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brglm2 provides tools for the estimation and inference from generalized linear models using various methods for bias reduction. brglm2 supports all generalized linear models supported in R, and provides methods for multinomial logistic regression (nominal responses), adjacent category models (ordinal responses), and negative binomial regression (for potentially overdispered count responses).

Reduction of estimation bias is achieved by solving either the mean-bias reducing adjusted score equations in Firth (1993) and Kosmidis & Firth (2009) or the median-bias reducing adjusted score equations in Kenne et al (2017), or through the direct subtraction of an estimate of the bias of the maximum likelihood estimator from the maximum likelihood estimates as prescribed in Cordeiro and McCullagh (1991). Kosmidis et al (2020) provides a unifying framework and algorithms for mean and median bias reduction for the estimation of generalized linear models.

In the special case of generalized linear models for binomial and multinomial responses (both ordinal and nominal), the adjusted score equations 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). See, Kosmidis & Firth (2021) for the proof of the latter result in the case of mean bias reduction for logistic regression (and, for more general binomial-response models where the likelihood is penalized by a power of the Jeffreys’ invariant prior).

The core model fitters are implemented by the functions brglm_fit() (univariate generalized linear models), brmultinom() (baseline category logit models for nominal multinomial responses), bracl() (adjacent category logit models for ordinal multinomial responses), and brnb() for negative binomial regression.


Install the current version from CRAN:


or the development version from github:

# install.packages("remotes")
remotes::install_github("ikosmidis/brglm2", ref = "develop")


Estimation of binomial-response GLMs with separated data

Below we follow the example of Heinze and Schemper (2002) and fit a logistic regression model using maximum likelihood (ML) to analyze data from a study on endometrial cancer (see ?brglm2::endometrial for details and references).

data("endometrial", package = "brglm2")
modML <- glm(HG ~ NV + PI + EH, family = binomial("logit"), data = endometrial)
#> Call:
#> glm(formula = HG ~ NV + PI + EH, family = binomial("logit"), 
#>     data = endometrial)
#> Deviance Residuals: 
#>      Min        1Q    Median        3Q       Max  
#> -1.50137  -0.64108  -0.29432   0.00016   2.72777  
#> Coefficients:
#>               Estimate Std. Error z value Pr(>|z|)    
#> (Intercept)    4.30452    1.63730   2.629 0.008563 ** 
#> NV            18.18556 1715.75089   0.011 0.991543    
#> PI            -0.04218    0.04433  -0.952 0.341333    
#> EH            -2.90261    0.84555  -3.433 0.000597 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> (Dispersion parameter for binomial family taken to be 1)
#>     Null deviance: 104.903  on 78  degrees of freedom
#> Residual deviance:  55.393  on 75  degrees of freedom
#> AIC: 63.393
#> Number of Fisher Scoring iterations: 17

The ML estimate of the parameter for NV is actually infinite, as can be quickly verified using the detectseparation R package

# install.packages("detectseparation")
update(modML, method = "detect_separation")
#> Implementation: ROI | Solver: lpsolve 
#> Separation: TRUE 
#> Existence of maximum likelihood estimates
#> (Intercept)          NV          PI          EH 
#>           0         Inf           0           0 
#> 0: finite value, Inf: infinity, -Inf: -infinity

The reported, apparently finite estimate r round(coef(summary(modML))["NV", "Estimate"], 3) for NV is merely due to false convergence of the iterative estimation procedure for ML. The same is true for the estimated standard error, and, hence the value 0.011 for the z-statistic cannot be trusted for inference on the size of the effect for NV.

As mentioned earlier, many of the estimation methods implemented in brglm2 not only return estimates with improved frequentist properties (e.g. asymptotically smaller mean and median bias than what ML typically delivers), but also estimates and estimated standard errors that are always finite in binomial (e.g. logistic, probit, and complementary log-log regression) and multinomial regression models (e.g. baseline category logit models for nominal responses, and adjacent category logit models for ordinal responses). For example, the code chunk below refits the model on the endometrial cancer study data using mean bias reduction.

