Introduction
-
Multifactorial design. Many interventions are evaluated, both individually and in combination with one another.
-
Scheduled and frequent Bayesian analyses and decision rules. Trial data are analysed and operating decisions are made, at scheduled instances, as efficacy information becomes available instead of waiting for the trial to finish [22], without compromising the integrity of the trial design or the statistical inference [18, 21]. Based on the results of scheduled analyses and prespecified decision rules, if an intervention is shown to be clearly efficacious, randomisation is will be ceased for that domain and the results declared publicly. Inferior interventions will be dropped from the study, possibly to be replaced with other promising candidates. Similarly, to conserve trial resources, an intervention arm may be stopped for statistical futility if there is a low probability of demonstrating efficacy [23].
Trial structure
Silo
Domain (
\(\boldsymbol{\mathcal{D}}\))
| ||||
---|---|---|---|---|
Backbone antibiotic (\({\boldsymbol D}_{\mathbf1}\)) | Adjunctive antibiotic (\({\boldsymbol D}_{\mathbf2}\)) | Early oral switch (\({\boldsymbol D}_{\mathbf3}\)) | ||
Strata (S) | MSSA (\(s=1\)) |
\(\begin{array}{c} D_{11}=\\ \left\{ \begin{array}{l} d_{11}:\text {(Flu)cloxacillin}* \\ d_{12}:\text {Cefazolin} \\ \end{array}\right. \\ {Non\text {-}inferiority} \end{array}\)
|
\(\begin{array}{c} D_{21}=\\ \left\{ \begin{array}{l} d_{21}:\text {No clindamycin}* \\ d_{22}:\text {Clindamycin} \\ \end{array}\right. \\ {Superiority} \end{array}\)
|
\(\begin{array}{c} D_{31}=\\ \left\{ \begin{array}{l} d_{31}:\text {Usual care}* \\ d_{32}:\text {Early oral switch} \\ \end{array}\right. \\ {Non\text {-}inferiority} \end{array}\)
|
PSSA (\(s=2\)) |
\(\begin{array}{c} D_{12} =\\ \left\{ \begin{array}{l} d_{11}:\text {(Flu)cloxacillin}* \\ d_{13}:\text {Penicillin} \\ \end{array}\right. \\ {Non\text {-}inferiority} \end{array}\)
|
\(\begin{array}{c} D_{22}=\\ \left\{ \begin{array}{l} d_{21}:\text {No clindamycin}* \\ d_{22}:\text {Clindamycin} \\ \end{array}\right. \\ {Superiority} \end{array}\)
|
\(\begin{array}{c} D_{32}=\\ \left\{ \begin{array}{l} d_{31}:\text {Usual care}* \\ d_{32}:\text {Early oral switch} \\ \end{array}\right. \\ {Non\text {-}inferiority} \end{array}\)
| |
MRSA (\(s=3\)) |
\(\begin{array}{c} D_{13}=\\ \left\{ \begin{array}{l} d_{14}:\text {Vancomycin}* \\ d_{15}:\text {Vancomycin} + \text {cefazolin} \\ \end{array}\right. \\ {Superiority} \end{array}\)
|
\(\begin{array}{c} D_{23}=\\ \left\{ \begin{array}{l} d_{21}:\text {No clindamycin}* \\ d_{22}:\text {Clindamycin} \\ \end{array}\right. \\ {Superiority} \end{array}\)
|
\(\begin{array}{c} D_{33}=\\ \left\{ \begin{array}{l} d_{31}:\text {Usual care}* \\ d_{32}:\text {Early oral switch} \\ \end{array}\right. \\ Non\text {-}inferiority \end{array}\)
|
Domains
Subdomain
Subgroups
Covariates
Regions and countries
Epochs
Statistical modelling
Models
General linear function
-
\(\alpha _{s,u}\)—for participants in silo s and subgroup u, the value for eligible reference interventions, covariate level \(z_k \notin Z^*_k\), region \(r \notin R^*\), and current epoch.
-
\(\beta _{s,u,d_{kj}}\)—for participants in silo s and subgroup u, the effect of intervention \(d_{kj}\in D_{ks}^*\) compared to the reference intervention. Note that only parameters referencing eligible domains are included, where eligible domains for participant i are indexed by \(k_E\).
-
\(\psi _{s,u,d_{kj},d_{kl}}\)—for participants in silo s and subgroup u, the effect of the interaction between an intervention \(d_{kj} \in D_{ks}^*\) with an intervention \(d_{lj} \in D_{ls}^*\), where k is less than l. Note that only parameters referencing eligible domains are included, where eligible domains for participant i are indexed by \(k_E\) (and likewise, \(l_E\)). Also note that the parameter is specific to two-way interactions only.
