# Statistical Analysis Methods Applied to Early Outpatient COVID-19 Treatment Case Series Data

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## Abstract

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## 1. Introduction

## 2. Methods—Part I: Frequentist Methods for Case Series Analysis

#### 2.1. Comparing Treatment Group against Expected Adverse Event Rate without Treatment

#### 2.2. Comments on the Proposed Hypothesis Testing Technique

#### 2.3. Selection Bias Mitigation and Selection Bias Thresholds

## 3. Methods—Part II: Bayesian Factor Analysis of Efficacy Thresholds

#### 3.1. Bayesian Factor and the False Positive Rate

#### 3.2. Application to Hypothesis Testing for Case Series

## 4. Results

#### 4.1. Review of the Zelenko, Procter and Raoult Case Series

#### 4.2. Tabular Summaries of the Zelenko, Procter, and Raoult Case Series

#### 4.3. Analysis of Mortality Rate Reduction Efficacy

#### 4.4. Analysis of Hospitalization Rate Reduction Efficacy

#### 4.5. Bayesian Analysis of Efficacy Thresholds

## 5. Discussion and Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

SARS-CoV-2 | Severe Acute Respiratory Syndrome Coronavirus 2 |

RCT | Randomized Controlled Trial |

RDRP | RNA Dependent RNA Polymerase |

EGCG | Epigallocatechin Gallate |

RSV | Respiratory Syncytial Virus |

DSZ | Derwand–Scholz–Zelenko |

PCR | Polymerase Chain Reaction |

IgG | Immunoglobulin G |

BMI | Body Mass Index |

CDC | Centers for Disease Control and Prevention |

TNF-$\alpha $ | Tumor Necrosis Factor Alpha |

IL-1$\beta $ | Interleukin-1-Beta |

IL-6 | Interleukin 6 |

## Appendix A. Exact Fisher Test in the Limit of an Infinite Control Group

## Appendix B. Calculation of the Selection Bias Thresholds

## Appendix C. Monotonicity of the Bayesian Factor

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**Figure 1.**This flowchart shows the suggested interactions between medical doctors, public health agencies, and the proposed statistical methodology that are needed, in order to implement an emergency epidemic or pandemic response that leverages the direct experience of frontline medical doctors treating their patients.

**Figure 2.**We plot the p-value calculated from an exact Fisher test that compares the treatment group from the DSZ study [2] (141 high-risk patients treated with 1 death) against an artificial control group with 3.8% mortality rate. Note that the exact p-value in the infinite control group limit should be 0.047, which is approached to three decimals when we get to control group size between 160,000 and 180,000.

**Figure 3.**Coverage probability for the Sterne interval [71] with sample sizes $N=20$ and $N=100$. The black curve corresponds to $N=20$ and the blue curve, which is situated below the black curve, corresponds to $N=100$. The coverage probabilities were calculated using $0.01$ increments on the horizontal axis.

**Figure 4.**Comparison of the coverage probability for the Clopper–Pearson interval [75] versus the Sterne interval [71] with sample size $N=100$. The black curve shows the coverage probability for the Clopper–Pearson interval, and the blue curve, which is situated below the black curve, shows the coverage probability for the Sterne interval. The coverage probabilities were calculated using $0.01$ increments on the horizontal axis.

**Figure 5.**Relationship between p-value and expected mortality rate for high-risk patients without early treatment, based on the case series data from Procter’s dataset of 869 high-risk patients [12]. The zigzag curve follows $p(N,a,x)$ given by Equation (2), whereas the smooth curve approximates the right tail terms in the p-value sum by replacing them with the left-tail terms on the horizontal axis.

**Figure 6.**Cumulative case fatality rate in the United States and France between April 2020 and November 2021.

**Table 1.**Case series list: The table lists the total number of patients, the subset of high-risk patients that were treated with a sequenced multidrug regimen, number of patients that were hospitalized, and number of deaths, for the following case series: Derwand–Scholtz–Zelenko study treatment group [2], Zelenko’s complete April 2020 data set [9], Zelenko’s complete June 2020 data set [10], Procter’s observational studies [11,12], and Raoult’s high-risk (older than 60) treatment group [13]. The table also lists the same data for the control group in the DSZ study [2], the untreated group in the Israeli study [61], and the control group in the Raoult study [13].

