Bias reduction in Binomial-response GLMs
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:
the bias-reduced estimator is second-order unbiased and has smaller variance than the maximum likelihood estimator and
the resulting estimates and their corresponding standard errors are always finite while the maximum likelihood estimates can be infinite (in situations where complete or quasi separation occurs); see Kosmidis & Firth (2021) for the proof of finiteness in logistic regression models.
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 |
family |
as in |
data |
as in |
weights |
as in |
subset |
as in |
na.action |
as in |
start |
as in |
etastart |
as in |
mustart |
as in |
offset |
as in |
control.glm |
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control |
same as in |
intercept |
as in |
model |
as in |
method |
the method to be used for fitting the model. The default
method is |
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
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x |
as in |
y |
as in |
contrasts |
as in |
control.brglm |
a list of parameters for controlling the fitting
process when |
... |
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:
Calculate the diagonal components of the hat matrix (see
gethats and hatvalues).
Obtain the pseudo-data representation at the current value of the
parameters (see modifications for more information).
Fit a local GLM, using glm.fit on the pseudo data.
Adjust the quadratic weights to agree with the original binomial totals.
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 |
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residuals |
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fitted.values |
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effects |
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R |
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rank |
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qr |
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family |
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linear.predictors |
as in |
deviance |
as in |
aic |
as in |
null.deviance |
as in |
iter |
as in |
weights |
as in |
prior.weights |
as in |
df.residual |
as in |
df.null |
as in |
y |
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converged |
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boundary |
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ModifiedScores |
the vector of the modified scores for the
parameters at the final iteration. If |
FisherInfo |
the Fisher information matrix evaluated at the
resulting estimates. Only available when |
hats |
the diagonal elements of the hat matrix. Only available
when |
nIter |
the number of iterations that were required until
convergence. Only available when |
cur.model |
a list with components |
model |
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call |
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formula |
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terms |
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data |
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offset |
as in |
control.glm |
as |
control.brglm |
the |
method |
the method used for fitting the model. |
contrasts |
as in |
xlevels |
as in |
pl |
logical having the same value with the |
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 asymptotically valid, because
the log-likelihood derivatives dominate the modification (in terms
of asymptotic order).
1. Supported methods for objects of class "brglm" are:
2. A similar implementation of the bias-reduction method could be done for every GLM, following Kosmidis (2007) (see also Kosmidis and Firth, 2009). 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, ioannis.kosmidis@warwick.ac.uk
Kosmidis I. and Firth D. (2021). Jeffreys-prior penalty, finiteness and shrinkage in binomial-response generalized linear models. Biometrika, 108, 71–82.
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.
Kosmidis, I. (2007). Bias reduction in exponential family nonlinear models. PhD Thesis, Department of Statistics, University of Warwick.
Kosmidis, I. and Firth, D. (2009). Bias reduction in exponential family nonlinear models. Biometrika 96, 793–804.
## 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)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.