The SNAP trial data will be analysed using Bayesian statistical methods, which combine probability distributions that summarise the state of knowledge independent of the observed data (a prior probability distribution) with the observed data model (through a likelihood function) to produce probability distributions that reflect an updated state of knowledge (a posterior probability distribution).

ModelsGeneral linear function

The SNAP trial core protocol and its appendices specify multiple different endpoints for analysis, including binary, continuous, time-to-event, and ordinal endpoints. Here, we define the general linear function that will be used to model these endpoints using the appropriate linking functions. Note that we use the notation s(i), u(i) to denote the silo and subgroup to which participant i belongs, respectively. Similarly for participant i, \(d_\) indicates the treatment received, \(D_k(i)\) indicates domain non-membership, \(z_\) indicates covariate level, r(i) and \(c_r(i)\) indicate region and country location, and t(i) indicates epoch during which they were randomised. The general linear function is defined as follows:

$$\begin f_i&= \alpha _ + \sum \limits _} \beta _} + \sum \limits _)} \psi _,d_} \nonumber \\&\quad +\sum \limits _} \gamma _ + \sum \limits _}\theta _} + \delta _ + \omega _ + \phi _. \end$$

(9)

The parameters described in (9) are defined, for a general endpoint, as follows:

\(\alpha _\)—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.

\(\beta _}\)—for participants in silo s and subgroup u, the effect of intervention \(d_\in D_^*\) compared to the reference intervention. Note that only parameters referencing eligible domains are included, where eligible domains for participant i are indexed by \(k_E\).

\(\psi _,d_}\)—for participants in silo s and subgroup u, the effect of the interaction between an intervention \(d_ \in D_^*\) with an intervention \(d_ \in D_^*\), 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.

\(\gamma _\)—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 \(k_I\).

\(\theta _}\)—the effect of covariate factor \(z_ \in Z^*_\), for all covariates \(Z_k\).

\(\delta _r\)—the effect of the region \(r \notin R^*\) in which the participant is located.

\(\omega _\)—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

The SNAP trial primary endpoint is binary, denoted \(y \in \\), and is modelled using Bernoulli distribution with a logistic link function such that:

$$\begin y_i&\sim }(\pi _i)\nonumber \\ \pi _i&= }^\left( f_i\right) , \end$$

(10)

where \(\pi\) is the probability of the event conditional on the terms described in (9).

We interpret the \(\alpha _\) in (10) as the log-odds of the 90-day all-cause mortality for eligible domain reference intervention in silo s and subgroup u for the reference covariates and the \(\beta _}\) as the log-odds ratio of the 90-day all-cause mortality, relative to the reference intervention, of domain interventions \(d_\) in silo s and subgroup u.

Prior distributions and model hierarchy

The following subsections outline the prior distribution structure for the parameters of the model using a Bernoulli distributed endpoint and logistic link function, including the primary model. The model for alternatively distributed endpoints will require an alternative prior structure, which will be described in detail in an openly available, domain-specific statistical analysis plan that will be published prior to the closure of a domain for terminal analysis. For clarity, a simplified graphical model is provided for the primary endpoint in Fig. 1. The following presentation is general; however, the fixed values chosen by investigators for the SNAP trial are described in later sections and in Table 2.

Fig. 1

Graphical model of primary endpoint (simplified to intervention parameters only). IG, inverse-gamma

Table 2 Prior distributions and model hierarchyReference log-odds

The log-odds for domain-specific interventions \(d_\) are defined for each silo s and for each subgroup u and are assigned independent normal prior distributions as follows:

$$\begin \alpha _ \sim \mathcal (a, b^2), \end$$

(11)

where a and b are fixed values set by the investigators to be weakly-informative for all silos and subgroups.

Effects of interventions

The log-odds ratios for domain-specific interventions \(d_\) are similarly defined for each silo s and for each subgroup u; however, the hierarchical prior structure depends on the a priori assumptions of exchangeability of these parameters across silos. Irrespective of the different exchangeability assumptions, the prior structure is always hierarchical in that it allows information between subgroups (i.e. adult and paediatric groups) to be shared. The different prior structures are summarised as follows:

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:

$$\begin \beta _}&\sim \mathcal (}}, \tau ^2_}})\nonumber \\ \mu _}}&\sim \mathcal (a,b^2)\nonumber \\ \tau ^2_}}&\sim \text (p,q), \end$$

(12)

where a, b, p, and q are fixed values set by the blinded investigators. We refer to this prior structure as ‘silo-specific’.

