brglmFit | R Documentation |
glm()
for reduced-bias estimation and
inferencebrglmFit()
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 |
a design matrix of dimension |
y |
a vector of observations of length |
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 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 quasi-complete separation
in multinomial regression). See, Kosmidis and Firth (2021) for a
proof for binomial-response GLMs with Jeffreys-prior penalties to
the log-likelihood, which is equivalent to mean bias reduction for
logistic regression. See, also,
detectseparation::detect_separation()
and
detectseparation::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.
The description of method
argument and the Fitting functions
section in glm()
gives information on supplying fitting
methods to glm()
.
fixed_totals
specifies 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 (2021). Jeffreys-prior penalty, finiteness and shrinkage in binomial-response generalized linear models. Biometrika, 108, 71-82. 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.
brglmControl()
, glm.fit()
, glm()
## 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 (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**, 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)
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