Study | Total | High-Risk | Hospitalizations | Deaths |
---|---|---|---|---|

Case series data from Refs. [2,9,10,11,12,13] | ||||

DSZ study [2] | 712 | 141 | 4 (2.8%) | 1 (0.7%) |

Zelenko 04/2020 [9] | 1450 | 405 | 6 (1.4%) | 2 (0.4%) |

Zelenko 06/2020 [10] | 2200 | 800 | 12 (1.5%) | 2 (0.25%) |

Procter I [11] | 922 | 320 | 6 (1.8%) | 1 (0.3%) |

Procter II [12] | ? | 869 | 20 (2.3%) | 2 (0.2%) |

Raoult [13] | 10429 | 1495 | 106 (7.0%) | 5 (0.3%) |

Control group data from Refs. [2,13,61] | ||||

DSZ control [2] | 377 | $<377$ | 58 (>15%) | 13 (>3.4%) |

Israeli control [61] | 4179 | $<4179$ | N/A | 143 (>3.4%) |

Raoult control [13] | 2114 | 520 | 38 (7.3%) | 11 (2%) |

**Table 2.**Exact Fisher test comparing the mortality rate reduction and hospitalization rate reduction between the high-risk patient treated group the DSZ study [2], Zelenko’s complete April 2020 data set [9], and Zelenko’s complete June 2020 data set [10] against the low risk and high-risk patient control groups in the DSZ study [2] and the Israeli study [61]. The p-values where there is a failure to establish 95% confidence are highlighted with bold font.

Study | Odds Ratio | 95% CI | p-Value |
---|---|---|---|

Exact Fisher tests on mortality rates | |||

DSZ study vs. DSZ control | 0.2 | 0.02–1.54 | 0.12 |

Zelenko 04/2020 vs. DSZ control | 0.13 | 0.03–0.61 | 0.003 |

Zelenko 06/2020 vs. DSZ control | 0.07 | 0.01–0.31 | ${10}^{-5}$ |

DSZ vs. Israeli control | 0.2 | 0.03–1.45 | 0.09 |

Zelenko 04/2020 vs. Israeli control | 0.14 | 0.03–0.57 | 0.0002 |

Zelenko 06/2020 vs. Israeli control | 0.07 | 0.02–0.28 | ${10}^{-9}$ |

Exact Fisher tests on hospitalization rates | |||

DSZ vs. DSZ control | 0.16 | 0.05–0.45 | 0.02 |

Zelenko 04/2020 vs. DSZ control | 0.08 | 0.03–0.19 | ${10}^{-13}$ |

Zelenko 06/2020 vs. DSZ control | 0.08 | 0.04–0.16 | ${10}^{-19}$ |

**Table 3.**Mortality and hospitalization rate reduction efficacy thresholds, defined as the upper end of the Sterne interval [71], corresponding to 95%, 99%, and 99.9% confidence, for the DSZ study treatment group [2], Zelenko’s complete April 2020 data set [9], Zelenko’s complete June 2020 data set [10], Procter’s observational studies [11,12], and Raoult’s high-risk (older than 60) treatment group [13]. In parenthesis, we also display the corresponding higher random selection bias thresholds.

