18  Matching-Adjusted Indirect Comparison

18.1 Introduction

Chapter 14 left us with a sharp diagnosis and no cure. The anchored indirect comparison of Bucher et al. (1997) is exact under transitivity (Definition 14.3, Theorem 14.1), and transitivity fails for a definite mechanism: an effect modifier (Definition 11.3), a patient characteristic such as disease severity or age that changes how much a treatment helps, whose distribution differs between the two trial populations makes the marginal \(B\)-versus-\(C\) effect in the \(BC\) trial differ from the same effect in the population where we wish to report \(A\) versus \(B\), and the subtraction then mixes populations (Theorem 16.1). Chapter 16 quantified the resulting bias and showed that a standard network meta-analysis cannot remove it from aggregate data alone. The remainder of the book is the study of what to do when one of the two trials hands us not just summary statistics but the full individual patient data (IPD), so that we can see, patient by patient, how the trial population is built. Population adjustment is the art of using that microscope to rebuild one trial’s population in the image of the other’s.

Remark. MAIC in one paragraph, with a concrete trial to hold onto. Throughout the chapter it helps to picture a specific comparison. Suppose \(A\) is a new immunotherapy, \(B\) is an established competitor drug, and \(C\) is standard chemotherapy, the common comparator that both of the other drugs were tested against. One trial randomized patients to \(A\) or \(C\); we call it the \(AC\) trial (trials are named by the pair of arms they contain, so “\(AC\)” means “the trial with arms \(A\) and \(C\)”). A different trial randomized a different group of patients to \(B\) or \(C\); we call it the \(BC\) trial. We own the full patient-level data for the \(AC\) trial but only a published table of averages for the \(BC\) trial. If the new drug \(A\) works especially well in sicker patients, and the \(BC\) trial happened to enroll sicker patients than the \(AC\) trial, then simply subtracting the two published treatment effects is unfair: it compares \(A\)’s effect in a healthier group against \(B\)’s effect in a sicker group. MAIC fixes this by giving each \(AC\) patient a weight, a multiplier that says how many patients like them we pretend to have, chosen so that the reweighted \(AC\) patients have the same average characteristics as the \(BC\) patients. We then recompute the \(A\)-versus-\(C\) effect in this reweighted, “\(BC\)-like” pseudo-population, so that both treatment effects finally describe the same kind of patient and may be compared. Every theorem in this chapter is a precise statement about how to choose those weights and what they cost. \(\blacktriangleleft\)

This chapter develops the first and most widely used population-adjustment method, matching-adjusted indirect comparison (MAIC), introduced into the health-technology-assessment literature by Signorovitch et al. (2010). The setup is the canonical anchored triangle of Chapter 14, now asymmetric in what we know. One trial, which we call \(AC\), compared an active treatment \(A\) against a common comparator \(C\) and reports IPD: for each of its \(n\) patients we have the baseline covariate vector \(\mathbf{x}_i\), the arm \(T_i\in\{A,C\}\), and the outcome \(Y_i\). The other trial, \(BC\), compared a second active treatment \(B\) against the same comparator \(C\) and reports only aggregate data (AgD): the mean covariate vector \(\bar{\mathbf{x}}_{BC}\), perhaps standard deviations, and a published relative effect \(\hat d_{BC}^{\mathrm{AgD}}\) with its standard error. We cannot move the \(BC\) patients into our analysis, but we can move our analysis into the \(BC\) population: reweight the \(AC\) IPD so that its weighted covariate moments equal those reported by \(BC\), then read off the \(A\)-versus-\(C\) effect in this reweighted, \(BC\)-like pseudo-population. The reweighted effect \(\hat d_{AC}^{\mathrm{MAIC}}\) and the published \(\hat d_{BC}^{\mathrm{AgD}}\) now live in the same population, so the Bucher subtraction \(\hat d_{AB}=\hat d_{AC}^{\mathrm{MAIC}}-\hat d_{BC}^{\mathrm{AgD}}\) is licensed.

