cpaic

Component-Based Population-Adjusted Indirect Comparison

cpaic is experimental research software for combining component network meta-analysis (cNMA) with population-adjusted indirect comparison (PAIC) in a disconnected treatment network. Shared treatment components can identify a bridge across the disconnection. IPD-bearing comparisons can then be adjusted for differences in measured effect modifiers, subject to the restrictions below.

Research use only. The methodology and implementation have not been validated for clinical, regulatory, reimbursement, or other decision use. A successful fit and clean sampler diagnostics do not validate the cross-sub-network additivity or transportability assumptions.

Standard network meta-analysis needs a connected network. When the network is disconnected (no common comparator links two sub-networks), it cannot be analyzed directly. Two ideas each solve half of the problem:

• Component NMA (Rücker et al. 2020) decomposes multi-component treatments into additive component effects. If sub-networks share components, those component effects bridge the gap; but the method uses aggregate data only and ignores effect-modifier imbalance.
• PAIC (STC, MAIC, ML-NMR) adjusts for effect-modifier imbalance using individual patient data (IPD), but assumes the network is already connected.

cpaic implements experimental ways to combine these ideas. The one-stage cmlnmr() model and the two-stage cstc() / cmaic() bridge have different estimands and different failure modes. They should not be described as interchangeable whole-network population adjustments.

Two senses of "anchored"

These are different words that happen to be spelled the same, and conflating them causes real confusion:

• Anchored PAIC (the NICE sense): the adjusted comparison runs through a common comparator arm within each trial, rather than comparing single arms. cmaic() and cstc() are anchored in this sense.
• Anchored cNMA (the Rücker/Wigle sense): an inactive component is fixed at zero in the component parameterization. Setting inactive = NULL gives the unanchored parameterization of Wigle & Béliveau (2022), in which every unit receives its own parameter and no anchor can be misspecified.

Bridging a disconnected network through shared components is not anchoring by a common comparator. It is identification through additivity.

Status

• Frequentist core. cnma_bridge() (reconnect via components), cmaic() (component MAIC), cstc() (component anchored STC). Binary, continuous, count, and survival outcomes.
• Bayesian cmlnmr() (component-additive ML-NMR), fitted with rstan by default or with cmdstanr on request. Binary, continuous, count, and survival outcomes, with separate conditional and target-standardized marginal reporting paths.

The four outcome-family implementations are under active development. They are not decision validated. Read the estimand and bridge limitations before interpreting a result.

Estimability

Reconnecting a network through shared components does not guarantee that the effects you want are estimable. A relative effect is uniquely estimable exactly when its contrast vector lies in the row space of the component design matrix X = B C (Wigle et al. 2026). Full column rank of X is sufficient but not necessary, so a rank-deficient network can still identify useful cross-sub-network contrasts.

This matters because both engines will otherwise hand back a confident-looking number for a contrast that carries no information: the frequentist fit through the Moore-Penrose pseudoinverse, and the Bayesian fit through the prior. cpaic checks every contrast and returns NA instead.

estimable_effects(net)     # which relative effects are identified?
cpaic_connectivity(net)    # rank, null space, bridging components

Installation

# install.packages("remotes")
remotes::install_github("choxos/cpaic")

cpaic builds on netmeta for the component-NMA engine and maicplus for the MAIC weights, both on CRAN. The component-additive ML-NMR models and their quasi-Monte-Carlo integration are implemented in the package itself, following Phillippo et al. (2020).

Those models are compiled when cpaic is installed, so cmlnmr() works with no further setup. It fits through rstan by default; pass backend = "cmdstanr" to use CmdStan instead, which tracks Stan releases more closely and is often faster, and which needs the cmdstanr package from https://stan-dev.r-universe.dev plus a CmdStan installation. The two fit the same models and agree up to Monte Carlo error; they do not share a random number stream, so the same seed gives different draws on each. Under rstan the chains run one after another unless you set options(mc.cores = ...), as usual for R.

Summarize a fit with posterior_summary() rather than the sampler object under fit$fit: that slot is an S4 stanfit on one backend and an R6 object on the other, and they share no accessors.