Study | 95% Threshold | 99% Threshold | 99.9% Threshold |
---|---|---|---|

Mortality rate efficacy thresholds | |||

DSZ study | 3.8% (9.2%) | 5.3% (12.8%) | 7.0% (14.6%) |

Zelenko 04/2020 | 1.8% (4.0%) | 2.4% (5.2%) | 2.9% (6.9%) |

Zelenko 06/2020 | 1.0% (2.0%) | 1.2% (2.7%) | 1.6% (3.7%) |

Procter I | 1.7% (4.1%) | 2.3% (5.8%) | 3.1% (7.8%) |

Procter II | 0.84% (1.82%) | 1.08% (2.46%) | 1.4% (3.37%) |

Raoult | 0.79% (1.40%) | 0.96% (1.87%) | 1.18% (2.46%) |

Hospitalization rate efficacy thresholds | |||

DSZ study | 7.0% (12.7%) | 8.8% (17.5%) | 10.6% (21.5%) |

Zelenko 04/2020 | 3.2% (5.4%) | 3.9% (7.2%) | 4.7% (9.5%) |

Zelenko 06/2020 | 2.7% (4.2%) | 3.0% (5.0%) | 3.5% (6.4%) |

Procter I | 4.1% (7.3%) | 4.9% (9.1%) | 5.9% (11.6%) |

Procter II | 3.6% (5.2%) | 4.0% (6.1%) | 4.5% (7.5%) |

**Table 4.**Crude case fatality rate data, in the absence of early outpatient treatment, based on early data from China as of 11 February 2020, and published on 30 March 2020. [60]. We highlight with bold font the high-risk age brackets with $\mathrm{CFR}\ge 1.0\%$.

Age | Deaths | Cases | CFR |
---|---|---|---|

10–19 | 0 | 416 | 0% |

20–29 | 7 | 3619 | 0.193% |

30–39 | 18 | 7600 | 0.237% |

40–49 | 38 | 8571 | 0.4% |

50–59 | 130 | 10,008 | 1.3% |

60–69 | 309 | 8583 | 3.6% |

70–79 | 312 | 3918 | 7.96% |

≥80 | 208 | 1408 | 14.8% |

≥60 | 829 | 13,909 | 5.96% |

Age | Italy CFR | China CFR |
---|---|---|

0–9 | 0% | 0% |

10–19 | 0% | 0.2% |

20–29 | 0% | 0.2% |

30–39 | 0.3% | 0.2% |

40–49 | 0.4% | 0.4% |

50–59 | 1.0% | 1.3% |

60–69 | 3.5% | 3.6% |

70–79 | 12.8% | 8.0% |

≥80 | 20.2% | 14.8% |

Comorbidity | Deaths | Cases | CFR |
---|---|---|---|

Comorbidity CFR from Chinese study [58] | |||

Cardiovascular disease | 92 | 873 | 10.5% |

Diabetes | 80 | 1102 | 7.3% |

Respiratory disease | 32 | 511 | 6.3% |

Hypertension | 161 | 2683 | 6% |

Cancer | 6 | 107 | 5.6% |

Comorbidity CFR from Israeli study [61] | |||

Cardiovascular disease | 87 | 518 | 16.7% |

Diabetes | 71 | 531 | 13% |

Respiratory disease | 23 | 361 | 6% |

Hypertension | 102 | 744 | 13.7% |

Cancer | 37 | 264 | 10% |

**Table 7.**Bayes factor (decimal logarithm) corresponding to the 95% efficacy threshold (Sterne interval [71]) for mortality and hospitalization rate reduction, using maximum untreated mortality rate ${p}_{2}$ for high-risk patients at ${p}_{2}\in \{0.02,0.05,0.10\}$ and maximum untreated hospitalization rate ${p}_{2}$ for high-risk patients at ${p}_{2}\in \{0.10,0.15,0.20\}$, for the DSZ study treatment group [2], Zelenko’s complete April 2020 data set [9], Zelenko’s complete June 2020 data set [10], Procter’s observational studies [11,12], and Raoult’s high-risk (older than 60) treatment group [13]. We highlight in bold font the Bayes factors that violate the condition $b({x}_{0},{p}_{2})\ge 2$.

Bayes factors at the mortality rate efficacy thresholds | ||||
---|---|---|---|---|