Where two or more silos share a subdomain, and the log-odds ratios are a priori considered be common across silos (i.e. \(\beta _} = \beta _}\) 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:

$$\begin \beta _}&\sim \mathcal (\mu _}}, \tau ^2_}})\nonumber \\ \mu _}}&\sim \mathcal (a,b^2)\nonumber \\ \tau ^2_}}&\sim \text (p,q), \end$$

(13)

where a, b, p, and q are fixed values set by the blinded investigators. We refer to this prior structure as ‘subdomain-fixed’.

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:

$$\begin \beta _}&\sim \mathcal (\mu _}}, \tau ^2_}})\nonumber \\ \mu _}}&\sim \mathcal (\xi _}}, \upsilon ^2_}})\nonumber \\ \xi _}}&\sim \mathcal (a,b^2)\nonumber \\ \tau ^2_}}&\sim \text (p,q)\nonumber \\ \upsilon ^2_}}&\sim \text (p*,q*), \end$$

(14)

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’.

Effects of domain eligibility

The log-odds ratios for domain ineligibility are assigned independent normal prior distributions such that:

$$\begin \gamma _ \sim \mathcal (a,b^2), \end$$

(15)

where a and b are fixed values set by the blinded investigators.

Effects of two-way interactions

The log-odds ratios for two-way interactions are assigned independent normal prior distributions such that:

$$\begin \psi _,d_} \sim \mathcal (a,b^2), \end$$

(16)

where a and b are fixed values set by the blinded investigators and k is less than \(k'\). Where an interaction is considered a priori implausible, the interaction terms are set simply to zero.

Effects of regions and countries

The log-odds ratios for regions are assigned normal distributions such that:

$$\begin \delta _r \sim \mathcal (a,b^2), \end$$

(17)

where a and b are fixed values set by the investigators. The log-odds ratio of country are nested within region hierarchically and are treated as exchangeable within region such that:

$$\begin \omega _&\sim \mathcal (0, \tau _r^2)\nonumber \\ \tau _r^2&\sim \text (p,q) \end$$

(18)

where p, and q are fixed values set by the blinded investigators.

Effects of covariates

The log-odds ratios for covariate factors are assigned independent normal prior distributions such that:

$$\begin \theta _} \sim \mathcal (a,b^2), \end$$

(19)

where a and b are fixed values set by the blinded investigators.

Effects of epoch

Temporal trends will be accounted for by splitting the trial sample into separate cohorts defined by contiguous 26 week epochs and using first-order dynamic normal linear model within (9) and broadly suggested by [26, 27]:

$$\begin \phi _&=0\nonumber \\ \phi _&\sim \mathcal (\phi _t, \tau _t^2)\nonumber \\ \tau _t^2&\sim \text (p,q) \end$$

(20)

where p, and q are fixed values set by the blinded investigators.

Furthermore, an additional sensitivity analysis may be performed to evaluate all outcomes of interest using cohorts that are restricted to participants who were randomised concurrently among the available interventions at the time and removing the \(\phi_t\) parameter from the model.

Exploratory analyses

Additional analyses are described in the SNAP statistical appendix and relevant domain-specific appendices that include parameters for the interactions between interventions and other covariates to enable the comparative effectiveness by covariates as a pre-planned exploratory analysis.

Computational methods

The joint posterior probability distributions of the model parameters described in the preceding sections are analytically intractable, and therefore computational Bayesian methods will be used for the data analyses. We will use Markov-chain Monte Carlo (MCMC) methods implemented in 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 \(\hat\) , and graphical representations of the MCMC chains [30].

Missing data

Death from S. aureus bloodstream infection predominantly occurs prior to hospital discharge, therefore, participants that are discharged alive from hospital but are lost to follow-up prior to day 90 (i.e. have missing outcome data) will be assumed to be alive at day 90, unless later information becomes available indicating a death in the 90-day period. A complete case strategy will be used for all other missing outcome data.

Concurrently randomised cohorts

By design, the way that participants are assigned to treatments in adaptive randomised trials changes over time, which can lead to treatment-outcome confounding in certain cases. For when that confounder is based on calendar time, we have included epoch modelling as one countermeasure to confounding (see the ‘Epochs’ section). An additional potential for confounding may occur when a participant is unable to be randomised to a particular intervention within a subdomain. Reasons why might include contraindication or site unavailability for that particular intervention. Currently, the SNAP trial only randomises participants among two treatments per subdomain, and therefore the primary model is sufficient. In the event that the participant is to be randomised among more than two treatments, the SNAP study team may include additional model parameters to capture the ineligibility of participants to particular interventions.

As a further safeguard against treatment-outcome confounding arising from changing randomisation schemes, at the final analysis of a domain, we intend to perform and report sensitivity analyses of the primary model outcomes of each embedded fixed design corresponding to concurrently randomised cohorts, as recommended [31]. A concurrently randomised cohort is a subset of trial participants who had the same treatments available and all had the same chance of receiving those treatments (through randomisation) over the same time period.