# Analyzing County-Level COVID-19 Vaccination Rates in Texas: A New Lindley Regression Model

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. The Proposal Model

#### 2.2. Properties

#### 2.3. Quantile Function

#### 2.4. Moments

**Theorem 1.**

#### 2.5. Estimation

#### 2.6. Simulations

## 3. The GOLLL Regression Model

#### 3.1. Definition

#### 3.2. Simulations of the Regression Model

#### 3.3. Model Checking

## 4. Application

#### 4.1. COVID-19 Vaccination Rates on County-Level

- VR: Population rate with complete primary series of COVID-19 vaccination (response variable);
- HP: Total number of hospitals reporting vaccination;
- PR: Poverty rate (percentage of individuals with income below the poverty line);
- MS: Metropolitan status ($0=$ non-metropolitan, $1=$ metropolitan);
- HR: High school completion rate (proportion of individuals aged 25 and above who have completed high school or its equivalent);
- BA: Broadband access (percentage of households that have access to broadband internet);
- HT: Heart disease rate (percentage of individuals that have chronic heart disease).

#### 4.2. Results New Regression

#### 4.3. Diagnostic and Residual Analysis

- 83th: Gaines county with VR: $0.222$, HP: 1, PR: $0.142$, MS: 0, HR: $0.62$, BA: $0.80$ and HT: $0.063$;
- 151th: Loving county with VR: $0.189$, HP: 0, PR: $0.186$, MS: 0, HR: $0.97$, BA: $0.97$ and HT: $0.05$;
- 176th: Newton county with VR: $0.251$, HP: 0, PR: $0.206$, MS: 1, HR: $0.81$, BA: $0.75$ and HT: $0.105$.

## 5. Discussion

- All variables are statistically significant at a significance level of $5\%$;
- The HP variable shows a slight negative estimate, and this negative change is statistically significant;
- The PR variable is significant, and its estimate is negative. COVID-19 increased poverty and inequality worldwide [35,36]. Individuals living in poverty may lack access to reliable transportation, face barriers to accessing healthcare facilities, and have limited resources for paying out-of-pocket costs associated with vaccination [37,38]. The study of [39] revealed the lack of access to the COVID-19 vaccine in the lowest county’s poverty rates across the American state of Illinois. Other study [40] showed a strong negative correlation with poverty and vaccine coverage in the 189 countries’ research. This can result in lower vaccination rates among populations living in poverty, which is supported by data from the proposed model and prior studies;
- The MS variable has a negative estimate, which indicates that the vaccination rate is lower in metropolitan urban areas. The differences in vaccination rates between urban and rural communities are likely driven by various factors, such as differences in access to healthcare resources, vaccine distribution challenges, and mainly vaccine hesitancy [41]. Patterns in COVID-19 vaccination coverage by urbanity are addressed by [30]. It indicated lower vaccination rates in rural than urban areas, which corroborates with the study; Further, the study of [42] presented disparities in COVID-19 vaccination coverage between urban and rural counties and explained it by educational attainment, healthcare infrastructure, and Trump vote share.
- The HR variable is significant with a positive coefficient. Thus, counties with higher high school graduation rates tend to have higher vaccination rates as well, which can be attributed to more access to accurate information regarding vaccines to access better healthcare and vaccination services [43]. Other studies [44,45,46] revealed that high school is a key difference in coverage, access, and hesitancy vaccination;
- The BA variable has a positive estimate, which shows the internet has played a significant role in the COVID-19 vaccination effort. Websites and social media platforms have been used to disseminate information about vaccine availability, eligibility, and safety. The study’s results suggest that counties with greater access to broadcast media have a higher COVID-19 vaccination rate, which highlights the disparities in access to the internet and technology among some communities. This finding is consistent with the research presented in [47]. Alternative studies [48,49], showed that lack of internet access is a barrier to vaccination. In New York City and some counties in North Carolina, the COVID-19 vaccine hesitancy increases if there is difficulty accessing the internet;
- The HT variable has a highly positive estimate. Several studies [50,51,52] have demonstrated the heightened risk of individuals with chronic heart disease contracting and experiencing severe symptoms from COVID-19, as well as increased rates of hospitalization and mortality. For these reasons, many states in the US have implemented targeted outreach efforts to ensure that these populations have access to the vaccine. Hence, the study’s results indicate that counties with high rates of chronic heart disease have a correspondingly higher rate of vaccination. This finding highlights the importance of the government’s focus on prioritizing at-risk populations [53]. Subsequent studies [54,55], illustrated the efficacy and safety of the COVID-19 vaccine based on the presence of comorbidities, including heart disease.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

