Description Usage Arguments Details Author(s) References See Also Examples
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 biasreducing adjusted scores approach in Firth (1993) and
Kosmidis & Firth (2009), the median biasreduction adjusted scores
approach in Kenne Pagui et al. (2017), the correction of the asymptotic
bias in Cordeiro & McCullagh (1991), the mixed biasreduction
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 meanbias reduction, resembles
an iterative correction of the asymptotic bias of the Fisher
scoring iterates.
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  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.rproject.org/package=enrichwith/vignettes/bias.html).
Kosmidis and Firth (2010) describe a parallel quasi NewtonRaphson
iteration with the same stationary point.
In the special case of generalized linear models for binomial,
Poisson and multinomial responses, the adjusted score equation
approaches for type = "AS_mixed"
, type = "AS_mean"
,
and type = "AS_median"
(see below for what methods each
type
corresponds) return estimates with improved frequentist
properties, that are also always finite, even in cases where the
maximum likelihood estimates are infinite (e.g. complete and
quasicomplete separation in multinomial regression). See, Kosmidis
and Firth (2021) for a proof for binomialresponse GLMs with
Jeffreysprior penalties to the loglikelihood, which is equivalent
to mean bias reduction for logistic regression. See, also,
detect_separation
and
check_infinite_estimates
for
prefit and postfit 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 biasreducing 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 biasreducing 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 biasreducing 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 userspecified
link functions, mis
links, and power
links, but excluding quasi
,
quasipoisson
and quasibinomial
families.
The description of method
argument and the Fitting
functions
section in glm
gives information on
supplying fitting methods to glm
.
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
baselinecategory 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 (2021). Jeffreysprior penalty, finiteness and shrinkage in binomialresponse generalized linear models. *Biometrika*, **108**, 7182 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**, 4359 doi: 10.1007/s11222019098606
Cordeiro G M, McCullagh P (1991). Bias correction in generalized linear models. *Journal of the Royal Statistical Society. Series B (Methodological)*, **53**, 629643 doi: 10.1111/j.25176161.1991.tb01852.x
Firth D (1993). Bias reduction of maximum likelihood estimates. *Biometrika*. **80**, 2738 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**, 793804 doi: 10.1093/biomet/asp055
Kosmidis I, Firth D (2010). A generic algorithm for reducing bias in parametric estimation. *Electronic Journal of Statistics*, **4**, 10971112 doi: 10.1214/10EJS579
Kosmidis I (2014). Bias in parametric estimation: reduction and useful sideeffects. *WIRE Computational Statistics*, **6**, 185196 doi: 10.1002/wics.1296
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 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100  ## 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 biasreduced fit:
lizardsBR_mean < glm(cbind(grahami, opalinus) ~ height + diameter +
light + time, family = binomial(logit), data = lizards,
method = "brglmFit")
# Median biasreduced 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 (2021) 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**, 846870
data("coalition", package = "brglm2")
# The maximum likelihood fit with log link
coalitionML < glm(duration ~ fract + numst2, family = Gamma, data = coalition)
# The mean biasreduced fit
coalitionBR_mean < update(coalitionML, method = "brglmFit")
# The biascorrected fit
coalitionBC < update(coalitionML, method = "brglmFit", type = "correction")
# The median biascorrected 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 componentwise median unbiased). The Wald tests for
## anorexBC and anorexBR_mean differ from anorexML
## because the biasreduced 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 biasreduced 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)

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