cond.rsm: Approximate Conditional Inference in Regression-Scale Models

Description

Performs approximate conditional inference on a scalar parameter of interest in regression-scale models. The output is stored in an object of class marg.

Usage

## S3 method for class 'rsm':
cond(object, offset, formula = NULL, family = NULL, 
     dispersion = NULL, data = sys.frame(sys.parent()), pts = 20, 
     n = max(100, 2*pts), tms = 0.6, from = NULL, to = NULL, 
     control = glm.control(...), trace = FALSE, ...)

Arguments

object

a rsm object; for instance the result of a call to rsm.

offset

either the covariate occurring in the model formula whose coefficient represents the parameter of interest or scale if the parameter of interest is the scale parameter. In the first case, the variable may be numerical or a two

formula

a formula expression (only if no rsm object is defined).

family

a family.rsm object defining the error distribution (only if no rsm object is defined). See rsm.families for the families supported.

dispersion

argument only to be used if no rsm object is defined. If NULL, the scale parameter is taken to be unknown. If known, the numerical value can be passed. The default is NULL. Huber's least favourable

data

an optional data frame in which to interpret the variables occurring in the formula (only if no rsm object is defined).

pts

number of output points (minimum 10) that are calculated exactly; the default is 20.

approximate number of output points (minimum 50) produced by the spline interpolation. The default is the maximum between 100 and twice pts.

tms

defines the range MLE +/- tms * s.e. where cubic spline interpolation is replaced by polynomial interpolation. The default is 0.6.

from

starting value of the sequence that contains the values of the parameter of interest for which output points are calculated exactly. The default is MLE - 3.5 * s.e.

ending value of the sequence that contains the values of the parameter of interest for which output points are calculated exactly. The default is MLE + 3.5 * s.e.

control

a list of iteration and algorithmic constants that control the rsm fit. See glm.control for their names and default values.

trace

if TRUE, iteration numbers will be printed.

...

additional arguments, such as weights, subset, control etc. used by the rsm fitting routine if the rsm object is defined through formula and family.

Value

The returned value is an object of class marg; see marg.object for details.

Details

This function is a method for the generic function cond for class rsm. It can be invoked by calling cond for an object of the appropriate class, or directly by calling cond.rsm regardless of the class of the object. cond.rsm has also to be used if the rsm object is not provided throught the object argument but specified by formula and family.

The function cond.rsm implements several small sample asymptotic methods for approximate conditional inference in regression-scale models. Approximations for both the modified/marginal log likelihood function and approximate conditional/marginal tail probabilities are available (see marg.object for details). Attention is restricted to a scalar parameter of interest, either a regression coefficient or the scale parameter. In the first case, the associated covariate may be either numerical or a two-level factor.

Approximate conditional (or equivalently marginal) inference is performed by either updating a fitted regression-scale model or defining the model formula and family. All approximations are calculated exactly for pts equally spaced points ranging from from to to. A spline interpolation is used to extend them over the whole interval of interest, except for the range of values defined by MLE +/- tms * s.e. where the spline interpolation is replaced by a higher order polynomial interpolation. This is done in order to avoid numerical instabilities which are likely to occur for values of the parameter of interest close to the MLE. Results are stored in an object of class marg. Method functions like print, summary and plot can be used to examine the output or represent it graphically. Components can be extracted using coef, formula and family.

Main references for the methods considered are the papers by Barndorff-Nielsen (1991), DiCiccio, Field and Fraser (1990) and DiCiccio and Field (1991). The theory and statistics used are summarized in Brazzale (2000, Chapters 2 and 3). More details of the implementation are given in Brazzale (1999; 2000, Section 6.3.1).

References

Barndorff-Nielsen, O. E. (1991) Modified signed log likelihood ratio. Biometrika, 78, 557--564.

Brazzale, A. R. (1999) Approximate conditional inference for logistic and loglinear models. J. Comput. Graph. Statist., 8, 653--661.

Brazzale, A. R. (2000) Practical Small-Sample Parametric Inference. Ph.D. Thesis N. 2230, Department of Mathematics, Swiss Federal Institute of Technology Lausanne.

DiCiccio, T. J., Field, C. A. and Fraser, D. A. S. (1990) Approximations of marginal tail probabilities and inference for scalar parameters. Biometrika, 77, 77--95.

DiCiccio, T. J. and Field, C. A. (1991) An accurate method for approximate conditional and Bayesian inference about linear regression models from censored data. Biometrika, 78, 903--910.

Examples

Run this code

## Sea Level Data
data(venice)
attach(venice)
Year <- 1:51/51
c11 <- cos(2*pi*1:51/11) ; s11 <- sin(2*pi*1:51/11)
c19 <- cos(2*pi*1:51/18.62) ; s19 <- sin(2*pi*1:51/18.62)
##
## quadratic model fitted to the sea level, includes 18.62-year 
## astronomical tidal cycle and 11-year sunspot cycle
venice.rsm <- rsm(sea ~ Year + I(Year^2) + c11 + s11 + c19 + s19, 
                  family = extreme)
names(coef(venice.rsm))
## "(Intercept)"  "Year"  "I(Year^2)"  "c11"  "s11"  "c19"  "s19"      
##
## variable of interest: quadratic term
venice.marg <- cond(venice.rsm, I(Year^2))
##
detach()

## House Price Data
data(houses)
houses.rsm <- rsm(price ~ ., family = student(5), data = houses)
##
## parameter of interest: scale parameter
houses.marg <- cond(houses.rsm, scale)

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