Study | 95% threshold | log Bayes factors | ||

${p}_{2}=0.02$ | ${p}_{2}=0.05$ | ${p}_{2}=0.1$ | ||

DSZ study | 3.8% | N/A | 1.38 | 1.99 |

Zelenko 04/2020 | 1.8% | 1.17 | 2.04 | 2.45 |

Zelenko 06/2020 | 1.0% | 2.06 | 2.66 | 3.02 |

Procter I | 1.7% | 1.28 | 2.07 | 2.47 |

Procter II | 0.84% | 1.92 | 2.48 | 2.82 |

Raoult | 0.79% | 1.91 | 2.45 | 2.79 |

Bayes factors at the hospitalization rate efficacy thresholds | ||||

Study | 95% threshold | log Bayes factors | ||

${p}_{2}=0.10$ | ${p}_{2}=0.15$ | ${p}_{2}=0.20$ | ||

DSZ study | 7.0% | 1.30 | 1.71 | 1.92 |

Zelenko 04/2020 | 3.2% | 2.00 | 2.24 | 2.39 |

Zelenko 06/2020 | 2.7% | 2.24 | 2.47 | 2.61 |

Procter I | 4.1% | 1.89 | 2.15 | 2.32 |

Procter II | 3.6% | 1.98 | 2.23 | 2.39 |

**Table 8.**Comparison of the 95% confidence efficacy threshold (Sterne interval [71]) for mortality and hospitalization rate reduction with the Bayes factor efficacy thresholds at log Bayes = 2, using maximum untreated mortality rate ${p}_{2}$ for high-risk patients at ${p}_{2}\in \{0.02,0.05,0.10\}$ and maximum untreated hospitalization rate ${p}_{2}$ for high-risk patients at ${p}_{2}\in \{0.10,0.15,0.20\}$, for the DSZ study treatment group [2], Zelenko’s complete April 2020 data set [9], Zelenko’s complete June 2020 data set [10], Procter’s observational studies [11,12], and Raoult’s high-risk (older than 60) treatment group [13]. We highlight in bold font the Bayesian thresholds that exceed the frequentist thresholds.

Mortality rate Bayesian efficacy thresholds | ||||
---|---|---|---|---|

Study | 95% | log Bayes = 2 thresholds | ||

threshold | ${p}_{2}=2$% | ${p}_{2}=5$% | ${p}_{2}=10$% | |

DSZ study | 3.8% | N/A | N/A | 3.9% |

Zelenko 04/2020 | 1.8% | N/A | 1.8% | 1.5% |

Zelenko 06/2020 | 1.0% | 1.0% | 0.8% | 0.6% |

Procter I | 1.7% | N/A | 1.9% | 1.3% |

Procter II | 0.84% | 0.87% | 0.7% | 0.6% |

Raoult | 0.79% | 0.82% | $<0.7$% | $<0.7$% |

Hospitalization rate Bayesian efficacy thresholds | ||||

Study | 95% | log Bayes = 2 thresholds | ||

threshold | ${p}_{2}=10$% | ${p}_{2}=15$% | ${p}_{2}=20$% | |

DSZ study | 7.0% | 9.5% | 7.8% | 7.2% |

Zelenko 04/2020 | 3.2% | 3.2% | 3.0% | 2.9% |

Zelenko 06/2020 | 2.7% | 2.6% | 2.5% | 2.4% |

Procter I | 4.1% | 4.3% | 4.0% | 3.7% |

Procter II | 3.6% | 3.7% | 3.5% | 3.4% |

**Table 9.**Bayes factor (decimal logarithm) corresponding to the 99% efficacy threshold (Sterne interval [71]) for mortality and hospitalization rate reduction, using maximum untreated mortality rate ${p}_{2}$ for high-risk patients at ${p}_{2}\in \{0.02,0.05,0.10\}$ and maximum untreated hospitalization rate ${p}_{2}$ for high-risk patients at ${p}_{2}\in \{0.10,0.15,0.20\}$, for the DSZ study treatment group [2], Zelenko’s complete April 2020 data set [9], Zelenko’s complete June 2020 data set [10], Procter’s observational studies [11,12], and Raoult’s high-risk (older than 60) treatment group [13]. We highlight in bold font the Bayes factors that violate the condition $b({x}_{0},{p}_{2})\ge 2.7$.