Quick start

library(cpaic)

# A disconnected network: sub-network {Placebo, A, B} and isolated
# sub-network {A+B, A+B+C, A+B+D}, bridged by the shared components A and B.
net <- cpaic_network(cpaic_bin_agd, ipd = cpaic_bin_ipd, sm = "OR",
                     family = "binomial", ipd_covariates = "x1",
                     inactive = "Placebo")

# Which effects can this network actually identify?
cpaic_connectivity(net)
estimable_effects(net)

# 1. Connect only (aggregate component NMA)
cnma_bridge(net)

# 2. Experimental two-stage STC bridge. This network retains AgD edges, so
#    explicit opt-in is required because the bridge mixes populations.
cstc(net, target = c(x1 = 0), effect_modifiers = "x1",
     allow_experimental_bridge = TRUE)

# 3. Experimental two-stage MAIC bridge. On the OR scale, marginal weighted
#    contrasts also need not obey component additivity.
cmaic(net, target = c(x1 = 0), effect_modifiers = "x1",
      allow_experimental_bridge = TRUE)

By default, cstc() and cmaic() stop when retained aggregate contrasts would be combined with target-specific adjusted contrasts. cmaic() also stops by default for non-Gaussian families because a marginal weighted contrast is not generally additive across components on a nonlinear scale. allow_experimental_bridge = TRUE records the override in the fitted object and emits a warning. It does not make the estimands coherent.

For the Bayesian model, conditional relative effects vary with the effect modifiers because theta_t(x) = C_t' (beta + Gamma x). A one-row newdata value is interpreted as a vector of target covariate means. Because this contrast is linear in x on the link scale, it is an average conditional link-scale effect at those means.

fit <- cmlnmr(ipd, agd, effect_modifiers = "x1", inactive = "Placebo")
# Average conditional link-scale effects at target mean x1 = 0.3
relative_effects(fit, newdata = data.frame(x1 = 0.3))
component_effects(fit, newdata = data.frame(x1 = 0.3))

Marginal reporting uses a separate target-distribution interface. It averages treatment-specific outcomes within every posterior draw, then forms the requested contrast. For example, an empirical binary target distribution can be supplied as actual 0/1 rows with weights:

target_population <- data.frame(x1 = c(0, 1))
target_weights <- c(0.7, 0.3)

# Binomial marginal risk ratio. Replace "study_id" with a study retained by fit.
marginal_effects(
  fit, target = target_population, weights = target_weights,
  measure = "risk_ratio", baseline_study = "study_id"
)

The donor study supplies the fitted intercept needed for absolute risks. The target rows remain a separate input. Gaussian mean differences and Poisson rate ratios do not need a donor baseline because the common intercept cancels.

An empirical target is a data frame with exactly the fitted effect modifiers. Its finite, nonnegative weights are normalized and zero-weight rows are discarded. Bernoulli modifiers must be represented by actual 0/1 rows. A summary target is a named list of means, sds, and margins, with optional latent-scale cor and deterministic Sobol' n_int. Continuous modifiers can use normal, gamma, lognormal, or beta margins independently; fitted Bernoulli modifiers must remain Bernoulli. Independent Bernoulli strata are enumerated exactly, while continuous margins use deterministic integration nodes. Summary targets have moment-fidelity gates for Bernoulli strata and non-normal means and SDs. The Bernoulli prevalence tolerance is max(1e-8, 0.1 * min(p, 1 - p)); non-normal means must be within 0.1 target SD and non-normal SDs within 15 percent relative error. A failed gate stops with a request for more nodes or an empirical target. Mixed or non-normal multivariable targets also require resolved pairwise correlations to agree within 0.05 of a high-resolution deterministic reference grid. Target-distribution uncertainty is not propagated, and these checks do not constitute a formal overlap or positivity diagnostic.

For survival, predictions start at model time zero and are not landmark or delayed-entry conditioned. times must lie within the observed follow-up support of the selected donor study, which may be shorter than the global support. Survival and risk ratios, piecewise-exponential RMST ratios, and time-specific marginal hazard ratios are calculated on the log scale with stable log-sum-exp operations before optional back-transformation. RMST measures require a piecewise-exponential donor baseline and are evaluated analytically over its intervals. M-spline fits support marginal survival, risk, and time-specific marginal hazard measures, but not RMST. Nonlinear marginal effects are treatment-level contrasts; the package does not report marginal component effects or rankings.

relative_effects(), marginal_effects(), league_table(), component_effects(), estimable_effects(), additivity_test(), effective_sample_size(), forest() and plot() summarize and visualize a fit.

Validation status

The package does not yet have a tracked, independently reproducible validation suite that supports decision use. Local exploratory simulations are useful for finding failure modes, but they are not release evidence and are not summarized as performance claims here. At minimum, future validation must cover overlap, model misspecification, partial IPD coverage, nonlinear estimands, multi-arm studies, survival reconstruction, and violations of cross-sub-network additivity.

Limitations

The following apply to any result produced by this package.

