# Repeated Measurements Models for Continuous Variables with Frailty or Serial Dependence `kalseries` is designed to handle repeated measurements models with time-varying covariates. The distributions have two extra parameters as compared to the functions specified by `intensity` and are generally longer tailed than those distributions. Dependence among observations on a unit can be through gamma or power variance family frailties (a type of random effect), with or without autoregression, or one of two types of serial dependence over time. ```r kalseries( response = NULL, times = NULL, intensity = "exponential", depend = "independence", mu = NULL, shape = NULL, density = FALSE, ccov = NULL, tvcov = NULL, torder = 0, interaction = NULL, preg = NULL, ptvc = NULL, pintercept = NULL, pshape = NULL, pinitial = 1, pdepend = NULL, pfamily = NULL, delta = NULL, transform = "identity", link = "identity", envir = parent.frame(), print.level = 0, ndigit = 10, gradtol = 1e-05, steptol = 1e-05, fscale = 1, iterlim = 100, typsize = abs(p), stepmax = 10 * sqrt(p %*% p) ) ``` ## Arguments - `response`: A list of two column matrices with responses and corresponding times 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. - `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`. - `intensity`: The form of function to be put in the Pareto distribution. Choices are exponential, Weibull, gamma, normal, logistic, Cauchy, log normal, log logistic, log Cauchy, log Student, inverse Gauss, and gen(eralized) logistic. (For definitions of distributions, see the corresponding [dpqr]distribution help.) - `depend`: Type of dependence. Choices are `independence`, `Markov`, `serial`, and `frailty`. - `mu`: A regression function for the location parameter 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. Give the initial estimates in `preg` if there are no time-varying covariates and in `ptvc` if there are. - `shape`: A regression function for the shape parameter 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. It must yield one value per observation. - `density`: If TRUE, the density of the function specified in `intensity` is used instead of the intensity. - `ccov`: A vector or matrix containing time-constant 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 or `mu` is given. - `tvcov`: A list of matrices with time-varying covariate values, observed at the event times in `response`, for each individual (one column per variable), one matrix or dataframe of such covariate values, or an object of class, `tvcov` (created by `tvctomat`). If a time-varying covariate is observed at arbitrary time, `gettvc` can be used to find the most recent values for each response and create a suitable list. If response has class, `repeated`, the covariates must be supplied as a Wilkinson and Rogers formula unless none are to be used or `mu` is given. - `torder`: The 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`. - `preg`: Initial parameter estimates for the regression model: intercept, one for each covariate in `ccov`, and `torder` plus sum(`interaction`). If `mu` is a formula or function, the parameter estimates must be given here only if there are no time-varying covariates. If `mu` is a formula with unknown parameters, their estimates must be supplied either in their order of appearance in the expression or in a named list. - `ptvc`: Initial parameter estimates for the coefficients of the time-varying covariates, as many as in `tvcov`. If `mu` is a formula or function, the parameter estimates must be given here if there are time-varying covariates present. - `pintercept`: The initial estimate of the intercept for the generalized logistic intensity. - `pshape`: An initial estimate for the shape parameter of the intensity function (except exponential intensity). If `shape` is a function or formula, the corresponding initial estimates. If `shape` is a formula with unknown parameters, their estimates must be supplied either in their order of appearance in the expression or in a named list. - `pinitial`: An initial estimate for the initial parameter. With `frailty` dependence, this is the frailty parameter. - `pdepend`: An initial estimate for the serial dependence parameter. For `frailty` dependence, if a value is given here, an autoregression is fitted as well as the frailty. - `pfamily`: An optional initial estimate for the second parameter of a two-parameter power variance family mixture instead of the default gamma mixture. This yields a gamma mixture as `family -> 0`, an inverse Gauss mixture for `family = 0.5`, and a compound distribution of a Poisson-distributed number of gamma distributions for `-1 < family < 0`. - `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. If the response has been pretransformed, this must be multiplied by the Jacobian. This transformation cannot contain unknown parameters. For example, with a log transformation, `delta=1/y`. The jacobian is calculated automatically for the transform option. Ignored