-
\(\gamma _{D_k}\)—for all participants, the effect of ineligibility for domain \(D_k\). Note that only parameters referencing ineligible domains are included, where ineligible domains for participant i are indexed by \(k_I\).
-
\(\theta _{Z_k,z_{kj}}\)—the effect of covariate factor \(z_{kj} \in Z^*_{k}\), for all covariates \(Z_k\).
-
\(\delta _r\)—the effect of the region \(r \notin R^*\) in which the participant is located.
-
\(\omega _{c_r}\)—the effect of the country (that is nested within region) in which the participant i is located.
-
\(\phi _t\)—for participants the current 26 week epoch, where epochs are contiguous, the effect of time t.
Primary endpoint model
Prior distributions and model hierarchy
Parameters | Normal\(\boldsymbol(\boldsymbol a\boldsymbol,\boldsymbol b^{\mathbf2}\boldsymbol)\)
| Inv-Gamma(p, q) | ||
---|---|---|---|---|
a | b | p | q | |
Reference incidences (11) | ||||
\(\alpha _{s,u}\)
| -2 | 10 | – | – |
Effects of intervention | ||||
Silo-specific (12) | ||||
\(\mu _{{s,d_{kj}}}\)
| 0 | 1 | – | – |
\(\tau ^2_{{s,d_{kj}}}\)
| – | – | 1 | 1/16 |
Subdomain-fixed (13) | ||||
\(\mu _{{d_{kj}}}\)
| 0 | 1 | – | – |
\(\tau ^2_{{d_{kj}}}\)
| – | – | 1 | 1/16 |
Subdomain-exchangeable (14) | ||||
\(\tau ^2_{u,d_{kj}}\)
| – | – | 1/10 | 1/400 |
\(\xi _{{d_{kj}}}\)
| 0 | 1 | – | – |
\(\upsilon ^2_{{d_{kj}}}\)
| – | – | 1 | 1/16 |
Effects of domain eligibility (15) | ||||
\(\gamma _{D_k}\)
| 0 | 1 | – | – |
Effects of two-way interactions (16) | ||||
\(\psi _{s,u,d_{kj}, d_{k'j}}\)
| 0 | 1 | – | – |
\(\delta _r\)
| 0 | 10 | – | – |
\(\tau ^2_r\)
| – | – | 1 | 1/16 |
Effects of covariates (19) | ||||
\(\theta _{Z_k,z_k}\)
| 0 | 10 | – | – |
Effects of epoch (20) | ||||
\(\tau ^2_t\)
| – | – | 1/4 | 1/10 |
Reference log-odds
Effects of interventions
-
Where a silo has a unique subdomain (i.e. it exists only for a single silo), the log-odds ratios where the intervention is silo-specific are modelled as normally distributed such that all subgroups have the same silo-specific mean and variance:where a, b, p, and q are fixed values set by the blinded investigators. We refer to this prior structure as ‘silo-specific’.$$\begin{aligned} \beta _{s,u,d_{kj}}&\sim \mathcal {N}({\mu_{s, d_{kj}}}, \tau ^2_{{s,d_{kj}}})\nonumber \\ \mu _{{s, d_{kj}}}&\sim \mathcal {N}(a,b^2)\nonumber \\ \tau ^2_{{s,d_{kj}}}&\sim \text {Inv--Gamma}(p,q), \end{aligned}$$(12)
-
Where two or more silos share a subdomain, and the log-odds ratios are a priori considered be common across silos (i.e. \(\beta _{s,u,d_{kj}} = \beta _{u,d_{kj}}\) for all s), the silo- and subgroup-specific log-odds ratio for an intervention may be modelled as normally distributed with an intervention-specific mean and variance:where a, b, p, and q are fixed values set by the blinded investigators. We refer to this prior structure as ‘subdomain-fixed’.$$\begin{aligned} \beta _{u,d_{kj}}&\sim \mathcal {N}(\mu _{{d_{kj}}}, \tau ^2_{{d_{kj}}})\nonumber \\ \mu _{{d_{kj}}}&\sim \mathcal {N}(a,b^2)\nonumber \\ \tau ^2_{{d_{kj}}}&\sim \text {Inv--Gamma}(p,q), \end{aligned}$$(13)
-
Where two or more silos share a subdomain, and the log-odds ratios are a priori considered to be exchangeable, the log-odds ratios are modelled as normally distributed with a subgroup-specific mean and variance, with the subgroup-specific mean modelled as normally distributed with an intervention-specific mean and variance:where a, b, p, q, \(p*\), and \(q*\) are fixed values set by the investigators. This hierarchical prior structure ensures that the effect estimates for interventions in each silo will be ‘shrunk’ toward one another. Note that \(p*\) and \(q*\) are not necessarily equal to p and q, respectively. We refer to this prior structure as ‘subdomain-exchangeable’.$$\begin{aligned} \beta _{s,u,d_{kj}}&\sim \mathcal {N}(\mu _{{u, d_{kj}}}, \tau ^2_{{u,d_{kj}}})\nonumber \\ \mu _{{u, d_{kj}}}&\sim \mathcal {N}(\xi _{{d_{kj}}}, \upsilon ^2_{{d_{kj}}})\nonumber \\ \xi _{{d_{kj}}}&\sim \mathcal {N}(a,b^2)\nonumber \\ \tau ^2_{{u,d_{kj}}}&\sim \text {Inv--Gamma}(p,q)\nonumber \\ \upsilon ^2_{{d_{kj}}}&\sim \text {Inv--Gamma}(p*,q*), \end{aligned}$$(14)
