**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) and adjacent category models (ordinal 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 (2020) 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).

Install the current version from CRAN:

`install.packages("brglm2")`

or the development version from github:

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

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

for details and references).

```
library("brglm2")
data("endometrial", package = "brglm2")
modML <- glm(HG ~ NV + PI + EH, family = binomial("probit"), data = endometrial)
summary(modML)
#>
#> Call:
#> glm(formula = HG ~ NV + PI + EH, family = binomial("probit"),
#> data = endometrial)
#>
#> Deviance Residuals:
#> Min 1Q Median 3Q Max
#> -1.47007 -0.67917 -0.32978 0.00008 2.74898
#>
#> Coefficients:
#> Estimate Std. Error z value Pr(>|z|)
#> (Intercept) 2.18093 0.85732 2.544 0.010963 *
#> NV 5.80468 402.23641 0.014 0.988486
#> PI -0.01886 0.02360 -0.799 0.424066
#> EH -1.52576 0.43308 -3.523 0.000427 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> (Dispersion parameter for binomial family taken to be 1)
#>
#> Null deviance: 104.90 on 78 degrees of freedom
#> Residual deviance: 56.47 on 75 degrees of freedom
#> AIC: 64.47
#>
#> 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")
library("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 0.014 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("probit"),
#> data = endometrial, method = "brglm_fit")
#>
#> Deviance Residuals:
#> Min 1Q Median 3Q Max
#> -1.4436 -0.7016 -0.3783 0.3146 2.6218
#>
#> Coefficients:
#> Estimate Std. Error z value Pr(>|z|)
#> (Intercept) 1.91460 0.78877 2.427 0.015210 *
#> NV 1.65892 0.74730 2.220 0.026427 *
#> PI -0.01520 0.02089 -0.728 0.466793
#> EH -1.37988 0.40329 -3.422 0.000623 ***
#> ---
#> 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: 57.587 on 75 degrees of freedom
#> AIC: 65.587
#>
#> Number of Fisher Scoring iterations: 4
```

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.

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).

**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” and can be watched online at this link.

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.