The mathematics is the mathematics of one optimization problem. Infinitely many weight vectors achieve covariate balance; MAIC selects the one that distorts the sample as little as possible, where “as little as possible” is made precise by the entropy, equivalently the Kullback-Leibler divergence from uniform weighting. The unique solution has the exponential form \[ w_i \;=\; \exp\!\big(\alpha+\boldsymbol{\beta}^\top(\mathbf{x}_i-\bar{\mathbf{x}}_{BC})\big), \] and the parameters \((\alpha,\boldsymbol{\beta})\) are pinned down by the balance constraints. We derive this form from a Lagrangian and the Karush-Kuhn-Tucker (KKT) conditions (Theorem 18.1), characterize exactly when finite weights exist via convex duality and the geometry of the covariate convex hull (Theorem 18.2), and show that the same weights are the inverse probability-of-selection weights of a logistic trial-membership model (Theorem 18.3), which places MAIC squarely inside the inverse-probability-weighting theory of Chapter 10 (Theorem 10.3) and the transportability theory of Chapter 12 (Definition 12.3).

We then pay the bill. Reweighting buys covariate balance at the cost of precision, and the cost is governed by the effective sample size \(\mathrm{ESS}=(\sum_i w_i)^2/\sum_i w_i^2\), which never exceeds \(n\) and equals \(n\) only when the weights are uniform (Theorem 18.6, the one result in this chapter whose Coq proof we cite). The variance of a weighted estimate is inflated by the factor \(n/\mathrm{ESS}\) (Theorem 18.7), and when the two populations barely overlap this factor explodes. Worse, no choice of finite weights can manufacture information about a region of covariate space that the IPD never visited: MAIC cannot extrapolate beyond the convex hull of its own covariates (Theorem 18.8). Finally we put honest standard errors on the estimator by treating the whole procedure, weights and outcomes together, as an M-estimator with a sandwich variance (Theorem 18.9, using Definition 5.11) and by establishing bootstrap validity (Theorem 18.10).

Two consistency theorems sit at the center, one for each anchoring regime. Anchored MAIC (Theorem 18.4) is consistent for the target-population \(A\)-versus-\(B\) contrast under conditional constancy of relative effects (Definition 17.4), the shared-effect-modifier assumption (Definition 17.6), balance of every imbalanced effect modifier, and overlap; it is the MAIC instance of the general anchored population-adjustment theorem of Chapter 17 (Theorem 17.1). Unanchored MAIC (Theorem 18.5) discards the comparator and is consistent only under the far stronger conditional constancy of absolute effects (Definition 17.5) and balance of every imbalanced prognostic variable as well as every effect modifier; it is the MAIC instance of Theorem 17.2, and it is rarely defensible. Throughout we are scrupulous about a distinction the early literature blurred: because the weights are estimated from the data, MAIC estimators are consistent, not finite-sample unbiased. A fully worked two-covariate example, in which every weight, the effective sample size, and the adjusted contrast can be checked by hand, closes the development. This chapter corresponds to results F.1 to F.12 of the proof catalogue.

Throughout, \(g\) is the link function (Definition 7.2) on whose scale effects are reported and modeled, \(\mu_t(\mathbf{x})=\mathbb{E}[Y(t)\mid\mathbf{X}=\mathbf{x}]\) is the conditional mean potential outcome under treatment \(t\), and \(\tau(\mathbf{x})=\mu_A(\mathbf{x})-\mu_C(\mathbf{x})\) is the conditional average treatment effect (CATE). The populations in play are \(\mathcal{P}_{AC}\) (the IPD trial’s population, with covariate density \(f_{AC}\)) and \(\mathcal{P}_{BC}\) (the comparator trial’s population, with density \(f_{BC}\) and mean \(\bar{\mathbf{x}}_{BC}\)); for the unanchored development a general target \(\mathcal{P}^{*}\) appears. We use the relative-effect notation \(d_{ab(\mathcal{P})}\) of Notation.

Remark. A reading path for the applied reader. The logical order below (define the weights, prove they exist, prove they are consistent, count their cost) is not the only reading order. If you want to see MAIC work before studying why, read the plain-language setup (Section 18.2), then the small worked example in Section 18.3, then skip ahead to the effective sample size (Section 18.11) and the full worked example (Section 18.16), and return to the two hardest sections, the convex-duality existence proof (Section 18.7) and the sandwich and bootstrap standard errors (Section 18.14, Section 18.15), afterward. Every section opens with a one-line plain reading of what it establishes, so you can navigate by those. \(\blacktriangleleft\)

18.2 The two-trial setup and the matching idea

We fix the data structures once and for all.

