Network Meta-Analytic Predictive (NAP) Priors

Description

The NAPrior package facilitates the implementation of the Network Meta-Analytic Predictive (NAP) prior framework, specifically designed to address changes in the Standard of Care (SoC) during ongoing randomized controlled trials (RCTs). The framework synthesizes in-trial data from both pre- and post-SoC change periods by leveraging external trial data—specifically the head-to-head comparisons between the original and new SoC that established the new SoC—to bridge the two phases of evidence.

To ensure robust inference, the package implements two robustified priors: (i) the mixture NAP (mNAP), which incorporates a noninformative prior via a fixed mixing weight, and (ii) the elastic NAP (eNAP), which adaptively adjusts mixing weight and information borrowing based on the consistency between direct and indirect evidence. The NAP framework is fully prespecifiable, easy to calibrate, and computationally straightforward. This package provides a comprehensive toolkit to calibrate eNAP tuning parameters, generate NAP priors, simulate operating characteristics, and obtain posterior distributions.

Installation

System requirements

The NAPrior package fits Bayesian models using JAGS via the R2jags interface.
Please ensure that JAGS is installed on your system before running model functions.

• macOS: brew install jags
  
• Ubuntu/Debian: sudo apt-get install jags
  
• Windows: https://mcmc-jags.sourceforge.io/

Install and load

install.packages("devtools")
devtools::install_github("EstravenZZZ/NAPrior")

library(NAPrior)

Quick start

Consider a scenario in which the SoC changes mid-trial in a registrational or pivotal RCT that was initially designed to compare an experimental treatment \(E\) against an SoC \(C_1\) using 1:1 randomization of \(2N\) patients. At the outset, a total of \(2n_1\) patients are randomized between \(E\) and \(C_1\). Subsequently, a mid-trial SoC change occurs, and another \(2n_2\) patients are randomized between \(E\) and the new SoC \(C_2\). As a result, the trial data consist of two components: data directly comparing \(E\) versus \(C_1\) (denoted as \(D_{E,C_1}\)) and data directly comparing \(E\) versus \(C_2\) (denoted as \(D_{E,C_2}\)). Meanwhile, summary statistics from external trial(s) motivated SoC change is also available (denoted as \(D_{C_2,C_2}\)). The primary objective of the RCT is to evaluate the treatment effect of \(E\) versus \(C_2\), denoted as \(\theta_{E, C_2}\). For estimating \(\theta_{E, C_2}\), \(D_{E,C_2}\) provides direct evidence while \(D_{E,C_1}\) and \(D_{C_2,C_2}\) provide indirect evidence.

Let \(y_{E,C_2}\), \(y_{E,C_1}\), and \(y_{C_2,C_1}\) denote the estimated log hazard ratios (log-HRs) calculated from the datasets \(D_{E, C_2}\), \(D_{E, C_1}\) and \(D_{C_2, C_1}\), respectively, with sampling variances \(s^2_{E,C_2}\), \(s^2_{E,C_1}\), and \(s^2_{C_2,C_1}\). These estimates can be derived, for example, using the Cox proportional hazards model. Following standard meta-analytic convention, we assume that \(s^2_{E,C_2}\), \(s^2_{E,C_1}\), and \(s^2_{C_2,C_1}\) are known. This package accommodates external data (\(D_{C_2,C_2}\)) from either a single trial and multiple trials by allowing users to supply a scalar or a vector for y_C2C1 and s_C2C1 arguments accordingly.

1) Calibrate the tuning parameters for eNAP prior

The eNAP uses dynamic weight based on scaled Bucher test statistic \(Z\), to quantify the extent of consistency between direct and indirect evidence, and mapped to the mixing weight using an elastic function:

\[w(Z) = \{ 1 + \exp\bigl(a + b \log(Z + 1)\bigr)\}^{-1},\]

where Bucher test statistic

\[Z = n_{eff}^{-1/4}\times\frac{ \bigl| y_{E,C_2} - (y_{E,C_1} - y_{C_2,C_1}) \bigr| }{ s^2_{E,C_2} + s^2_{E,C_1} + s^2_{C_2,C_1} }, \] where \(n_{eff}=(s^2_{E,C_2} + s^2_{E,C_1} + s^2_{C_2,C_1})^{-1}.\)

To calibrate the tuning parameters \((a, b)\), the user provides:

• \(\delta\): A clinically meaningful difference on the log-HR scale; differences beyond this threshold indicate strong inconsistency between direct and indirect evidence.

