ssym (version 1.5.1)

ssym.l2: Fitting Censored Semi-parametric Log-symmetric Regression Models

Description

ssym.l2 is used to fit a semi-parametric regression model suitable for analysis of data sets in which the response variable is continuous, strictly positive, asymmetric and right-censored. Under this setup, both median and skewness of the response variable distribution are explicitly modeled through semi-parametric functions, whose nonparametric components may be approximated by natural cubic splines or P-splines.

Usage

ssym.l2(formula, family, xi, data, epsilon, maxiter, subset, link.mu, link.phi,
        local.influence, spec, std.out)

Arguments

formula

a symbolic description of the systematic component of the model to be fitted. See details for further information.

family

a description of the (log) error distribution to be used in the model. Supported families include Normal, Student, Contnormal, Powerexp, Hyperbolic, Slash,

xi

a numeric value or numeric vector that represents the extra parameter of the specified error distribution.

data

an optional data frame, list or environment containing the variables in the model.

epsilon

an optional positive value, which represents the convergence criterion. Default value is 1e-07.

maxiter

an optional positive integer giving the maximal number of iterations for the estimating process. Default value is 1e03.

subset

an optional expression specifying a subset of individuals to be used in the fitting process.

link.mu

an optional character that specifies the link function of the median submodel.

link.phi

an optional character that specifies the link function of the skewness submodel.

local.influence

logical. If TRUE, local influence measures under two perturbation schemes are calculated. Default is FALSE.

spec

character. The smoothing parameter is estimated by minimizing a overall goodness-of-fit criterion such as AIC or BIC. spec is an optional string to specify the goodness-of-fit measure to be used. Default value is AIC

std.out

logical. If FALSE, just a reduced set of attributes is returned by the model-fitting function. Default is TRUE

Value

• theta.mu
  a vector of parameter estimates associated with the median submodel.
• theta.phi
  a vector of parameter estimates associated with the skewness (or the relative dispersion) submodel.
• vcov.mu
  approximate variance-covariance matrix associated with the median submodel.
• vcov.phi
  approximate variance-covariance matrix associated with the skewness (or the relative dispersion) submodel.
• lambdas.mu
  estimate of the smoothing parameter(s) associated with the nonparametric part of the median submodel.
• lambdas.phi
  estimate of the smoothing parameter(s) associated with the nonparametric part of the skewness (or the relative dispersion) submodel.
• gle.mu
  degrees of freedom associated with the nonparametric part of the median submodel.
• gle.phi
  degrees of freedom associated with the nonparametric part of the skewness (or the relative dispersion) submodel.
• deviance.mu
  a vector with the individual contributions to the deviance associated with the median submodel.
• deviance.phi
  a vector with the individual contributions to the deviance associated with the skewness (or the relative dispersion) submodel.
• mu.fitted
  a vector with the fitted values of the (in log-scale) median submodel.
• phi.fitted
  a vector with the fitted values of the skewness (or the relative dispersion) submodel.
• lpdf
  a vector of individual contributions to the log-likelihood function.

Details

The argument formula comprises of three parts (separated by the symbols "~" and "|"), namely: censoring status and observed response variable (in log-scale) in a object of class Surv, predictor of the median submodel (having logarithmic link) and predictor of the skewness (or the relative dispersion) submodel (having logarithmic link). An arbitrary number of nonparametric effects may be specified in the predictors. These effects are specified to be approximated by natural cubic splines or P-splines using the functions ncs() or psp(), respectively. The iterative estimation process is based on the Gauss-Seidel, Newton-Raphson and backfitting algorithms. Because some distributions such as log-Student-t, log-contaminated-normal, log-slash and log-hyperbolic may be obtained as a power mixture of the log-normal distribution, the expectation-constrained maximization (ECM) algorithm is applied in those cases to obtain a more efficient iterative process of parameter estimation. The smoothing parameter is chosen by minimizing the AIC or BIC criteria. The function ssym.l2() calculates overall goodness-of-fit statistics, deviance-type residuals for both submodels, as well as local influence measures under the case-weight and response perturbation schemes.

References

Vanegas, L.H. and Paula, G.A. (2015a) A Semiparametric Approach for Joint Modeling of Median and Skewness. TEST 24, 110-135. Vanegas, L.H. and Paula, G.A. (2015b) Log-symmetric distributions: statistical properties and parameter estimation. Brazilian Journal of Probability and Statistics (to appear)

See Also

ssym.nl, ssym.l

Examples

###################################################################################
################ Lung Cancer Trial - a log-Student model ##########################
###################################################################################
#data("veteran", package="survival")
#fit <- ssym.l2(Surv(log(time), 1-status) ~ karno| karno, data = veteran,
#              family="Student", xi=4.5)
#summary(fit)
#
########################### Extra parameter ###########################
#extra.parameter(fit,3,10)
#
################## Graph of deviance-type residuals ##################
#plot(fit)

####################################################################################
########## Primary biliary cirrhosis - a Birnbaum-Saunders-t model #################
####################################################################################
#data("pbc", package="survival")
#pbc2 <- as.data.frame(pbc[!is.na(pbc$edema) & !is.na(pbc$stage)& !is.na(pbc$bili),])
#fit <- ssym.l2(Surv(log(time),ifelse(status>=1,0,1) ) ~ factor(edema) + stage +
#               ncs(bili), data = pbc2, family="Sinh-t", xi=c(0.65,3))
#summary(fit)			   
#
########################### Extra parameter ###########################
#extra.parameter(fit,c(0.6,3),c(0.9,5))
#
################## Graph of the nonparametric effect ##################
#np.graph(fit, which=1, exp=TRUE)
#
################## Graph of deviance-type residuals ##################
#plot(fit)

####################################################################################
################ Baboons Data - a log-power-exponential model   ####################
####################################################################################
########################## left-censored observations ##############################
####################################################################################

#data("Baboons", package="ssym")
#fit <- ssym.l2(Surv(-log(t),cs) ~ 1, data=Baboons, family="Powerexp", xi=-0.35)
#
################## Graph of deviance-type residuals ##################
#plot(fit)