The cpaic statistical framework • cpaic

Research use only. cpaic is experimental. Its methodology and implementation have not been validated for clinical, regulatory, reimbursement, or other decision use.

This vignette describes the statistical framework behind cpaic. The companion mathematical-foundations document (shipped with the development sources) gives the full derivations; here we summarize the model and show how each piece maps to a function.

Two layers

cpaic targets networks that are disconnected: the treatments split into two or more sub-networks with no common comparator, so standard network meta-analysis cannot compare across the gap. Two layers solve the two distinct problems.

Connection layer (component network meta-analysis). Multi-component treatments are decomposed into additive component effects. When sub-networks share components, the component effects bridge the gap.
Adjustment layer (population adjustment). Where individual patient data (IPD) are available, each evidence edge is corrected for between-study imbalance in effect modifiers, using anchored STC, MAIC, or ML-NMR.

The component model can identify a cross-gap contrast, but scientific transportability is an additional assumption. A two-stage result is not a coherent single-target population adjustment when target-specific IPD contrasts are mixed with retained AgD contrasts from their original study populations.

The additive component model

Let $\delta$ be the vector of observed relative effects (one per comparison), $B$ the edge-incidence (contrast) matrix mapping comparisons to treatments, and $C$ the treatment-by-component matrix with $C_{tc} = 1$ if treatment $t$ contains component $c$ . Treatment effects are additive in the component effects $\beta$ , $\theta = C\beta, \qquad \delta = B\theta = BC\beta = X\beta,$ with $X = BC$ the component design matrix. With inverse-variance weights $W$ , the component effects are estimated by weighted least squares $\hat\beta = (X^\top W X)^{+} X^\top W d, \qquad \mathrm{Cov}(\hat\theta) = C (X^\top W X)^{+} C^\top,$ where $(\cdot)^{+}$ is the Moore-Penrose inverse and $d$ the data vector (Rücker et al. 2020). The additivity assumption is checked with a Cochran $Q$ statistic. This is implemented in cnma_bridge(), a wrapper around netmeta::discomb().

Connecting a disconnected network

A full set of component effects is identifiable when $\mathrm{rank}(X) = K$ , the number of components. Even in a rank-deficient design, a particular contrast can be identifiable when its contrast vector lies in the row space of $X$ . cpaic_connectivity() detects the sub-networks and reports the component-design diagnostics; estimable_effects() checks the requested contrasts.

net <- cpaic_network(cpaic_bin_agd, sm = "OR", inactive = "Placebo")
cpaic_connectivity(net)
#> cpaic connectivity
#>   Connected network: FALSE
#>   Sub-networks:      2
#>     [1] 3 treatments
#>     [2] 3 treatments
#>   Bridging components: A, B
#>     (components that OCCUR in more than one sub-network; occurrence is
#>      not identifiability, homogeneity, or influence for any contrast.)
#>   Component design:  rank(X) = 4 / 4 components -> all component effects identified
#>   Estimable effects: 5 / 5 vs Placebo

When identifiable, cnma_bridge() reconstructs the relative effects across the gap from the component effects.

component_effects(cnma_bridge(net))
#>   component  estimate        se     lower    upper statistic      pval
#> 1         A 0.5000000 1.1922140 -1.836697 2.836697 0.4193878 0.6749328
#> 2         B 0.4000000 1.1922140 -1.936697 2.736697 0.3355102 0.7372402
#> 3         C 0.7170248 0.9734562 -1.190914 2.624964 0.7365763 0.4613800
#> 4         D 0.3250136 0.9728622 -1.581761 2.231788 0.3340798 0.7383193

Anchored simulated treatment comparison (cSTC)

For each IPD study, cstc() fits an outcome regression with treatment main effects, prognostic main effects, and treatment-by-effect-modifier interactions, with the effect modifiers centered at supplied target covariate means. On the link scale, $g\{E(y \mid \text{arm } t, x)\} = \mu + \beta_t + \gamma_t^\top (x - \bar x_{\text{target}}) + \dots$ so the treatment coefficient $\beta_t$ is the conditional link-scale contrast at the supplied target covariate means (the interaction term vanishes at $x = \bar x_{\text{target}}$ ). Because this contrast is linear in the effect modifiers on the link scale, it is also the average conditional link-scale contrast at those means. It is not an outcome-scale marginal standardization. The model is implemented natively.

