# Using Random Effect Models to Produce Robust Estimates of Death Rates in COVID-19 Data

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

## 3. Technical Details

#### 3.1. Choice of Lag

#### 3.2. Choice of Number of Mass Points

#### 3.3. Clustering and MAP Rule

## 4. Results

#### 4.1. Robust Rates for All Countries

#### 4.2. Finding Clusters of Countries

#### 4.3. Case Study 1: San Marino Data in 2021

#### 4.4. Case Study 2: Saudi Arabia Data in 2021

## 5. Discussion

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

BIC | Bayesian Information Criterion |

EM | Expectation-Maximization |

MAP | Maximum a posteriori |

NaN | Not a Number |

NPML | Nonparametric maximum likelihood |

## References

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**Figure 1.**

**Top**: daily COVID-19 cases (

**left**) and deaths (

**right**) in San Marino in 2021;

**middle**: fitted daily COVID-19 rates (

**left**) and death rates (

**right**) in San Marino in 2021;

**bottom**: fitted versus raw case rates (

**left**) and death rates (

**right**) in San Marino in 2021.

**Figure 2.**

**Top**: daily COVID-19 cases (

**left**) and deaths (

**right**) in Saudi Arabia in 2021;

**middle**: fitted daily COVID-19 rates (

**left**) and death rates (

**right**) in Saudi Arabia in 2021;

**bottom**: fitted versus raw case rates (

**left**) and death rates (

**right**) in Saudi Arabia in 2021.

**Table 1.**Lag from onset of symptoms to death for COVID-19 patients and the total numbers of deaths in each study.

Study ID | Region | Sample Size | Lag (Days) | Deaths (Count) |
---|---|---|---|---|

Zhou et al. (2020) [17] | China, Wuhan | 191 | 18.5 | 53 |

Ruan et al. (2020) [18] | China, Wuhan | 150 | 18.0 | 68 |

Jin et al. (2020) [19] | China, Wuhan | 1056 | 13.0 | 37 |

Chen et al. (2020) [20] | China, Wuhan | 50 | 13.0 | 50 |

Verity et al.(2020) [1] | China, mainland | 3665 | 17.8 | NA |

Harrison et al. (2020) [21] | UK | 7802 | 7 | 7802 |

Harrison et al. (2021) [22] | UK | 1026 | 21 | 236 |

Faes et al. (2020) [23] | Belgium | 14,618 | 9 | 1534 |

Hawryluk et al. (2020) [24] | Brazil | 1,557,000 | 15.2 | NA |

Marschner (2021) [25] | Australia | 6235 | 18.1 | 816 |

Lefrancq et al. (2021) [26] | France | 198,846 | 19.0 | 33,269 |

Asirvatham et al. (2021) [27] | India | 1761 | 4.0 | 1710 |

Mehta et al. (2021) [28] | India | 346 | 9.0 | 76 |

Location | Population | Cases | Fitted | Raw | Fitted | Deaths | Fitted | Raw | Fitted |
---|---|---|---|---|---|---|---|---|---|

Cases | Case Rate | Case Rate | Deaths | Death Rate | Death Rate | ||||

Afghanistan | 40,099,462 | 83 | 91.839 | 0.0000021 | 0.0000023 | 1 | 0.359 | 0.0120482 | 0.0039064 |

Albania | 2,854,710 | 219 | 229.240 | 0.0000767 | 0.0000803 | 0 | 0.251 | 0.0000000 | 0.0010937 |

Algeria | 44,177,969 | 8 | 2.766 | 0.0000002 | 0.0000001 | 0 | 0.005 | 0.0000000 | 0.0019397 |

Andorra | 79,034 | 0 | 0.013 | 0.0000000 | 0.0000002 | 0 | 0.000 | NaN | 0.0019828 |

Angola | 34,503,774 | 0 | 2.160 | 0.0000000 | 0.0000001 | 0 | 0.004 | NaN | 0.0019489 |

Anguilla | 15,753 | 0 | 0.005 | 0.0000000 | 0.0000003 | 0 | 0.000 | NaN | 0.0019829 |

Antigua/Barbuda | 93,220 | 0 | 0.014 | 0.0000000 | 0.0000002 | 0 | 0.000 | NaN | 0.0019828 |

