Negative binomial regression using **brglm2**

In brglm2: Bias Reduction in Generalized Linear Models

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brnb

The brglm2 R package provides the brnb() function for fitting negative binomial regression models (see @agresti:15, Section 7.3, for a recent account on negative binomial regression models) using either maximum likelihood or any of the various bias reduction and adjusted estimating functions methods provided by brglmFit() (see ?brglmFit for resources).

This vignette demonstrates the use of brnb() and of the associated methods, using the case studies in @kenne:20.

Ames salmonella data

@magolin:89 provide data from an Ames salmonella reverse mutagenicity assay. The response variable corresponds to the number of revertant colonies observed (freq) on each of three replicate plates (plate), and the covariate (dose) is the dose level of quinoline on the plate in micro-grams. The code chunk below sets up a data frame with the data from replicate 1 in @magolin:89[, Table 1].

freq <- c(15, 16, 16, 27, 33, 20,
          21, 18, 26, 41, 38, 27,
          29, 21, 33, 60, 41, 42)
dose <- rep(c(0, 10, 33, 100, 333, 1000), 3)
plate <- rep(1:3, each = 6)
(salmonella <- data.frame(freq, dose, plate))

The following code chunks reproduces @kenne:20[, Table 2] by estimating the negative binomial regression model with log link and model formula

ames_f <- freq ~ dose + log(dose + 10)

using the various estimation methods that brnb() supports.

Maximum likelihood estimation

library("brglm2")
ames_ML <- brnb(ames_f, link = "log", data = salmonella,
                transformation = "identity",  type = "ML")
## Estimated regression and dispersion parameters
est <- coef(ames_ML, model = "full")
## Estimated standard errors for the regression parameters
sds <- sqrt(c(diag(ames_ML$vcov.mean), ames_ML$vcov.dispersion))
round(cbind(est, sds), 4)

Bias reduction

Asymptotic mean-bias correction

The following code chunks updates the model fit using asymptotic mean-bias correction for estimating the model parameters

ames_BC <- update(ames_ML, type = "correction")
## Estimated regression and dispersion parameters
est <- coef(ames_BC, model = "full")
## Estimated standard errors for the regression parameters
sds <- sqrt(c(diag(ames_BC$vcov.mean), ames_BC$vcov.dispersion))
round(cbind(est, sds), 4)

Mean-bias reducing adjusted score equations

The corresponding fit using mean-bias reducing adjusted score equations is

ames_BRmean <- update(ames_ML, type = "AS_mean")
## Estimated regression and dispersion parameters
est <- coef(ames_BRmean, model = "full")
## Estimated standard errors for the regression parameters
sds <- sqrt(c(diag(ames_BRmean$vcov.mean), ames_BRmean$vcov.dispersion))
round(cbind(est, sds), 4)

Median-bias reducing adjusted score equations

The corresponding fit using median-bias reducing adjusted score equations is

ames_BRmedian <- update(ames_ML, type = "AS_median")
## Estimated regression and dispersion parameters
est <- coef(ames_BRmedian, model = "full")
## Estimated standard errors for the regression parameters
sds <- sqrt(c(diag(ames_BRmedian$vcov.mean), ames_BRmedian$vcov.dispersion))
round(cbind(est, sds), 4)

Mixed bias reducing adjusted score equations

As is done in @kosmidis:2019[, Section 4] for generalized linear models, we can exploit the Fisher orthogonality of the regression parameters and the dispersion parameter and use a composite bias reduction adjustment to the score functions. Such an adjustment delivers mean-bias reduced estimates for the regression parameters and a median-bias reduced estimate for the dispersion parameter. The resulting estimates of the regression parameters are invariant in terms of their mean bias properties under arbitrary contrasts, and that of the dispersion parameter is invariant in terms of its median bias properties under monotone transformations.

Fitting the model using mixed-bias reducing adjusted score equations gives

ames_BRmixed <- update(ames_ML, type = "AS_mixed")
## Estimated regression and dispersion parameters
est <- coef(ames_BRmixed, model = "full")
## Estimated standard errors for the regression parameters
sds <- sqrt(c(diag(ames_BRmixed$vcov.mean), ames_BRmixed$vcov.dispersion))
round(cbind(est, sds), 4)

The differences between reduced-bias estimation and maximum likelihood are particularly pronounced for the dispersion parameter. Improved estimation of the dispersion parameter results to larger estimated standard errors than maximum likelihood. Hence, the estimated standard errors based on the maximum likelihood estimates appear to be smaller than they should be, which is also supported by the simulation results in @kenne:20[, Section 5].

Relevant resources

?brglmFit and ?brglm_control contain quick descriptions of the various bias reduction methods supported in brglm2. The iteration vignette describes the iteration and gives the mathematical details for the bias-reducing adjustments to the score functions for generalized linear models.

Citation

If you found this vignette or brglm2, in general, useful, please consider citing brglm2 and the associated paper. You can find information on how to do this by typing citation("brglm2").

References

brglm2 documentation built on Oct. 12, 2023, 1:07 a.m.