# Applying Benford’s Law to Monitor Death Registration Data: A Management Tool for the COVID-19 Pandemic

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Methods and Empirical Procedure

#### 2.1. Description of Benford’s Law

_{(n)}is the probability of a number having the first non-zero digit n.

#### 2.2. Chi-Square Test

^{2}(Chi-square) test. Through the χ

^{2}test we tested whether the n entries in a set of data were compatible with BL (Equation (2)). That is to say, we tested the null hypothesis for the first digit probabilities, ${p}_{i}={P}_{r}$ (${D}_{1}=i)$. Thus, we tested the hypothesis specified below [38].

_{0}) that the first digit is the same as expected on BL basis. Hence, the chi-square test points to those sets of numbers in which we must look into the possible causes of noncompliance with BL, and are those for which we can reject the H

_{0}.

#### 2.3. Sensitivity Analysis Steps

^{2}test, we designed a sensitivity analysis following the steps detailed below. As the observed figures are random, and that randomization depends on chance, we ran this sensitivity analysis to validate results.

- Step 1.
- First, the series of observed values were modified by random perturbations assuming that:
- (i)
- such a disturbance was unintentional;
- (ii)
- the applied perturbations were independent of each other;
- (iii)
- the perturbation size varied over a 20% range, and within that range any possible outcome was equally likely. This assumption implies consideration of the uniform probability distribution taking values within the interval [−0.1, +0.1]. Denoted as U [−0.1; +0.1].

- Step 2.
- From the observed mortality rate of a specific AC, an arbitrarily large set of alternative series with a generated perturbation was obtained through a Montecarlo simulation. Specifically, we generated 1000 replications for each series. Therefore, given the observed series $\left\{{x}_{1}^{obs},{x}_{2}^{obs},{x}_{3}^{obs}\dots {x}_{n}^{obs}\right\}$, we obtained the ith series modified as ${x}_{k}^{i}={x}_{k}^{obs}\xb7\left(1+{u}_{k}^{i}\right),k=1,\dots .,n$ where $i=$ 1, …, 1000, $\{{{u}_{k}^{i}\}}_{k=1}^{n}$ are n values obtained by simulation from the distribution U [−0.1; +0.1].
- Step 3.
- The BL test was applied to each series ${i}_{0},\{{u}_{k}^{{i}_{0}}{\}}_{k=1}^{n}$ generated synthetically, by calculating the statistics distance of χ
^{2}and the p-value test for that series. Then, we obtained 1000 synthetic series, with their 1000 p-values $\{{p}_{1}^{i},{p}_{2}^{i},\dots ,{p}_{n}^{i}{\}}_{i=1}^{1000}$ and their 1000 χ^{2}distances.

^{2}. In addition, we calculated quantiles of α-order for those p-values, ${q}_{\alpha}$.

- Step 4.
- From ${q}_{\alpha}$ it was possible to obtain the equivalent of a confidence interval that allowed validation of the decision of BL fulfillment with the observed data. That is to say, our goal was to check if the decision for observed data could be kept for data with perturbations. Then, we set a ${q}_{1-\alpha}$ value and took a decision according to the scheme displayed in Table 1:

## 3. Data and Source

## 4. Results

^{2}test) and the COVID mortality rate by ACs. Among those regions for which we reject the hypothesis that BL is fulfilled, two kinds of interpretation can be offered. The majority of the ACs for which we reject the H

_{0}are ranked at the top of the mortality rate ranking (above the Spanish rate). In these cases, the explanation for the deviation from BL relates to mistakes in the registration of the daily number of deaths, or an uncontrolled pandemic crisis providing skyrocketing figures. The region with the largest χ

^{2}value is Catalonia, that is to say, the one with the largest deviation from what was expected according to BL. In fact, Catalonia rectified up to approximately 20% of the data initially supplied to the Ministry, confirming that there had been errors in the registry or in the counting of cases.

^{2}test.

## 5. Discussion

## 6. Conclusions

_{0}are ranked at the top of the mortality rate ranking (above the Spanish rate). In these cases, the explanation for the deviation from BL relates to mistakes in the registration of the daily number of deaths (this is the case with Cataluña, Navarra and Madrid among others). These mistakes in data registration may be due to delays in information reporting (events on a specific day may be recorded afterwards), human error and differences in counting or recording criteria, among others. In fact, anomalous figures, such as the case of Catalonia, have been often reported in the press. In this AC, one of the main recording errors was the delay in reporting and recording information (sometimes attributing to a single day death cases from previous days) [44].