• The bridging assumption is untestable. Reconnecting a disconnected network requires component effects (and, under population adjustment, component by effect-modifier interactions) to be constant across sub-networks. There is by construction no cross-gap evidence to test this against. additivity_test() reports the fit of the additive model within the observed evidence; a large p-value is not a license to bridge.
• A two-stage bridge can mix populations. cmaic() and cstc() adjust the edges that have IPD. Retained aggregate contrasts remain on their original study-population estimands. Combining both types does not produce a coherent single-target network merely because the component model can solve the algebra. The default is to stop; allow_experimental_bridge = TRUE is an explicit research override, not a validity correction.
• Estimands differ across the methods. cstc() returns a conditional effect on the link scale at the target covariate means; cmaic() returns a marginal weighted contrast. On a nonlinear scale a marginal contrast is not generally component-additive, so feeding cMAIC edges into an additive bridge can be methodologically incoherent even with perfect matching. Non-Gaussian cMAIC bridges therefore require the experimental override.
• Conditional and marginal cML-NMR summaries are separate estimands. A one-row newdata profile reports an average conditional link-scale contrast. marginal_effects() requires a target distribution and standardizes absolute outcomes before forming treatment-level contrasts. Nonlinear marginal effects are not component-additive, so marginal component effects and marginal treatment rankings are not reported.
• Some marginal measures borrow a study baseline. Binomial effects, Poisson rate differences, and every survival measure transport a selected fitted study intercept or baseline hazard to the target population. This is an explicit assumption. Target-distribution uncertainty is not propagated, and study-arm random effects are set to zero for population-mean predictions. Survival predictions begin at model time zero, not at a landmark or delayed entry time, and prediction times are restricted to the selected donor study's observed follow-up support.
• Target checks are not overlap diagnostics. Empirical and summary targets are checked for finite values, fitted-margin support, Bernoulli strata, and summary moment fidelity. The package does not estimate a formal overlap or positivity statistic, so extrapolation risk remains an analyst-level responsibility.
• Survival conditions on the supplied rows. cmlnmr() evaluates the exact analytic individual survival likelihood (events plus right, left, and interval censoring, and delayed entry), integrated over each aggregate arm's covariate distribution, with a study-specific piecewise-exponential or continuous cubic M-spline baseline. It is exact only conditional on the supplied (or reconstructed) pseudo-individual rows, the chosen baseline basis, and the finite numerical integration; uncertainty from reconstructing pseudo-IPD out of a published Kaplan-Meier curve is not propagated, and proportional hazards is assumed. marginal_effects() can report standardized survival and risk contrasts and time-specific marginal hazard ratios within the fitted follow-up support. Restricted mean survival time contrasts are available for piecewise-exponential baselines. M-spline fits retain the pointwise survival, risk, and time-specific hazard measures. These outputs additionally require transport of a selected study's baseline hazard.
• Survival interfaces have different status contracts. The two-stage cstc() and cmaic() network input accepts only 0 for right censoring and 1 for an event. cmlnmr() separately uses 0 right-censored, 1 event, 2 left-censored, and 3 interval-censored rows, with reconstructed rows required for aggregate survival arms. The package reports proportional-hazard contrasts through relative_effects(). A scalar marginal hazard ratio is not reported because the standardized hazard ratio generally varies with time.
• cmlnmr() treatment effects may be fixed or random. trt_effects = "random" adds study-arm heterogeneity with a single common tau, which is one exchangeability assumption, not a component-, sub-network-, or era-specific heterogeneity structure.

Documentation

The introductory and methods vignettes ship with the package. Five archived outcome examples are excluded until they are regenerated against the current estimand labels, safeguards, and API. The standalone manual in the development-only documentation/ tree is not public package guidance.

References

• Rücker G, Petropoulou M, Schwarzer G (2020). Network meta-analysis of multicomponent interventions. Biometrical Journal 62(3):808-821.
• Wigle A, Béliveau A (2022). Bayesian unanchored additive models for component network meta-analysis. Statistics in Medicine 41(22):4444-4466.
• Wigle A, Béliveau A, Nikolakopoulou A, Lin L (2026). Creating treatment and component hierarchies in component network meta-analysis.
• Efthimiou O, et al. (2022). A Bayesian model for combining aggregate and individual participant data in component network meta-analysis. Statistics in Medicine 41(14):2586-2606.
• Phillippo DM, et al. (2020). Multilevel network meta-regression for population-adjusted treatment comparisons. JRSS A 183(3):1189-1210.

License

GPL-3