Examples 2

Clinical Trial of Pfizer-BioNTech COVID Vaccine (2020)

In December 2020, Pfizer and BioNTech successfully obtained a US FDA Emergency Use Authorization (EUA) to begin distributing their two-dose vaccine (BNT162b2) for SARS-CoV-2 ("COVID-19"). This example deals with the statistical approach delineated in the official protocol. The data analyses were reported in the formal FDA Briefing Document and in Polack, et al., (2020). Safety and efficacy of the BNT162b2 mRNA COVID-19 vaccine. NEJM.

In brief, this placebo-controlled, observer-blinded, efficacy study randomly assigned persons 16 years of age or older in a 1:1 ratio to receive two doses, 21 days apart, of either placebo or the candidate, BNT162b2 vaccine, which was based the mRNA approach, as Moderna used, but not Janssen (Johnson and Johnson) or AstraZeneca. The primary efficacy endpoint was the time of laboratory-confirmed COVID-19.

To their credit, Pfizer-BioNTech used the Bayesian beta-binomial model to assess their vaccine's efficacy. We delineate their approach and focus on their choice for the prior beta distribution, which we show to be ill-suited. We then use BetaBinomial() to define a prior tailored to the research mission and to re-analyze the incidence counts on Table 2 of Polack, et al. and also in Table 6 of the FDA Briefing Document, summarized here in Table 2.1.

Table 2.1. From Table 2 of Polack, et al. and in Table 6 of the FDA Briefing
   Document.
****************************************************************************
Group:    BNT162b2 Placebo
   N randomized (without evidence of infection):    18,198 18,325
   Overall surveillance time (person-years): 2,214    2,222
Overall surveillance time ratio (OSTR = 2214/2222): 0.996
n not censored before 7 days after Dose 2:    17,411 17,511
Average surveillance time (years): 0.127    0.127
   Number of cases confirmed infected: 8    162
Incidence rate (8/2214, 162/2222):    0.0036 0.0729
   Incidence rate ratio (IRR = 0.0036/0.0729):    0.0496
Vaccine Efficacy (VE = 1 - IRR):    0.9504
****************************************************************************
Reported posterior Bayesian results for vaccine efficacy (VE):
<> 95% probability (credible) interval, [0.903, 0.976].
<> Prob[VE > 0.30 | 8 vaccine, 162 placebo] > 0.9999.

The Pfizer-BioNTech (P-B) prior for IRR. In their reports, Pfizer-BioNTech stated they used the transform

P = IRR/(1 + IRR) and set P ~ beta(0.700102, 1.0) as its prior distribution. However, evidence below shows that they used P = OSTR*IRR/(1 + OSTR*IRR) with OSTR being the ratio of the groups' surveillance times, OSTR = 2214/2222 = 0.996. This OSTR is so close to 1.0, because the randomization was 1:1 and the numbers of confirmed COVID cases were so small relative to the numbers randomized. Likewise, the sequential at-risk ratios, NARR[j], (covered here) would have not deviated much from 1.0.

Pfizer-BioNTech gave no rational for the curious value of a0 = 0.700102 (6 decimals!). They described this prior as being "minimally informative" and "pessimistic." We can use DP.IRR() and DescribeIRR() to challenge that claim.

> DP.IRR(0.700102, 1.0, "2.2", OSTR=0.996,
+    irr=c(0,0.001,0.05,0.5,0.7,1,2,10^10),
+    names.irr=c("0.000", "0.001", "0.050", "0.500",
+    "0.700", "1.000" ,"2.000", "10^10"))

Table 2.2. Densities & Cum. Prob's of IRR ~ betaIRR(0.70, 1.00, OSTR=0.996).
*****************************************************************************
   irr: 0.000 0.001 0.050 0.500 0.700 1.000 2.000 10^10
   Density(irr)    Inf 5.532 1.578 0.432 0.316 0.216 0.088 0.000
Prob[IRR < irr]    0.000 0.008 0.118 0.463 0.536 0.615 0.752 1.000
*****************************************************************************

