# Negative binomial regression using brglm2

#### Euloge Clovis Kenne Pagui, Ioannis Kosmidis

#### 12 June 2021

# brnb

The brglm2 R package provides the `brnb()` function for fitting negative binomial regression models (see Agresti (2015), 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 Pagui, Salvan, and Sartori (2020).

# Ames salmonella data

Margolin, Kim, and Risko (1989) 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 Margolin, Kim, and Risko (1989, 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))
    #>    freq dose plate
    #> 1    15    0     1
    #> 2    16   10     1
    #> 3    16   33     1
    #> 4    27  100     1
    #> 5    33  333     1
    #> 6    20 1000     1
    #> 7    21    0     2
    #> 8    18   10     2
    #> 9    26   33     2
    #> 10   41  100     2
    #> 11   38  333     2
    #> 12   27 1000     2
    #> 13   29    0     3
    #> 14   21   10     3
    #> 15   33   33     3
    #> 16   60  100     3
    #> 17   41  333     3
    #> 18   42 1000     3

The following code chunks reproduces Kenne Pagui, Salvan, and Sartori (2020, 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)
    #>                          est    sds
    #> (Intercept)           2.1976 0.3246
    #> dose                 -0.0010 0.0004
    #> log(dose + 10)        0.3125 0.0879
    #> identity(dispersion)  0.0488 0.0281

## 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)
    #>                          est    sds
    #> (Intercept)           2.2098 0.3482
    #> dose                 -0.0010 0.0004
    #> log(dose + 10)        0.3105 0.0947
    #> identity(dispersion)  0.0626 0.0328

### 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)
    #>                          est    sds
    #> (Intercept)           2.2155 0.3515
    #> dose                 -0.0010 0.0004
    #> log(dose + 10)        0.3092 0.0956
    #> identity(dispersion)  0.0647 0.0334

## 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)
    #>                          est    sds
    #> (Intercept)           2.2114 0.3592
    #> dose                 -0.0010 0.0004
    #> log(dose + 10)        0.3091 0.0978
    #> identity(dispersion)  0.0692 0.0350

## Mixed bias reducing adjusted score equations

As is done in Kosmidis, Kenne Pagui, and Sartori (2020, 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)
    #>                          est    sds
    #> (Intercept)           2.2170 0.3591
    #> dose                 -0.0010 0.0004
    #> log(dose + 10)        0.3088 0.0978
    #> identity(dispersion)  0.0693 0.0350

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 Pagui, Salvan, and Sartori (2020, 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

Agresti, A. 2015. Foundations of Linear and Generalized Linear Models. Wiley Series in Probability and Statistics. Wiley.
Kenne Pagui, E. C., A. Salvan, and N. Sartori. 2020. “Accurate Inference in Negative Binomial Regression.” Eprint arXiv:2011.02784. https://arxiv.org/abs/2011.02784.
Kosmidis, Ioannis, Euloge Clovis Kenne Pagui, and Nicola Sartori. 2020. “Mean and Median Bias Reduction in Generalized Linear Models.” Statistics and Computing 30: 43–59. https://doi.org/10.1007/s11222-019-09860-6.
Margolin, Barry H., Byung Soo Kim, and Kenneth J. Risko. 1989. “The Ames Salmonella/Microsome Mutagenicity Assay: Issues of Inference and Validation.” Journal of the American Statistical Association 84 (407): 651–61. https://doi.org/10.1080/01621459.1989.10478817.