R: Robust Fitting Generalized Linear Models using Weighted...
wle.glm
R Documentation
Robust Fitting Generalized Linear Models using Weighted Likelihood
Description
wle.glm is used to robustly fit generalized linear models, specified by
giving a symbolic description of the linear predictor and a
description of the error distribution.
Usage
wle.glm(formula, family = binomial, data, weights, subset,
na.action, start = NULL, etastart, mustart, offset,
control = list(glm = glm.control(...), wle = wle.glm.control()),
model = TRUE, method = "wle.glm.fit", x = FALSE, y = TRUE,
contrasts = NULL, dist.method = "euclidean", ...)
wle.glm.fit(x, y, weights = NULL, wle.weights = rep(1, NROW(y)),
start = NULL, etastart = NULL, mustart = NULL, offset = rep(0, NROW(y)),
family = gaussian(), control = list(glm=glm.control(),
wle=wle.glm.control()), dist.method='euclidean',
intercept = TRUE, dispersion = NULL)
## S3 method for class 'wle.glm'
weights(object, type = c("prior", "working", "wle"), root="all", ...)
Arguments
formula
an object of class "formula" (or one that
can be coerced to that class): a symbolic description of the
model to be fitted. The details of model specification are given
under ‘Details’.
family
a description of the error distribution and link
function to be used in the model. This can be a character string
naming a family function, a family function or the result of a call
to a family function. (See family for details of
family functions.)
data
an optional data frame, list or environment (or object
coercible by as.data.frame to a data frame) containing
the variables in the model. If not found in data, the
variables are taken from environment(formula),
typically the environment from which wle.glm is called.
weights
an optional vector of ‘prior weights’ to be used
in the fitting process. Should be NULL or a numeric vector.
subset
an optional vector specifying a subset of observations
to be used in the fitting process.
na.action
a function which indicates what should happen
when the data contain NAs. The default is set by
the na.action setting of options, and is
na.fail if that is unset. The ‘factory-fresh’
default is na.omit. Another possible value is
NULL, no action. Value na.exclude can be useful.
start
starting values for the parameters in the linear predictor.
etastart
starting values for the linear predictor.
mustart
starting values for the vector of means.
offset
this can be used to specify an a priori known
component to be included in the linear predictor during fitting.
This should be NULL or a numeric vector of length equal to
the number of cases. One or more offset terms can be
included in the formula instead or as well, and if more than one is
specified their sum is used. See model.offset.
control
a list with two components of parameters for
controlling the fitting process. The first component (glm)
is set using the function glm.control while the
second component (wle) is set using the function
wle.glm.control and it is used to set the parameters
regarding the behaviour of the robust method. See the documentation
of these functions for details.
model
a logical value indicating whether model frame
should be included as a component of the returned value.
method
the method to be used in fitting the model.
The default method "wle.glm.fit" uses iteratively reweighted
least squares (IWLS). The only current alternative is
"model.frame" which returns the model frame and does no fitting.
x, y
For wle.glm:
logical values indicating whether the response vector and model
matrix used in the fitting process should be returned as components
of the returned value.
For wle.glm.fit: x is a design matrix of dimension
n * p, and y is a vector of observations of length n.
contrasts
an optional list. See the contrasts.arg
of model.matrix.default.
dist.method
distance method passed to dist to measure the
distance between predictor rows.
intercept
logical. Should an intercept be included in the
null model?
dispersion
numeric or NULL. If provided used as starting value.
object
an object inheriting from class "wle.glm".
type
character, partial matching allowed. Type of weights to
extract from the fitted model object.
root
character ("all") or a number. For which solutions the
weights are reported.
wle.weights
For wle.glm.fit these are weights used in the iterative algorithm evaluated at each step by wle.glm.weights.
...
For wle.glm: arguments to be passed by default to
glm.control: see argument control.
For weights:
further arguments passed to or from other methods.
Details
A typical predictor has the form response ~ terms where
response is the (numeric) response vector and terms is a
series of terms which specifies a linear predictor for
response. For binomial and quasibinomial
families the response can also be specified as a factor
(when the first level denotes failure and all others success) or as a
two-column matrix with the columns giving the numbers of successes and
failures. A terms specification of the form first + second
indicates all the terms in first together with all the terms in
second with any duplicates removed.
