elliptic: Nonlinear Multivariate Elliptically-contoured Repeated...
In swihart/growth: Multivariate Normal and Elliptically-Contoured Repeated Measurements Models
Description
Usage
Arguments
Details
Value
Methods (by generic)
Author(s)
References
See Also
Examples
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
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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 class 'elliptic'
deviance(object, ...)
## S3 method for class 'elliptic'
fitted(object, recursive = FALSE, ...)
## S3 method for class 'elliptic'
residuals(object, recursive = FALSE, ...)
## S3 method for class '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 = (s * (1 - f^2)) /
(sqrt(2) * f). 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|<=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(2rho min(t_i,t_j)+exp(-rho t_i) +exp(-rho
t_j)-1 -exp(rho|ti-t_j|))/2rho^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.
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.
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
Author(s)
J.K. Lindsey
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
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# 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")
swihart/growth documentation built on May 30, 2019, 9:38 p.m.
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Description Usage Arguments Details Value Methods (by generic) Author(s) References See Also Examples
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
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 | 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 class 'elliptic'
deviance(object, ...)
## S3 method for class 'elliptic'
fitted(object, recursive = FALSE, ...)
## S3 method for class 'elliptic'
residuals(object, recursive = FALSE, ...)
## S3 method for class '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, |
model |
The model to be fitted for the location. Builtin choices are
(1) |
distribution |
Multivariate |
times |
When |
dose |
A vector of dose levels for the |
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, |
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,
|
nest |
When |
torder |
When the |
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 |
transform |
Transformation of the response variable: |
link |
Link function for the location: |
autocorr |
The form of the autocorrelation function: |
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 |
preg |
Initial parameter estimates for the regression model. Only
required for |
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 |
varfn |
The builtin variance (dispersion) function has the
variance/dispersion proportional to a function of the location: pvar*v(mu) =
|
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, |
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, |
twins |
Only possible when there are two observations per individual
(e.g. twin data). If TRUE and |
envir |
Environment in which model formulae are to be interpreted or a
data object of class, |
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, |
... |
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, |
digits |
number of digits to print. |
correlation |
logical; print correlations. |
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.
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
Author(s)
J.K. Lindsey
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
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 | # 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")
|
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