Fitting function for glm for reduced-bias estimation and inference
brglmFit is a fitting method for glm
that fits generalized linear models using implicit and explicit
bias reduction methods (Kosmidis, 2014), and other penalized
maximum likelihood methods. Currently supported methods include the
mean bias-reducing adjusted scores approach in Firth (1993) and
Kosmidis & Firth (2009), the median bias-reduction adjusted scores
approach in Kenne Pagui et al. (2017), the correction of the asymptotic
bias in Cordeiro & McCullagh (1991), the mixed bias-reduction
adjusted scores approach in Kosmidis et al (2020), maximum
penalized likelihood with powers of the Jeffreys prior as penalty,
and maximum likelihood. Estimation is performed using a quasi
Fisher scoring iteration (see vignette("iteration",
"brglm2")), which, in the case of mean-bias reduction, resembles
an iterative correction of the asymptotic bias of the Fisher
scoring iterates.
brglmFit( x, y, weights = rep(1, nobs), start = NULL, etastart = NULL, mustart = NULL, offset = rep(0, nobs), family = gaussian(), control = list(), intercept = TRUE, fixed_totals = NULL, singular.ok = TRUE ) brglm_fit( x, y, weights = rep(1, nobs), start = NULL, etastart = NULL, mustart = NULL, offset = rep(0, nobs), family = gaussian(), control = list(), intercept = TRUE, fixed_totals = NULL, singular.ok = TRUE )
x |
|
y |
|
weights |
an optional vector of ‘prior weights’ to be used
in the fitting process. Should be |
start |
starting values for the parameters in the linear
predictor. If |
etastart |
applied only when start is not
|
mustart |
applied only when start is not |
offset |
this can be used to specify an a priori known
component to be included in the linear predictor during fitting.
This should be |
family |
a description of the error distribution and link
function to be used in the model. For |
control |
a list of parameters controlling the fitting
process. See |
intercept |
logical. Should an intercept be included in the null model? |
fixed_totals |
effective only when |
singular.ok |
logical. If |
A detailed description of the supported adjustments and the quasi
Fisher scoring iteration is given in the iteration vignette (see,
vignette("iteration", "brglm2") or Kosmidis et al, 2020). A
shorter description of the quasi Fisher scoring iteration is also
given in one of the vignettes of the *enrichwith* R package (see,
https://cran.r-project.org/package=enrichwith/vignettes/bias.html).
Kosmidis and Firth (2010) describe a parallel quasi Newton-Raphson
iteration with the same stationary point.
In the special case of generalized linear models for binomial,
Poisson and multinomial responses, the adjusted score equations
approach returns estimates with improved frequentist properties,
that are also always finite, even in cases where the maximum
likelihood estimates are infinite (e.g. complete and quasi-complete
separation in multinomial regression). See, Kosmidis and Firth
(2020) for a proof for binomial-response GLMs with Jeffreys-pior
penalties to the log-likelihood, which is equivalent to mean bias
reduction for logistic regression. See, also,
detect_separation and
check_infinite_estimates for pre-fit and post-fit
methods for the detection of infinite estimates in binomial
response generalized linear models.
The type of score adjustment to be used is specified through the
type argument (see brglmControl for
details). The available options are
type = "AS_mixed": the mixed bias-reducing score adjustments in
Kosmidis et al (2020) that result in mean bias reduction for the
regression parameters and median bias reduction for the dispersion
parameter, if any; default.
type = "AS_mean": the mean bias-reducing score adjustments
in Firth, 1993 and Kosmidis & Firth, 2009. type = "AS_mixed"
and type = "AS_mean" will return the same results when
family is binomial or poisson, i.e. when the
dispersion is fixed)
type = "AS_median": the median-bias reducing score
adjustments in Kenne Pagui et al. (2017)
type = "MPL_Jeffreys": maximum penalized likelihood
with powers of the Jeffreys prior as penalty.
type = "ML": maximum likelihood
type = "correction": asymptotic bias correction, as in
Cordeiro & McCullagh (1991).
The null deviance is evaluated based on the fitted values using the
method specified by the type argument (see
brglmControl).
The family argument of the current version of
brglmFit can accept any combination of family
objects and link functions, including families with user-specified
link functions, mis links, and power
links, but excluding quasi,
quasipoisson and quasibinomial
families.
fixed_totals to specify groups of observations for which the
sum of the means of a Poisson model will be held fixed to the
observed count for each group. This argument is used internally in
brmultinom and bracl for
baseline-category logit models and adjacent category logit models,
respectively.
brglm_fit is an alias to brglmFit.
