

carma(growth)                                R Documentation

_C_o_n_t_i_n_u_o_u_s _A_R_M_A _f_o_r _U_n_e_q_u_a_l_l_y _S_p_a_c_e_d _R_e_p_e_a_t_e_d _M_e_a_s_u_r_e_m_e_n_t_s

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

     `carma' is designed to handle a polynomial within sub-
     ject design matrix with unequally spaced observations
     which can be at different times for different subjects.
     The origin of time is taken as the mean time of all the
     subjects. The within subject errors are assumed to be
     independent Gaussian or have a continuous time
     ARMA(p,q) Gaussian structure with the option to include
     measurement error.  The between subject random coeffi-
     cients are assumed to have an arbitrary covariance
     matrix. The fixed effect design matrix is a polynomial
     of equal or higher order than the within subject design
     matrix. This matrix can be augmented by covariates mul-
     tiplied by polynomial design matrices of any order up
     to the order of the first partition of the design
     matrix. The method is based on exact maximum likelihood
     using the Kalman filter to calculate the likelihood.

     For clustered (non-longitudinal) data, where only ran-
     dom effects will be fitted, the `times' may be any
     strictly increasing sequence distinguishing the
     responses on an individual.

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

_U_s_a_g_e_:

     carma(response, ccov=NULL, times=NULL, torder=0, interaction,
             transform="identity", arma=c(0,0,0), parma=NULL, pre=NULL,
             position=NULL, iopt=T, resid=T, delta=NULL, print.level=0,
             iterlim=100, typsiz=abs(p), ndigit=10, gradtol=0.00001,
             fscale=1, stepmax=10*sqrt(p%*%p), steptol=0.00001)

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

response: A list of two column matrices with response values
          and times for each individual, one matrix or
          dataframe of response values, or an object of
          either class, response (created by `restovec') or
          repeated (created by `rmna').

    ccov: A matrix of columns of baseline covariates with
          one row per individual, a model formula using vec-
          tors of the same size, or an object of class,
          tccov (created by `tcctomat'). If response has
          class, repeated, the covariates must be supplied
          as a Wilkinson and Rogers formula unless none are
          to be used.

   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 neces-
          sary if equally spaced. Ignored if response has
          class, response or repeated.

  torder: Order of the polynomial in time to be fitted.

interaction: Vector indicating order of interactions of
          covariates with time.

transform: Transformation of the response variable: `iden-
          tity', `exp', `square', `sqrt', or `log'.

    arma: Vector of three values: order of AR, order of MA,
          binary indicator for presence of measurement
          error. Not required for an AR(1) if an initial
          estimate is supplied. If only one value is sup-
          plied, it is assumed to be the order of the AR.

   parma: Initial estimates of ARMA parameters. For example,
          with `arma=c(1,0,0)', an AR(1), the parameter is
          `parma[1]=log(theta)', where `theta' is the posi-
          tive, continuous time autoregressive coefficient.
          The finite step autoregression coefficient for a
          step of length `delta' is then
          `alpha=exp(-delta*theta)' i.e.
          `alpha=exp(-delta*exp(parma[1]))'.

     pre: Initial estimates of random effect parameters.

position: Two column matrix with rows giving index positions
          of random effects in the covariance matrix.

    iopt: TRUE if optimization should be performed.

   resid: TRUE if residuals to be calculated.

   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 dec-
          imals, delta=0.01. Ignored if response has class,
          response or repeated.

  others: Arguments controlling `nlm'.

_V_a_l_u_e_:

     A list of class `carma' is returned. For any ARMA of
     order superior to an AR(1), the (complex) roots of the
     characteristic equation are printed out; see Jones and
     Ackerson (1991) for their use in calculation of the
     covariance function.

_A_u_t_h_o_r_(_s_)_:

     R.H. Jones and J.K. Lindsey

_R_e_f_e_r_e_n_c_e_s_:

     Jones, R. H. and Ackerson, L. M. (1991) Serial correla-
     tion in unequally spaced longitudinal data. Biometrika,
     77, 721-731.

     Jones, R.H. (1993) Longitudinal Data Analysis with
     Serial Correlation: A State-space Approach. Chapman and
     Hall

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

     `elliptic', `gar', `glmm', `gnlmm', `iprofile',
     `kalseries', `plot.residuals', `profile', `potthoff',
     `read.list', `restovec', `rmna', `tcctomat', `tvc-
     tomat'.

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

     y <- matrix(rnorm(40),ncol=5)
     x1 <- gl(2,4)
     x2 <- gl(2,1,8)
     # independence with time trend
     carma(y, ccov=~x1, torder=2)
     # AR(1)
     carma(y, ccov=~x1, torder=2, arma=c(1,0,0), parma=-0.5)
     carma(y, ccov=~x1, torder=3, interact=3, arma=c(1,0,0), parma=-1)
     # ARMA(2,1)
     carma(y, ccov=~x1+x2, interact=c(2,0), torder=3,arma=c(2,1,0),
             parma=c(0.3,2,0.7))
     # random intercept
     carma(y, ccov=~x1+x2, interact=c(2,0), torder=3, pre=-0.4,
             position=c(1,1))
     # random coefficients
     carma(y, ccov=~x1+x2, interact=c(2,0), torder=3, pre=c(-0.4,0.1),
             position=rbind(c(1,1),c(2,2)))

