brglm {brglm} | R Documentation |
Fits binomial-response GLMs using the bias-reduction method developed in Firth (1993) for the removal of the leading (O(n^{-1})) term from the asymptotic expansion of the bias of the maximum likelihood estimator. Fitting is performed using pseudo-data representations, as described in Kosmidis (2007, Chapter 5). For estimation in binomial-response GLMs, the bias-reduction method is an improvement over traditional maximum likelihood because:
brglm(formula, family = binomial, data, weights, subset, na.action, start = NULL, etastart, mustart, offset, control.glm = glm.control1(...), model = TRUE, method = "brglm.fit", pl = FALSE, x = FALSE, y = TRUE, contrasts = NULL, control.brglm = brglm.control(...), ...) brglm.fit(x, y, weights = rep(1, nobs), start = NULL, etastart = NULL, mustart = NULL, offset = rep(0, nobs), family = binomial(), control = glm.control(), control.brglm = brglm.control(), intercept = TRUE, pl = FALSE)
formula |
as in glm . |
family |
as in glm . brglm currently
supports only the "binomial" family with links
"logit" , "probit" , "cloglog" , "cauchit" . |
data |
as in glm . |
weights |
as in glm . |
subset |
as in glm . |
na.action |
as in glm . |
start |
as in glm . |
etastart |
as in glm . |
mustart |
as in glm . |
offset |
as in glm . |
control.glm |
control.glm replaces the control
argument in glm but essentially does the same job. It
is a list of parameters to control glm.fit . See the
documentation of glm.control1 for details. |
control |
same as in glm . Only available to
brglm.fit . |
intercept |
as in glm . |
model |
as in glm . |
method |
the method to be used for fitting the model. The default
method is "brglm.fit" , which uses either the modified-scores
approach to estimation or maximum penalized likelihood (see
the pl argument below). The standard
glm methods "glm.fit" for maximum likelihood and
"model.frame" for returning the model frame without any
fitting, are also accepted. |
pl |
a logical value indicating whether the
model should be fitted using maximum penalized likelihood, where the
penalization is done using Jeffreys invariant prior, or using the
bias-reducing modified scores. It is only used when
method = "brglm.fit" . The default value is FALSE (see also the
Details section). |
x |
as in glm . |
y |
as in glm . |
contrasts |
as in glm . |
control.brglm |
a list of parameters for controlling the fitting
process when method = "brglm.fit" . See documentation of
brglm.control for details. |
... |
further arguments passed to or from other methods. |
brglm.fit
is the workhorse function for fitting the model using
either the bias-reduction method or maximum penalized likelihood. If
method = "glm.fit"
, usual maximum likelihood is used via
glm.fit
.
The main iteration of brglm.fit
consists of the following
steps:
gethats
and hatvalues
).
modifications
for more information).
glm.fit
on the pseudo data.
Iteration is repeated until either the iteration limit has been reached
or the sum of the absolute values of the modified scores is less than
some specified positive constant (see the br.maxit
and
br.epsilon
arguments in brglm.control
).
The default value (FALSE
) of pl
, when method = "brglm.fit"
,
results in estimates that are free of any O(n^{-1}) terms in the asymptotic expansion of their bias. When
pl = TRUE
bias-reduction is again achieved but generally not at
such order of magnitude. In the case of logistic regression the value of
pl
is irrelevant since maximum penalized likelihood and the
modified-scores approach coincide for natural exponential families (see
Firth, 1993).
For other language related details see the details section in
glm
.
brglm
returns an object of class "brglm"
. A
"brglm"
object inherits first from "glm"
and then from
"lm"
and is a list containing the following components:
coefficients |
as in glm . |
residuals |
as in glm . |
fitted.values |
as in glm . |
effects |
as in glm . |
R |
as in glm . |
rank |
as in glm . |
qr |
as in glm . |
family |
as in glm . |
linear.predictors |
as in glm . |
deviance |
as in glm . |
aic |
as in glm (see Details). |
null.deviance |
as in glm . |
iter |
as in glm . |
weights |
as in glm . |
prior.weights |
as in glm . |
df.residual |
as in glm . |
df.null |
as in glm . |
y |
as in glm . |
converged |
as in glm . |
boundary |
as in glm . |
ModifiedScores |
the vector of the modified scores for the
parameters at the final iteration. If pl = TRUE they are the
derivatives of the penalized likelihood at the final iteration. |
FisherInfo |
the Fisher information matrix evaluated at the
resultant estimates. Only available when method = "brglm.fit" . |
hats |
the diagonal elements of the hat matrix. Only available
when method = "brglm.fit" |
nIter |
the number of iterations that were required until
convergence. Only available when method = "brglm.fit" . |
cur.model |
a list with components ar and at which
contains the values of the additive modifications to the responses
(y ) and to the binomial totals (prior.weights ) at
the resultant estimates (see modifications for more
information). Only available when method = "brglm.fit" . |
model |
as in glm . |
call |
as in glm . |
formula |
as in glm . |
terms |
as in glm . |
data |
as in glm . |
offset |
as in glm . |
control.glm |
as control in the result of
glm . |
control.brglm |
the control.brglm argument that was passed to
brglm . Only available when method = "brglm.fit" . |
method |
the method used for fitting the model. |
contrasts |
as in glm . |
xlevels |
as in glm . |
pl |
logical having the same value with the pl
argument passed to brglm . Only available when method =
"brglm.fit" . |
1. It is not advised to use methods associated with model comparison
(add1
, drop1
,
anova
, etc.) on objects of class
"brglm"
. Model comparison when estimation is performed using
the modified scores or the penalized likelihood is an on-going
research topic and will be implemented as soon as it is concluded.