> DescribeIRR(parms=matrix(c(0.700102, 1, 0.996),
+ ncol=3, byrow=T), table="2.3")

Table 2.3. Descriptive measures, IRR ~ betaIRR(a, b | OSTR).
******************************************************************************
a b OSTR Mean Median Mode PI95 Spread95
Case 1 0.70 1.00 0.996 9.314 0.594 0.000 [0.005, 27.265] 27.259
******************************************************************************

Key points:

The protocol correctly states that P ~ beta(0.700102, 1.0) has a mean of 0.700102/(0.700102 + 1.0) = 0.4118, but this is irrelevant, because for any constant k > 0, P ~ beta(k*0.700102) has the same mean. Besides, we care about the distribution of IRR, not P. For k = 1.0, 1.4, 2.0, and 3.0, the means of IRR ~ betaIRR(k*0.700102, k | OSTR=0.996) are 9.31, 2.40, 1.40, 1.05. (Interestingly, for k = 100,000,000, the mean and median of IRR ~ betaIRR(k*0.700102, k | OSTR=0.996) are both 0.700102!

The mean of 9.3 may indicate a pessimistic prior, but the central tendency of IRR ~ betaIRR(a, b | OSTR) is better quantified by its median. The median of betaIRR(0.700102, 1.0 | OSTR=0.996) is 0.594, which is clearly optimistic.

The stated goal of the trial was to infer that VE exceeds 30%, i.e., that IRR is less than 0.70. For IRR ~ betaIRR(0.700102, 1.0 | OSTR=0.996), Prob[IRR < 0.70] = pbeta(P.IRR(0.70,0.996), 0.700102, 1) = 0.54.

Thus, IRR ~ beta(0.700102, 1.0 | OSTR=0.0.996) is not a "minimally informative" and "pessimistic" prior for this problem. On the contrary, it is both informative and optimistic in that it that produces posterior distributions that have a greater chance of inferring IRR < 0.70 (or any other threshold). Such priors give frequentists a worthy argument to criticize Bayesian analyses.

Let's use BetaBimomial() with the P-B prior to obtain the posterior distribution based on observing 8 cases in the vaccine group versus 162 in the placebo group. Note how the Trans and TransName arguments dictate that the results for P are reported in terms of IRR = P/((1-P)*OSTR), where OSTR is defined before the function call. The obtained posterior 95% probability interval for, [0.024, 0.097], agrees perfectly with that reported for VE by Pfizer-BioNTech, [0.903, 0.976].

> OSTR <- 0.996 # ratio of total surveillance times, 2214/2222
> irr <- c(0.01, 0.05, 0.10, 0.20, 0.35, 0.50, 0.70, 1.00)
> BetaBinomial(A=0.700102, B=1.0, # P-B prior
+    Trans="P/((1-P)*OSTR)", TransName="IRR",
+    CP.points=P.IRR(irr,OSTR),
+    M=c(8, 162))

Transformation: t(P) = IRR(P) = P/((1-P)*OSTR)

Prior: P ~ beta(a0 = 0.700102, b0 = 1)
   This prior distribution for IRR(P) is not unimodal.

Posterior: P ~ beta(a1 = 0.700102 + 8 = 8.70, b1 = 1 + 162 = 163.00)

Median and 95% PI and its spread, IRR(P) = P/((1-P)*OSTR), P ~ beta().
**********************************************************************
Median Prob. Interval PI Spread
Prior 0.594 [0.005, 27.265]    27.3
Posterior 0.052 [0.024, 0.097] 0.073
**********************************************************************

