Advanced Search
Article Contents
Article Contents

A numerical method to compute Fisher information for a special case of heterogeneous negative binomial regression

  • * Corresponding author

    * Corresponding author 
The first author is partially supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. PolyU 25301115)
Abstract Full Text(HTML) Figure(1) / Table(1) Related Papers Cited by
  • Negative binomial regression has been widely applied in various research settings to account for counts with overdispersion. Yet, when the gamma scale parameter, $ \nu $, is parameterized, there is no direct algorithmic solution to the Fisher Information matrix of the associated heterogeneous negative binomial regression, which seriously limits its applications to a wide range of complex problems. In this research, we propose a numerical method to calculate the Fisher information of heterogeneous negative binomial regression and accordingly develop a preliminary framework for analyzing incomplete counts with overdispersion. This method is implemented in R and illustrated using an empirical example of teenage drug use in America.

    Mathematics Subject Classification: Primary: 62J12; Secondary: 49M15.


    \begin{equation} \\ \end{equation}
  • 加载中
  • Figure 1.  Time complexity m for achieving relative errors

    Table 1.  Heterogeneous negative-binomial regression analysis of lifetime marijuana use among American youth (Number of observations = 8,874). Data source: the 2012 wave of the Monitoring the Future study

    Coefficient Coefficient Z value 95% confidence interval
    Covariates for estimating µ
    Intercept 0.677*** 0.183 3.696 [0.318, 1.036]
    10th graders 1.551*** 0.153 10.145 [1.251, 1.850]
    12th graders 2.002*** 0.168 11.927 [1.673, 2.331]
    Male 1.268*** 0.125 10.143 [1.023, 1.513]
    African American -0.796*** 0.149 -5.361 [-1.087, -0.505]
    Metropolitan areas 0.148 0.150 0.983 [-0.147, 0.442]
    Covariates for estimating ν
    Intercept -3.627*** 0.082 -44.331 [-3.787, -3.466]
    10th graders 0.972*** 0.068 14.374 [0.839, 1.104]
    12th graders 1.332*** 0.074 18.018 [1.188, 1.477]
    Male -0.006 0.051 -0.107 [-0.106, 0.095]
    African American 0.268*** 0.077 3.480 [0.117, 0.418]
    Metropolitan areas 0.117 . 0.063 1.844 [-0.007, 0.240]
    Goodness of fit
    AIC 18400 BIC 18480
    McFadden’s R2 0.04828 McFadden’s adjusted R2 0.04703
    Note: ***p<0.001 ** p<0.01 * p<0.05 . P<0.1
     | Show Table
    DownLoad: CSV
  • [1] P. D. Allison and R. P. Waterman, Fixed–effects negative binomial regression models, Sociol. Methodol., 32 (2002), 247-265.  doi: 10.1111/1467-9531.00117.
    [2] B. M. BolkerM. E. BrooksC. J. ClarkS. W. GeangeJ. R. PoulsenM. H. H. Stevens and J. S. S. White, Generalized linear mixed models: a practical guide for ecology and evolution, Trends Ecol. Evol., 24 (2009), 127-135.  doi: 10.1016/j.tree.2008.10.008.
    [3] A. C. Cameron and  P. K. TrivediRegression analysis of count data, vol. 53, Cambridge University Press, 2013.  doi: 10.1017/CBO9781139013567.
    [4] A. C. Cameron and F. A. Windmeijer, R-squared measures for count data regression models with applications to health-care utilization, J. Busin. Econ. Statist., 14 (1996), 209-220. 
    [5] B. Efron and D. V. Hinkley, Assessing the accuracy of the maximum likelihood estimator: observed versus expected Fisher information, Biometrika, 65 (1978), 457-487.  doi: 10.1093/biomet/65.3.457.
    [6] S. Ehsan SaffariR. Adnan and W. Greene, Hurdle negative binomial regression model with right censored count data, SORT Statist. Oper. Res. Trans., 36 (2012), 0181-194. 
