Simplex Generalized Linear Model Regression Function

Share:

Description

Regression Analysis of Proportional Data Using Various Types of Simplex Models

Usage

1
2
3
4
5
6
simplexreg(formula, data, subset, na.action, 
   	link = c("logit", "probit", "cloglog", "neglog"), corr = "Ind", id = NULL, 
   	control = simplexreg.control(...), model = TRUE, y = TRUE, x = TRUE, ...)
	
simplexreg.fit(y, x, z = NULL, t = NULL, link = "logit", corr = "Ind",
   	id = NULL, control = simplexreg.control())

Arguments

formula

a symbolic description of the model to be fitted(of type y ~ x or y ~ x | z | t. The Details are given under 'Details').

data

an optional data frame, list or environment containing variables in formula and id.

subset, na.action

arguments controlling formula processing via model.frame

link

type of link function to the mean. Currently, "logit"(logit function), "probit"(probit function), "cloglog"(complementary log-log function), "neglog"(negative log function) are supported.

corr

the covariance structure, chosen from "Ind"(independent structure), "Exc"(exchangeability) and "AR1"(AR(1)), see Details

id

a factor identifies the clusters when gee = TRUE. The length of id should be the same as the number of observations. y, x, z, t are assumed to be sorted in accordance with clusters specified by id

control

a list of control argument via simplexreg.control

model

a logical value indicating whether model frame should be included as a component of the return value

y, x

For simplexreg:logical values indicating whether response vector and covariates modelling the mean parameter should be returned as components of the returned value For simplexreg.fit:x is the design matrix and y is the response vector

z

regressor matrix modelling the dispersion parameter

t

time covariate in the correlation structure, see Details

...

argument passed to simplexreg.control

Details

Outcomes of continuous proportions arise in many applied areas. Such data could be properly modelled using simplex regression. See also simplex. The mean and dispersion parameters are linked to set of regressors. Regression analysis of the simplex model is implemented in simplexreg. If corr = "Ind", simplex generalized regression model is employed. Estimations is performed by maximum likelihood via Fisher scoring technique.

Apart from including generalized simplex regression models, this function also provides users with generalized estimating equations (GEE) techniques to model longitudinal proportional response. Exchangeability and AR(1) structures are available. Parameter estimation and residual analysis are involved.

We employ the specification approach designed in the fitting model function betareg of beta regression in the package betareg. As for simplex regression models, assuming the dispersion is homogeneous, the response is linked to a linear predictor described by y ~ x1 + x2 using a link function. Four types of function are available linking the regressors to the mean. However, for dispersion, the link function is restricted to logarithm function. When modeling dispersion, the regressor modelling the dispersion parameter should be specified in a formula form of type y ~ x1 + x2 | z1 + z2 where z1 and z2 are linked to the dispersion parameter σ^2.

Model specification is a bit complicated when it comes to modelling longitudinal proportional response. Song et. al (2004) proposed a marginal simplex model consists of three components, the population-average effects, the pattern of dispersion and the correlation structure. Let the percentage responses for the ith subject be y_{ij}, observed at time t_{ij}. If corr = "AR1", the working covariance matrix of y_{ij}, j = 1, 2, ..., n_i, is

{exp(α * |t_{ik} - t_{ij}|)}_{kj}

where α < 0 and exp(α) is the lag-1 autocorrelation. If corr = "Exc", the covariance matrix will be (1 - exp(α)) I + exp(α) 1 where I is the identity matrix while 1 the matrix with all elements being equal to one.

For homogeneous dispersion, the formula is supposed to be of the form y ~ x1 + x2 | 1 | t where t is the time covariate. Otherwise, the formula will be of the form y ~ x1 + x2 | z1 + z2 | t.

Value

fixef

estimates of coefficients modelling the mean as well as the standard deviation

dispar

estimates of coefficients modelling dispersion as well as the standard deviation

Dispersion

estimate of the dispersion parameter

appstdPerr

approximated standard deviations of the regression coefficients

stdPerr

exact standard deviations of the regression coefficients

meanmu

estimate of mean parameter

adjvar

adjusted dependent variable s_i. Details could be found in McCullagh and Nelder (1989)

stdscor

standardised score residuals. Details can be found in Song et al. (2004)

predict

predicted values of g(μ_i) where g is the link function and μ_i the mean parameter

loglike

value of maximum log-likelihood function

deviance

value of deviance

call

the original function call

formula

the original formula

terms

a list with elements "mean" and "dispersion" containing term object for the model

levels

a list with elements "mean" and "dispersion" containing levels of categorical regressors

link

type of function linking to the mean

type

type = "homo" for homogeneous dispersion while type = "hetero" for heterogeneous dispersion

model

the full model frame (if model = TRUE)

y

response vector (if y = TRUE)

x

a list with elements mean, dispersion, time and id containing corresponding variables

n

number of proportional observations

iter

number of Fisher iterations

...

argument passed to simplexreg.control

Author(s)

Zhenguo Qiu, Peng Zhang and Chengchun Shi

References

Barndorff-Nielsen, O.E. and Jorgensen, B. (1991) Some parametric models on the simplex. Journal of Multivariate Analysis, 39: 106–116

Jorgensen, B. (1997) The Theory of Dispersion Models. London: Chapman and Hall

McCullagh, P and Nelder J. (1989) Generalized Linear Models. London: Chapman and Hall

Song, P. and Qiu, Z. and Tan, M. (2004) Modelling Heterogeneous Dispersion in Marginal Models for Longitudinal Proportional Data. Biometrical Journal, 46: 540–553

Zhang, P. and Qiu, Z. and Shi, C. (2016) simplexreg: An R Package for Regression Analysis of Proportional Data Using the Simplex Distribution. Journal of Statistical Software, 71: 1–21

See Also

simplex

Examples

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
# GLM models
data("sdac", package = "simplexreg")
sim.glm1 <- simplexreg(rcd~ageadj+chemo, link = "logit", 
  data = sdac)
sim.glm2 <- simplexreg(rcd~ageadj+chemo|age, link = "logit", 
  data = sdac)

# GEE models
data("retinal", package = "simplexreg")
sim.gee1 <- simplexreg(Gas~LogT+LogT2+Level|1|Time, link = "logit", 
  corr = "Exc", id = ID, data = retinal)
sim.gee2 <- simplexreg(Gas~LogT+LogT2+Level|LogT+Level|Time, 
  link = "logit", corr = "AR1", id = ID, data = retinal)