Table 2 Posterior Bayesian estimates of the 28-day mortality rate in the Covidicus trial using the simplest Bayes models according to the approach. CrI: Credibility Interval

Ignoring external data

    Simple Bayes

Beta(1,1)

\(\theta\)

0.2407

0.1653

0.3253

Incorporating external data

    Combining data

Beta(96,230)

\(\theta\)

0.2801

0.2388

0.3233

    Modifying the prior

    - Based on quantiles

Beta(77,185)

\(\theta\)

0.2771

0.2327

0.3239

    - Based on shrinkage intensity,\(m=10\)

Beta(2.9,7.1)

\(\theta\)

0.2408

0.0711

0.5927

    - Based on shrinkage intensity,\(m=100\)

Beta(29.3,70.7)

\(\theta\)

0.2637

0.2086

0.3856

    Fixed Power Priors, EB estimate

\(\theta\)

0.2725

0.2208

0.3240

    Random Power Priors

Beta(1,1)

\(\theta\)

0.2479

0.1770

0.3176

\(a_0 \sim\)Beta(1,1)

0.1955

0.0001

0.6212

\(\theta\)

0.2390

0.1675

0.3192

\(a_0 \sim\)Beta(1/2,1/2)

0.1072

0.0001

0.4887

\(\theta\)

0.2473

0.1697

0.3160

\(a_0 \sim\)Exp(100)

0.1930

0.0001

0.6232

    Hierarchical Bayesian Models

Beta(1,1)

\(\theta\)

0.2944

0.2480

0.3457

Beta(1/2,1/2)

\(\theta\)

0.2934

0.2450

0.3440

    Pocock’s bias Approach

N(0, 0.03)

\(\delta _h\)

-0.00035

-0.0629

0.0549

\(\theta\)

0.2770

0.2355

0.3194

N(0.07, 0.03)

\(\delta _h\)

-0.0697

0.00706

0.1249

\(\theta\)

0.2770

0.2365

0.3194

\(N(-0.07,0.03)\)

\(\delta _h\)

-0.0703

-0.13296

-0.0151

\(\theta\)

0.2770

0.2365

0.3194