short superior to long?

Needs to be multiplied by R above? All DAIR durations are non-randomised, so only applicable if received revision.

Should all of these be silo specific? E.g. one-stage better/worse in early silo doesn't mean one-stage better in chronic silo. Is DAIR preferred for early, but revision preferred for chronic? Is that preference due to perceived efficacy? Reason for doing one-stage in early silo may be different than reason for doing one-stage in chronic, i.e. different patient mix. Would this introduce too much complexity.

For an early silo patient who has randomisation revelead for all domains but receives non-randomised DAIR and no rifampicin?

Late silo?

Yes, I used average conditional log-odds ratio, as that is the more basic quantity. It turns out that, under the assumed model, the average conditional log-odds ratio (over S) is equal to the conditional log-odds ratios weighted by the distribution. But its not obvious to me that that relationship always holds for any model (it clearly wouldn't if it wasn't a logistic model, but even for any logistic model it's not immediately obvious to me that it always holds, but it probably does?).

Plus infinitely many other weighted combinations. When we say revision versus DAIR, which comparison do we care about? "average" revision (50% long/50% short duration?) vs DAIR?

Yes, if we want to avoid having to marginalise over every variable in the model. Most of the above was just to run through and check the intuition that the beta_1 + beta_2 E[S] was a reasonable summary parameter to use and to distinguish between planned revision type and revision vs. DAIR. Point of the rest of the doc was to check this still makes sense once duration and other domains are brought in.

Yes, so this is a simple summary of the model parameters for comparing revision to DAIR. It is the conditional (on all model covariates) log-odds ratio for revision vs DAIR averaged over type of revision. Say for every patient in the population for whom a one-stage is preferred, we assigned them all to DAIR or revision (which will be one-stage). Then the condtional log-odds ratio is beta_1. If we do the same for everyone for whom two-stage is preferred the conditional log-odds ratio is beta_1 + beta_2. The average conditional log-odds ratio over all patients in the population is beta_1 + beta_ * E[S].

As opposed to considering the marginal log-odds ratio (or risk difference) in which we are marginalizing over every covariate in the model.

In terms of decision making, there is not really any difference (if all we care about is direction of effect), but the former is more easily calculated and perhaps more easily interpreted (e.g. for the marginal log-odds ratio, if site/surgeon is in the model, how do we want to average over them? Integrate out random effects (even if we know a Normal is a poor model for the actual population)? Average over sites included in the sample? Average over surgeons included in the sample? Weighted by how often they appear in the sample? Why should any of these generalize any more than the simpler conditional effect? Perhaps it's simpler to reason about trade-offs if considering an unconditional effect?

I don't really have a preference. Just noting that we could look at either. Could look at the completely unconditional comparison or the average conditional one. The one comparison that we can't look at is the completely conditional comparison because we need to integrate out revision type.

Yes, just noting that the average conditional log odds ratio has the same functional form irrespective of adjustment for S.

Yes, full sample. In the below I'm just being explicit that these things aren't necessarily equal, they only are equal because R is randomised.

I mean that the first model is comparing revision to DAIR (alpha_1) whereas the second model is comparing a specific revision type to DAIR (e.g. beta_1 and beta_1 + beta_2), which is not a randomsied comparison).

Just that "total" revision vs. DAIR is the only randomised comparison we can make. We might wish we could compare two-stage vs. one-stage or two-stage vs DAIR, but we can't under the design restrictions.

As per meeting, mean the revision type eventually received, not the underlying preference for a given revision type.

Given that we assume the effect of revision is conditional on the nature of that revision, it's not clear to me what "DAIR vs. Revision" means for late infection silo,

Just thinking through it for myself...

The options are:

• A = DAIR -> B = 12w
• A = Revision(one) -> B in {12w, 6w}
• A = Revision(two) -> B in {12w, 7d}
• C in (no rifampicin, rifampicin)

Currently, (just focusing on surgery/duration, and making 12w the reference for all groups rather than 7w/6d) cell parameters as specified in the model are:

$$ \begin{matrix} \text{DAIR} \ \text{Revision(one), 12w} \ \text{Revision(two), 12w} \ \text{Revision(one), 6w} \ \text{Revision(two), 7d} \end{matrix} \begin{pmatrix} \alpha \ \alpha + \beta_A \ \alpha + \beta_A \ \alpha + \beta_A + \beta_{B1} \ \alpha + \beta_A + \beta_{B2} \end{pmatrix} $$

So, effect of one-stage + 12 w assumed equal to the effect of two-stage + 12w, then revision type specific deviations from those. And "DAIR vs Revision" (beta_A) is really DAIR vs. weighted average of one-stage 12w and two-stage 12w, i.e. ignores the duration options.

I'm guessing this is the only randomised comparison we can make: a weighted average of one/two + 12w is the "default" revision.

As an alternative, I assume it's plausible that one-stage 12w and two-stage 12w differ due to clinician selection of one/two stage. So there may be preference (or maybe it just makes things messy) to have something like

$$ \begin{matrix} \text{DAIR} \ \text{Revision(one), 12w} \ \text{Revision(two), 12w} \ \text{Revision(one), 6w} \ \text{Revision(two), 7d} \end{matrix} \begin{pmatrix} \alpha \ \alpha + \beta_{A1} \ \alpha + \beta_{A2} \ \alpha + \beta_{A1} + \beta_{B1} \ \alpha + \beta_{A2} + \beta_{B2} \end{pmatrix} $$

Noting that \beta_{A1} and \beta_{A2} aren't "causal" in the sense that any differences could just be due to selection bias rather than differences in effectiveness of one/two stage.

The effect of revision versus DAIR will depend on what "revision" means. We can't just compare one-stage 12 weeks to DAIR vs 12 weeks because surgeon's choose who gets one-stage. Only comparison that seems to make sense is weighted combination of one/two stage, with weight as observed in the trial. I think that comparison makes sense, but maybe not.

Assume that $$p_{A1}$$ is the proportion randomised to revision who are selected to receive one-stage and $$1 - p_{A1}$$ the proportion selected to receive two-stage. Then the comparison for any revision versus DAIR might be taken to mean

$$ p_{A1}(0.5\beta_{A1} + 0.5(\beta_{A1} + \beta_{B1}))\ + (1 - p_{A1})(0.5\beta_{A2} + 0.5(\beta_{A2} + \beta_{B2})) $$

which just explicitly allows for the differences. Or some other combination of groups, where we assume that selection of one/two stage in the trial is the same in the population. Presumably though there are issues in interpreting such a comparison as "causal", unless also adjust for factors determining one/two stage selection.

The \beta_{A1} and \beta_{A2} are necessary for estimation of the duration effects, unless willing to assume no differences between one/two stage 12w.

necessary?