• \(t_1\): The target posterior mixing weights under exact consistency, typically set close to 1 (e.g., \(t_1 = 0.999\)).

• \(t_0\): The target posterior mixing weights under strongly inconsistency \(Z_\delta=n_{eff}^{-1/4}\times \delta/(s^2_{E,C_2} + s^2_{E,C_1} + s^2_{C_2,C_1})\), typically set close to 0 (e.g., \(t_0 = 0.05\))
  These inputs calibrate \((a, b)\) such that the updated posterior weight \(w'\) satisfies \(w'(Z_0) = t_1\) and \(w'(Z_\delta) = t_0\). A companion plot method is provided for visualizing the resulting posterior updated weight; simply call plot() on the returned object to view the weighting function relative to evidence consistency..

tuned_ab <- tune_param_eNAP(
  y_C2C1=c(-0.4,-0.5,-0.5),
  s_EC2  = 0.12^2,                          # Var(E:C2)
  s_EC1  = 0.16^2,                          # Var(E:C1)
  s_C2C1 = c(0.12^2, 0.11^2, 0.15^2),       # vector => multiple external trials
  delta  = 0.5,                             # clinically meaningful difference on log-HR
  t1     = 0.999,                           # near full borrowing at Z = 0
  t0     = 0.05                             # near zero borrowing at Z(delta)
)
#> Warning in tune_param_eNAP(y_C2C1 = c(-0.4, -0.5, -0.5), s_EC2 = 0.12^2, :
#> Please consider exact solution and provide log-HR for E-C2, E-C1 and C2-C1
c(tuned_ab$a,tuned_ab$b)
#> [1] -1.903302 12.605520
plot(tuned_ab)

2) Construct NAP priors

To construct an mNAP or eNAP prior, use the NAP_prior() function. This requires summary statistics for the indirect evidence (\(D_{E,C_1}\) and \(D_{C_2,C_2}\)), and a selection for the weighting method (“adaptive” for eNAP or “fixed” for mNAP). For mNAP, a fixed mixing weight must specified (a weight of 1 results in the NAP prior, assuming full transitivity). For eNAP, the function requiresthe tuning parameters \((a,b)\). A companion plot() method is available to visualize the resulting prior distribution.

Since the dynamic weight for eNAP depends on direct evidence (\(D_{E,C_2}\)) that may not yet be observed (e.g., at the time of the SoC change), NAP_prior() offers two operational modes:

• When direct evidence:
  If an assumed log-HR and sampling variance for \(D_{E,C_2}\) are provided, the function calculates the specific dynamic weight and resulting prior under those assumptions.

• When direct evidence:
  If no assumed direct evidence is provided, the function returns the informative (NAP) and vague components separately, allowing for later synthesis.

The function returns an object of class "NAP_prior", which serves as the input for subsequent simulations or posterior data analyses.

NAP

NAP_test1 <- NAP_prior(
  weight_mtd = "fixed", w = 1,            # w = 1 ⇒ informative component only (NAP)
  y_EC1  = -0.36, s_EC1  = 0.16^2,
  y_C2C1 = -0.30, s_C2C1 = 0.14^2       # single external → FE
)
NAP_test1$table
#>               NAP (Informative) Vague
#> Mixing Weight        1.00000000     0
#> Mean                -0.05999666     0
#> Variance             0.04519896  1000
#> ESS (events)        88.49761046    NA
plot(NAP_test1)

