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

Xin Guo; Qiang Fu; Yue Wang; Kenneth C. Land

doi:10.3934/cpaa.2020187

Article Contents

2020, Volume 19, Issue 8: 4179-4189. Doi: 10.3934/cpaa.2020187

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

1.
Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, China
2.
Department of Sociology, The University of British Columbia, V6T 1Z1, Vancouver, BC, Canada
3.
Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, China
4.
Department of Sociology and Social Science Research Institute, Duke University, 27708, Durham, NC, USA

^* Corresponding author
^* Corresponding author

Received: August 31, 2019

Revised: October 31, 2019

Published: May 2020

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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Abstract

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.

Keywords:

Regression analysis,
incomplete counts,
overdispersion,
heterogeneous negative binomial regression,
Fisher information,
gamma scale parameter.

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

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Figure 1. Time complexity m for achieving relative errors

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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]
10^th graders	1.551^***	0.153	10.145	[1.251, 1.850]
12^th 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]
10^th graders	0.972^***	0.068	14.374	[0.839, 1.104]
12^th 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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