${\mathrm{A}}^{*}$ | Anderson Darling |

AE | average estimate |

AL | average estimate length |

cdf | cumulative distribution function |

COVID-19 | corona virus disease 2019 |

CP | coverage probability |

L | Lindley distribution |

EL | exponentiated Lindley distribution |

GCD | generalized Cook distance |

GL | gamma-Lindley distribution |

GOLL-G | generalized odd log-logistic distribution |

GOLLL | generalized odd log-logistic Lindley distribution |

KS | Kolmorogov-Sminorv |

KwE | Kumaraswamy Lindley distribution |

LD | loglikelihood distance |

LR | likelihood ratio |

MLE | maximum likelihood estimate |

MSE | mean squared error |

OLLL | odd log-logistic Lindley distribution |

probability distribution function | |

T-X | transformer-transformer generator |

${\mathrm{W}}^{*}$ | Cramér-von Misses |

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**Figure 1.**GOLLL pdf. (

**a**) Varying $\alpha $, fixed $\theta $ and $\lambda $ (

**b**) Varying $\theta $, fixed $\alpha $ and $\lambda $. (

**c**) Varying $\lambda $, fixed $\alpha $ and $\theta $.

**Figure 2.**GOLLL hrf (

**a**) Varying $\alpha $, fixed $\theta $ and $\lambda $ (

**b**) Varying $\theta $, fixed $\alpha $ and $\lambda $. (

**c**) Varying $\lambda $, fixed $\alpha $ and $\theta $.

**Figure 4.**Plots of the biases of the estimates. (

**a**) $\widehat{\alpha}$. (

**b**) $\widehat{\theta}$. (

**c**) ${\widehat{\beta}}_{0}$. (

**d**) ${\widehat{\beta}}_{1}$.

**Figure 5.**Plots of the MSEs of the estimates. (

**a**) $\widehat{\alpha}$. (

**b**) $\widehat{\theta}$. (

**c**) ${\widehat{\beta}}_{0}$. (

**d**) ${\widehat{\beta}}_{1}$.

**Figure 6.**Plots of the ALs of the estimates. (

**a**) $\widehat{\alpha}$. (

**b**) $\widehat{\theta}$. (

**c**) ${\widehat{\beta}}_{0}$. (

**d**) ${\widehat{\beta}}_{1}$.

**Figure 11.**(

**a**) Deviance residual plot (Circles—residuals and Lines—Bands of three standard deviations). (

**b**) Normal probability plot of ${r}_{D}$’s with envelope.

**Figure 12.**Profile log-likelihood functions from the fitted GOLLL regression model to COVID-19 data with $95\%$ confidence intervals. Parameters: (

**a**) ${\widehat{\beta}}_{0}$. (

**b**) ${\widehat{\beta}}_{1}$. (

**c**) ${\widehat{\beta}}_{2}$. (

**d**) ${\widehat{\beta}}_{3}$. (

**e**) ${\widehat{\beta}}_{4}$. (

**f**) ${\widehat{\beta}}_{5}$. (

**g**) ${\widehat{\beta}}_{6}$.

$\mathit{\alpha}$ | $\mathit{\theta}$ | Sub-Model |
---|---|---|

- | 1 | Generalized log-logistic family [14] |

1 | - | Proportional reversed hazard rate family [15] |

1 | 1 | Baseline |

Scenario 1 | |||||||||
---|---|---|---|---|---|---|---|---|---|

Par | n = 50 | n = 100 | n = 200 | ||||||

AE | Bias | MSE | AE | Bias | MSE | AE | Bias | MSE | |

$\alpha $ | 0.561 | 0.061 | 0.126 | 0.507 | 0.007 | 0.010 | 0.538 | 0.0038 | 0.061 |

$\theta $ | 1.019 | 0.269 | 0.588 | 0.778 | 0.028 | 0.035 | 0.883 | 0.133 | 0.238 |

$\lambda $ | 1.529 | 0.279 | 0.919 | 1.280 | 0.030 | 0.074 | 1.389 | 0.139 | 0.420 |

Par | n = 400 | n = 800 | n = 1000 | ||||||

AE | Bias | MSE | AE | Bias | MSE | AE | Bias | MSE | |

$\alpha $ | 0.515 | 0.015 | 0.025 | 0.560 | 0.060 | 0.212 | 0.511 | 0.011 | 0.013 |

$\theta $ | 0.822 | 0.072 | 0.099 | 1.280 | 0.530 | 1.467 | 0.778 | 0.028 | 0.043 |

$\lambda $ | 1.328 | 0.078 | 0.194 | 1.774 | 0.524 | 1.719 | 1.278 | 0.028 | 0.087 |

Scenario 2 | |||||||||

Par | n = 50 | n = 100 | n = 200 | ||||||

AE | Bias | MSE | AE | Bias | MSE | AE | Bias | MSE | |

$\alpha $ | 1.466 | 0.016 | 0.534 | 1.457 | 0.007 | 0.040 | 1.453 | 0.003 | 0.249 |

$\theta $ | 0.393 | 0.143 | 0.214 | 0.255 | 0.005 | 0.002 | 0.308 | 0.058 | 0.047 |

$\lambda $ | 1.694 | 0.744 | 4.627 | 0.981 | 0.031 | 0.066 | 1.282 | 0.332 | 1.280 |