Remark. The asymmetric anchored triangle. The index trial \(AC\) supplies IPD \(\{(\mathbf{x}_i,T_i,Y_i)\}_{i=1}^{n}\) on \(n\) patients drawn from \(\mathcal{P}_{AC}\), with \(\mathbf{x}_i\in\mathbb{R}^p\) a vector of baseline covariates, \(T_i\in\{A,C\}\) the randomized arm, and \(Y_i\) the outcome. Randomization within the index trial means \(T\perp(Y(A),Y(C))\mid\mathbf{X}\) holds with the conditioning vacuous, that is, treatment assignment is independent of the potential outcomes even without conditioning on \(\mathbf{X}\) (randomization guarantees the independence outright, so we do not need to adjust for covariates to make the two arms comparable within the \(AC\) trial); consequently each arm mean identifies the corresponding marginal potential-outcome mean of \(\mathcal{P}_{AC}\) (Theorem 9.3, Proposition 14.1). The comparator trial \(BC\) supplies AgD from \(\mathcal{P}_{BC}\): the covariate mean \(\bar{\mathbf{x}}_{BC}\in\mathbb{R}^p\) (and possibly second moments), and a relative effect \(\hat d_{BC}^{\mathrm{AgD}}\) with standard error. The estimand is the \(A\)-versus-\(B\) contrast in a stated target population; in the anchored case the natural target is \(\mathcal{P}_{BC}\), the population of the trial we cannot re-analyze. \(\blacktriangleleft\)

The naive anchored comparison \(\hat d_{AC}^{(AC)}-\hat d_{BC}^{\mathrm{AgD}}\) uses the index trial’s \(A\)-versus-\(C\) effect as measured in \(\mathcal{P}_{AC}\), whereas the comparator effect is measured in \(\mathcal{P}_{BC}\); if an effect modifier is distributed differently across the two, the subtraction mixes populations and is biased (Theorem 16.1). MAIC removes the mismatch by recomputing the index effect in \(\mathcal{P}_{BC}\). Because we have the index IPD but only \(BC\)’s covariate summaries, the only handle we have on \(\mathcal{P}_{BC}\) is its reported moments, and the only operation available is to reweight the index patients. The entire method is the disciplined choice of those weights.

What a weight means, in patient terms. Before any formula, fix the intuition for a weight \(w_i\). Think of \(w_i\) as the number of target-population patients that index patient \(i\) is asked to stand in for. A patient whose covariate profile is common in the comparator population, for example a sicker patient when the comparator trial was sicker, is a good representative of many \(BC\) patients and receives a large weight; a patient whose profile is rare in the comparator population contributes little and receives a small weight. Uniform weights \(w_i=1\) mean every patient stands only for themselves, which is the original, unadjusted trial. Reweighting therefore does not add or invent patients; it only shifts emphasis among the patients we already have, upweighting the ones who look like the target and downweighting the ones who do not. This is why a weight can never be negative, and why, as we will see in Section 18.11, concentrating the emphasis on a few patients is exactly what makes the effective sample size shrink.

One distinction, imported from Chapters 11 and 17, runs through the whole chapter and decides which covariates we must balance. It is worth restating in plain language once here.

Covariate type	Plain meaning	What it changes	Must MAIC balance it?
Effect modifier (EM)	changes how much the treatment helps	the treatment contrast \(\tau(\mathbf{x})\)	Yes if imbalanced (anchored and unanchored)
Prognostic variable (PV)	changes the outcome regardless of treatment	the outcome level \(\mu_t(\mathbf{x})\), equally in every arm	Only in the unanchored case (it cancels when anchored)

The reason a pure prognostic variable can be ignored in the anchored case, but not the unanchored case, is the single most important structural fact in the chapter; we prove it in Section 18.9 and Section 18.10 and preview it in the worked example of Section 18.3.

18.3 A first numerical example: four patients, one covariate

Before any optimization theory, it is worth seeing the whole method run once on numbers small enough to check in your head. Everything in this preview, the exponential weight form, the effective sample size, the bias correction, is derived rigorously in the sections that follow; here we simply exhibit the answer so that the later derivations feel like explanations of something already touched. We reuse this example at each transition.