100q% quantiles of IRR(P) = P/((1-P)*OSTR).
*************************************************************************
q: 0.005 0.025 0.050 0.250 0.500 0.750 0.950 0.975 0.995
Prior 0.001 0.005 0.014 0.161 0.594 1.976 13.208 27.265 139.730
Posterior 0.018 0.024 0.027 0.040 0.052 0.065 0.088 0.097 0.115
*************************************************************************
Prior: P ~ beta(0.700102, 1)
Posterior: P ~ beta(8.70, 163.00)

Prob[t(P) < t(p)], where t(P) = IRR(P) = P/((1-P)*OSTR).
******************************************************************
t(p): 0.010 0.050 0.100 0.200 0.350 0.500 0.700 1.000
Prior    0.039 0.118 0.186 0.285 0.388 0.463 0.536 0.615
Posterior    0.000 0.463 0.981 1.000 1.000 1.000 1.000 1.000
******************************************************************
Prior: P ~ beta(0.700102, 1)
Posterior: P ~ beta(8.70, 163.00)

To reiterate, these results (and those in Example 3 and 4) are approximate. They assume something impossible to be true, that the ratio of the numbers subjects at risk ("under surveillance") in the two groups was always OSTR when every COVID event occurred, i.e., NARR[j] = OSTR. However, this must have been nearly true, because the numbers of COVID events were so small relative the sample sizes and the rates of "random" study dropouts were probably equal between the groups. When this is so. a randomization of 1:1 will yield an OSTR around 1.0; a 2:1 randomization will yield an OSTR around 2.0 Another Rfunc, IRR.Bayes(), is being developed to use all 170 NARR[j] values instead of OSTR.

Let's now consider two justifiable priors for this problem. They are quite different.

A null, diffuse (ND) prior for IRR. Here we seek to fit IRR ~ betaIRR(a0, b0 | OSTR=0.996) such that it is unimodal, has median 1.0, and is extremely diffuse with a 1% quantile of 0.01. This is easily done using BetaBinomial(), however, note that the specifications for the median and a Q100% quantile, P.Q, are supplied with respect to P, not IRR, as as executed using the P.IRR() function given here.

> OSTR <- 0.996
> irr <- c(0.01, 0.05, 0.10, 0.20, 0.35, 0.50, 0.70, 1.00)
> ND <- BetaBinomial(Median=P.IRR(1.0,OSTR),
+    P.Q=P.IRR(0.01,OSTR), Q=0.01,
+    Trans="P/((1-P)*OSTR)", TransName="IRR",
+    CP.points=P.IRR(irr,OSTR),
+    M=c(8, 162))

Transformation: t(P) = IRR(P) = P/((1-P)*OSTR)

Solution for prior: P ~ beta(a0 = 1.0001000, b0 = 1.0027778)

Fit of beta(a0, b0) to requested Median (IRR_0.50) and IRR_0.01.
*****************************************************************
   Median    IRR_0.01
As requested: 1.000    0.0100
As fit: 1.000    0.0101
*****************************************************************

Prior: P ~ beta(a0 = 1.00, b0 = 1.00)
Posterior: P ~ beta(a1 = 1.00 + 8 = 9.00, b1 = 1.00 + 162 = 163.00)

Median and 95% PI and its spread, IRR(P) = P/((1-P)*OSTR), P ~ beta().
**********************************************************************
Median Prob. Interval PI Spread
Prior 1.00 [0.026, 38.752]    38.7
Posterior 0.054 [0.025, 0.099] 0.074
**********************************************************************

100q% quantiles of IRR(P) = P/((1-P)*OSTR).
*************************************************************************
q: 0.005 0.025 0.050 0.250 0.500 0.750 0.950 0.975 0.995
Prior 0.005 0.026 0.053 0.334 1.000 2.997 18.912 38.752 196.893
Posterior 0.019 0.025 0.029 0.042 0.054 0.067 0.091 0.099 0.118
*************************************************************************
Prior: P ~ beta(1.00, 1.00)
Posterior: P ~ beta(9.00, 163.00)

Table 2.4 displays some densities and cumulative probabilities.

