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A simple and useful regression model for fitting count data

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Abstract

We present a novel regression model for count data where the response variable is BerG-distributed using a new parameterization of this distribution, which is indexed by mean and dispersion parameters. An attractive feature of this model lies in its potential to fit count data when overdispersion, equidispersion, underdispersion, or zero inflation (or deflation) is indicated. The advantage of our new parameterization and approach is the straightforward interpretation of the regression coefficients in terms of the mean and dispersion as in generalized linear models. The maximum likelihood method is used to estimate the model parameters. Also, we conduct hypothesis tests for the dispersion parameter and consider residual analysis. Simulation studies are conducted to empirically evidence the properties of the estimators, the test statistics, and the residuals in finite-sized samples. The proposed model is applied to two real datasets on wildlife habitat and road traffic accidents, which illustrates its capabilities in accommodating both over- and underdispersed count data. This paper contains Supplementary Material.

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Authors and Affiliations

  1. Departamento de Estatística, Universidade Federal do Rio Grande do Norte, Natal, RN, 59078-970, Brazil

    Marcelo Bourguignon & Rodrigo M. R. de Medeiros

Authors
  1. Marcelo Bourguignon
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  2. Rodrigo M. R. de Medeiros
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Supplementary Information

Below is the link to the electronic supplementary material.

11749_2022_801_MOESM1_ESM.pdf

The Supplementary Material includes additional details and calculations of the score function, the Hessian matrix, and the Fisher information matrix of the BerG regression model. (151KB)

A. Chi-square Q–Q plots of the score, Wald, likelihood ratio, and gradient statistics in the second simulation study

A. Chi-square Q–Q plots of the score, Wald, likelihood ratio, and gradient statistics in the second simulation study

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Q–Q plots of the score statistic according to each sample size (row) and number of parameters tested (column)

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figure 13

Q–Q plots of the Wald statistic according to each sample size (row) and number of parameters tested (column)

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figure 14

Q–Q plots of the likelihood ratio statistic according to each sample size (row) and number of parameters tested (column), where the first column corresponds to \(q = 1\) and the second, \(q = 3\)

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Fig. 15
figure 15

Q–Q plots of the gradient statistic according to each sample size (row) and number of parameters tested (column), where the first column corresponds to \(q = 1\) and the second, \(q = 3\)

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Bourguignon, M., de Medeiros, R.M.R. A simple and useful regression model for fitting count data. TEST 31, 790–827 (2022). https://doi.org/10.1007/s11749-022-00801-6

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  • DOI: https://doi.org/10.1007/s11749-022-00801-6

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