Bayes factors at the mortality rate efficacy thresholds | ||||
---|---|---|---|---|

Study | 99% threshold | log Bayes factors | ||

${p}_{2}=0.02$ | ${p}_{2}=0.05$ | ${p}_{2}=0.1$ | ||

DSZ study | 5.3% | N/A | N/A | 2.70 |

Zelenko 04/2020 | 2.4% | N/A | 2.81 | 3.27 |

Zelenko 06/2020 | 1.2% | 2.53 | 3.21 | 3.57 |

Procter I | 2.3% | N/A | 2.72 | 3.17 |

Procter II | 1.08% | 2.55 | 3.17 | 3.53 |

Raoult | 0.96% | 2.57 | 3.16 | 3.51 |

Bayes factors at the hospitalization rate efficacy thresholds | ||||

Study | 99% threshold | log Bayes factors | ||

${p}_{2}=0.10$ | ${p}_{2}=0.15$ | ${p}_{2}=0.20$ | ||

DSZ study | 8.8% | 1.83 | 2.42 | 2.67 |

Zelenko 04/2020 | 3.9% | 2.75 | 3.00 | 3.17 |

Zelenko 06/2020 | 3.0% | 2.77 | 3.00 | 3.16 |

Procter I | 4.9% | 2.55 | 2.85 | 3.02 |

Procter II | 4.0% | 2.63 | 2.89 | 3.05 |

**Table 10.**Bayes factor (decimal logarithm) corresponding to the 99.9% efficacy threshold (Sterne interval [71]) for mortality and hospitalization rate reduction, using maximum untreated mortality rate ${p}_{2}$ for high-risk patients at ${p}_{2}\in \{0.02,0.05,0.10\}$ and maximum untreated hospitalization rate ${p}_{2}$ for high-risk patients at ${p}_{2}\in \{0.10,0.15,0.20\}$, for the DSZ study treatment group [2], Zelenko’s complete April 2020 data set [9], Zelenko’s complete June 2020 data set [10], Procter’s observational studies [11,12], and Raoult’s high-risk (older than 60) treatment group [13]. We highlight in bold font the Bayes factors that violate the condition $b({x}_{0},{p}_{2})\ge 3.7$.

Bayes factors at the mortality rate efficacy thresholds | ||||
---|---|---|---|---|

Study | 99.9% threshold | log Bayes factors | ||

${p}_{2}=0.02$ | ${p}_{2}=0.05$ | ${p}_{2}=0.1$ | ||

DSZ study | 7.0% | N/A | N/A | 3.51 |

Zelenko 04/2020 | 2.9% | N/A | 3.47 | 4.00 |

Zelenko 06/2020 | 1.6% | 3.43 | 4.34 | 4.73 |

Procter I | 3.1% | N/A | 3.59 | 4.16 |

Procter II | 1.4% | 3.38 | 4.15 | 4.53 |

Raoult | 1.18% | 3.49 | 4.16 | 4.52 |

Bayes factors at the hospitalization rate efficacy thresholds | ||||

Study | 99.9% threshold | log Bayes factors | ||

${p}_{2}=0.10$ | ${p}_{2}=0.15$ | ${p}_{2}=0.20$ | ||

DSZ study | 10.6% | N/A | 3.17 | 3.49 |

Zelenko 04/2020 | 4.7% | 3.68 | 3.97 | 4.15 |

Zelenko 06/2020 | 3.5% | 3.75 | 4.00 | 4.16 |

Procter I | 5.9% | 3.45 | 3.80 | 3.99 |

Procter II | 4.5% | 3.54 | 3.82 | 3.99 |

**Table 11.**Comparison of the 99% confidence efficacy threshold (Sterne interval [71]) for mortality and hospitalization rate reduction with the Bayes factor efficacy thresholds at log Bayes = 2.7, using maximum untreated mortality rate ${p}_{2}$ for high-risk patients at ${p}_{2}\in \{0.02,0.05,0.10\}$ and maximum untreated hospitalization rate ${p}_{2}$ for high-risk patients at ${p}_{2}\in \{0.10,0.15,0.20\}$, for the DSZ study treatment group [2], Zelenko’s complete April 2020 data set [9], Zelenko’s complete June 2020 data set [10], Procter’s observational studies [11,12], and Raoult’s high-risk (older than 60) treatment group [13]. We highlight in bold font the Bayesian thresholds that exceed the frequentist thresholds.