Effects of domain eligibility
Effects of two-way interactions
Effects of regions and countries
Effects of covariates
Effects of epoch
Exploratory analyses
Computational methods
stan
, a probabilistic programming language [28], to numerically compute the joint posterior distributions of the parameters for each model based on the likelihood functions for the models and prior parameter distributions. Specifically, we will sample from the posterior distribution of each parameter of the model by using the Hamiltonian Monte Carlo algorithm implemented within stan
, called from the R
software environment [29]. For each parameter, we will run three or more MCMC chains in parallel for a ‘burn-in’ phase and sampling phase for as many iterations that are sufficient for the SNAP trial analytic team to be confident in the inference. Convergence of the MCMC chains will be assessed by the SNAP trial analytic team via the effective sample size, the scale reduction \(\hat{R}\) , and graphical representations of the MCMC chains [30].Missing data
Concurrently randomised cohorts
Randomisation
General principles
Response-adaptive randomisation
Ineligible or unavailable domains
Scheduled updates and decision rules
Data source
Posterior summaries
Decision rules
Superiority
Non-inferiority
Frequency of scheduled updates
Introducing new interventions
Trial conclusions and reporting
Changes to prespecified analyses
Initial implementation
Trial structure
-
The primary endpoint is all-cause mortality at 90 days after platform entry.
-
Each subdomain initially consists of two interventions only.
-
The subdomains of each silo within the backbone antibiotic domain are unique.
-
For the adjunctive and early oral switch domains, subdomains are the same across silos.
-
Only participants who have clinically stable disease and the ability to absorb or adhere to oral regimens, at day 7 for uncomplicated disease or day 14 for complicated disease will be eligible for the early oral switch domain.
Primary model
Prior distributions and model hierarchy
Decision rule thresholds
Trial operating characteristics
-
A maximum sample size of 7000 (comprising 6000 adults and 1000 children)
-
Proportion of patients in silos: MSSA (64%), PSSA (16%), and MRSA (20%)
-
Reference mortality rates for adults and children in each silo, respectively, for participants who were not eligible for early oral switch: MSSA (15%, 2.2%), PSSA (15%, 2.2%), MRSA (20%, 3.5%)
-
Proportion of participants eligible for early oral switch domain, for adults and children (respectively), at day 7 (10%, 45%) and day 14 (60%, 30%). Note that eligibility for early oral switch modified the reference mortality rates (as detailed in Additional file 1)
-
Scenario 1 refers to the ‘null’ scenario in which the odds ratios were set equal to 1 where superiority was the main objective and 1.2 where non-inferiority was the main objective
-
Scenario 2 refers to the ‘moderate’ treatment effect scenario where the odds ratio was set to 0.75 for all scenarios
-
For the backbone antibiotic domain, decisions are based upon the ‘silo-specific’ effects of interventions (see (12)). For the adjunctive antibiotic domain, decisions are based upon the ‘subdomain-fixed’ effects of interventions (see (13)). For the early oral switch domain, decisions are based upon the ‘subdomain-exchangeable’ effects of interventions (see (14))
Scenario | Strata | Non-inferiority | Superiority | ||||
---|---|---|---|---|---|---|---|
Backbone | Adjunctive | EOS | Backbone | Adjunctive | EOS | ||
1 | MSSA | 0.02 | – | 0.07 | 0.00 | 0.07 | – |
PSSA | 0.06 | – | 0.03 | 0.01 | 0.07 | – | |
MRSA | – | – | 0.04 | 0.01 | 0.07 | – | |
2 | MSSA | 0.99 | – | 0.95 | 0.77 | 0.93 | – |
PSSA | 0.61 | – | 0.78 | 0.29 | 0.93 | – | |
MRSA | – | – | 0.84 | 0.46 | 0.93 | – |