Par | n = 400 | n = 800 | n = 1000 | ||||||

AE | Bias | MSE | AE | Bias | MSE | AE | Bias | MSE | |

$\alpha $ | 1.431 | −0.019 | 0.097 | 1.546 | 0.096 | 1.399 | 1.444 | −0.006 | 0.047 |

$\theta $ | 0.271 | 0.021 | 0.007 | 0.608 | 0.358 | 0.855 | 0.258 | 0.008 | 0.002 |

$\lambda $ | 1.087 | 0.137 | 0.251 | 2.556 | 1.606 | 13.543 | 1.007 | 0.057 | 0.085 |

Statistics | |||||||
---|---|---|---|---|---|---|---|

Variable | Mean | Median | SD | Skewness | Kurtosis | Min. | Max. |

VR | 0.483 | 0.452 | 0.132 | - 1.485 | 6.021 | 0.189 | 0.950 |

HP | 1.717 | 1.000 | 4.282 | - 6.666 | 56.447 | 0.000 | 45.000 |

PR | 0.161 | 0.152 | 0.061 | -1.022 | 4.878 | 0.026 | 0.395 |

HR | 0.818 | 0.830 | 0.085 | −2.056 | 12.509 | 0.220 | 0.970 |

BA | 0.769 | 0.770 | 0.084 | −0.388 | 3.301 | 0.480 | 0.970 |

HT | 0.082 | 0.082 | 0.017 | -0.248 | 2.906 | 0.045 | 0.134 |

Model | Parameters | ${W}^{*}$ | ${A}^{*}$ | KS | ||
---|---|---|---|---|---|---|

$\mathrm{GOLLL}\left(\alpha ,\theta ,\lambda \right)$ | 1.490 | 18.003 | 7.814 | 0.306 | 2.076 | 0.059 |

(0.0002) | (0.0001) | (0.0003) | (0.338) | |||

OLLL($\alpha ,\lambda $) | 4.985 | 1 | 2.084 | 0.444 | 2.774 | 0.067 |

(0.264) | (-) | (0.025) | (0.202) | |||

EL($\theta ,\lambda $) | 1 | 35.255 | 9.127 | 0.321 | 2.275 | 0.097 |

(-) | (5.861) | (0.400) | (0.015) | |||

L($\lambda $) | 1 | 1 | 2.640 | 0.702 | 4.480 | 0.450 |

(-) | (-) | (0.136) | (<0.0001) | |||

BL($a,b,\lambda $) | 30.015 | 1.810 | 7.245 | 0.397 | 2.696 | 0.079 |

(7.324) | (0.503) | (1.155) | (0.082) | |||

KwL($a,b,\lambda $) | 20.113 | 2.156 | 6.676 | 0.493 | 3.235 | 0.081 |

(4.372) | (0.514) | (0.764) | (0.069) | |||

GL($a,b,\lambda $) | 7.807 | 0.005 | 0.307 | 0.793 | 5.013 | 0.126 |

(0.751) | (<0.001) | (0.022) | (<0.0001) |

Models | Statistic w | p-Value |
---|---|---|

GOLLL vs. OLLL | 5.194 | <0.0227 |

GOLLL vs. EL | 15.127 | <0.0001 |

GOLLL vs. L | 498.805 | <0.0001 |

Parameter | Estimate | SE | p-Value |
---|---|---|---|

$\gamma $ | $\phantom{-}1.017$ | 0.347 | $\phantom{<}0.004$ |

HP | −0.010 | 0.003 | $\phantom{<}0.003$ |

PR | −0.524 | 0.243 | $\phantom{<}0.032$ |

MS | −0.149 | 0.031 | $<$0.001 |

HR | $\phantom{-}0.408$ | 0.197 | $\phantom{<}0.039$ |

BA | $\phantom{-}0.637$ | 0.223 | $\phantom{<}0.005$ |

HT | $\phantom{-}2.275$ | 0.995 | $\phantom{<}0.023$ |

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**MDPI and ACS Style**

da Costa, N.S.S.; de Lima, M.d.C.S.; Cordeiro, G.M.
Analyzing County-Level COVID-19 Vaccination Rates in Texas: A New Lindley Regression Model. *COVID* **2023**, *3*, 1761-1780.
https://doi.org/10.3390/covid3120122

**AMA Style**

da Costa NSS, de Lima MdCS, Cordeiro GM.
Analyzing County-Level COVID-19 Vaccination Rates in Texas: A New Lindley Regression Model. *COVID*. 2023; 3(12):1761-1780.
https://doi.org/10.3390/covid3120122

**Chicago/Turabian Style**

da Costa, Nicollas S. S., Maria do Carmo S. de Lima, and Gauss M. Cordeiro.
2023. "Analyzing County-Level COVID-19 Vaccination Rates in Texas: A New Lindley Regression Model" *COVID* 3, no. 12: 1761-1780.
https://doi.org/10.3390/covid3120122