Mortality rate Bayesian efficacy thresholds | ||||
---|---|---|---|---|

Study | 99% | log Bayes = 2.7 thresholds | ||

threshold | ${p}_{2}=2$% | ${p}_{2}=5$% | ${p}_{2}=10$% | |

DSZ study | 5.3% | N/A | N/A | 5.3% |

Zelenko 04/2020 | 2.4% | N/A | 2.4% | 2.0% |

Zelenko 06/2020 | 1.2% | 1.3% | 1.1% | 0.9% |

Procter I | 2.3% | N/A | 2.3% | 1.9% |

Procter II | 1.08% | 1.14% | 0.92% | 0.80% |

Raoult | 0.96% | 1.0% | 0.86% | 0.77% |

Hospitalization rate Bayesian efficacy thresholds | ||||

Study | 99% | log Bayes = 2.7 thresholds | ||

threshold | ${p}_{2}=10$% | ${p}_{2}=15$% | ${p}_{2}=20$% | |

DSZ study | 8.8% | N/A | 9.5% | 8.9% |

Zelenko 04/2020 | 3.9% | N/A | 3.7% | 3.5% |

Zelenko 06/2020 | 3.0% | 3.0% | 2.9% | 2.8% |

Procter I | 4.9% | 5.1% | 4.8% | 4.6% |

Procter II | 4.0% | 4.1% | 3.9% | 3.8% |

**Table 12.**Comparison of the 99.9% confidence efficacy threshold (Sterne interval [71]) for mortality and hospitalization rate reduction with the Bayes factor efficacy thresholds at log Bayes = 3.7, using maximum untreated mortality rate ${p}_{2}$ for high-risk patients at ${p}_{2}\in \{0.02,0.05,0.10\}$ and maximum untreated hospitalization rate ${p}_{2}$ for high-risk patients at ${p}_{2}\in \{0.10,0.15,0.20\}$, for the DSZ study treatment group [2], Zelenko’s complete April 2020 data set [9], Zelenko’s complete June 2020 data set [10], Procter’s observational studies [11,12], and Raoult’s high-risk (older than 60) treatment group [13]. We highlight in bold font the Bayesian thresholds that exceed the frequentist thresholds.

Mortality rate Bayesian efficacy thresholds | ||||
---|---|---|---|---|

Study | 99.9% | log Bayes = 3.7 thresholds | ||

threshold | ${p}_{2}=2$% | ${p}_{2}=5$% | ${p}_{2}=10$% | |

DSZ study | 7.0% | N/A | N/A | 7.4% |

Zelenko 04/2020 | 2.9% | N/A | 3.1% | 2.7% |

Zelenko 06/2020 | 1.6% | 1.8% | 1.4% | 1.3% |

Procter I | 3.1% | N/A | 3.2% | 2.8% |

Procter II | 1.4% | 1.53% | 1.26% | 1.14% |

Raoult | 1.18% | 1.23% | 1.08% | 1.01% |

Hospitalization rate Bayesian efficacy thresholds | ||||

Study | 99.9% | log Bayes = 3.7 thresholds | ||

threshold | ${p}_{2}=10$% | ${p}_{2}=15$% | ${p}_{2}=20$% | |

DSZ study | 10.6% | N/A | 11.9% | 11.1% |

Zelenko 04/2020 | 4.7% | 4.8% | 4.5% | 4.4% |

Zelenko 06/2020 | 3.5% | 3.5% | 3.4% | 3.3% |

Procter I | 5.9% | 6.2% | 5.8% | 5.7% |

Procter II | 4.5% | 4.6% | 4.5% | 4.4% |

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## Share and Cite

**MDPI and ACS Style**

Gkioulekas, E.; McCullough, P.A.; Zelenko, V.
Statistical Analysis Methods Applied to Early Outpatient COVID-19 Treatment Case Series Data. *COVID* **2022**, *2*, 1139-1182.
https://doi.org/10.3390/covid2080084

**AMA Style**

Gkioulekas E, McCullough PA, Zelenko V.
Statistical Analysis Methods Applied to Early Outpatient COVID-19 Treatment Case Series Data. *COVID*. 2022; 2(8):1139-1182.
https://doi.org/10.3390/covid2080084

**Chicago/Turabian Style**

Gkioulekas, Eleftherios, Peter A. McCullough, and Vladimir Zelenko.
2022. "Statistical Analysis Methods Applied to Early Outpatient COVID-19 Treatment Case Series Data" *COVID* 2, no. 8: 1139-1182.
https://doi.org/10.3390/covid2080084