> OSTR <- 0.996
> DP.IRR(ND$a0, ND$b0, "2.4",
+    irr=c(0, 0.001, 0.05, 0.50, 0.70, 1 ,2 , 10^10),
+    names.irr=c("0.000", "0.001", "0.05", "0.50",
+    "0.70", "1" ,"2", "10^10"), OSTR=OSTR)

Table 2.4. Densities & Cum. Prob's of IRR ~ betaIRR(1.00, 1.00, OSTR=0.996).
*****************************************************************************
   irr: 0.000 0.001 0.05 0.50 0.70 1 2 10^10
   Density(irr)    0.000 0.996 0.906 0.445 0.346 0.250 0.111 0.000
Prob[IRR < irr]    0.000 0.001 0.048 0.333 0.412 0.500 0.667 1.000
*****************************************************************************

This prior distribution, IRR ~ betaIRR(1.0001, 1.0028 | OSTR=0.996), has these properties:

It has a null median, 1.00.
Its 95% probability interval of [0.026, 38.8] is extremely wide.
If m1 and m2 are the numbers of subjects who become COVID positive in the vaccine and placebo groups, respectively, then the nominal posterior distribution is IRR ~ betaIRR(1.0001 + m1, 1.0028 + m2), which allows the data to quickly dominate the prior. As also seen in Example 1, such priors make Bayesian analyses operate like frequentist analyses. However, the Bayesian schema leads to more straightforward inferential statements about the research questions.

This prior does have one arguably troublesome feature. The stated goal of this trial was to infer that IRR is less than 0.70. Yet, Prob[IRR < 0.70] = 0.412 before any data are collected. Notice also that Prob[IRR < 0.50] = 0.33. Looking at it this way, this prior actually injects a substantial amount of prior optimism into the analysis. Such calculations should be reported in the protocol and in subsequent trial reports.

We see that 8 vaccine cases vs. 162 placebo cases yields posterior results that firmly support efficacy. Note how BetaBinomial() accepts the desired transformation, IRR = P/((1-P)*OSTR). The posterior median and 95% probability interval are 0.054 and [0.025, 0.099]. Prob[IRR < 0.10 | 8 vaccine vs 162 placebo] = 0.976. Such direct inference is a main advantage of Bayesianism.

A null, skeptical (NS) prior for IRR. First, what is the difference between being "pessimistic" and being "skeptical?"

pessimistic: having a lack of hope that something good will happen.
skeptical: not easily convinced; having doubts or reservations.

A tenet of the Scientific Method is that investigators design and carry out their studies working optimistically yet maintaining a healthy skepticism about the True State of Nature. This changes only when good data, properly analyzed, "confirm" (weaker than "prove") what is very likely to be true. When testing hypotheses, Bayesian methods with skeptical priors incorporate this formally into the data analysis, something that frequentist null hypothesis testing (using p-values) does not.

Again, we seek a prior that gives IRR a median of 1.0. The stated goal of this trial was to infer that VE > 0.30, i.e., IRR < 0.70. To formally embed prudent scientific skepticism into the analysis, consider setting the 5% quantile for IRR at 0.70, i.e., Prob[IRR < 0.70 | before data] = 0.05. In other words, the prior probability is only a 5% that the vaccine's efficacy exceeds 0.30. Then we challenge the study to make Prob[IRR < 0.70 | after data] sufficiently high or low to infer whether or not the vaccine is efficacious. This aligns the analysis to the specific research question.

What is the IRR ~ betaIRR(a0, b0 | OSTR=0.996) prior distribution that has median of 1.00 and a 5% quantile of 0.70? And how do 8 vaccine cases vs. 162 placebo cases change that prior?

