Complex decisions based on multivariate outcomes in Bayesian world

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I’ve run into something of a challenge for a Bayesian analysis I’m preparing. Briefly, the challenge goes something like this:

  • A flu vaccine has antigens for three strains.
  • We want to evaluate whether—I’m going to simplify this a little bit—high dose results in better immunogenicity than standard dose.
  • I want to be able to construct a “declaration” rule a priori based on a posterior probability of an event that looks something like this (although I welcome feedback on whether this is advisable or not):
    • Geometric mean ratio (comparing high dose to standard) > 1.0 for all antigens, OR
    • Geometric mean ratio (comparing high dose to standard) > 1.25 for two antigens and > 0.8 for all antigens, OR
    • Geometric mean ratio (comparing high dose to standard) > 1.5 for one antigen and > 0.9 for all antigens

Ostensibly, I would use a posterior probability threshold of something like 0.90 or 0.95. I pulled out the numbers 1.25, 0.8, 1.5, and 0.9 somewhat arbitrarily but I imagine the logic is apparent: if we sacrifice superiority on one antigen, then the threshold for the other should certainly be higher than “ratio > 1.” If we’re going to sacrifice superiority on two antigens, then the threshold for such a declaration should be more stringent and/or the bar for the other antigen should be yet higher.

But…I honestly am not aware of whether there is a principled way to develop these numbers or if there is a way to build an “equivalence class” of decision rules that takes a whole bunch of combinations into account at once. Has anyone done something like this?

Ultimately, we will have 95% credible intervals on the geometric mean ratio individually, but I’m trying to do something principled toward a decision rule in advance if it’s at all possible. Any advice on choosing thresholds and/or making the above proposal stronger is appreciated.

DISCLAIMER: I’m not asking to get into a debate about immunogenicity—I am well aware that HAI titers do not translate to protection from severe complications from infection; this is a phase 2 immunogenicity and safety trial and not an effectiveness trial.

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Why not lean into DOE here, and use a factorial design to estimate the response surface?

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Thanks for your reply. I should clarify a couple of things:

  • We are not in the design stage—the study was already designed and data collection is ongoing.
  • All flu vaccines contain these three antigens, so we’re not in a situation where we’re trying to assess multiple independently manipulable interventions or their interactions.

The idea is more that we want a concrete rule that we can pre-specify that would allow us to conclude overall superiority of the high dose relative to the standard dose, both of which contain antigen from all three strains.

Overall superiority includes a lot of different scenarios, including—most obviously—the one in which all three are conclusively better…but also other scenarios including superiority on two and non-inferiority on another. The question is about how to formulate an event like that—one based on a complicated series of unions and intersections—using thresholds/cutoffs that are formed in the most principled way.

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Principled formation of clinically/public health relevant assertions is a great way to go and is in sharp contrast to the informal way that everyone tries to put together evidence from multiple simple assertions. To me the only question is whether there is a consensus about how the assertions are formed and the thresholds they use. Sometimes it’s beneficial to find 10 experts without vested interests and have them provide independent opinions about the assertions, then see if there is consensus. Occasionally averaging opinions that disagree is still OK also.

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My point had been to try the 8 formulations \{low, high\}^3. But too late now, of course.