Argentina | 45,276,780 | 0 | 2.835 | 0.0000000 | 0.0000001 | 0 | 0.005 | NaN | 0.0019386 |

Armenia | 2,790,974 | 0 | 0.183 | 0.0000000 | 0.0000001 | 0 | 0.000 | NaN | 0.0019801 |

Aruba | 106,536 | 0 | 0.016 | 0.0000000 | 0.0000001 | 0 | 0.000 | NaN | 0.0019828 |

Australia | 25,921,089 | 32,895 | 36,211.746 | 0.0012690 | 0.0013970 | 62 | 75.118 | 0.0018848 | 0.0020744 |

Austria | 8,922,082 | 5233 | 5305.067 | 0.0005865 | 0.0005946 | 5 | 4.154 | 0.0009555 | 0.0007830 |

Azerbaijan | 10,312,992 | 21 | 23.619 | 0.0000020 | 0.0000023 | 0 | 0.040 | 0.0000000 | 0.0016807 |

Bahamas | 407,906 | 34 | 33.705 | 0.0000834 | 0.0000826 | 0 | 0.054 | 0.0000000 | 0.0015891 |

Bahrain | 1,463,265 | 2078 | 2044.180 | 0.0014201 | 0.0013970 | 0 | 1.553 | 0.0000000 | 0.0007595 |

Bangladesh | 169,356,251 | 874 | 973.659 | 0.0000052 | 0.0000057 | 2 | 1.425 | 0.0022883 | 0.0014636 |

Barbados | 281,200 | 145 | 146.826 | 0.0005156 | 0.0005221 | 0 | 0.173 | 0.0000000 | 0.0011807 |

Belarus | 9,578,168 | 0 | 0.600 | 0.0000000 | 0.0000001 | 0 | 0.001 | NaN | 0.0019734 |

Belgium | 11,611,420 | 0 | 0.727 | 0.0000000 | 0.0000001 | 0 | 0.001 | NaN | 0.0019714 |

Belize | 400,031 | 277 | 238.169 | 0.0006924 | 0.0005954 | 1 | 0.433 | 0.0036101 | 0.0018169 |

**Table 3.**Raw and fitted case and death rates, on Tuesday, 21 June 2022, relative to cases 14 days earlier (i.e., Tuesday 7th of June). Population and death counts omitted since they are the same as in Table 2.

Location | Cases | Fitted Cases | Raw CASE Rate | Fitted Case Rate | Fitted Deaths | Raw Death Rate | Fitted Death Rate |
---|---|---|---|---|---|---|---|

Afghanistan | 53 | 65.833 | 0.0000013 | 0.0000016 | 0.439 | 0.0120482 | 0.0066713 |

Albania | 53 | 54.046 | 0.0000186 | 0.0000189 | 0.070 | 0.0000000 | 0.0012918 |

Algeria | 4 | 3.914 | 0.0000001 | 0.0000001 | 0.007 | 0.0000000 | 0.0018068 |

Andorra | 0 | 0.017 | 0.0000000 | 0.0000002 | 0.000 | NaN | 0.0018613 |

Angola | 0 | 3.056 | 0.0000000 | 0.0000001 | 0.006 | NaN | 0.0018185 |

Anguilla | 0 | 0.007 | 0.0000000 | 0.0000004 | 0.000 | NaN | 0.0018614 |

Antigua and Barbuda | 0 | 0.019 | 0.0000000 | 0.0000002 | 0.000 | NaN | 0.0018612 |

Argentina | 0 | 4.010 | 0.0000000 | 0.0000001 | 0.007 | NaN | 0.0018054 |

Armenia | 0 | 0.273 | 0.0000000 | 0.0000001 | 0.001 | NaN | 0.0018576 |

Aruba | 0 | 0.021 | 0.0000000 | 0.0000002 | 0.000 | NaN | 0.0018612 |

Australia | 33,223 | 33,031.390 | 0.0012817 | 0.0012743 | 48.139 | 0.0018848 | 0.0014574 |

Austria | 2183 | 2284.134 | 0.0002447 | 0.0002560 | 3.953 | 0.0009555 | 0.0017305 |