mNAP with a fixed weight of 0.5

mNAP_test1 <- NAP_prior(
  weight_mtd = "fixed", w = 0.50,         # fixed mixture weight
  y_EC1  = -0.36, s_EC1  = 0.16^2,
  y_C2C1 = -0.30, s_C2C1 = 0.14^2,        # single external → FE
  tau0   = 1000
)
mNAP_test1$table
#>               NAP (Informative) Vague
#> Mixing Weight        0.50000000 5e-01
#> Mean                -0.05999666 0e+00
#> Variance             0.04519896 1e+03
#> ESS (events)        88.49761046    NA
plot(mNAP_test1,main="mNAP prior")

eNAP without direct data \(D_{E,C_2}\) provided

eNAP_test1 <- NAP_prior(
  weight_mtd = "adaptive",
  a = tuned_ab$a, b = tuned_ab$b,         # from calibration
  y_EC1  = -0.36, s_EC1  = 0.16^2,                 # E:C1 (current, pre-change)
  y_C2C1 = c(-0.28, -0.35, -0.31),                 # C2:C1 (external trials)
  s_C2C1 = c(0.12^2, 0.11^2, 0.15^2),
  tau0   = 1000                                    # vague variance
)
eNAP_test1$table
#>                 NAP (Informative) Vague
#> Mixing Weight                 TBD   TBD
#> Mean          -0.0437723254666285     0
#> Variance       0.0306875093991601  1000
#> ESS (events)     130.346192255976  <NA>

eNAP with direct data \(D_{E,C_2}\) provided

eNAP_test2 <- NAP_prior(
  weight_mtd = "adaptive",
  a = tuned_ab$a, b = tuned_ab$b,         # from calibration
  y_EC1  = -0.36, s_EC1  = 0.16^2,                 # E:C1 (current, pre-change)
  y_C2C1 = c(-0.28, -0.35, -0.31),                 # C2:C1 (external trials)
  s_C2C1 = c(0.12^2, 0.11^2, 0.15^2),
  y_EC2  = 0, s_EC2 = 0.2^2,
  tau0   = 1000                                    # vague variance
)
eNAP_test2$table
#>               NAP (Informative)        Vague
#> Mixing Weight        0.70600021    0.2939998
#> Mean                -0.04377233    0.0000000
#> Variance             0.03068751 1000.0000000
#> ESS (events)       130.34619226           NA
plot(eNAP_test2,main="eNAP prior")

3) Simulate operating characteristics

With previously obtained object from NAP_prior() function, use NAP_oc() function to simulate operating characteristics.

set.seed(123)
# Run JAGS quietly
oc <- NULL
invisible(capture.output(
  oc <- NAP_oc(
    NAP_prior   = eNAP_test1,
    theta_EC2   = 0.00,                     # true log-HR for E:C2
    n_EC2       = 400,                      # total N for the post-SoC-change period
    lambda      = 1,                        # randomization ratio for the post-SoC-change period
    sim_model   = "Weibull",                # "Exponential" or "Weibull"
    model_param = c(shape = 1.2, rate = 0.05),
    nsim        = 100,
    iter        = 3000,
    chains      = 4
  ),
  type = "output"   # capture JAGS console output
))

