Comments on the Bernoulli Distribution and Hilbe’s Implicit Extra-Dispersion
For decades, conventional wisdom maintained that binary 0–1 Bernoulli random variables cannot contain extra-binomial variation. Taking an unorthodox stance, Hilbe actively disagreed, especially for correlated observation instances, arguing that the universally adopted diagnostic Pearson or deviance...
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| Format: | Article |
| Language: | English |
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MDPI AG
2024-03-01
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| Series: | Stats |
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| Online Access: | https://www.mdpi.com/2571-905X/7/1/16 |
| _version_ | 1827304992108707840 |
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| author | Daniel A. Griffith |
| author_facet | Daniel A. Griffith |
| author_sort | Daniel A. Griffith |
| collection | DOAJ |
| description | For decades, conventional wisdom maintained that binary 0–1 Bernoulli random variables cannot contain extra-binomial variation. Taking an unorthodox stance, Hilbe actively disagreed, especially for correlated observation instances, arguing that the universally adopted diagnostic Pearson or deviance dispersion statistics are insensitive to a variance anomaly in a binary context, and hence simply fail to detect it. However, having the intuition and insight to sense the existence of this departure from standard mathematical statistical theory, but being unable to effectively isolate it, he classified this particular over-/under-dispersion phenomenon as implicit. This paper explicitly exposes his hidden quantity by demonstrating that the variance in/deflation it represents occurs in an underlying predicted beta random variable whose real number values are rounded to their nearest integers to convert to a Bernoulli random variable, with this discretization masking any materialized extra-Bernoulli variation. In doing so, asymptotics linking the beta-binomial and Bernoulli distributions show another conventional wisdom misconception, namely a mislabeling substitution involving the quasi-Bernoulli random variable; this undeniably is not a quasi-likelihood situation. A public bell pepper disease dataset exhibiting conspicuous spatial autocorrelation furnishes empirical examples illustrating various features of this advocated proposition. |
| first_indexed | 2024-04-24T17:48:33Z |
| format | Article |
| id | doaj.art-9886b022de5d4c8d98263273fe7b7475 |
| institution | Directory Open Access Journal |
| issn | 2571-905X |
| language | English |
| last_indexed | 2024-04-24T17:48:33Z |
| publishDate | 2024-03-01 |
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| spelling | doaj.art-9886b022de5d4c8d98263273fe7b74752024-03-27T14:05:06ZengMDPI AGStats2571-905X2024-03-017126928310.3390/stats7010016Comments on the Bernoulli Distribution and Hilbe’s Implicit Extra-DispersionDaniel A. Griffith0School of Economic, Political and Policy Sciences, The University of Texas at Dallas, Richardson, TX 75080, USAFor decades, conventional wisdom maintained that binary 0–1 Bernoulli random variables cannot contain extra-binomial variation. Taking an unorthodox stance, Hilbe actively disagreed, especially for correlated observation instances, arguing that the universally adopted diagnostic Pearson or deviance dispersion statistics are insensitive to a variance anomaly in a binary context, and hence simply fail to detect it. However, having the intuition and insight to sense the existence of this departure from standard mathematical statistical theory, but being unable to effectively isolate it, he classified this particular over-/under-dispersion phenomenon as implicit. This paper explicitly exposes his hidden quantity by demonstrating that the variance in/deflation it represents occurs in an underlying predicted beta random variable whose real number values are rounded to their nearest integers to convert to a Bernoulli random variable, with this discretization masking any materialized extra-Bernoulli variation. In doing so, asymptotics linking the beta-binomial and Bernoulli distributions show another conventional wisdom misconception, namely a mislabeling substitution involving the quasi-Bernoulli random variable; this undeniably is not a quasi-likelihood situation. A public bell pepper disease dataset exhibiting conspicuous spatial autocorrelation furnishes empirical examples illustrating various features of this advocated proposition.https://www.mdpi.com/2571-905X/7/1/16Bernoullibetabeta-binomialHilbelogistic regressionspatial autocorrelation |
| spellingShingle | Daniel A. Griffith Comments on the Bernoulli Distribution and Hilbe’s Implicit Extra-Dispersion Stats Bernoulli beta beta-binomial Hilbe logistic regression spatial autocorrelation |
| title | Comments on the Bernoulli Distribution and Hilbe’s Implicit Extra-Dispersion |
| title_full | Comments on the Bernoulli Distribution and Hilbe’s Implicit Extra-Dispersion |
| title_fullStr | Comments on the Bernoulli Distribution and Hilbe’s Implicit Extra-Dispersion |
| title_full_unstemmed | Comments on the Bernoulli Distribution and Hilbe’s Implicit Extra-Dispersion |
| title_short | Comments on the Bernoulli Distribution and Hilbe’s Implicit Extra-Dispersion |
| title_sort | comments on the bernoulli distribution and hilbe s implicit extra dispersion |
| topic | Bernoulli beta beta-binomial Hilbe logistic regression spatial autocorrelation |
| url | https://www.mdpi.com/2571-905X/7/1/16 |
| work_keys_str_mv | AT danielagriffith commentsonthebernoullidistributionandhilbesimplicitextradispersion |