Example 18.1 (A one-covariate MAIC you can check by hand) The \(AC\) trial (new immunotherapy \(A\) versus chemotherapy \(C\)) enrolled four patients, described by a single standardized covariate \(x\), disease severity, coded so that \(x=+1\) is “severe” and \(x=-1\) is “mild.” Two patients are severe and two are mild, so the index-trial mean severity is \(\bar x_{AC}=\tfrac14(1+1-1-1)=0\). The comparator \(BC\) trial (competitor \(B\) versus the same chemotherapy \(C\)) reports, in its published table, a higher mean severity \(\bar x_{BC}=0.5\): its patients were sicker on average. Severity is an effect modifier: the identity-scale benefit of \(A\) over \(C\) is larger in sicker patients, specifically \(\mu_C(x)-\mu_A(x)=4+2x\), with arm means \(\mu_C(x)=10+x\) and \(\mu_A(x)=6-x\).

The matching. We seek nonnegative weights \(w_i\) making the weighted mean severity equal the target \(0.5\). By symmetry the two severe patients share a weight \(w_{+}\) and the two mild patients a weight \(w_{-}\), and the balance condition \[ \frac{2w_{+}(+1)+2w_{-}(-1)}{2w_{+}+2w_{-}}=0.5 \quad\Longleftrightarrow\quad \frac{w_{+}-w_{-}}{w_{+}+w_{-}}=0.5 \quad\Longleftrightarrow\quad w_{+}=3\,w_{-} \] forces the severe patients to carry three times the weight of the mild ones. Normalizing the four weights to sum to \(n=4\) gives \(w_{-}=0.5\) and \(w_{+}=1.5\): \[ w_{\text{severe}}=1.5,\qquad w_{\text{mild}}=0.5 . \] Each severe patient now stands in for one and a half patients, each mild patient for half a patient, and the weighted sample looks, on average, like the sicker \(BC\) population. One checks the balance directly: the weighted mean severity is \(\tfrac14(1.5+1.5-0.5-0.5)=\tfrac{2}{4}=0.5\), as required.

The exponential form appears on its own. Write the two weights as \(w(x)=e^{\alpha+\beta x}\). Then \(w_{+}/w_{-}=e^{2\beta}=3\), so \(\beta=\tfrac12\ln 3\approx 0.549\), and \(w_{-}=e^{\alpha-\beta}=0.5\) gives \(e^{\alpha}=0.5\sqrt3\approx 0.866\). Thus, with no assumption imposed beyond “balance the mean with the least lopsided weights,” the weights came out log-linear in the covariate, \(w(x)=e^{-0.144+0.549\,x}\). Section Section 18.6 explains why this exponential shape is forced, not chosen.

The precision cost. The effective sample size (Definition 18.3) is \[ \mathrm{ESS}=\frac{(\sum_i w_i)^2}{\sum_i w_i^2} =\frac{4^2}{2(1.5)^2+2(0.5)^2}=\frac{16}{5}=3.2,\qquad \frac{\mathrm{ESS}}{n}=0.8 . \] Matching one covariate cost us the equivalent of \(0.8\) of a patient: four real patients now carry only as much statistical information as \(3.2\) equally weighted ones, so the variance of a weighted arm mean is inflated by \(n/\mathrm{ESS}=1.25\) and its standard error by \(\sqrt{1.25}\approx 1.12\), a \(12\%\) increase (Theorem 18.7).

The payoff: an honest bias correction. In the original \(AC\) sample the estimated benefit of \(A\) over \(C\) is \(\mu_C-\mu_A\) averaged with equal weights, namely \(4+2\bar x_{AC}=4\). After reweighting to the comparator population the estimated benefit is \[ \hat d_{AC}^{\mathrm{MAIC}} =\underbrace{\tfrac14\big(1.5\cdot 11+1.5\cdot 11+0.5\cdot 9+0.5\cdot 9\big)}_{\hat\mu_C^{\mathrm{MAIC}}=10.5} -\underbrace{\tfrac14\big(1.5\cdot 5+1.5\cdot 5+0.5\cdot 7+0.5\cdot 7\big)}_{\hat\mu_A^{\mathrm{MAIC}}=5.5} =5, \] exactly the true comparator-population benefit \(4+2\bar x_{BC}=4+2(0.5)=5\). MAIC moved the estimate from \(4\) to \(5\) because the sicker comparator population is one in which \(A\)’s benefit is genuinely larger. If we had instead subtracted the raw \(AC\) benefit of \(4\) from the published \(BC\) benefit, we would have carried a \(1\)-unit bias into the indirect comparison, the very transitivity failure of Theorem 16.1. \(\blacktriangleleft\)

