Table 1 Prior distributions for probabilistic bias analysis.

From: Substantial underestimation of SARS-CoV-2 infection in the United States

Distribution

Minimum (lower bound)

Mean

Maximum (upper bound)

Shape 1

Shape 2

P(S1|tested)

0.00

0.93

1.00

20.00

1.40

P(S1|untested)

0.00

0.03

0.15

1.18

45.97

α

0.80

0.90

1.00

49.73

5.53

β

0.002

0.15

0.40

2.21

12.53

P(S0|test+)

0.25

0.42

0.70

6.00

9.00

Sensitivity (Se)

0.65

0.86

1.00

4.20

1.05

Specificity (Sp)

0.9998

0.9999

1.00

4998.50

0.25

  1. P(S1|tested) is the probability of having moderate to severe symptoms among tested individuals, and P(S1|untested) is the analogous probability among untested individuals. We defined α and β as random variables describing the ratio of P(test + |S1, untested) and P(test + |S0, untested) to the empirical state-level estimate P(test + |tested). P(S0|test+) is the probability of having mild or no symptoms among individuals who tested positive.
  2. Detailed descriptions of each prior distribution including cited literature are given in the Methods section. For truncated Beta distributions (those with lower and upper bounds not equal to 0 and 1), the mean was calculated via numerical integration. Distributions truncated to region (a,b] are defined by modifying the untruncated density function \(f(x)\) to be: \(\frac{{f(x)}}{{F\left( b \right) - F(a)}}\), where \(F(x)\) is the distribution function, such that the truncated density integrates to 1 over that region. The values for P(S0|test+) give the distribution prior to Bayesian melding.
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