Description Usage Arguments Details Value Methods (by generic) Author(s) References See Also Examples
elliptic
fits special cases of the multivariate
ellipticallycontoured 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.
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 = 1e05, steptol = 1e05, 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, ...)

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 timeconstant 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 timevarying 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 timeconstant
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. 
With two levels of nesting, the first is the individual and the second will consist of clusters within individuals.
For clustered (nonlongitudinal) data, where only random effects will be
fitted, times
are not necessary.
This function is designed to fit linear and nonlinear models with timevarying covariates observed at arbitrary time points. A continuoustime AR(1) and zero, one, or two levels of nesting can be handled. Recall that zero correlation (all zeros offdiagonal 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
.
A list of class elliptic
is returned that contains all of the
relevant information calculated, including error codes.
deviance
: Deviance method
fitted
: Fitted method
residuals
: Residuals method
print
: Print method
J.K. Lindsey
Lindsey, J.K. (1999) Multivariate ellipticallycontoured distributions for repeated measurements. Biometrics 55, 12771280.
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.
carma
, dpowexp
,
dskewlaplace
, finterp
,
gar
, gettvc
,
gnlmix
, glmm
,
gnlmm
, gnlr
,
iprofile
, kalseries
,
mprofile
, potthoff
,
read.list
, restovec
,
rmna
, tcctomat
,
tvctomat
.
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 userspecified 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 userspecified formula
# independence
elliptic(conc, model=~exp(absorptionvolume)*
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(absorptionvolume)*
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(absorptionvolume)*
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(b1b2)*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(absorptionvolume)*
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(absorptionvolume)*
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(b1b2)*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(absorptionvolume)*
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(b1b2)*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(absorptionvolume)*
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(b1b2)*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 squareroot 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 userspecified 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 userspecified formula
# independence
elliptic(y, model=~exp(yinf)*(1+((exp(yinfy0))^b41)*
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(yinfy0))^b41)*
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(yinfy0))^b41)*
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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