net_ipd <- cpaic_network(cpaic_bin_agd, ipd = cpaic_bin_ipd, sm = "OR",
                         family = "binomial", ipd_covariates = "x1",
                         inactive = "Placebo")
component_effects(cstc(net_ipd, target = c(x1 = 0),
                       effect_modifiers = "x1",
                       allow_experimental_bridge = TRUE))
#> Warning: cstc() cannot form a decision-grade component bridge:
#>   - retained aggregate-only edge(s) remain in their own study populations: S1: A vs Placebo; S2: B vs Placebo; S5: A+B+C vs A+B+D
#> Use cmlnmr() for a joint model, restrict the analysis to a design in which every edge is adjusted and the estimand is additive, or set `allow_experimental_bridge = TRUE` only for explicitly exploratory sensitivity work.
#>   component  estimate        se        lower     upper statistic        pval
#> 1         A 0.5000000 0.2563324 -0.002402322 1.0024023  1.950592 0.051105590
#> 2         B 0.4000000 0.2563324 -0.102402322 0.9024023  1.560474 0.118647988
#> 3         C 0.4896667 0.2406290  0.018042458 0.9612910  2.034944 0.041856471
#> 4         D 0.6408956 0.2317142  0.186744196 1.0950470  2.765889 0.005676788

Anchored matching-adjusted indirect comparison (cMAIC)

cmaic() reweights each IPD study so that its effect-modifier moments match the supplied target moments, using entropy-balancing weights $w_i = \exp(\tilde x_i^\top \alpha)$ with $\tilde x_i$ the centered effect modifiers (Phillippo et al. 2020). The effective sample size is $\mathrm{ESS} = (\sum_i w_i)^2 / \sum_i w_i^2$ . The weighted within-study contrasts, with bootstrap standard errors that propagate the weighting uncertainty, are then combined through the component model.

That last step is an additional methodology constraint. A marginal weighted contrast is not generally additive across treatment components on a nonlinear link scale. Also, if AgD-only edges remain, those contrasts still refer to their original study populations. cMAIC therefore stops by default outside the Gaussian identity-link case and whenever retained AgD edges would be mixed in. The experimental override does not repair either estimand mismatch.

fit_maic <- cmaic(net_ipd, target = c(x1 = 0), effect_modifiers = "x1",
                  n_boot = 100, seed = 1,
                  allow_experimental_bridge = TRUE)
#> Warning: cmaic() cannot form a decision-grade component bridge:
#>   - retained aggregate-only edge(s) remain in their own study populations: S1: A vs Placebo; S2: B vs Placebo; S5: A+B+C vs A+B+D
#>   - cMAIC estimates marginal binomial contrasts, which are not generally additive in the component design on a nonlinear link scale
#> Use cmlnmr() for a joint model, restrict the analysis to a design in which every edge is adjusted and the estimand is additive, or set `allow_experimental_bridge = TRUE` only for explicitly exploratory sensitivity work.
effective_sample_size(fit_maic)
#>       S3       S4 
#> 207.4202 358.1461

The unification

For a single anchored comparison, the contrast algebra resembles a Bucher indirect comparison. A network bridge combines many adjusted and unadjusted contrasts through $\hat\beta = (X^\top W X)^{+} X^\top W d$ . That algebra does not establish that every input contrast shares the same target estimand. It also does not test the required constancy of component effects and effect-modifier interactions across disconnected sub-networks (Rücker et al. 2021).

Component-additive ML-NMR

cmlnmr() places the component structure inside multilevel network meta-regression: the relative effect of an arm is $C\beta$ rather than a free per-treatment parameter, and aggregate arms are fitted by integrating the individual model over each study’s covariate distribution. Disconnected sub-networks share the component parameters, so the network is connected by construction. All four outcome families are supported; survival uses a proportional-hazards model with a flexible baseline (piecewise-exponential by default, or a smooth M-spline), set through cut_points and baseline. Its analytic row-level likelihood handles events, right/left/interval censoring, and delayed entry. Aggregate survival arms require reconstructed event and censoring rows; event counts plus person-time are rejected. The finite quasi-Monte Carlo covariate integration is approximate. Aggregate covariates are integrated with a Gaussian copula whose correlation is estimated from the individual patient data. A component whose effect-modifier interaction is informed only by aggregate data is weakly identified and relies on the regression prior.