summary(update(modML, method = "brglm_fit"))
#> Call:
#> glm(formula = HG ~ NV + PI + EH, family = binomial("logit"), 
#>     data = endometrial, method = "brglm_fit")
#> Deviance Residuals: 
#>     Min       1Q   Median       3Q      Max  
#> -1.4740  -0.6706  -0.3411   0.3252   2.6123  
#> Coefficients:
#>             Estimate Std. Error z value Pr(>|z|)    
#> (Intercept)  3.77456    1.48869   2.535 0.011229 *  
#> NV           2.92927    1.55076   1.889 0.058902 .  
#> PI          -0.03475    0.03958  -0.878 0.379914    
#> EH          -2.60416    0.77602  -3.356 0.000791 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> (Dispersion parameter for binomial family taken to be 1)
#>     Null deviance: 104.903  on 78  degrees of freedom
#> Residual deviance:  56.575  on 75  degrees of freedom
#> AIC:  64.575
#> Type of estimator: AS_mixed (mixed bias-reducing adjusted score equations)
#> Number of Fisher Scoring iterations: 6

A quick comparison of the output from mean bias reduction to that from ML reveals a dramatic change in the z-statistic for NV, now that estimates and estimated standard errors are finite. In particular, the evidence against the null of NV not contributing to the model in the presence of the other covariates being now stronger.

See ?brglm_fit and ?brglm_control for more examples and the other estimation methods for generalized linear models, including median bias reduction and maximum penalized likelihood with Jeffreys’ prior penalty. Also do not forget to take a look at the vignettes (vignette(package = "brglm2")) for details and more case studies.

Improved estimation of the exponential of regression parameters

See, also ?expo for a method to estimate the exponential of regression parameters, such as odds ratios from logistic regression models, while controlling for other covariate information. Estimation can be performed using maximum likelihood or various estimators with smaller asymptotic mean and median bias, that are also guaranteed to be finite, even if the corresponding maximum likelihood estimates are infinite. For example, modML is a logistic regression fit, so the exponential of each coefficient is an odds ratio while controlling for other covariates. To estimate those odds ratios using the correction* method for mean bias reduction (see ?expo for details) we do

expoRB <- expo(modML, type = "correction*")
#> Call:
#> expo.glm(object = modML, type = "correction*")
#>              Estimate Std. Error     2.5 %  97.5 %
#> (Intercept) 20.671826  33.136511  0.893142 478.451
#> NV           8.496974   7.825240  1.397511  51.662
#> PI           0.965089   0.036795  0.895602   1.040
#> EH           0.056848   0.056344  0.008148   0.397
#> Type of estimator: correction* (explicit mean bias correction with a multiplicative adjustment)

The odds ratio between presence of neovasculation and high histology grade (HG) is estimated to be 8.497, while controlling for PI and EH. So, for each value of PI and EH, the estimated odds of high histology grade are about 8.5 times higher when neovasculation is present. An approximate 95% interval for the latter odds ratio is (1.4, 51.7) providing evidence of association between NV and HG while controlling for PI and EH. Note here that, the maximum likelihood estimate of the odds ratio is not as useful as the correction* estimate, because it is  + ∞ with an infinite standard error (see previous section).

Solving adjusted score equations using quasi-Fisher scoring

The workhorse function in brglm2 is brglm_fit (or equivalently brglmFit if you like camel case), which, as we did in the example above, can be passed directly to the method argument of the glm function. brglm_fit implements a quasi Fisher scoring procedure, whose special cases result in a range of explicit and implicit bias reduction methods for generalized linear models for more details). Bias reduction for multinomial logistic regression (nominal responses) can be performed using the function brmultinom, and for adjacent category models (ordinal responses) using the function bracl. Both brmultinom and bracl rely on brglm_fit.

The iteration vignette and Kosmidis et al (2020) present the iteration and give mathematical details for the bias-reducing adjustments to the score functions for generalized linear models.

The classification of bias reduction methods into explicit and implicit is as given in Kosmidis (2014).

References and resources

brglm2 was presented by Ioannis Kosmidis at the useR! 2016 international conference at University of Stanford on 16 June 2016. The presentation was titled “Reduced-bias inference in generalized linear models”.

Motivation, details and discussion on the methods that brglm2 implements are provided in

Kosmidis, I, Kenne Pagui, E C, Sartori N. (2020). Mean and median bias reduction in generalized linear models. Statistics and Computing 30, 43–59.

Code of Conduct

Please note that the brglm2 project is released with a Contributor Code of Conduct. By contributing to this project, you agree to abide by its terms.