A specification of the form first:second indicates the the set
of terms obtained by taking the interactions of all terms in
first with all terms in second. The specification
first*second indicates the cross of first and
second. This is the same as first + second +
first:second.
The terms in the formula will be re-ordered so that main effects come
first, followed by the interactions, all second-order, all third-order
and so on: to avoid this pass a terms object as the formula.
Non-NULLweights can be used to indicate that different
observations have different dispersions (with the values in
weights being inversely proportional to the dispersions); or
equivalently, when the elements of weights are positive
integers w_i, that each response y_i is the mean of
w_i unit-weight observations. In case of binomial GLM prior weights
CAN NOT be used to give the number of trials when the response is the
proportion of successes; in this situation please submit the response
variable as two columns (first column successes, second column unsuccesses).
They would rarely be used for a Poisson GLM.
wle.glm.fit is the workhorse function: it is not normally
called directly but can be more efficient where the response vector
and design matrix have already been calculated. However, this function
needs starting values and does not look for possible multiple roots in the system of equations.
If more than one of etastart, start and mustart
is specified, the first in the list will be used. It is often
advisable to supply starting values for a quasi family,
and also for families with unusual links such as gaussian("log").
All of weights, subset, offset, etastart
and mustart are evaluated in the same way as variables in
formula, that is first in data and then in the
environment of formula.
For the background to warning messages about ‘fitted probabilities
numerically 0 or 1 occurred’ for binomial GLMs, see Venables &
Ripley (2002, pp. 197–8).
Multiple roots may occur if the asymptotic weights are used or in the
case of continuous models. The function implements the bootstrap root
serach approach described in Markatou, Basu and Lindsay (1998) in order
to find these roots.
Value
wle.glm returns an object of class inheriting from
"wle.glm".
The function summary (i.e., summary.wle.glm) can
be used to obtain or print a summary of the results and the function
anova (i.e., anova.wle.glm.root)
to produce an analysis of variance table.
The generic accessor functions coefficients,
effects, fitted.values and residuals can be used to
extract various useful features of the value returned by wle.glm.
weights extracts a vector of weights, one for each case/root in the fit (after subsetting and na.action).
An object of class "wle.glm" is a (variable length) list
containing at least the following components:
root1 which is a list with the following components:
coefficients
a named vector of coefficients
residuals
the working residuals, that is the residuals
in the final iteration of the IWLS fit. Since cases with zero
weights are omitted, their working residuals are NA.
fitted.values
the fitted mean values, obtained by transforming
the linear predictors by the inverse of the link function.
rank
the numeric rank of the fitted linear model.
family
the family object used.
linear.predictors
the linear fit on link scale.
deviance
up to a constant, minus twice the maximized
log-likelihood. Where sensible, the constant is chosen so that a
saturated model has deviance zero.
aic
Akaike's An Information Criterion, minus twice the
maximized log-likelihood plus twice the number of coefficients (so
assuming that the dispersion is known).
null.deviance
The deviance for the null model, comparable with
deviance. The null model will include the offset, and an
intercept if there is one in the model. Note that this will be
incorrect if the link function depends on the data other than
through the fitted mean: specify a zero offset to force a correct
calculation.
iter
the number of iterations of IWLS used.
weights
the working weights, that is the weights
in the final iteration of the IWLS fit.
prior.weights
the weights initially supplied, a vector of
1s if none were.
df.residual
the residual degrees of freedom.
df.null
the residual degrees of freedom for the null model.
y
if requested (the default) the y vector
used. (It is a vector even for a binomial model.)
x
if requested, the model matrix.
model
if requested (the default), the model frame.
converged
logical. Was the IWLS algorithm judged to have converged?
boundary
logical. Is the fitted value on the boundary of the
attainable values?
wle.weights
final (robust) weights based on the WLE approach.
wle.asymptotic
logicals. If TRUE asymptotic weight based on Anscombe residual is used for the corresponding observation.
In addition, non-empty fits will have components qr,
R, qraux, pivot
and effects relating to the final weighted linear fit.
and the following components:
family
the family object used.
call
the matched call.
formula
the formula supplied.
terms
the terms object used.
data
the data argument.
offset
the offset vector used.
control
the value of the control argument used.
method
the name of the fitter function used, currently always
"wle.glm.fit".
contrasts
(where relevant) the contrasts used.
xlevels
(where relevant) a record of the levels of the factors
used in fitting.
tot.sol
the number of solutions found.
not.conv
the number of starting points that does not converge after the max.iter (defined using wle.glm.control) iterations are reached.
na.action
(where relevant) information returned by
model.frame on the special handling of NAs.
