elliptic               package:growth               R Documentation

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_D_e_s_c_r_i_p_t_i_o_n:

     `elliptic' fits a special case of the multivariate
     elliptically-contoured distribution, called the multivariate power
     exponential distribution. It includes the multivariate normal
     (power=1), the multivariate Laplace (power=0.5), and the
     multivariate uniform (power -> infinity) distributions as special
     cases.

     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, the `times' may be any strictly increasing
     sequence distinguishing the responses on an individual.

     It 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. Factor variables cannot be used and
     parameters must be scalars. (See `finterp'.)

     When an AR(1) of `exponential' form and/or a single random
     intercept is estimated for the multivariate normal distribution,
     marginal and individual profiles can be plotted using `profile'
     and `iprofile' and residuals with `plot.residuals'.

_U_s_a_g_e:

     elliptic(response, model="linear", distribution="elliptic",
             times=NULL, dose=NULL, ccov=NULL, tvcov=NULL, nest=NULL,
             torder=0, interaction=NULL, transform="identity",
             link="identity", autocorr="exponential", pell=NULL,
             preg=rep(1,4), pvar=var(y), varfn=NULL, pre=NULL, par=NULL,
             delta=NULL, shfn=F, common=F, envir=sys.frame(sys.parent()),
             print.level=0, gradtol=0.00001, typsiz=abs(theta),
             stepmax=10*sqrt(theta%*%theta), steptol=0.00001,
             iterlim=100, ndigit=10, fscale=1)

_A_r_g_u_m_e_n_t_s:

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').

   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: If `elliptic', a multivariate elliptically-contoured
          distribution is fitted unless `pell' is NULL, in which case a
          multivariate normal distribution is fitted. If `Student  t',
          a multivariate Student t distribution is fitted and a value
          must be given for `pell'.

   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 supplied 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 supplied 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.

    pell: Initial estimate of the power parameter of the multivariate
          elliptically-contoured distribution or of the degrees of
          freedom parameter of the multivariate Student t distribution.
          If missing and `distribution' is `elliptic', the multivariate
          normal distribution is used.

    preg: Initial parameter estimates for the regression model. Only
          required for `linear model' if the `link' is not the
          `identity' or a variance function is fitted.

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

   varfn: The builtin variance function has the variance 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, yielding a
          vector the same length as the number of observations.

     pre: Zero, one or two parameter estimates for the variance
          components, depending on the number of levels of nesting.

     par: If supplied, an initial estimate for the autocorrelation
          parameter.

   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 function depends on the mean
          function. The name of this mean function must be the last
          argument of the variance function.

  common: If TRUE, `mu' and `varfn' 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.

   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'.

_V_a_l_u_e:

     A list of class `elliptic' is returned.

_A_u_t_h_o_r(_s):

     J.K. Lindsey

_S_e_e _A_l_s_o:

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

_E_x_a_m_p_l_e_s:

     # 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
     times <- 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])*times)-exp(-exp(p[1])*times)))
     shape <- function(p) exp(p[1]-p[2])*times*dose*exp(-exp(p[1])*times)
     conc <- matrix(rnorm(40,mu(log(c(1,0.3,0.2))),sqrt(shape(log(c(0.1,0.4))))),
             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 <- 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 elliptical 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)
     # 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")
     # 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 elliptical 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)
     # 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")
     # or with user-specified formula
     # independence
     elliptic(conc, model=~exp(absorption-volume)*
             dose/(exp(absorption)-exp(elimination))*
             (exp(-exp(elimination)*times)-exp(-exp(absorption)*times)),
             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)*times)-exp(-exp(absorption)*times)),
             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)*times)-exp(-exp(absorption)*times)),
             preg=list(absorption=log(0.5),elimination=log(0.4),volume=log(0.1)),
             varfn=~exp(b1-b2)*times*dose*exp(-exp(b1)*times),
             par=0.1, pvar=list(b1=log(0.5),b2=log(0.2)))
     # multivariate elliptical distribution
     elliptic(conc, model=~exp(absorption-volume)*
             dose/(exp(absorption)-exp(elimination))*
             (exp(-exp(elimination)*times)-exp(-exp(absorption)*times)),
             preg=list(absorption=log(0.5),elimination=log(0.4),volume=log(0.1)),
             varfn=~exp(b1-b2)*times*dose*exp(-exp(b1)*times),
             par=0.1, pvar=list(b1=log(0.5),b2=log(0.2)), pell=1)
     # multivariate Student t distribution
     elliptic(conc, model=~exp(absorption-volume)*
             dose/(exp(absorption)-exp(elimination))*
             (exp(-exp(elimination)*times)-exp(-exp(absorption)*times)),
             preg=list(absorption=log(0.5),elimination=log(0.4),volume=log(0.1)),
             varfn=~exp(b1-b2)*times*dose*exp(-exp(b1)*times),
             par=0.1, pvar=list(b1=log(0.5),b2=log(0.2)), pell=5,
             distribution="Student t")
     #
     # generalized logistic regression with square-root transformation
     # and square  link
     times <- 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])*times))^(-1/p[4])}
     y <- matrix(rnorm(40,sqrt(mu(c(2,1.5,0.05,-2))),0.05)^2,ncol=20,byrow=T)
     y[,2:20] <- y[,2:20]+0.5*(y[,1:19]-matrix(mu(c(2,1.5,0.05,-2)),
             ncol=20,byrow=T)[,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)*times))^(-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)*times))^(-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)*times))^(-1/b4),
             preg=list(y0=2,yinf=1,b3=0.1,b4=-1),
             trans="sqrt", link="square", pre=1, par=0.1)
     #
     # multivariate elliptical and Student t distributions for outliers
     y <- matrix(rcauchy(40,mu(c(2,1.5,0.05,-2)),0.05),ncol=20,byrow=T)
     y[,2:20] <- y[,2:20]+0.5*(y[,1:19]-matrix(mu(c(2,1.5,0.05,-2)),
             ncol=20,byrow=T)[,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 elliptic
     elliptic(y, model="logistic", preg=c(1,1,0.1,-1), pell=1)
     elliptic(y, model="logistic", preg=c(1,1,0.1,-1), par=0.5, pell=1)
     # 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")