if response has class, `response` or `repeated`. - `transform`: Transformation of the response variable: `identity`, `exp`, `square`, `sqrt`, or `log`. - `link`: Link function for the mean: `identity`, `exp`, `square`, `sqrt`, or `log`. - `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. - `fscale`: Arguments for nlm. - `iterlim`: Arguments for nlm. - `typsize`: Arguments for nlm. - `stepmax`: Arguments for nlm. ## Returns A list of classes `kalseries` and `recursive` is returned. ## Details By default, a gamma mixture of the distribution specified in `intensity` is used, as the conditional distribution in the `Markov` and `serial` dependence models, and as a symmetric multivariate (random effect) model for `frailty` dependence. For example, with a Weibull `intensity` and `frailty` dependence, this yields a multivariate Burr distribution and with `Markov` or `serial` dependence, univariate Burr conditional distributions. If a value for `pfamily` is used, the gamma mixture is replaced by a power variance family mixture. 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`.) Marginal and individual profiles can be plotted using `mprofile` and `iprofile` and residuals with `plot.residuals`. ## Examples ```r treat <- c(0,0,1,1) tr <- tcctomat(treat) dose <- matrix(rpois(20,10), ncol=5) dd <- tvctomat(dose) y <- restovec(matrix(rnorm(20), ncol=5), name="y") reps <- rmna(y, ccov=tr, tvcov=dd) # # normal intensity, independence model kalseries(y, intensity="normal", dep="independence", preg=1, pshape=5) ## Not run: # random effect kalseries(y, intensity="normal", dep="frailty", preg=1, pinitial=1, psh=5) # serial dependence kalseries(y, intensity="normal", dep="serial", preg=1, pinitial=1, pdep=0.1, psh=5) # random effect and autoregression kalseries(y, intensity="normal", dep="frailty", preg=1, pinitial=1, pdep=0.1, psh=5) # # add time-constant variable kalseries(y, intensity="normal", dep="serial", pinitial=1, pdep=0.1, psh=5, preg=c(1,0), ccov=treat) # or equivalently kalseries(y, intensity="normal", mu=~treat, dep="serial", pinitial=1, pdep=0.1, psh=5, preg=c(1,0)) # or kalseries(y, intensity="normal", mu=~b0+b1*treat, dep="serial", pinitial=1, pdep=0.1, psh=5, preg=c(1,0), envir=reps) # # add time-varying variable kalseries(y, intensity="normal", dep="serial", pinitial=1, pdep=0.1, psh=5, preg=c(1,0), ccov=treat, ptvc=0, tvc=dose) # or equivalently, from the environment dosev <- as.vector(t(dose)) kalseries(y, intensity="normal", mu=~b0+b1*rep(treat,rep(5,4))+b2*dosev, dep="serial", pinitial=1, pdep=0.1, psh=5, ptvc=c(1,0,0)) # or from the reps data object kalseries(y, intensity="normal", mu=~b0+b1*treat+b2*dose, dep="serial", pinitial=1, pdep=0.1, psh=5, ptvc=c(1,0,0), envir=reps) # try power variance family instead of gamma distribution for mixture kalseries(y, intensity="normal", mu=~b0+b1*treat+b2*dose, dep="serial", pinitial=1, pdep=0.1, psh=5, ptvc=c(1,0,0), pfamily=0.1, envir=reps) # first-order one-compartment model # data objects for formulae dose <- c(2,5) dd <- tcctomat(dose) times <- matrix(rep(1:20,2), nrow=2, byrow=TRUE) tt <- tvctomat(times) # vector covariates for functions dose <- c(rep(2,20),rep(5,20)) times <- rep(1:20,2) # functions 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) # response conc <- matrix(rgamma(40,shape(log(c(0.01,1))), scale=mu(log(c(1,0.3,0.2))))/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 <- restovec(ifelse(conc>0,conc,0.01)) reps <- rmna(conc, ccov=dd, tvcov=tt) # # constant shape parameter kalseries(reps, intensity="gamma", dep="independence", mu=mu, ptvc=c(-1,-1.1,-1), pshape=1.5) # or kalseries(reps, intensity="gamma", dep="independence", mu=~exp(absorption-volume)* dose/(exp(absorption)-exp(elimination))* (exp(-exp(elimination)*times)-exp(-exp(absorption)*times)), ptvc=list(absorption=-1,elimination=-1.1,volume=-1), pshape=1.2) # add serial dependence kalseries(reps, intensity="gamma", dep="serial", pdep=0.9, mu=~exp(absorption-volume)* dose/(exp(absorption)-exp(elimination))* (exp(-exp(elimination)*times)-exp(-exp(absorption)*times)), ptvc=list(absorption=-1,elimination=-1.1,volume=-1), pshape=0.2) # time dependent shape parameter kalseries(reps, intensity="gamma", dep="independence", mu=mu, shape=shape, ptvc=c(-1,-1.1,-1), pshape=c(-3,0)) # or kalseries(reps, intensity="gamma", dep="independence", mu=~exp(absorption-volume)* dose/(exp(absorption)-exp(elimination))* (exp(-exp(elimination)*times)-exp(-exp(absorption)*times)), ptvc=list(absorption=-1,elimination=-1.1,volume=-1), shape=~exp(b1-b2)*times*dose*exp(-exp(b1)*times), pshape=list(b1=-3,b2=0)) # add serial dependence kalseries(reps, intensity="gamma", dep="serial", pdep=0.5, mu=~exp(absorption-volume)* dose/(exp(absorption)-exp(elimination))* (exp(-exp(elimination)*times)-exp(-exp(absorption)*times)), ptvc=list(absorption=-1,elimination=-1.1,volume=-1), shape=~exp(b1-b2)*times*dose*exp(-exp(b1)*times), pshape=list(b1=-3,b2=0)) ## End(Not run) ``` ## Author(s) J.K. Lindsey

kalseries function