By the summer of 2017, ZOVID-17 (ZOgaVIrusDisease; fictitious) had infected over 300 billion people worldwide in the previous nine months. The acute phase of the illness is usually mild and lasts only 2-7 days. but still with no effective treatment options, around 3% were experiencing severe illness that caused long-term organ damage, and, in rare cases, death.

In June 2021, MarWes Pharmaceuticals began conducting randomized clinical trials to test their experimental ZOVID vaccine, VM17W19, versus placebo. The primary incidence event (IE) was lab confirmed ZOVIC infection (ZOVIC+). The formal protocol, vetted by the U.S. Food and Drug Administration (FDA), called for using a Bayesian schema to conjoin a small Phase I/II trial with a subsequent Phase III trial.

Phase I/II trial. In order to rapidly acquire a sufficient sample size in the VM17W19 group for comprehensive safety studies (not dealt with here), this protocol called for recruiting subjects from high-risk populations and first randomized in a 2:1 ratio, VM17W19 to placebo. Subjects and almost all researchers were blinded. (The blind was broken on an as-truly-needed basis to evaluate serious adverse events.) After at least two ZOVID+ events occurred over both groups, the randomization ratio was changed 1:1. The trial was planned to end when at least three more ZOVID+ events occurred, totaling five events (2 + 3 = 5) over both groups. However, a sixth ZOVID+ event occurred on the same day (Day 30) as the fifth, so six ZOVID+ events occurred among the 509 VM17W19 and 295 placebo subjects randomized by that time. All six ZOVID+ events were placebo subjects. A basic summary of these results was provided to all subjects, and because no serious safety signals were raised, placebo subjects were offered the VM17W19 vaccine.

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Phase III trial. Subjects in this huge, multi-center trial were recruited through public announcements and thus had a substantially lower base infection rate than those in the Phase I/II trial. The randomization ratio began at 1:1, but by the 40th ZOVID+ event (not counting the six events in the Phase I/II trial), the efficacy and safety results were so strongly positive that the trial's Data Monitoring Committee changed this to 3:1, VM17W19 to placebo. As per protocol, the dataset was locked for the first analysis when the 60th ZOVID+ event occurred. finish this later. These results were publicly shared, and were so finish this later.

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Assaying vaccine efficacy. Why does the word "assay" appear in the title above, instead of "test"? When considered fairly, the majority of research questions are not testing hypotheses. In general, statistical plans are better formed as estimating some focal parameter and quantifying the uncertainty about that estimate. This applies even when the research question is a specific hypothesis. Bayesianism enables us customize the statistical approach so that it aligns tightly with posing and answering our research questions. In this case, MarWes sought to both estimate the efficacy of VM17W19 and test whether it exceeded some minimal threshold.

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The focal parameter here is the incidence rate ratio, IRR = IR.V/IR.P, where IR.V and IR.P are the incidence rates for VM17W19 and placebo. If VM17W19 has no affect at all, positive or negative, then IRR = 1.0. Note that the commonly used parameter, vaccine efficacy, is merely VE = 1 - IRR. The study protocol stated that the primary statistical goal was to infer that VE exceeded 30%, an effect equivalent to IRR being less than 0.70.

On frequentism. The common frequentist approach would be to calculate the p-value associated with the point null hypothesis, H0: IRR = 0.70. Two points:

• The logic behind this rests on the shaky premise that the null hypothesis is true. Thus, if IRR has little or no chance of being exactly 0.70 (or extremely close to it), then the logic falls apart.

• Testing H0: IRR = 0.70 is equivalent to simultaneously testing two directional null hypotheses, H0': IRR ≥ 0.70 yielding the p-value p', and H0'': IRR ≤ 0.70 yielding p''. The p-value for H0: IRR = 0.70 is p = 2*min(p', p''). But the sole statistical question here is whether IRR is less than 0.70, which aligns with testing H0': IRR ≥ 0.70. Thus, p' is the p-value of interest. This is computed exactly below along with it's exact 95% confidence interval, which has the form [0.00, UCL].

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The case for Bayesianism. MarWes choose to use the Bayesian approach embodied within IRR.Bayes(), because it offered distinct advantages for "assaying" the efficacy of VM17W19.

• The exact value of this study's IRR is unknowable, but Bayesianism let them model it as a random variable. Plotting its distribution visually illustrated the overall story. Numerical descriptions were provided by the median (50% quantile), and 2.5% and 97.5% quantiles of this IRR distribution.

