Model specification

Primary outcome

The primary outcome is treatment success at 12 months post platform entry, defined as all of:

1. Alive
2. Clinical cure (no clinical or microbiological evidence of infection)
3. No ongoing use of antibiotics for the index joint; and
4. The prosthesis present after the initial management strategy is complete (destination prosthesis) still in place

referred to as ‘treatment success’.

Silos

Different cohorts enter inform different parts of the experiment.

Early infection

Patients with early stage infection are not revealed to the surgical domain. The surgical intervention will usually be dair for which 12 weeks of backbone antibiotics are recommended. There are, however, instances where early stage infection patients will receive either one-stage or two-stage revision. These patients will be able to enter the randomised backbone duration domain (one-stage, but not two-stage) and the extended prophylaxis domain (two-stage, but not one-stage). Early silo patients will also enter the choice domain if they are eligible to do so.

Late infection

Patients with late infection can enter all domains with some restrictions. They are randomised to dair vs revision in the surgical domain. Patients allocated to revision will receive a one or two-stage procedure based on self-selection. Both the planned surgery (one-stage/two-stage) and the surgery actually performed should be captured – the former should be recorded at the time of randomisation. Patients receiving one-stage receive randomised backbone antibiotics. Patients receiving two-stage receive randomised extended prophylaxis. Late silo patients will also enter the choice domain if they are eligible to do so.

Chronic infection

Patients with chronic stage infection are not randomised into the surgical domain. Like the early silo cohort, they can enter into the antibiotic backbone domain and the extended prophylaxis domain based on the type of surgery they receive. Chronic silo patients will also enter the choice domain if they are eligible to do so.

Causal structure

The treatment options and outcome status are minimally dependent on silo membership, randomisation and self selection. We assume the following DAGs as a simplified representation of how patients reach their treatment status across the domains.

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flowchart LR
  ER(E<sub>R) --> RA(R<sub>A) 
  ER --> SR(S<sub>R)
  ED(E<sub>D) --> DA(D<sub>A) 
  SR --> RA 
  SD(S<sub>D) --> DA 
  R --> RA
  RA --> RP(R<sub>P) 
  SRP(S<sub>R<sub>P) --> RP
  RA --> Y
  RP --> ED
  D --> DA(D<sub>A)
  DA --> Y
  F --> FA(F<sub>A)
  EF(E<sub>F) --> FA
  SF(S<sub>F) --> FA 
  FA --> Y
  UER((U<sub>E<sub>R)) -.-> ER
  UED((U<sub>E<sub>D)) -.-> ED
  UEF((U<sub>E<sub>F)) -.-> EF
  UF((U<sub>F)) -.-> F
  URP((U<sub>R<sub>P)) -.-> SRP
  UD((U<sub>D)) -.-> D
  USR((U<sub>S<sub>R)) -.-> SR
  UR((U<sub>R)) -.-> R
  USD((U<sub>S<sub>D)) -.-> SD
  USF((U<sub>S<sub>F)) -.-> SF
  UY((U<sub>Y)) -.-> Y

Figure 1: Assumed causal diagram representing the (simplified) processes on successful treatment

where the following definitions apply:

• \(E_R\) reveal for surgical domain - 0: no, 1: yes
• \(E_D\) reveal for duration domain - 0: no, 1: yes
• \(E_F\) reveal for choice domain - 0: no, 1: yes
• \(R\) randomised surgery - 0: DAIR, 1: revision
• \(S_R\) revision type preference (pre-randomisation) - 0: DAIR, 1: one-stage, 2: two-stage
• \(R_A\) allocated surgery - 0: DAIR, 1: one-stage, 2: two-stage
• \(R_P\) performed surgery - 0: DAIR, 1: revision
• \(S_{R_P}\) revision type performed (post-randomisation) - 0: dair, 1: one-stage, 2: two-stage
• \(D\) randomised duration - 0: long, 1: short (for one-stage), 0: short, 1: long (for two-stage)
• \(D_A\) allocated duration - 0: long, 1: short, 2: other
• \(S_D\) selected duration - 0: long, 1: short, 2: other (sometimes duration will be selected rather than randomised)
• \(F\) randomised choice - 0: norif, 1: rif
• \(F_A\) allocated choice - 0: norif, 1: rif, 2: other
• \(S_F\) selected choice - 0: norif, 1: rif, 2: other
• \(Y\) treatment success - 0: no, 1: yes

Research questions

The questions of interest are as follows:

For the surgical domain we evaluate whether revision is superior to dair and futility for superiority. This applies only to the late silo cohort.

For the backbone duration domain, we evaluate whether short duration (6 wk) is non-inferior to long duration (12 wk) antibiotic treatment. This is applicable only to participants that receive one-stage revision. We also evaluate whether the non-inferiority decision is futile in the sense that it appears unlikely that non-inferiority will ever be established.

For ethe extended prophylaxis domain, we evaluate whether a long duration (12 wk) is superior to a short duration (7 days) of treatment. This is applicable only to participants that receive two-stage revision (the extended prophylaxis follows the second operation). We also evaluate whether the superiority decision is futile in the sense that it appears unlikely that superiority will ever be established.

For the choice domain we evaluate whether rifampicin is superior to no rifampicin. This is applicable to the participants that are enter into the choice domain. We also evaluate whether the superiority decision is futile in the sense that it appears unlikely that superiority will ever be established.

Model specification

For each silo \(l\) and site of infection \(j\) we therefore simulate the probability of treatment success as:

\[ \begin{aligned} \text{logit}(\pi) &= \mu + \lambda_s + \rho_j + \phi_{l} + \sum_{d=1}^{D} x_d^\top \vec{\beta_d} + \zeta_{r,v} + \tau_t + z^\top \vec{\omega} \end{aligned} \tag{1}\]

• \(\mu\) grand mean log-odds of treatment success; it serves as a reference from which all other effects deviate
• \(\lambda_s\) change associated with membership silo \(s\)
• \(\rho_j\) change associated with site of infection (joint) \(j\)
• \(\phi_{l}\) preference for surgical approach under revision type \(l\) with elements for non-randomised treatment, one-stage and two-stage
• \(\vec{\beta_d}\) change associated treatment allocation with domain \(d\)
• \(\zeta_{r,v}\) change associated with site \(v\) nested within region \(r\)
• \(\tau_t\) change associated with randomisation period \(t\)
• \(\omega\) parameters associated with baseline factors

The trial data will be modelled as above with decisions made on the basis of the joint posterior, but an additional analysis model run with treatment by site of infection (hip/knee) interactions to characterise and report treatment heterogeneity.

Priors

todo

Treatment effects

For the surgical domain, the effect of interest is an average conditional log-odds ratio and is applicable to the late silo. This amounts to a weighted combination of \(\beta\) parameters associated with one and two stage revision. If we let \(\beta_1\) correspond to the effect of one-stage revision and \(\beta_2\) correspond to the effect of two-stage revision (both being for the cohort receiving randomised surgical intervention), then we would have:

\[ \begin{aligned} \Delta_R = \beta_1 \mathbb{E}[S_{R_P} = 1 \land R = 1] + \beta_2 \mathbb{E}[S_{R_P} = 2 \land R = 1] \end{aligned} \]

where the \(S_{R_P}\) denotes the surgery received and \(R\) indicates randomised surgical intervention (revision).