> OSTR <- 0.996
> irr <- c(0.01, 0.05, 0.10, 0.20, 0.35, 0.50, 0.70, 1.00)
> NS <- BetaBinomial(Median=P.IRR(1.0,OSTR),
+    P.Q=P.IRR(0.70,OSTR), Q=0.05,
+    Trans="P/((1-P)*OSTR)", TransName="IRR",
+    CP.points=P.IRR(irr,OSTR),
+    M=c(8, 162))

Transformation: t(P) = IRR(P) = P/((1-P)*OSTR)

Solution for prior: P ~ beta(a0 = 42.9, b0 = 43.1)

Fit of beta(a0, b0) to requested Median (IRR_0.50) and IRR_0.05.
*****************************************************************
   Median    IRR_0.05
As requested: 1.000 0.700
As fit: 1.000 0.700
*****************************************************************

Prior: P ~ beta(a0 = 42.9, b0 = 43.1)
Posterior: P ~ beta(a1 = 42.9 + 8 = 50.9, b1 = 43.1 + 162 = 205.1)

Median and 95% PI and its spread, IRR(P) = P/((1-P)*OSTR), P ~ beta().
**********************************************************************
Median Prob. Interval PI Spread
Prior 1.000 [0.653, 1.530] 0.877
Posterior 0.248 [0.181, 0.334] 0.154
**********************************************************************

100q% quantiles of IRR(P) = P/((1-P)*OSTR).
*********************************************************************
q: 0.005 0.025 0.050 0.250 0.500 0.750 0.950 0.975 0.995
Prior 0.571 0.653 0.700 0.864 1.000 1.157 1.428 1.530 1.752
Posterior 0.163 0.181 0.190 0.223 0.248 0.275 0.319 0.334 0.366
*********************************************************************
Prior: P ~ beta(42.9, 43.1)
Posterior: P ~ beta(50.9, 205.1)

Prob[t(P) < t(p)], where t(P) = IRR(P) = P/((1-P)*OSTR).
******************************************************************
t(p): 0.010 0.050 0.100 0.200 0.350 0.500 0.700 1.000
Prior    0.000 0.000 0.000 0.000 0.000 0.001 0.050 0.500
Posterior    0.000 0.000 0.000 0.090 0.988 1.000 1.000 1.000
******************************************************************
Prior: P ~ beta(42.9, 43.1)
Posterior: P ~ beta(50.9, 205.1)

> NS$cprob[1,5]
0.350
1.025158e-06

BetaBinomial() finds that P ~ beta(42.9, 43.1) fits the desired quantiles perfectly to three decimals. Using this prior is akin to having started the study with 86 positive COVID cases almost evenly divided between vaccine and placebo groups, which is considerable nullness to overcome for a sample size of 170. Nevertheless, the posterior distribution, IRR ~ betaIRR(42.9 + 8, 43.1 + 162 | OSTR=0.996), has a median of 0.248 and 95% PI of [0.181, 0.334]. Prob[IRR < 0.35 | no data] is virtually 0.00 (< 0.0000002), but Prob[IRR < 0.35 | 8 vs. 162] = 0.988. Such statistical cogency is compelling.

Comparing the P-B, ND, and NS priors. Comparing the P-B and ND priors is simple. They produce essentially the posterior distribution, but ND is based on clear, defensible logic. Choosing between the ND and NS priors depends on whether you want the data to have maximal influence (ND) or you want to incorporate scientific skepticism directly into the analysis (NS). These are starkly different philosophies and choosing between them—or using some other logic—should be situational, that is, study dependent. To be sure, however, whatever is decided should be done at the planning stage and delineated in the formal study protocol.

Incorporating more data (Bayesian sequential analysis). Using this schema allows us to "automagically" incorporate additional COVID events into the analysis to see how the distribution of IRR shifts and tightens to summarize the revised state of knowledge about the vaccine's efficacy.

The following is science fiction. Suppose the actual study (now called Phase 1) continued on (Phase 2) to have 181 more subjects develop COVID, 6 from the vaccine group and 175 from the placebo group. The overall surveillance time ratio in Phase 2 was OSTR = 1.015.