The default reporting path evaluates $(C_t-C_u)'(\beta + \Gamma x)$ at a one-row vector of supplied covariate means. Since this expression is linear in $x$ , it is an average conditional link-scale effect at those means. A separate marginal_effects() path accepts empirical target rows or a named Gaussian-copula target distribution. Empirical rows must contain exactly the fitted effect modifiers; finite nonnegative weights are normalized and zero-weight rows are discarded. Bernoulli modifiers use actual 0/1 rows. Summary targets provide named means, SDs, and margins, with optional latent-scale correlation and deterministic Sobol’ integration. Continuous modifiers can use normal, gamma, lognormal, or beta margins independently, while fitted Bernoulli modifiers remain Bernoulli. Independent Bernoulli strata are enumerated exactly. Summary integration stops when the requested Bernoulli strata or non-normal moments fail their moment-fidelity gates, rather than silently reporting a distorted target. 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. Mixed or non-normal multivariable targets also require resolved pairwise correlations to agree within 0.05 of a high-resolution deterministic reference grid.

The estimator averages treatment-specific outcomes within each posterior draw before forming odds, risk, rate, survival, time-specific hazard, or restricted mean survival time contrasts. Binomial, rate-difference, and survival measures require an explicit donor study intercept or baseline hazard. Survival predictions start at model time zero and are not landmark or delayed-entry conditioned. Prediction times must remain within the observed follow-up support of the selected donor study. Survival and risk ratios, piecewise-exponential RMST ratios, and time-specific marginal hazard ratios are formed on the log scale with stable log-sum-exp calculations 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. These target checks are not a formal overlap or positivity diagnostic. Nonlinear marginal effects remain treatment-level contrasts, not marginal component effects or rankings.

# Fits with rstan by default; backend = "cmdstanr" is the alternative.
fit <- cmlnmr(ipd, agd, effect_modifiers = "x1", inactive = "Placebo",
              family = "binomial")
component_effects(fit)

# Marginal risk ratio over an empirical target distribution.
marginal_effects(
  fit, target = data.frame(x1 = c(0, 1)), weights = c(0.7, 0.3),
  measure = "risk_ratio", baseline_study = "study_id"
)

Assumptions and caveats

Additivity of component effects. additivity_test() can diagnose lack of fit where evidence exists, but it cannot validate an unobserved cross-gap bridge.
Identifiability: a disconnected network is bridgeable only when cpaic_connectivity() reports identifiable = TRUE.
Cross-population transportability of effect modifiers (the standard population-adjustment assumption), here extended to constancy of component effects across sub-networks.
Two-stage estimand mixing: retained AgD contrasts remain tied to their study populations. The default gate stops this mixed bridge unless the caller explicitly opts into research-only behavior.
Nonlinearity and non-collapsibility: MAIC targets a marginal weighted contrast and STC a conditional link-scale contrast. Marginal contrasts are not generally component-additive on nonlinear links.
cML-NMR reporting: average conditional effects at covariate means and marginal effects over target distributions are separate estimands. Marginal standardization is treatment-level and does not preserve component additivity on nonlinear scales. Measures requiring absolute outcomes transport a selected study intercept or baseline hazard. Survival predictions start at model time zero, not a landmark or delayed-entry time, and use the donor study’s observed follow-up support. No formal overlap diagnostic is estimated.

References

Phillippo, David M., Sofia Dias, A. E. Ades, et al. 2020. “Multilevel Network Meta-Regression for Population-Adjusted Treatment Comparisons.” Journal of the Royal Statistical Society: Series A 183 (3): 1189–210. https://doi.org/10.1111/rssa.12579.

Rücker, Gerta, Maria Petropoulou, and Guido Schwarzer. 2020. “Network Meta-Analysis of Multicomponent Interventions.” Biometrical Journal 62 (3): 808–21. https://doi.org/10.1002/bimj.201800167.

Rücker, Gerta, Susanne Schmitz, and Guido Schwarzer. 2021. “Component Network Meta-Analysis Compared to a Matching Method in a Disconnected Network.” Biometrical Journal 63 (2): 447–61.