Azerbaijan | 0 | 0.925 | 0.0000000 | 0.0000001 | 0.002 | 0.0000000 | 0.0018483 |

Bahamas | 20 | 19.675 | 0.0000490 | 0.0000482 | 0.032 | 0.0000000 | 0.0016104 |

Bahrain | 997 | 1054.167 | 0.0006814 | 0.0007204 | 0.340 | 0.0000000 | 0.0003224 |

Bangladesh | 54 | 60.609 | 0.0000003 | 0.0000004 | 0.722 | 0.0022883 | 0.0119116 |

Barbados | 104 | 93.738 | 0.0003698 | 0.0003334 | 0.099 | 0.0000000 | 0.0010537 |

Belarus | 0 | 0.861 | 0.0000000 | 0.0000001 | 0.002 | NaN | 0.0018492 |

Belgium | 5944 | 6189.447 | 0.0005119 | 0.0005330 | 1.182 | NaN | 0.0001910 |

Belize | 224 | 212.846 | 0.0005600 | 0.0005321 | 0.638 | 0.0036101 | 0.0029986 |

**Table 4.**Matrix V for data from 21 June 2022 using the 14-day lag, along with MAP classifications (right column) and the associated masses and mass points (bottom two rows).

ℓ | 1 | 2 | 3 | 4 | $\widehat{\mathit{\ell}}$ |
---|---|---|---|---|---|

Afghanistan | 0.020 | 0.174 | 0.230 | 0.577 | 4 |

Albania | 0.313 | 0.369 | 0.172 | 0.145 | 2 |

Algeria | 0.259 | 0.324 | 0.174 | 0.243 | 2 |

Andorra | 0.254 | 0.320 | 0.174 | 0.252 | 2 |

Angola | 0.258 | 0.323 | 0.174 | 0.245 | 2 |

Anguilla | 0.254 | 0.320 | 0.174 | 0.252 | 2 |

Antigua and Barbuda | 0.254 | 0.320 | 0.174 | 0.252 | 2 |

Argentina | 0.259 | 0.324 | 0.174 | 0.243 | 2 |

Armenia | 0.254 | 0.320 | 0.174 | 0.251 | 2 |

Aruba | 0.254 | 0.320 | 0.174 | 0.252 | 2 |

Australia | 0.000 | 1.000 | 0.000 | 0.000 | 2 |

Austria | 0.000 | 0.839 | 0.161 | 0.000 | 2 |

Azerbaijan | 0.255 | 0.321 | 0.174 | 0.250 | 2 |

Bahamas | 0.277 | 0.340 | 0.175 | 0.208 | 2 |

Bahrain | 0.746 | 0.247 | 0.007 | 0.000 | 1 |

Bangladesh | 0.000 | 0.025 | 0.099 | 0.875 | 4 |

Barbados | 0.351 | 0.392 | 0.164 | 0.093 | 2 |

Belarus | 0.255 | 0.321 | 0.174 | 0.250 | 2 |

Belgium | 1.000 | 0.000 | 0.000 | 0.000 | 1 |

Belize | 0.054 | 0.397 | 0.348 | 0.201 | 2 |

${\widehat{c}}_{\ell}$ | −8.564 | −6.531 | −5.458 | −4.251 | |

${\widehat{q}}_{\ell}$ | 0.254 | 0.320 | 0.174 | 0.252 |

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**MDPI and ACS Style**

Almohaimeed, A.; Einbeck, J.; Qarmalah, N.; Alkhidhr, H.
Using Random Effect Models to Produce Robust Estimates of Death Rates in COVID-19 Data. *Int. J. Environ. Res. Public Health* **2022**, *19*, 14960.
https://doi.org/10.3390/ijerph192214960

**AMA Style**

Almohaimeed A, Einbeck J, Qarmalah N, Alkhidhr H.
Using Random Effect Models to Produce Robust Estimates of Death Rates in COVID-19 Data. *International Journal of Environmental Research and Public Health*. 2022; 19(22):14960.
https://doi.org/10.3390/ijerph192214960

**Chicago/Turabian Style**

Almohaimeed, Amani, Jochen Einbeck, Najla Qarmalah, and Hanan Alkhidhr.
2022. "Using Random Effect Models to Produce Robust Estimates of Death Rates in COVID-19 Data" *International Journal of Environmental Research and Public Health* 19, no. 22: 14960.
https://doi.org/10.3390/ijerph192214960