Objects of class "wle.glm" are normally of class
"wle.glm".
If a binomialwle.glm model was specified by
giving a two-column response, the weights returned by
prior.weights are
the total numbers of cases (factored by the supplied case weights) and
the component y of the result is the proportion of successes.
In case of multiple roots (i.e. tot.sol > 1) then objects of the
same form as root1 are reported with names root2,
root3 and so on until tot.sol.
Warnings
Since in a model selection procedure and/or on an ANOVA table the weights of the WLE procedure must be that of the FULL model (and not that of the actual model) statistics on degrees of freedom, deviance and AIC are valid only if this is the FULL model.
Author(s)
Claudio Agostinelli and Fatemah Al-quallaf
References
Agostinelli, C. (1998) Inferenza statistica robusta basata sulla
funzione di verosimiglianza pesata: alcuni sviluppi,
Ph.D Thesis, Department of Statistics, University of Padova.
Agostinelli, C. and Markatou, M., (1998) A one-step robust estimator
for regression based on the weighted likelihood reweighting scheme,
Statistics & Probability Letters, Vol. 37, n. 4, 341-350.
Agostinelli, C. and Markatou, M. (2001) Test of hypotheses based on
the Weighted Likelihood Methodology, Statistica Sinica,
vol. 11, n. 2, 499-514.
Agostinelli, C. and Al-quallaf, F. (2009) Robust inference in
Generalized Linear Models. Manuscript in preparation.
Dobson, A. J. (1990)
An Introduction to Generalized Linear Models.
London: Chapman and Hall.
Hastie, T. J. and Pregibon, D. (1992)
Generalized linear models.
Chapter 6 of Statistical Models in S
eds J. M. Chambers and T. J. Hastie, Wadsworth & Brooks/Cole.
Markatou, M., Basu, A. and Lindsay, B.G. (1998) Weighted likelihood
estimating equations with a bootstrap root search. Journal of
the American Statistical Association, 93:740-750.
McCullagh P. and Nelder, J. A. (1989)
Generalized Linear Models.
London: Chapman and Hall.
Venables, W. N. and Ripley, B. D. (2002)
Modern Applied Statistics with S.
New York: Springer.
See Also
anova.wle.glm.root, summary.wle.glm, etc. for
wle.glm methods,
and the generic functions anova, summary,
effects, fitted.values,
and residuals.
wle.lm for robust non-generalized linear models
for ‘general’ linear models.
Examples
## Dobson (1990) Page 93: Randomized Controlled Trial :
counts <- c(18,17,15,20,10,20,25,13,12)
outcome <- gl(3,1,9)
treatment <- gl(3,3)
print(d.AD <- data.frame(treatment, outcome, counts))
wle.glm.D93 <- wle.glm(counts ~ outcome + treatment, family=poisson(), x=TRUE, y=TRUE)
wle.glm.D93
anova(extractRoot(wle.glm.D93))
summary(wle.glm.D93)
## Not run:
## Support for gaussian family not provided yet!
## an example with offsets from Venables & Ripley (2002, p.189)
utils::data(anorexia, package="MASS")
anorex.2 <- wle.glm(Postwt ~ Prewt + Treat + offset(Prewt),
family = gaussian, data = anorexia)
anorex.2
summary(anorex.2)
## End(Not run)
## Not run:
# Gamma family is not yet implemented!
# A Gamma example, from McCullagh & Nelder (1989, pp. 300-2)
clotting <- data.frame(
u = c(5,10,15,20,30,40,60,80,100),
lot1 = c(118,58,42,35,27,25,21,19,18),
lot2 = c(69,35,26,21,18,16,13,12,12))
wlot1 <- wle.glm(lot1 ~ log(u), data=clotting, family=Gamma,
control=list(glm=glm.control(), wle=wle.glm.control(use.asymptotic=1)))
wlot2 <- wle.glm(lot2 ~ log(u), data=clotting, family=Gamma,
control=list(glm=glm.control(), wle=wle.glm.control(use.asymptotic=1)))
wlot1
wlot2
summary(wlot1)
summary(wlot2)
## End(Not run)