The cpaic statistical framework • cpaic

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 output is an indirect comparison that is both connected and population-adjusted.

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 disconnected network can be bridged only if the shared components make the component effects identifiable, that is $\mathrm{rank}(X) = K$ , the number of components. cpaic_connectivity() detects the sub-networks, lists the bridging components, and reports identifiability.

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
#>   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 a target population. 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 population-adjusted contrast at the target (the interaction term vanishes at $x = \bar x_{\text{target}}$ ). This is the anchored generalization of regression-based standardization; it is implemented natively because mlumr::stc() targets the unanchored two-trial case.

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"))
#>   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 distribution matches the target population, 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.

fit_maic <- cmaic(net_ipd, target = c(x1 = 0), effect_modifiers = "x1",
                  n_boot = 100, seed = 1)
effective_sample_size(fit_maic)
#>       S3       S4 
#> 207.4202 358.1461

The unification

For a single IPD edge anchored on a common comparator, combining the adjusted contrast with the aggregate comparator contrast through the component model is exactly a Bucher indirect comparison. The component model generalizes this to a network: many adjusted and unadjusted contrasts are combined simultaneously through $\hat\beta = (X^\top W X)^{+} X^\top W d$ , and the bridge supplies the contrasts that are otherwise unavailable across the disconnection. The key assumption is that component effects (and their effect-modifier interactions) are constant across 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. 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.

# Requires cmdstanr; see ?cmlnmr.
fit <- cmlnmr(ipd, agd, effect_modifiers = "x1", inactive = "Placebo",
              family = "binomial")
component_effects(fit)

Assumptions and caveats

Additivity of component effects (test with additivity_test(); add interaction terms if violated).
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.
Non-collapsibility: MAIC targets a marginal effect and STC a conditional effect, so for noncollapsible measures they answer slightly different questions.

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.