2. The use of Akaike's information criterion (AIC) for
model selection when method = "brglm.fit"
is
controversial. AIC was developed under the assumptions that (i)
estimation is by maximum likelihood and (ii) that estimation is
carried out in a parametric family of distributions that contains
the “true” model. At least the first assumption is not valid
when using method = "brglm.fit"
. However, since the MLE is
asymptotically unbiased, asymptotically the modified-scores
approach is equivalent to maximum likelihood. A more appropriate
information criterion seems to be Konishi's generalized information
criterion (see Konishi & Kitagawa, 1996, Sections 3.2 and 3.3), which
will be implemented in a future version.
1. Supported methods for objects of class "brglm"
are:
print
print.brglm
.summary
summary.brglm
.coefficients
"glm"
class.vcov
"glm"
class.predict
"glm"
class.residuals
"glm"
class."glm"
2. A similar implementation of the bias-reduction method could be done for every GLM, following Kosmidis (2007) (see also Kosmidis and Firth, 2008). The full set of families and links will be available in a future version. However, bias-reduction is not generally beneficial as it is in the binomial family and it could cause inflation of the variance (see Firth, 1993).
3. Basically, the differences between maximum likelihood, maximum penalized likelihood and the modified scores approach are more apparent in small sample sizes, in sparse data sets and in cases where complete or quasi-complete separation occurs. Asymptotically (as n goes to infinity), the three different approaches are equivalent to first order.
4. When an offset is not present in the model, the modified-scores based estimates are usually smaller in magnitude than the corresponding maximum likelihood estimates, shrinking towards the origin of the scale imposed by the link function. Thus, the corresponding estimated asymptotic standard errors are also smaller.
The same is true for the maximum penalized likelihood estimates when for example, the logit (where the maximum penalized likelihood and modified-scores approaches coincide) or the probit links are used. However, generally the maximum penalized likelihood estimates do not shrink towards the origin. In terms of mean-value parameterization, in the case of maximum penalized likelihood the fitted probabilities would shrink towards the point where the Jeffreys prior is maximized or equivalently where the quadratic weights are simultaneously maximized (see Kosmidis, 2007).
5. Implementations of the bias-reduction method for logistic
regressions can also be found in the logistf package. In
addition to the obvious advantage of brglm
in the range of
link functions that can be used ("logit"
, "probit"
,
"cloglog"
and "cauchit"
), brglm
is also more
efficient computationally. Furthermore, for any user-specified link
function (see the Example section of family
), the
user can specify the corresponding pseudo-data representation to be
used within brglm
(see modifications
for
details).
Ioannis Kosmidis, I.Kosmidis@warwick.ac.uk
Bull, S. B., Lewinger, J. B. and Lee, S. S. F. (2007). Confidence intervals for multinomial logistic regression in sparse data. Statistics in Medicine 26, 903–918.
Firth, D. (1992) Bias reduction, the Jeffreys prior and {GLIM}. In Advances in GLIM and statistical modelling: Proceedings of the GLIM 92 conference, Munich, Eds. L.~Fahrmeir, B.~Francis, R.~Gilchrist and G.Tutz, pp. 91–100. New York: Springer.
Firth, D. (1992) Generalized linear models and Jeffreys priors: An iterative generalized least-squares approach. In Computational Statistics I, Eds. Y. Dodge and J. Whittaker. Heidelberg: Physica-Verlag.
Firth, D. (1993). Bias reduction of maximum likelihood estimates. Biometrika 80, 27–38.
Heinze, G. and Schemper, M. (2002). A solution to the problem of separation in logistic regression. Statistics in Medicine 21, 2409–2419.
Konishi, S. and Kitagawa, G. (1996). Generalised information criteria in model selection. Biometrika 83, 875–890.
Kosmidis, I. (2007). Bias reduction in exponential family nonlinear models. PhD Thesis, Department of Statistics, University of Warwick.
Kosmidis, I. and Firth, D. (2008). Bias reduction in exponential family nonlinear models. CRiSM Working Paper Series 8-5, http://www2.warwick.ac.uk/fac/sci/statistics/crism/research/2008/paper08-05.
## Begin Example data(lizards) # Fit the GLM using maximum likelihood lizards.glm <- brglm(cbind(grahami, opalinus) ~ height + diameter + light + time, family = binomial(logit), data=lizards, method = "glm.fit") # Now the bias-reduced fit: lizards.brglm <- brglm(cbind(grahami, opalinus) ~ height + diameter + light + time, family = binomial(logit), data=lizards, method = "brglm.fit") lizards.glm lizards.brglm # Other links update(lizards.brglm, family = binomial(probit)) update(lizards.brglm, family = binomial(cloglog)) update(lizards.brglm, family = binomial(cauchit)) # Using penalized maximum likelihood update(lizards.brglm, family = binomial(probit), pl = TRUE) update(lizards.brglm, family = binomial(cloglog), pl = TRUE) update(lizards.brglm, family = binomial(cauchit), pl = TRUE)