The full development below repeats this loop, moment balance, exponential weights, effective sample size, consistency, honest standard errors, in generality. Section 18.16 carries a two-covariate version of this same example (a severe/mild covariate and a purely prognostic one) all the way to the anchored indirect comparison, and shows there that balancing the extra prognostic covariate drops the effective sample size further, from \(3.2\) to \(2.56\), for no reduction in bias, our first glimpse of the central MAIC trade-off.

18.4 Method-of-moments covariate balance

The defining requirement of MAIC is that the weighted index covariate moments equal the comparator’s reported moments. We state the first-moment version, which is the workhorse, and then say precisely what “moment” can mean.

Definition 18.1 (Method-of-moments covariate balance) Let \(\{\mathbf{x}_i\}_{i=1}^{n}\subset\mathbb{R}^p\) be the index-trial covariate vectors and let \(\bar{\mathbf{x}}_{BC}\in\mathbb{R}^p\) be the comparator-trial covariate mean. A weight vector \(\mathbf{w}=(w_1,\dots,w_n)\) with \(w_i\ge 0\) and \(\sum_i w_i>0\) balances the first moments of the covariates if the weighted index mean equals the target mean, \[ \frac{\sum_{i=1}^{n} w_i\,\mathbf{x}_i}{\sum_{i=1}^{n} w_i}\;=\;\bar{\mathbf{x}}_{BC}, \tag{18.1}\] equivalently, writing \(\boldsymbol{\Delta}_i:=\mathbf{x}_i-\bar{\mathbf{x}}_{BC}\) for the centered covariates, \[ \sum_{i=1}^{n} w_i\,\boldsymbol{\Delta}_i\;=\;\mathbf{0}. \tag{18.2}\] More generally, given a feature map \(\boldsymbol{\phi}:\mathbb{R}^p\to\mathbb{R}^q\) and target feature means \(\bar{\boldsymbol{\phi}}_{BC}\), the weights balance the features \(\boldsymbol{\phi}\) if \(\sum_i w_i(\boldsymbol{\phi}(\mathbf{x}_i)-\bar{\boldsymbol{\phi}}_{BC})=\mathbf{0}\). Taking \(\phi\)-components equal to the squared and cross-product covariates \(x_jx_k\) balances the second moments, hence the covariance matrix; taking spline or indicator bases balances the corresponding functionals. \(\blacktriangleleft\)

Reader translation. Covariate balance is the whole goal: after weighting, the average patient in the reweighted \(AC\) sample should match the average patient reported for \(BC\). Equation Equation 18.1 says exactly that for the mean of each covariate; the centered form Equation 18.2 just subtracts the target so the requirement reads “the weighted centered covariates average to zero.” In the running Example 18.1 there was one covariate and one balance equation, and it pinned severity’s weighted mean to \(0.5\).

What a feature map is, concretely. A feature map \(\boldsymbol{\phi}\) is just a list of derived quantities we compute from each patient’s covariates and then also insist on balancing. The plainest choice is \(\boldsymbol{\phi}(\mathbf{x})=\mathbf{x}\), which balances the means, as above. If we append the square of a covariate, say \(\boldsymbol{\phi}(\mathbf{x})=(x_1,\,x_1^2)\), then balancing the second component forces the weighted average of \(x_1^2\) to equal the target’s, and since the variance is \(\overline{x_1^2}-\bar x_1^{\,2}\), matching both \(\bar x_1\) and \(\overline{x_1^2}\) matches the variance of \(x_1\) as well as its mean. For a two-patient toy with \(x_1\in\{-1,+1\}\), the raw values are \(\pm1\) and their squares are both \(1\); appending \(x_1^2\) adds the constraint “weighted average of \(1\) equals the reported \(\overline{x_1^2}\),” which is only satisfiable if the target’s second moment is also \(1\), a concrete reminder that the comparator must actually report the moment you wish to balance. Balancing more features means matching a richer summary of the two populations, at a precision cost we quantify in Section 18.11.

