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README.Rmd
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---
output: github_document
---
<!-- README.md is generated from README.Rmd. Please edit that file -->
```{r, include = FALSE}
knitr::opts_chunk$set(
collapse = TRUE,
comment = "#>",
fig.path = "man/figures/README-",
out.width = "100%"
)
```
# brglm2 <img src="man/figures/hex_brglm2.svg" width="320" align="right">
<!-- badges: start -->
[](https://cran.r-project.org/package=brglm2)
[](https://github.com/ikosmidis/brglm2/actions/workflows/R-CMD-check.yaml)
[](https://app.codecov.io/gh/ikosmidis/brglm2?branch=master)
[](https://www.gnu.org/licenses/gpl-3.0.en.html)
[](https://contributor-covenant.org/version/2/1/CODE_OF_CONDUCT.html)
<!-- badges: end -->
[**brglm2**](https://github.com/ikosmidis/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)](https://doi.org/10.1093/biomet/80.1.27) and [Kosmidis & Firth
(2009)](https://doi.org/10.1093/biomet/asp055) or the median-bias
reducing adjusted score equations in [Kenne et al
(2017)](https://doi.org/10.1093/biomet/asx046), 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)](https://www.jstor.org/stable/2345592). [Kosmidis et al
(2020)](https://doi.org/10.1007/s11222-019-09860-6) 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)](https://doi.org/10.1093/biomet/asaa052) 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.
## Installation
Install the current version from CRAN:
```r
install.packages("brglm2")
```
or the development version from github:
```r
# install.packages("remotes")
remotes::install_github("ikosmidis/brglm2", ref = "develop")
```
## Example
### Estimation of binomial-response GLMs with separated data
Below we follow the example of [Heinze and Schemper
(2002)](https://doi.org/10.1002/sim.1047) 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).
```{r, echo = TRUE, eval = TRUE}
library("brglm2")
data("endometrial", package = "brglm2")
modML <- glm(HG ~ NV + PI + EH, family = binomial("logit"), data = endometrial)
summary(modML)
```
The ML estimate of the parameter for `NV` is
actually infinite, as can be quickly verified using the
[**detectseparation**](https://cran.r-project.org/package=detectseparation) R package
```{r, echo = TRUE, eval = TRUE}
# install.packages("detectseparation")
library("detectseparation")
update(modML, method = "detect_separation")
```
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 `r round(coef(summary(modML))["NV", "z value"], 3)` 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.
```{r, echo = TRUE, eval = TRUE}
summary(update(modML, method = "brglm_fit"))
```
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
```{r, echo = TRUE, eval = TRUE}
expoRB <- expo(modML, type = "correction*")
expoRB
```
The odds ratio between presence of neovasculation and high histology
grade (`HG`) is estimated to be `r round(coef(expoRB)["NV"], 3)`, while
controlling for PI and EH. So, for each value of `PI` and `EH`, the
estimated odds of high histology grade are about `r round(coef(expoRB)["NV"], 1)` times higher when neovasculation is present. An approximate 95\% interval for the latter odds ratio is
`r paste0("(", paste(round(expoRB$ci["NV",], 1), collapse = ", "), ")")`
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 $+\infty$ with an infinite standard error (see previous section).
## Solving adjusted score equations using quasi-Fisher scoring
The workhorse function in **brglm2** is
[`brglm_fit`](https://github.com/ikosmidis/brglm2/blob/master/R/brglmFit.R)
(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](https://en.wikipedia.org/wiki/Scoring_algorithm) 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](https://cran.r-project.org/package=brglm2/vignettes/iteration.html)
and [Kosmidis et al
(2020)](https://doi.org/10.1007/s11222-019-09860-6) 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)](https://doi.org/10.1002/wics.1296).
## References and resources
**brglm2** was presented by [Ioannis
Kosmidis](https://www.ikosmidis.com) 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*](https://doi.org/10.1007/s11222-019-09860-6) *30*, 43–59.
## Code of Conduct
Please note that the **brglm2** project is released with a [Contributor Code of Conduct](https://contributor-covenant.org/version/2/1/CODE_OF_CONDUCT.html). By contributing to this project, you agree to abide by its terms.