# Now show only the summary in the knitted doc
summary(oc)
#>       rep           post_mean           post_sd          low95.2.5%     
#>  Min.   :  1.00   Min.   :-0.23056   Min.   :0.08741   Min.   :-0.4183  
#>  1st Qu.: 25.75   1st Qu.:-0.07669   1st Qu.:0.08890   1st Qu.:-0.2514  
#>  Median : 50.50   Median :-0.01206   Median :0.08984   Median :-0.1847  
#>  Mean   : 50.50   Mean   :-0.01427   Mean   :0.09018   Mean   :-0.1903  
#>  3rd Qu.: 75.25   3rd Qu.: 0.04285   3rd Qu.:0.09053   3rd Qu.:-0.1336  
#>  Max.   :100.00   Max.   : 0.17198   Max.   :0.09931   Max.   :-0.0133  
#>    hi95.97.5%       prob_E_better     prior_weight      post_weight    
#>  Min.   :-0.04547   Min.   :0.0360   Min.   :0.02636   Min.   :0.5758  
#>  1st Qu.: 0.09562   1st Qu.:0.3186   1st Qu.:0.23272   1st Qu.:0.9745  
#>  Median : 0.16137   Median :0.5507   Median :0.47493   Median :0.9914  
#>  Mean   : 0.16286   Mean   :0.5505   Mean   :0.46112   Mean   :0.9626  
#>  3rd Qu.: 0.21818   3rd Qu.:0.8079   3rd Qu.:0.68668   3rd Qu.:0.9967  
#>  Max.   : 0.37403   Max.   :0.9922   Max.   :0.86832   Max.   :0.9997  
#>    sigma_hat     
#>  Min.   :0.1144  
#>  1st Qu.:0.1246  
#>  Median :0.1301  
#>  Mean   :0.1307  
#>  3rd Qu.:0.1372  
#>  Max.   :0.1482

4) Calculate posterior

To obtain posterior posterior of target estimand \(\theta_{E,C_2}\), pass the object generated by the NAP_prior() function, along with the observed direct evidence (log_HR and sampling variance from \(D_{E,C2}\)) to the NAP_posterior() function.

res <- NAP_posterior(
  NAP_prior = eNAP_test1,
  y_EC2     = -0.20,
  s_EC2     = 0.12^2,
  iter      = 4000,
  chains    = 4
)
#> Compiling model graph
#>    Resolving undeclared variables
#>    Allocating nodes
#> Graph information:
#>    Observed stochastic nodes: 3
#>    Unobserved stochastic nodes: 6
#>    Total graph size: 34
#> 
#> Initializing model

res$posterior_sum   # mean, sd, 95% CI, prob_E_better, prior weight, post weight
#>    post_mean   post_sd      low95       hi95 prob_E_better prior_weight
#> 1 -0.1941428 0.1180834 -0.4267569 0.03765138       0.95075  0.006383881
#>   post_weight sigma_hat
#> 1       0.339 0.3632943
res$enap_prior      # actual eNAP prior at analysis time with realized dynamic weight
#>               NAP (Informative)        Vague
#> Mixing Weight       0.006383881    0.9936161
#> Mean               -0.043772325    0.0000000
#> Variance            0.030687509 1000.0000000
#> ESS (events)      130.346192256           NA
# res$jags_fit is stored but does not print by default

Core functions

• tune_param_eNAP() — Calibrates the eNAP tuning parameters \((a, b)\) based on user-defined consistency targets.
• NAP_prior() — Generates mNAP and eNAP prior objects using indirect evidence and specified weighting methods.
• NAP_posterior() — Computes the posterior distribution of the target estimand by combining a NAP_prior object with observed direct evidence (\(D_{E,C2}\)).
• NAP_oc() — Simulates operating characteristics.

JAGS model specification

The package includes default BUGS/JAGS model strings as data objects:

• jags_model_FE
• jags_model_RE

You may pass
- a path to a .txt model file (e.g., model = "path/to/model_RE.txt"), or
- a string containing BUGS/JAGS code (e.g., model = jags_model_RE).

Internally, models are routed through .model_path_from(), which writes text to a temporary file and passes it to R2jags::jags().

Tips & diagnostics

• Inputs: Sampling variances must be positive and finite.
• Stability: If tuning parameters \(|a|\) or \(b\) reaches internally pre-specified upper limits (by defualt 5 and 50), respectively, recommend reducing \(t_1\), increasing \(t_0\), or enlarging \(\delta\) to enhance the reliability of posterior inference.
• Reproducibility: Set a seed before simulations (set.seed()).
• JAGS messages: Lines about “resolving undeclared variables” are normal. Most JAGS errors arise from name or length mismatches in the data or model.

Citing

If you use NAPrior in publications, please cite both the methodological paper and this package (full citation to be added once available).

Maintainer

Chunyi Zhang (czhang12@mdanderson.org)