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Since the outbreak of the novel coronavirus disease (COVID19) in December 2019, the coronavirus has spread all over the world at an unprecedented rate. The transmissibility of the coronavirus from asymptomatic patients to healthy individuals has received enormous attention. An important study using COVID19 data from the city of Ningbo, China, was carried out to estimate and compare the transmission rates of the coronavirus by the symptomatic and asymptomatic patients. However, in the original analysis, the usual chisquare tests were unduly used for some contingency tables with small cell counts including zero, which may violate the assumptions for the chisquare test.
We reanalyze the data from the city of Ningbo with more appropriate statistical methods to draw more reliable and sound conclusions on the transmission rates of the coronavirus by the symptomatic and asymptomatic patients.
We excluded the cases associated with the superspreader and adopted a more appropriate statistical method, including the permutation test and the Fisher exact test, to reanalyze the COVID19 data from the city of Ningbo.
After excluding the cases related to the superspreader, the Fisher exact test yields a
Through a more indepth and comprehensive statistical analysis of the Ningbo data, we concluded that there is no difference in the transmission rates of coronavirus between symptomatic and asymptomatic patients.
Since the outbreak of the novel coronavirus disease (COVID19) in December 2019, the coronavirus has spread all over the world at an unprecedented rate. By May 21, 2020, more than 200 countries and territories have been affected by COVID19, with a total of more than 5 million confirmed cases and over 330,000 deaths [
During the disease incubation period, a percentage of coronavirus carriers may have no symptoms or minimal symptoms and thus often go undetected. These covert coronavirus carriers may not even be aware of the infection themselves but would be confirmed as positive cases if tested using the reverse transcriptase polymerase chain reaction (RTPCR). If the percentage of asymptomatic carriers is large and if their transmissibility of coronavirus is as high as the symptomatic cases, this would pose a great threat to the public health worldwide. Therefore, it is critical to determine the percentage and the transmissibility of asymptomatic coronavirus carriers in the population.
There has been some work in the literature on the estimation of the asymptomatic proportion of COVID19 cases. Based on the infected cases on the Diamond Princess cruise ship, the asymptomatic ratio was estimated to be 0.179 with a 95% Bayesian credible interval of 0.1550.202 [
There has been evidence for transmission of coronavirus from asymptomatic carriers. It was reported that the viral load detected in the asymptomatic patients was similar to that in the symptomatic patients, which suggests the potential transmissibility of asymptomatic carriers [
Chen et al [
The following is the permutation test algorithm:
However, in their original statistical analysis [
We adopted a permutation test to determine the difference in the average numbers of contacts by the symptomatic and asymptomatic cases. The permutation test algorithm gives the details of the permutation test, and
The diagram for the resampling step in the permutation test, where the lengths of segments are randomly generated corresponding to the number of close contacts for each individual patient.
To allow for small cell counts including zero in the contingency table, the Fisher exact tests [
Without making any assumptions on the data, the Fisher exact test simply adopts the hypergeometric distribution to calculate the exact probability of the observed data in the table. For example, as shown in
The
A typical 2×2 contingency table.
Group  Infected  Uninfected 
Number of close contacts of symptomatic cases 


Number of close contacts of asymptomatic cases 


To gain more insight into the Ningbo data, we further calculated the odds ratio between the symptomatic and asymptomatic groups as well as the corresponding confidence interval. For a 2×2 contingency table with cell counts (
If the estimated transmission rate with sample size
Both confidence intervals for the odds ratio and the transmission rate are based on normal approximation, and in the Ningbo data, the sample sizes for computation of these confidence intervals are reasonably large.
From January 21 to March 6, 2020, there were 157 symptomatic cases and 30 asymptomatic cases in the Ningbo COVID19 data [
The histograms of the average numbers of contacts by the symptomatic and asymptomatic cases (top panel) and the difference after the permutation in the average numbers of contacts by the symptomatic and asymptomatic cases in the permutation test (bottom panel). The red vertical line indicates the observed difference in the average number of contacts between symptomatic and asymptomatic cases, which lies at the far end of the null distribution.
Under the Fisher exact test, we consider two scenarios: (1) to combine the numbers of symptomatic and asymptomatic cases as the total number of infected patients, leading to a 2×2 table; or (2) to separate them, leading to a 2×3 table, as shown in the primary analysis of close contacts section of
From the results summarized in
Analysis of the transmission rates through close contacts by the symptomatic and asymptomatic cases of the coronavirus disease in Ningbo after removing all the cases associated with the superspreader.
Analysis  Close contacts, n  Infected  Uninfected, n  


Symptomatic cases, n  Asymptomatic cases, n 

Combined^{a}  Separate^{b}  

.84 (.37)^{d}  .11 (.08)  

Symptomatic cases  1904 (97)^{c}  79 (28)  15 (4)  1810 (65) 



Asymptomatic cases  146  3  3  140 



Total  2050 (97)  82 (28)  18 (4)  1950 (65) 



<.001  <.001  

Family  268  37  10  221 



Relatives  400  13  6  381 



Friends  153  23  1  129 



Coworkers  57  2  0  55 



Medical  79  0  0  79 



Others  1093  7  1  1085 



Total  2050  82  18  1950 



<.001  <.001  

Daily activities  1048  69  14  965 



Transportation  167  1  2  164 



Medical contact  297  4  0  293 



Other contact  538  8  2  528 



Total  2050  82  18  1950 


^{a}Combined means
^{b}Separate means
^{c}The numbers in the parentheses are associated with the superspreader.
^{d}
The estimated odds ratio, transmission rates, and their difference between symptomatic and asymptomatic cases as well as the corresponding 95% confidence intervals are all presented in
The transmission rates under different relationships with the infected cases are significantly different with both
Primary analysis with the estimated rates and 95% CIs.
Variable  Odds ratio (95% CI)  Transmission rate of symptomatic cases, (95% CI)  Transmission rate of asymptomatic cases (95% CI)  Difference of transmission rates (95% CI) 
With superspreader cases  1.568 (0.6793.620)  0.063 (0.0530.075)  0.041 (0.0170.091)  0.022 (–0.016 to 0.059) 
Without superspreader cases  1.212 (0.5222.815)  0.049 (0.0400.060)  0.041 (0.0170.091)  0.008 (–0.029 to 0.046) 
In summary, we provided a more indepth analysis of the Ningbo COVID19 data to examine the difference in the transmissibility of the coronavirus for symptomatic and asymptomatic patients. The conclusion remains the same, that there is no statistically significant difference in the transmissibility of the coronavirus between symptomatic and asymptomatic patients, but our evidence for no difference appears to be stronger with larger
As the proportion of asymptomatic carriers in the total infected cases is considerably high [
However, our analysis only focuses on the data from the city of Ningbo, China, and the sample size is small. Therefore, the generalization of our findings to a larger and more diverse population is limited. More work is warranted to study the transmissibility of coronavirus by the asymptomatic coronavirus carriers.
coronavirus disease
reverse transcriptase polymerase chain reaction
None declared.