Based on the null skeptical (NS) prior employed above, the posterior median and 5% quantile for Phase 1 were 0.248 and 0.190, respectively. Using these as the prior median and 5% quantile for Phase 2 leads to:

> OSTR <- 1.015
> irr <- seq(0.05,0.30,0.05)
> BetaBinomial(Median=P.IRR(0.248,OSTR),
+    P.Q=P.IRR(0.190,OSTR), Q=0.05,
+    Trans="P/((1-P)*OSTR)", TransName="IRR",
+    CP.points=P.IRR(irr,OSTR),
+    M=c(6, 175))

Transformation: t(P) = IRR(P) = P/((1-P)*OSTR)

Solution for prior: P ~ beta(a0 = 50.6, b0 = 200.2)

Fit of beta(a0, b0) to requested Median (IRR_0.50) and IRR_0.05.
*****************************************************************
   Median    IRR_0.05
As requested: 0.248 0.190
As fit: 0.248 0.190
*****************************************************************

Prior: P ~ beta(a0 = 50.6, b0 = 200.2)
Posterior: P ~ beta(a1 = 50.6 + 6 = 56.6, b1 = 200.2 + 175 = 375.2)

Median and 95% PI and its spread, IRR(P) = P/((1-P)*OSTR), P ~ beta().
**********************************************************************
Median Prob. Interval PI Spread
Prior 0.248 [0.180, 0.335] 0.155
Posterior 0.148 [0.111, 0.194] 0.083
**********************************************************************

100q% quantiles of IRR(P) = P/((1-P)*OSTR).
*********************************************************************
q: 0.005 0.025 0.050 0.250 0.500 0.750 0.950 0.975 0.995
Prior 0.162 0.180 0.190 0.223 0.248 0.276 0.319 0.335 0.367
Posterior 0.101 0.111 0.116 0.134 0.148 0.163 0.186 0.194 0.211
*********************************************************************
Prior: P ~ beta(50.6, 200.2)
Posterior: P ~ beta(56.6, 375.2)

Prob[t(P) < t(p)], where t(P) = IRR(P) = P/((1-P)*OSTR).
**********************************************************
t(p): 0.050 0.100 0.150 0.200 0.250 0.300
Prior    0.000 0.000 0.001 0.091 0.520 0.891
Posterior    0.000 0.004 0.538 0.986 1.000 1.000
**********************************************************
Prior: P ~ beta(50.6, 200.2)
Posterior: P ~ beta(56.6, 375.2)

The Phase 2 data lower the IRR median to 0.148 and tighten the 95% PI by 46% (1 - 0.083/0.155). This demonstrates how seamlessly the method incorporates new data, and how easily the essential results can be communicated. Using this Bayesian approach avoids using "alpha spending" rules (e.g., the Lan–DeMets method) to control for multiple frequentist testing as data accumulate.

This illustrates a highly successful "March of Science." It began with such high scientific skepticism that Prob[IRR < 0.20 | no data] was near 0.00. The Phase 1 data (actual) took this to Prob[IRR < 0.20 | 8 vs. 162] = 0.090, a solid first leg of the March, but not yet confirmatory of 80% or better vaccine efficacy. The Phase 2 data (fictitious) gave us Prob[IRR < 0.20 | 8+6 vs. 162+175] = 0.986.

A simple text diagram displays the story in terms of Prob[IRR < 0.20]. Table 2.7 summarizes how the median, 95% PI, and 95% PI spread for IRR ~ betaIRR(a, b | OSTR) changed.