Equation Equation 18.1 is a system of constraints, not a recipe: for \(p<n-1\) it has an infinite affine family of nonnegative solutions whenever it has one at all. The next section selects a single solution. First, two points about which moments to balance, because the choice is where the scientific assumptions enter.

Remark. Which moments? The link scale decides. Balancing first moments suffices for unbiased population adjustment exactly on scales where the quantity being averaged is affine in the matched features. On the identity scale with a conditional mean \(\mu_t(\mathbf{x})=a_0+\mathbf{a}^\top\mathbf{x}\) linear in the covariates, the marginal mean is \(a_0+\mathbf{a}^\top\bar{\mathbf{x}}\), a function of the first moments alone, and matching means matches the marginal mean exactly. On a nonlinear link such as the logit or the log hazard, the conditional contrast involves \(g^{-1}(\eta(\mathbf{x}))\), whose population average depends on higher moments and on the joint covariate distribution, so first-moment matching is in general not sufficient; one balances second moments, or functional moments, or (with a copula assumption, Chapter 22) the joint distribution. The scale dependence of which covariates even count as effect modifiers is the subject of Example 17.1; the practical recommendation of Phillippo et al. (2016) is to balance at least the first moments of every imbalanced effect modifier, adding higher moments as the link and the data demand. \(\blacktriangleleft\)

The previous remark is abstract; because most health-technology work is on the odds-ratio or hazard-ratio scale, it deserves numbers. The next example shows two populations with an identical mean covariate yet different average risks on the logit scale, so that matching the mean alone leaves a residual bias.

Example 18.2 (On the logit scale, matching the mean is not enough) Let the outcome be binary with inverse-logit (expit) mean \(g^{-1}(\eta)=1/(1+e^{-\eta})\), and consider a covariate \(x\) (say a standardized biomarker) that enters the linear predictor for treatment \(A\) as \(\eta_A(x)=1+x\). Compare two populations with the same mean covariate but different spread:

• Population I places all its mass at \(x=0\) (mean \(0\), variance \(0\)).
• Population II places half its mass at \(x=-1\) and half at \(x=+1\) (mean \(0\), variance \(1\)).

Matching the mean covariate treats these two populations as interchangeable; MAIC with first-moment balancing cannot tell them apart. Yet their marginal (population-average) risks under \(A\) differ: \[ \begin{aligned} \text{Population I:}\quad & g^{-1}(1)=\tfrac{1}{1+e^{-1}}=0.731,\\ \text{Population II:}\quad & \tfrac12\,g^{-1}(0)+\tfrac12\,g^{-1}(2) =\tfrac12(0.500)+\tfrac12(0.881)=0.690 . \end{aligned} \] The population-average risk depends on the spread of \(x\), not only its mean, because \(g^{-1}\) is curved. Now add a control arm \(C\) with \(\eta_C(x)=0.5+x\), so the conditional log odds ratio of \(A\) versus \(C\) is the constant \((1+x)-(0.5+x)=0.5\) at every \(x\); on this scale \(x\) is purely prognostic, not an effect modifier. Even so, the marginal log odds ratios differ between the two populations, \[ \log\mathrm{OR}_{\mathrm{I}}=0.500,\qquad \log\mathrm{OR}_{\mathrm{II}}=0.407, \] (the second from marginal risks \(0.690\) under \(A\) and \(0.598\) under \(C\)). Two lessons follow. First, on a nonlinear link, first-moment matching does not equate marginal outcomes; to make Populations I and II agree one must also match the variance, which is why Definition 18.1 allows balancing \(x^2\). Second, even matching the whole covariate distribution does not by itself equate marginal odds ratios, because the marginal contrast also depends on the baseline risk through the curvature of \(g^{-1}\); this is non-collapsibility (Theorem 11.2), and it is why the anchored consistency proof (Section 18.9) needs a separate scale-alignment step on odds- and hazard-ratio scales. \(\blacktriangleleft\)

Remark. Aggregation and ecological bias. Matching the index mean to the comparator mean presumes that the mean is the right summary, which is again a linearity statement. When the integrand is nonlinear, equating means while ignoring the spread induces the ecological bias of Definition 16.2: two populations with the same covariate mean but different variances have different marginal effects on a nonlinear scale. Second-moment balancing is the first-order correction, and multilevel network meta-regression (Chapter 20) is the systematic one. \(\blacktriangleleft\)

This result corresponds to F.1 in the proof catalogue.

