlmm, gnlr, iprofile, kalcount, kalseries, kalsurv, plot.residuals, profile, read.list, restovec, rmna, tcctomat, tvctomat.

Examples

# first-order one-compartment model
# data objects for formulae
dose <- c(2,5)
dd <- tcctomat(dose)
times <- matrix(rep(1:20,2), nrow=2, byrow=T)
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)
# response
conc <- matrix(rgamma(40,shape(log(c(0.1,0.4))),mu(log(c(1,0.3,0.2)))),
	ncol=20,byrow=T)
conc[,2:20] <- conc[,2:20]+0.5*(conc[,1:19]-matrix(mu(log(c(1,0.3,0.2))),
	ncol=20,byrow=T)[,1:19])
conc <- restovec(ifelse(conc>0,conc,0.01))
reps <- rmna(conc, ccov=dd, tvcov=tt)
# constant shape parameter
gar(conc, dist="gamma", times=1:20, mu=mu,
	preg=log(c(1,0.4,0.1)), pdepend=0.5, pshape=1)
# or
gar(conc, dist="gamma", times=1:20, mu=~exp(absorption-volume)*
	dose/(exp(absorption)-exp(elimination))*
	(exp(-exp(elimination)*times)-exp(-exp(absorption)*times)),
	preg=list(absorption=0,elimination=log(0.4),volume=log(0.1)),
	pdepend=0.5, pshape=1, envir=reps)
# (if the covariates contained NAs, reps would have to be used as
# response instead of conc)
#
# time dependent shape parameter
gar(conc, dist="gamma", times=1:20, mu=mu, shape=shape,
	preg=log(c(1,0.4,0.1)), pdepend=0.5, pshape=log(c(1,0.2)))
# or
gar(conc, dist="gamma", times=1:20, mu=~exp(absorption-volume)*
	dose/(exp(absorption)-exp(elimination))*
	(exp(-exp(elimination)*times)-exp(-exp(absorption)*times)),
	shape=~exp(b1-b2)*times*dose*exp(-exp(b1)*times),
	preg=list(absorption=0,elimination=log(0.4),volume=log(0.1)),
	pdepend=0.5, pshape=list(b1=0,b2=log(0.2)), envir=reps)


[Package Contents]