residuals.simplexreg: Extract residuals for simplexreg Objects

Description

Methods for extracting various types of residuals from simplex regression, from approximate Pearson residuals, standard Pearson residuals and standardise score residuals to adjusted dependent variable suggested by McCullagh and Nelder (1989). The first three can be used to examine mean-variance relation while the last aims to test the link function.

Usage

"residuals"(object, type = c("appstdPerr", "stdPerr", "stdscor", "adjvar"),  	...)

Arguments

object

fitted model object of class "simplexreg"

type

character specifying types of residuals:approximate Pearson residual (appstdPerr), standard Pearson residual (stdPerr), adjusted dependent variable $s_i$ (adjvar). Details are given in 'Details'

...

currently not used

Details

The Pearson residual takes the form $$r_i^P=\frac{y_i-\hat{\mu}_i}{\hat{\tau}_i}$$ where $\mu_i$ is the fitted mean parameter and details about calculation of $\tau$ is given in Jorgensen (1997). When the dispersion parameter $\sigma^2$ (see simplex) is large the variance of response approaches to $\mu(1-\mu)$ and this leads to the approximate Pearson residual $$r_i^a=\frac{y_i-\hat{\mu}_i}{\sqrt{\hat{\mu}_i(1-\hat{\mu}_i)}}$$ Plot of the standardised score residuals, $$r_i^S=\frac{u_i}{\sqrt{var(u_i)}}$$ where $u_i$ is the score function, can also detect model assumption violation. Details can be found in Song et al. (2004). The adjusted dependent variable suggested by McCullagh and Nelder (1989) could be employed as an informal check for the link function, $$s_i = g(\mu_i) + (\frac{3\sigma^4}{\mu_i(1-\mu_i)}+\frac{\sigma^2}{V(\mu_i)})^{-1/2} u(y_i;\mu_i)$$ where $u(y_i;\mu_i)$ and $V(\mu_i)$ are the score function and variance function.

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

Examples

Run this code

## fit the model
data("sdac", package="simplexreg")
sim.glm2 <- simplexreg(rcd~ageadj+chemo|age,
  link = "logit", data = sdac)
	
data("retinal", package = "simplexreg")
sim.gee2 <- simplexreg(Gas~LogT+LogT2+Level|LogT+Level|Time,
  link = "logit", corr = "AR1", id = ID, data = retinal)  

## extract the residuals
res <- residuals(sim.glm2, type = "stdPerr")
res <- residuals(sim.gee2, type = "adjvar")