18.5 The entropy-balancing objective

Among all weight vectors satisfying Equation 18.2, MAIC chooses the one closest to uniform. “Closest to uniform” is measured by negative entropy, equivalently by the Kullback-Leibler (KL) divergence from the uniform weight vector.

Definition 18.2 (The entropy-balancing problem) With \(\boldsymbol{\Delta}_i=\mathbf{x}_i-\bar{\mathbf{x}}_{BC}\) as in Definition 18.1, the entropy-balancing problem is the convex program \[ \min_{\mathbf{w}\in\mathbb{R}^n_{>0}}\;\; H(\mathbf{w}):=\sum_{i=1}^{n} w_i\log w_i \qquad\text{subject to}\qquad \sum_{i=1}^{n} w_i\,\boldsymbol{\Delta}_i=\mathbf{0},\quad \sum_{i=1}^{n} w_i=n. \tag{18.3}\] The normalization \(\sum_i w_i=n\) fixes the scale (any positive constant would do; the estimator and the effective sample size below are invariant to it). \(\blacktriangleleft\)

Reader translation. The constraint line (\(\sum_i w_i\boldsymbol{\Delta}_i=\mathbf 0\)) is the scientific requirement: make the reweighted sample match the target. The objective line (\(\min\sum_i w_i\log w_i\)) is the tie-breaker among the infinitely many ways to meet that requirement. Its plain meaning is “distort the original trial as little as possible.” We want to reweight just enough to fix the covariate mismatch and no more, because every extra distortion piles emphasis onto a few patients, overfits to their idiosyncrasies, and inflates the standard error. Entropy is the mathematical name for “how far the weights stray from equal”; minimizing it keeps the pseudo-population as close as it can be to the trial we actually ran while still hitting the target. In the running Example 18.1, of all the weightings that pull the mean severity up to \(0.5\), entropy balancing chose the mildest one, \((1.5,1.5,0.5,0.5)\).

Remark. Why this objective is the KL divergence from uniform. Write the normalized weights \(p_i=w_i/\sum_j w_j\), so \(\sum_i p_i=1\). The KL divergence of \(\mathbf{p}\) from the uniform distribution \(\mathbf{u}=(1/n,\dots,1/n)\) is \[ \mathrm{KL}(\mathbf{p}\,\|\,\mathbf{u})=\sum_{i=1}^{n} p_i\log\frac{p_i}{1/n} =\sum_{i=1}^{n} p_i\log p_i+\log n. \tag{18.4}\] Under the normalization \(\sum_i w_i=n\) we have \(p_i=w_i/n\). Substituting this into each term and expanding the logarithm of a quotient, \[ \begin{aligned} \sum_i p_i\log p_i &=\sum_i \frac{w_i}{n}\log\frac{w_i}{n} =\sum_i\frac{w_i}{n}\big(\log w_i-\log n\big)\\ &=\frac1n\underbrace{\sum_i w_i\log w_i}_{H(\mathbf{w})} -\frac{\log n}{n}\underbrace{\sum_i w_i}_{=\,n} =\frac1n H(\mathbf{w})-\log n . \end{aligned} \] Adding the \(+\log n\) from Equation 18.4 cancels the trailing \(-\log n\), leaving \(\mathrm{KL}(\mathbf{p}\,\|\,\mathbf{u})=\tfrac1n H(\mathbf{w})\). Minimizing \(H\) is exactly minimizing the KL divergence from uniform weighting: the maximum-entropy principle made into an estimator. Equivalently, \(H(\mathbf{w})\) is the generalized I-divergence \(\sum_i(w_i\log w_i-w_i+1)\) from the all-ones vector once the constant \(\sum_i(1-w_i)=n-n=0\) is added. \(\blacktriangleleft\)

