carma                 package: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 subject 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 coefficients 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 multiplied 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 random 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 vectors 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 necessary 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: `identity', `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 supplied, 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 positive,
          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 decimals, 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 correlation 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', `tvctomat'.

_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)))