    [7] K. V. Finn, Patterns of alcohol and marijuana use at school, J. Res. Adol., 16 (2006), 69-77. 
    [8] R. A. Fisher, The negative binomial distribution, Ann. Eugen., 11 (1941), 182-187. 
    [9] Q. FuX. Guo and K. C. Land, A Poisson-multinomial mixture approach to grouped and right-censored counts, Commun. Statist. Theory Meth., 47 (2018), 427-447.  doi: 10.1080/03610926.2017.1303736.
    [10] Q. Fu, X. Guo and K. C. Land, Optimizing count responses in surveys: A machine-learning approach, Sociol. Meth. Res., (2018). doi: 10.1177/0049124117747302.
    [11] Q. FuK. C. Land and V. L. Lamb, Bullying victimization, socioeconomic status and behavioral characteristics of 12th graders in the united states, 1989 to 2009: Repetitive trends and persistent risk differentials, Child Indi. Res., 6 (2013), 1-21.  doi: 10.1007/s12187-012-9152-8.
    [12] Q. FuK. C. Land and V. L. Lamb, Violent physical bullying victimization at school: has there been a recent increase in exposure or intensity? an age-period-cohort analysis in the united states, 1991 to 2012, Child Indi. Res., 9 (2016), 485-513. 
    [13] Q. FuC. WuH. LiuZ. Shi and J. Gu, Live like mosquitoes: Hukou, rural–urban disparity, and depression, Chin. J. Sociol., 4 (2018), 56-78. 
    [14] W. H. Greene, Accounting for excess zeros and sample selection in Poisson and negative binomial regression models, in NYU working paper no. EC-94-10.
    [15] R. M. Groves, F. J. Fowler Jr, M. P. Couper, J. M. Lepkowski, E. Singer and R. Tourangeau, Survey Methodology, vol. 561, John Wiley & Sons, 2011.
    [16] J. M. HilbeNegative Binomial Regression, 2$^nd$ edition, Cambridge University Press, Cambridge, 2011.  doi: 10.1017/CBO9780511973420.
    [17] R. A. Horn and  C. R. JohnsonMatrix analysis, 2$^nd$ edition, Cambridge University Press, Cambridge, 2013. 
    [18] L. D. JohnstonP. M. O'Malley and J. G. Bachman, Bachman, Monitoring the Future: National results on adolescent drug use: Overview of key findings, Focus, 1 (2003), 213-234. 
    [19] L. D. Johnston, P. M. O'Malley, R. A. Miech, J. G. Bachman and J. E. Schulenberg, Monitoring the future national survey results on drug use, 1975–2016: Overview, key findings on adolescent drug use, 2017. Available from: https://files.eric.ed.gov/fulltext/ED578534.pdf.
    [20] L. D. Johnston, P. M. O'Malley, R. A. Miech, J. G. Bachman and J. E. Schulenberg, Monitoring the Future national survey results on drug use, 1975-2016: Overview, key findings on adolescent drug use, Inst. Social Res..
    [21] F. Kunstner, L. Balles and P. Hennig, Limitations of the empirical Fisher approximation, preprint, arXiv: 1905.12558.
    [22] K. C. LandP. L. McCall and D. S. Nagin, A comparison of Poisson, negative binomial, and semiparametric mixed Poisson regression models: With empirical applications to criminal careers data, Sociol. Meth. Res., 24 (1996), 387-442. 
    [23] E. L. Lehmann and G. Casella, Theory of Point Estimation, 2$^{nd}$ edition, Springer Texts in Statistics, Springer-Verlag, New York, 1998.
    [24] L. R. PacekR. J. Malcolm and S. S. Martins, Race/ethnicity differences between alcohol, marijuana, and co-occurring alcohol and marijuana use disorders and their association with public health and social problems using a national sample, Amer. Addi., 21 (2012), 435-444. 
    [25] W. W. Piegorsch, Maximum likelihood estimation for the negative binomial dispersion parameter, Biometrics, 46 (1990), 863-867.  doi: 10.2307/2532104.
  • 加载中




Article Metrics

HTML views(483) PDF downloads(333) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint