Description Details Author(s) References See Also

Estimation and inference from 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.

In the special case of generalized linear models for binomial,
Poisson and multinomial responses (both nominal and ordinal), mean
and median bias reduction and maximum penalized likelihood 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 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). Estimation in all cases takes
place via a modified Fisher scoring algorithm, and S3 methods for
the construction of confidence intervals for the reduced-bias
estimates are provided.

The core model fitters are implemented by the functions
`brglm_fit`

(univariate generalized linear models),
`brmultinom`

(baseline category logit models for
nominal multinomial responses), and `bracl`

(adjacent
category logit models for ordinal multinomial responses).

The similarly named **brglm** R package can only handle generalized
linear models with binomial responses. Special care has been taken
when developing **brglm2** in order not to have conflicts when the
user loads **brglm2** and **brglm** simultaneously. The development
and maintenance of the two packages will continue in parallel,
until **brglm2** incorporates all **brglm** functionality and gets
an appropriate wrapper to the `brglm::brglm`

function.

Ioannis Kosmidis ioannis.kosmidis@warwick.ac.uk

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

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, 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

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

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