Regression Analysis of Proportional Data Using Various Types of Simplex Models

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())
``` |

`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 |

`subset, na.action` |
arguments controlling formula processing via |

`link` |
type of link function to the mean. Currently, |

`corr` |
the covariance structure, chosen from |

`id` |
a factor identifies the clusters when |

`control` |
a list of control argument via |

`model` |
a logical value indicating whether |

`y, x` |
For |

`z` |
regressor matrix modelling the dispersion parameter |

`t` |
time covariate in the correlation structure, see Details |

`...` |
argument passed to |

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 *i*th 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`

.

`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 |

`stdscor` |
standardised score residuals. Details can be found in Song |

`predict` |
predicted values of |

`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 |

`levels` |
a list with elements |

`link` |
type of function linking to the mean |

`type` |
type = |

`model` |
the full model frame (if |

`y` |
response vector (if |

`x` |
a list with elements |

`n` |
number of proportional observations |

`iter` |
number of Fisher iterations |

`...` |
argument passed to |

Zhenguo Qiu, Peng Zhang and Chengchun Shi

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

`simplex`

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)
``` |

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