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

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    * 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)
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  • 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.


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  • 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
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