Description

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.

Details

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.

Author(s)

Ioannis Kosmidis ioannis.kosmidis@warwick.ac.uk

References

Kosmidis I, Firth D (2020). Jeffreys-prior penalty, finiteness and shrinkage in binomial-response generalized linear models. *Biometrika* 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

See Also

brglm_fit, brmultinom, bracl