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
1 2 3 4 5 6 7 8 9 10 11 12 13 14  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 
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 
data 
an optional data frame, list or environment (or object
coercible by 
weights 
an optional vector of ‘prior weights’ to be used
in the fitting process. Should be 
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 
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 
control 
a list with two components of parameters for
controlling the fitting process. The first component ( 
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 
x, y 
For For 
contrasts 
an optional list. See the 
dist.method 
distance method passed to 
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 
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 
... 
For For 
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
twocolumn 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 reordered so that main effects come
first, followed by the interactions, all secondorder, all thirdorder
and so on: to avoid this pass a terms
object as the formula.
NonNULL
weights
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 unitweight 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 
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 
linear.predictors 
the linear fit on link scale. 
deviance 
up to a constant, minus twice the maximized loglikelihood. Where sensible, the constant is chosen so that a saturated model has deviance zero. 
aic 
Akaike's An Information Criterion, minus twice the maximized loglikelihood plus twice the number of coefficients (so assuming that the dispersion is known). 
null.deviance 
The deviance for the null model, comparable with

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

df.residual 
the residual degrees of freedom. 
df.null 
the residual degrees of freedom for the null model. 
y 
if requested (the default) the 
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 
In addition, nonempty fits will have components qr
,
R
, qraux
, pivot
and effects
relating to the final weighted linear fit.
and the following components:
family 
the 
call 
the matched call. 
formula 
the formula supplied. 
terms 
the 
data 
the 
offset 
the offset vector used. 
control 
the value of the 
method 
the name of the fitter function used, currently always

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 
na.action 
(where relevant) information returned by

Objects of class "wle.glm"
are normally of class
"wle.glm"
.
If a binomial
wle.glm
model was specified by
giving a twocolumn 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 Alquallaf
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 onestep robust estimator for regression based on the weighted likelihood reweighting scheme, Statistics \& Probability Letters, Vol. 37, n. 4, 341350.
Agostinelli, C. and Markatou, M. (2001) Test of hypotheses based on the Weighted Likelihood Methodology, Statistica Sinica, vol. 11, n. 2, 499514.
Agostinelli, C. and Alquallaf, 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, 93:740750.
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 nongeneralized linear models
for ‘general’ linear models.
Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39  ## 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. 3002)
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)
