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:
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
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
# 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
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.
?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.
Solving adjusted score equations using quasi-Fisher scoring
The workhorse function in brglm2 is
brglmFit if you like camel case), which, as we did in
the example above, can be passed directly to the
method argument of
brglm_fit implements a quasi Fisher
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 rely on
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” 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.