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Frequency distribution of the first digit of the number of deaths per day by COVID in Spain.

**Figure 3.**Frequency distribution of the first digit of the number of COVID deaths per day by AA.CC. (As in Figure 2, in this figure, the Y axes represent the frequency and X axes represent the first digit of the number of COVID deaths.).

Decision for Observed Data | If q_{0}_{:}_{95} > α for Data with Perturbations | If q_{0}_{:}_{95} < α for Data with Perturbations |
---|---|---|

H_{0} Fail to Reject | We can keep the decision | We cannot keep the decision |

H_{0} Reject | We cannot keep the decision | We can keep the decision |

ACs Code | Autonomous Communities (ACs) | χ^{2} ValueEstimator | χ^{2} Testp-Value | Mortality Rate | Mortality Rate Ranking |
---|---|---|---|---|---|

(×10^{5}) | |||||

9 | Cataluña | 291.947 | 0.000293 *** | 74.5 | 7 |

15 | Navarra | 217.510 | 0.005398 *** | 81.5 | 6 |

12 | Galicia | 214.966 | 0.005938 *** | 23.4 | 14 |

13 | Madrid | 195.582 | 0.012143 ** | 127.4 | 2 |

17 | La Rioja | 178.412 | 0.022448 ** | 116.2 | 4 |

7 | Castilla y León | 177.992 | 0.022782 ** | 117.2 | 3 |

11 | Extremadura | 169.880 | 0.030233 * | 49.2 | 10 |

8 | Castilla La Mancha | 164.449 | 0.036437 * | 143.4 | 1 |

2 | Aragón | 139.270 | 0.083687 * | 82.2 | 5 |

0 | Spain | 128.710 | 0.116364 | 60.9 | 9 |

1 | Andalucía | 118.706 | 0.157069 | 17.3 | 16 |

6 | Cantabria | 114.593 | 0.177005 | 36.0 | 11 |

10 | C. Valenciana | 98.378 | 0.276588 | 29.0 | 13 |

16 | País Vasco | 93.121 | 0.316654 | 70.9 | 8 |

4 | Baleares | 55.347 | 0.699181 | 19.8 | 15 |

3 | Asturias | 54.368 | 0.710025 | 32.8 | 12 |

14 | Murcia | 34.197 | 0.905324 | 10.0 | 17 |

_{0}at levels * 5%,** 3%, *** 1%.

AC | Initial Decision (for Observed Data) | q_{95%} | Final Decision (for Data with Perturbations) |
---|---|---|---|

Cataluña | Rejection | 0.00013 | Rejection |

Navarra | Rejection | 0.00610 | Rejection |

Madrid | Rejection | 0.00058 | Rejection |

La Rioja | Rejection | 0.00633 | Rejection |

Galicia | Rejection | 0.00583 | Rejection |

Castilla León | Rejection | 0.04743 | Rejection |

Spain | Fail to reject | 0.67138 | Fail to reject |

C. Valenciana | Fail to reject | 0.34756 | Fail to reject |

Andalucía | Fail to reject | 0.47747 | Fail to reject |

Cantabria | Fail to reject | 0.28642 | Fail to reject |

Baleares | Fail to reject | 0.33385 | Fail to reject |

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**MDPI and ACS Style**

Morillas-Jurado, F.G.; Caballer-Tarazona, M.; Caballer-Tarazona, V.
Applying Benford’s Law to Monitor Death Registration Data: A Management Tool for the COVID-19 Pandemic. *Mathematics* **2022**, *10*, 46.
https://doi.org/10.3390/math10010046

**AMA Style**

Morillas-Jurado FG, Caballer-Tarazona M, Caballer-Tarazona V.
Applying Benford’s Law to Monitor Death Registration Data: A Management Tool for the COVID-19 Pandemic. *Mathematics*. 2022; 10(1):46.
https://doi.org/10.3390/math10010046

**Chicago/Turabian Style**

Morillas-Jurado, Francisco Gabriel, María Caballer-Tarazona, and Vicent Caballer-Tarazona.
2022. "Applying Benford’s Law to Monitor Death Registration Data: A Management Tool for the COVID-19 Pandemic" *Mathematics* 10, no. 1: 46.
https://doi.org/10.3390/math10010046