"March of Science" text diagram, adding 6 vaccine vs. 175 placebo.
**********************************************************************
   Start ====1===> End Phase1 ===2==> End Phase 2
   Data:    none    8 vs. 162    6 vs. 175
   OSTR: (1:1 randomiz'n) 0.996 1.015
Prob[IRR < 0.20]: <0.0000002    0.090 0.986
**********************************************************************

Table 2.7. Summarizing Phase 1 and Phase 2 analyses.
******************************************************************************
Prior IRR Distribution    Posterior IRR Distribution
   ******************************** ********************************
Data Median    95% PI PI Spread Median    95% PI PI Spread
8, 172    1.000 [0.65, 1.53] 0.877    0.248 [0.18, 0.33] 0.155
+6, +175 0.248 [0.18, 0.34] 0.155    0.148 [0.11, 0.19] 0.083
******************************************************************************

What if the Phase 2 data were less indicative of a low IRR than the Phase 1 data? Suppose the 181 Phase 2 COVID events had 36 from the vaccine group and 145 from the placebo group.

> OSTR <- 1.015
> irr <- seq(0.05,0.30,0.05)
> BetaBinomial(Median=P.IRR(0.248,OSTR),
+    P.Q=P.IRR(0.190,OSTR), Q=0.05,
+    Trans="P/((1-P)*OSTR)", TransName="IRR",
+    CP.points=P.IRR(irr,OSTR),
+    M=c(36, 145))

Transformation: t(P) = IRR(P) = P/((1-P)*OSTR)

Solution for prior: P ~ beta(a0 = 50.6, b0 = 200.2)

Fit of beta(a0, b0) to requested Median (IRR_0.50) and IRR_0.05.
*****************************************************************
   Median    IRR_0.05
As requested: 0.248 0.190
As fit: 0.248 0.190
*****************************************************************

Prior: P ~ beta(a0 = 50.6, b0 = 200.2)
Posterior: P ~ beta(a1 = 50.6 + 36 = 86.6, b1 = 200.2 + 145 = 345.2)

Median and 95% PI and its spread, IRR(P) = P/((1-P)*OSTR), P ~ beta().
**********************************************************************
   Median Prob. Interval PI Spread
   Prior 0.248 [0.180, 0.335] 0.155
Posterior 0.247 [0.194, 0.311] 0.117
**********************************************************************

100q% quantiles of IRR(P) = P/((1-P)*OSTR).
*********************************************************************
q: 0.005 0.025 0.050 0.250 0.500 0.750 0.950 0.975 0.995
Prior 0.162 0.180 0.190 0.223 0.248 0.276 0.319 0.335 0.367
Posterior 0.179 0.194 0.201 0.227 0.247 0.267 0.299 0.311 0.333
*********************************************************************
Prior: P ~ beta(50.6, 200.2)
Posterior: P ~ beta(86.6, 345.2)

Prob[t(P) < t(p)], where t(P) = IRR(P) = P/((1-P)*OSTR).
**********************************************************
t(p): 0.050 0.100 0.150 0.200 0.250 0.300
Prior    0.000 0.000 0.001 0.091 0.520 0.891
Posterior    0.000 0.000 0.000 0.044 0.546 0.952
**********************************************************
Prior: P ~ beta(50.6, 200.2)
Posterior: P ~ beta(86.6, 345.2)

In this case, The median for IRR barely changed, but the 95% PI spread shrunk 25%. As a result, Prob[IRR < 0.20] was cut in half.

"March of Science" text diagram, adding 36 vaccine vs. 145 placebo.
**********************************************************************
   Start ====1===> End Phase1 ===2==> End Phase 2
   Data:    none    8 vs. 162 36 vs. 145
   OSTR: (1:1 randomiz'n) 0.996 1.015
Prob[IRR < 0.20]: <0.0000002 0.090 0.044
**********************************************************************

Table 2.8. Summarizing Phase 1 and Phase 2 analyses.
******************************************************************************
Prior IRR Distribution    Posterior IRR Distribution
   ******************************** ********************************
Data Median    95% PI PI Spread Median    95% PI PI Spread
8, 172    1.000 [0.65, 1.53] 0.877    0.248 [0.18, 0.33] 0.155
+36, +145 0.248 [0.18, 0.34] 0.155    0.247 [0.19, 0.31] 0.117
******************************************************************************