# Multivariate Tail Probabilities: Predicting Regional Pertussis Cases in Washington State

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## Abstract

**:**

## 1. Introduction

## 2. Density Ratio Model

## 3. Application: County-level Pertussis Cases in Washington State

#### 3.1. Univariate Analysis

#### 3.2. Multivariate Analysis

## 4. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Risk Factors of Pertussis Incidence

**Table A1.**County statistics and risk factors: Population Estimate, Average Household Size, Percent Hispanic, Pertussis Vaccine Coverage, Percent Population Below 5 years, Population Density, Rural/Urban, Socioeconomic Status (SES) as per 2017 estimates.

County | Population | Household | %Hispanic | %Vaccine | %Below5 | Density | Rural/Urban | SES |
---|---|---|---|---|---|---|---|---|

Grays Harbor | 72,490 | 2.43 | 9.8 | 80.7 | 5.5 | 14.78 | Mostly Rural | Mid |

Jefferson | 31,210 | 2.07 | 3.7 | 80.8 | 2.9 | 6.70 | Mostly Rural | Mid |

Clallam | 75,637 | 2.25 | 5.8 | 87.1 | 4.7 | 16.74 | Mostly Rural | Mid |

Clark | 474,381 | 2.69 | 8.7 | 84.7 | 6.2 | 290.74 | Semi-Urban | High |

Cowlitz | 106,805 | 2.52 | 8.4 | 94.1 | 6.2 | 36.13 | Semi-Urban | High |

Lewis | 78,320 | 2.52 | 9.7 | 91.5 | 5.9 | 12.56 | Mostly Rural | Mid |

King | 2,203,836 | 2.45 | 9.4 | 91.4 | 5.9 | 400.75 | Urban | Mid/High |

Snohomish | 802,089 | 2.68 | 9.7 | 90.7 | 6.4 | 147.82 | Semi-Urban | High |

Skagit | 125,860 | 2.55 | 17.8 | 90.4 | 6.1 | 28.03 | Semi-Urban | High |

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**Table 1.**Summary statistics of Jefferson, Cowlitz and Snohomish counties, WA. Q1 and Q3 are referred to 25th and 75th percentile respectively.

Statistics | Min. | Q1 | Median | Q3 | Max. | |
---|---|---|---|---|---|---|

County | ||||||

Jefferson | 0.00 | 0.00 | 1.00 | 6.50 | 30.00 | |

Cowlitz | 0.00 | 3.00 | 8.00 | 23.25 | 108.00 | |

Snohomish | 7.00 | 36.25 | 46.50 | 54.75 | 549.00 |

**Table 2.**Selected joint probability estimates non-obtainable from the empirical distribution and the corresponding 95% confidence intervals. Here, t represents the annual pertussis cases in Jefferson.

Probability | Estimate | 95% Confidence Interval |
---|---|---|

$P(t>30)$ | 0.0200 | (−0.0204, 0.0604) |

$P(t>40)$ | 0.0084 | (−0.0124, 0.0292) |

$P(t>50)$ | 0.0021 | (−0.0041, 0.0083) |

**Table 3.**Summary statistics of each county used in the multivariate analysis. Q1 and Q3 are referred to 25th and 75th percentile respectively.

Statistics | Min. | Q1 | Median | Q3 | Max. | |
---|---|---|---|---|---|---|

County | ||||||

Grays Harbor | 0.00 | 1.00 | 2.50 | 4.75 | 24.00 | |

Jefferson | 0.00 | 0.00 | 1.00 | 6.50 | 30.00 | |

Clallam | 0.00 | 1.00 | 2.00 | 4.75 | 25.00 | |

Clark | 3.00 | 20.25 | 33.50 | 85.00 | 326.00 | |

Cowlitz | 0.00 | 3.00 | 8.00 | 23.25 | 108.00 | |

Lewis | 0.00 | 2.00 | 5.00 | 10.75 | 71.00 | |

King | 38.00 | 115.00 | 141.00 | 194.25 | 785.00 | |

Snohomish | 7.00 | 36.25 | 46.50 | 54.75 | 549.00 | |

Skagit | 1.00 | 5.00 | 9.00 | 17.75 | 559.00 |

**Table 4.**AIC values for different choices of ${\mathit{h}}_{1}$ and ${\mathit{h}}_{2}$. A hyphen “-” indicates that ${\mathit{h}}_{k}\left(\mathit{x}\right)\equiv \mathit{0}$ and therefore ${g}_{0}$ and ${g}_{k}$ are identical for $k=1,2$.