• P-values were avoided. Instead, MarWes was able to directly ask and answer, What are Prob[IRR < 0.70], Prob[IRR < 0.50], and Prob[IRR < 0.20]?

• Both the Phase I/II and Phase III studies addressed the primary efficacy question using the same focal parameter, IRR. This Bayesian approach allowed the data from those studies to be fused into a single analysis.

• The data were analyzed under competing explicit, tailored models for what was then known and/or believed about the distribution of IRR when planning the study. Factored into this were issues related to what kinds of inferences the study needed to make.

>   Ex1a <- IRR.Bayes(DFrame=NULL,
+                     GroupNames=c("VM17W19","Placebo"),
+                     IRR0.50=1.00,
+                     FitDiffuse=TRUE,
+                     NUSR=2,
+                     PI.level=c(0.90,0.95,0.99),
+                     IRR.points=c(0.05,0.10,0.25,0.50,0.70,1.00))

Fit of initial prior, IRR ~ betaIRR(1.67, 1.00 | NUSR = 2.00)
*************************************************************
               IRR0.50 (Median)      IRR0.01
As requested:           1.0000       0.0337
      As fit:           0.9695       0.0337
*************************************************************

Medians and Probability Intervals
********************************************
                 Median            LPL-UPL
Level: 0.900      0.970    [0.099, 15.998]
       0.950      0.970    [0.061, 32.666]
       0.990      0.970   [0.022, 166.000]
********************************************

Quantiles of Prior Distribution of IRR
******************************************************************************
           Q:   0.005  0.025  0.050  0.250  0.500  0.750  0.950  0.975   0.995
Quantiles:      0.022  0.061  0.099  0.385  0.970  2.654 15.998 32.666 166.000
******************************************************************************

Prob[IRR < irr] for Prior Distribution of IRR
***************************************************************
                 irr:  0.050  0.100  0.250  0.500  0.700  1.000
Prob[IRR < irr]:       0.018  0.050  0.160  0.315  0.407  0.509
***************************************************************

IRR.GenData() was written to work with IRR.Bayes() and is included in the downloaded file. To be clear, deciphering what IRR.GenData() does in this example, and especially, how it does it, is not needed to learn how to use IRR.Bayes(). However, for those yearning to know, here are the essentials.

On any given day of the trial, $StartDay, IRR.GenData() created independent N.d cases, with N.d distributed as a Poisson variate with a mean recruitment of RecRate, such as 400 per day. These were randomized in a RanRatio:1 ratio, Groups[1] = "VM17W19" to Groups[2] = "Placebo", giving sample sizes for that day of n1.d and n2.d. The randomization ratio was initially 1:1, but changed to 3:1 after the first 50 ZOVID+ events were distributed 5:45, VM17W19 to placebo.

For each subject, two random variables were generated and rounded to the nearest integer.

• IEDay, the day being confirmed ZOVID+, was modeled as an exponential variate, which has a constant hazard rate. For placebo, this was 12 ZOVID+ cases per 100,000 surveillance days: H.placebo(t) = 0.00012. (In the United States, population 331.5 million, this is almost 40,000 per day.) The true constant hazard ratio was set at H.VM17W19(t)/H.placebo(t) = 0.10, making H.VM17W19(t) = 0.000012. Prob[IEday < 90.50 | placebo]  = 0.01080; and Prob[IEday < 90.50 | M17W19] = 0.00109.

• Cday, the day of random right (non-informative) censoring, was modeled as an exponential variate with rate 3 per 100,000 surveillance days: CensRate = 0.00003.

The data generation stopped on the day that the total number of confirmed ZOVID infections was at least 70. For this run (Seed = 987651), 14,845 subjects (52.1%) had received VM17W19 and 13,631 subjects had received placebo.
   
IRR.GenData() returns $EventDay and $EventType for each subject based on their IEday, Rday, and Cday values:

• $EventDay is the minimum of IEday and Cday.

• Regarding $EventType, if $EventTime = IEday, then $EventType = "ZOVID+". Else, $EventType = "censored".

• Finally, all subjects with $EventDay > StopDay (here, 70) are reset to have $EventDay = StopDay + 1 (here, 71), and EventType = "censored".

The returned data.frame has the common structure for a two-group time-to-event analysis with right censoring, one row per subject with these variables:
  $Group ... "VM17W19" or "Placebo"
  $StartDay ... day of study (beginning at 0) that subject was first treated
                and thus became under surveillance for ZOVID.
  $EventDay ... day of study that the subject was deemed to be ZOVID+.
  $EventType ... type of event, here "ZOVID+" or "censored".