Ioannis Kosmidis [aut, cre] ioannis.kosmidis@warwick.ac.uk, Euloge Clovis Kenne Pagui [ctb] kenne@stat.unipd.it
Kosmidis I, Firth D (2020). Jeffreys-prior penalty, finiteness and shrinkage in binomial-response generalized linear models. *Biometrika* doi: 10.1093/biomet/asaa052
Kosmidis I, Kenne Pagui E C, Sartori N (2020). Mean and median bias reduction in generalized linear models. *Statistics and Computing*, **30**, 43-59 doi: 10.1007/s11222-019-09860-6
Cordeiro G M, McCullagh P (1991). Bias correction in generalized linear models. *Journal of the Royal Statistical Society. Series B (Methodological)*, **53**, 629-643 doi: 10.1111/j.2517-6161.1991.tb01852.x
Firth D (1993). Bias reduction of maximum likelihood estimates. *Biometrika*. **80**, 27-38 doi: 10.2307/2336755
Kenne Pagui E C, Salvan A, Sartori N (2017). Median bias reduction of maximum likelihood estimates. *Biometrika*, **104**, 923–938 doi: 10.1093/biomet/asx046
Kosmidis I, Firth D (2009). Bias reduction in exponential family nonlinear models. *Biometrika*, **96**, 793-804 doi: 10.1093/biomet/asp055
Kosmidis I, Firth D (2010). A generic algorithm for reducing bias in parametric estimation. *Electronic Journal of Statistics*, **4**, 1097-1112 doi: 10.1214/10-EJS579
Kosmidis I (2014). Bias in parametric estimation: reduction and useful side-effects. *WIRE Computational Statistics*, **6**, 185-196 doi: 10.1002/wics.1296
## The lizards example from ?brglm::brglm
data("lizards")
# Fit the model using maximum likelihood
lizardsML <- glm(cbind(grahami, opalinus) ~ height + diameter +
light + time, family = binomial(logit), data = lizards,
method = "glm.fit")
# Mean bias-reduced fit:
lizardsBR_mean <- glm(cbind(grahami, opalinus) ~ height + diameter +
light + time, family = binomial(logit), data = lizards,
method = "brglmFit")
# Median bias-reduced fit:
lizardsBR_median <- glm(cbind(grahami, opalinus) ~ height + diameter +
light + time, family = binomial(logit), data = lizards,
method = "brglmFit", type = "AS_median")
summary(lizardsML)
summary(lizardsBR_median)
summary(lizardsBR_mean)
# Maximum penalized likelihood with Jeffreys prior penatly
lizards_Jeffreys <- glm(cbind(grahami, opalinus) ~ height + diameter +
light + time, family = binomial(logit), data = lizards,
method = "brglmFit", type = "MPL_Jeffreys")
# lizards_Jeffreys is the same fit as lizardsBR_mean (see Firth, 1993)
all.equal(coef(lizardsBR_mean), coef(lizards_Jeffreys))
# Maximum penalized likelihood with powers of the Jeffreys prior as
# penalty. See Kosmidis & Firth (2020) for the finiteness and
# shrinkage properties of the maximum penalized likelihood
# estimators in binomial response models
a <- seq(0, 20, 0.5)
coefs <- sapply(a, function(a) {
out <- glm(cbind(grahami, opalinus) ~ height + diameter +
light + time, family = binomial(logit), data = lizards,
method = "brglmFit", type = "MPL_Jeffreys", a = a)
coef(out)
})
# Illustration of shrinkage as a grows
matplot(a, t(coefs), type = "l", col = 1, lty = 1)
abline(0, 0, col = "grey")
## Another example from
## King, Gary, James E. Alt, Nancy Elizabeth Burns and Michael Laver
## (1990). "A Unified Model of Cabinet Dissolution in Parliamentary
## Democracies", _American Journal of Political Science_, **34**, 846-870
data("coalition", package = "brglm2")
# The maximum likelihood fit with log link
coalitionML <- glm(duration ~ fract + numst2, family = Gamma, data = coalition)
# The mean bias-reduced fit
coalitionBR_mean <- update(coalitionML, method = "brglmFit")
# The bias-corrected fit
coalitionBC <- update(coalitionML, method = "brglmFit", type = "correction")
# The median bias-corrected fit
coalitionBR_median <- update(coalitionML, method = "brglmFit", type = "AS_median")
## An example with offsets from Venables & Ripley (2002, p.189)
data("anorexia", package = "MASS")
anorexML <- glm(Postwt ~ Prewt + Treat + offset(Prewt),
family = gaussian, data = anorexia)
anorexBC <- update(anorexML, method = "brglmFit", type = "correction")
anorexBR_mean <- update(anorexML, method = "brglmFit")
anorexBR_median <- update(anorexML, method = "brglmFit", type = "AS_median")
## All methods return the same estimates for the regression
## parameters because the maximum likelihood estimator is normally
## distributed around the `true` value under the model (hence, both
## mean and component-wise median unbiased). The Wald tests for
## anorexBC and anorexBR_mean differ from anorexML
## because the bias-reduced estimator of the dispersion is the
## unbiased, by degree of freedom adjustment (divide by n - p),
## estimator of the residual variance. The Wald tests from
## anorexBR_median are based on the median bias-reduced estimator
## of the dispersion that results from a different adjustment of the
## degrees of freedom (divide by n - p - 2/3)
summary(anorexML)
summary(anorexBC)
summary(anorexBR_mean)
summary(anorexBR_median)
## endometrial data from Heinze & Schemper (2002) (see ?endometrial)
data("endometrial", package = "brglm2")
endometrialML <- glm(HG ~ NV + PI + EH, data = endometrial,
family = binomial("probit"))
endometrialBR_mean <- update(endometrialML, method = "brglmFit",
type = "AS_mean")
endometrialBC <- update(endometrialML, method = "brglmFit",
type = "correction")
endometrialBR_median <- update(endometrialML, method = "brglmFit",
type = "AS_median")
summary(endometrialML)
summary(endometrialBC)
summary(endometrialBR_mean)
summary(endometrialBR_median)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.