${\mathit{h}}_{1}$ | - | ${\mathit{x}}_{1}$ | ${\mathit{x}}_{2}$ | ${\mathit{x}}_{3}$ | $({\mathit{x}}_{1},{\mathit{x}}_{2})$ | $({\mathit{x}}_{1},{\mathit{x}}_{3})$ | $({\mathit{x}}_{2},{\mathit{x}}_{3})$ | $({\mathit{x}}_{1},{\mathit{x}}_{3},{\mathit{x}}_{3})$ | ||
---|---|---|---|---|---|---|---|---|---|---|

AIC | ||||||||||

${\mathit{h}}_{2}$ | ||||||||||

- | 553.03 | 554.32 | 552.37 | 554.39 | 554.19 | 556.22 | 554.22 | 556.11 | ||

${x}_{1}$ | 527.36 | 487.62 | 529.32 | 526.98 | 483.92 | 489.53 | 528.98 | 485.89 | ||

${x}_{2}$ | 525.03 | 524.09 | 516.98 | 525.37 | 518.98 | 525.56 | 518.94 | 520.94 | ||

${x}_{3}$ | 549.19 | 551.19 | 549.92 | 547.88 | 551.36 | 549.45 | 547.50 | 549.45 | ||

$({x}_{1},{x}_{2})$ | 523.36 | 485.04 | 515.77 | 522.57 | 485.22 | 487.03 | 517.24 | 487.17 | ||

$({x}_{1},{x}_{3})$ | 558.58 | 489.07 | 530.52 | 528.38 | 485.37 | 486.05 | 530.36 | 483.22 | ||

$({x}_{2},{x}_{3})$ | 527.03 | 526.08 | 518.97 | 526.34 | 520.97 | 526.85 | 520.93 | 522.92 | ||

$({x}_{1},{x}_{2},{x}_{3})$ | 524.91 | 486.51 | 517.25 | 524.33 | 486.71 | 483.32 | 519.19 | 485.22 |

**Table 5.**Selected joint probability estimates non-obtainable from the empirical distribution and the corresponding 95% confidence intervals. Here, $({t}_{1},{t}_{2},{t}_{3})$ represents the number of annual pertussis cases in (Grays Harbor, Jefferson, Clallam) respectively.

Probability | Estimate | 95% Confidence Interval |
---|---|---|

$P({t}_{1}>20,{t}_{2}\le 10,{t}_{3}\le 10)$ | $5.6511\times {10}^{-3}$ | $(-2.5641\times {10}^{-2}$, $3.6943\times {10}^{-2})$ |

$P({t}_{1}>10,{t}_{2}>10,{t}_{3}\le 10)$ | $8.1231\times {10}^{-3}$ | $(-2.9312\times {10}^{-2}$, $4.5558\times {10}^{-2})$ |

$P({t}_{1}>15,{t}_{2}>15,{t}_{3}\le 15)$ | $2.6609\times {10}^{-3}$ | $(-1.8843\times {10}^{-2}$, $2.4166\times {10}^{-2})$ |

$P({t}_{1}>25,{t}_{2}>20,{t}_{3}>10)$ | $2.6517\times {10}^{-7}$ | ($-1.8969\times {10}^{-4}$, $1.9010\times {10}^{-4}$) |

$P({t}_{1}>15,{t}_{2}>30,{t}_{3}>10)$ | $3.7789\times {10}^{-9}$ | ($-2.5683\times {10}^{-5}$, $2.5691\times {10}^{-5}$) |

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**MDPI and ACS Style**

Zhang, X.; Pyne, S.; Kedem, B.
Multivariate Tail Probabilities: Predicting Regional Pertussis Cases in Washington State. *Entropy* **2021**, *23*, 675.
https://doi.org/10.3390/e23060675

**AMA Style**

Zhang X, Pyne S, Kedem B.
Multivariate Tail Probabilities: Predicting Regional Pertussis Cases in Washington State. *Entropy*. 2021; 23(6):675.
https://doi.org/10.3390/e23060675

**Chicago/Turabian Style**

Zhang, Xuze, Saumyadipta Pyne, and Benjamin Kedem.
2021. "Multivariate Tail Probabilities: Predicting Regional Pertussis Cases in Washington State" *Entropy* 23, no. 6: 675.
https://doi.org/10.3390/e23060675