Remark. Why entropy and not some other distance. One could minimize \(\sum_i(w_i-1)^2\), or \(\sum_i|w_i-1|\), or any strictly convex separable penalty; each yields a different weight family. Entropy is canonical for four reasons that the next three sections make precise: it is strictly convex, so the optimum is unique (Theorem 18.1); its KKT conditions deliver weights of clean exponential form, log-linear in the covariates; its convex dual is an unconstrained smooth minimization whose geometry exactly characterizes feasibility (Theorem 18.2); and that exponential form coincides with the inverse-odds weights of a logistic selection model (Theorem 18.3), tying MAIC to the propensity-score machinery of Chapter 10. \(\blacktriangleleft\)

Remark. Entropy keeps any one patient from taking over. Two elementary features of \(h(w)=w\log w\) explain why it is the right penalty, and both are visible in its derivative \(h'(w)=\log w+1\). As \(w\to 0^{+}\), \(h'(w)\to-\infty\): there is an infinite pull away from zero, so entropy weights are automatically strictly positive and never need an explicit \(w\ge 0\) constraint to be enforced. As \(w\) grows, \(h'(w)=\log w+1\) keeps rising, so each further increase of an already-large weight is charged an ever-higher price; entropy therefore resists dumping the whole pseudo-population onto one or two patients. Contrast the squared penalty \((w-1)^2\), whose derivative \(2(w-1)\) is a straight line: it charges the same marginal price whether a weight is tiny or huge, and it has no barrier at zero, so it can even return negative “weights.” Numerically, the uniform weights \((1,1,1,1)\) have entropy \(H=0\), while the running example’s balanced weights \((1.5,1.5,0.5,0.5)\) have \(H=2(1.5\log 1.5)+2(0.5\log 0.5)\approx 0.52\): entropy is a scoreboard for how far the weights have strayed from equal, and MAIC keeps that score as low as the balance constraint permits. \(\blacktriangleleft\)

The strict convexity that drives everything is the elementary fact recorded next.

Lemma 18.1 (Strict convexity of the entropy) The function \(h(w)=w\log w\) is strictly convex on \((0,\infty)\), with \(h''(w)=1/w>0\), and extends continuously to \(h(0)=0\). Consequently \(H(\mathbf{w})=\sum_i h(w_i)\) is strictly convex on \(\mathbb{R}^n_{\ge 0}\), and its Hessian on \(\mathbb{R}^n_{>0}\) is the positive-definite diagonal matrix \(\operatorname{diag}(1/w_1,\dots,1/w_n)\).

Proof. On \((0,\infty)\) the map \(h(w)=w\log w\) is twice differentiable with \(h'(w)=\log w+1\) and \(h''(w)=1/w>0\), so \(h\) is strictly convex there; since \(w\log w\to 0\) as \(w\downarrow 0\), setting \(h(0)=0\) extends \(h\) continuously and preserves convexity on \([0,\infty)\). A finite sum of strictly convex functions of separate coordinates is strictly convex, and its Hessian is the diagonal matrix of the second derivatives \(h''(w_i)=1/w_i\), which is positive definite on the open positive orthant. The feasible set of Equation 18.3 is the intersection of an affine subspace with the nonnegative orthant, hence convex, so a minimizer, if one exists, is unique. \(\square\)

This result corresponds to F.2 in the proof catalogue.

18.6 The exponential weights via Lagrangian and KKT

We now solve Equation 18.3. The structure of the problem, a strictly convex objective with affine constraints, makes the KKT conditions both necessary and sufficient for global optimality, and the inequality constraints \(w_i\ge 0\) turn out inactive because the entropy’s derivative blows up at the boundary.

Why this matters, and a map of the argument. This is the theoretical heart of the method: it explains why the weights in Example 18.1 came out as \(e^{\alpha+\beta x}\) rather than any other shape. The exponential form is not a modeling choice we imposed; it is forced by asking for the least-distorting weights that balance the covariates. The proof has two moves. First, the Lagrangian and KKT conditions (the standard calculus of constrained optimization from Chapter 5) turn the balance-and-minimize problem into the stationarity equation \(\log w_i+1=\lambda+\boldsymbol{\beta}^\top\boldsymbol{\Delta}_i\), whose solution is exactly the exponential weight. Second, rather than search over the \(n\) weights \(\mathbf{w}\) directly, we recognize that the whole problem collapses to minimizing a single smooth convex function \(Q(\boldsymbol{\beta})\) of the \(p\) dual parameters, which is what MAIC software actually optimizes. Read the theorem as: “least-distorting balanced weights must be exponential, and finding them is one small convex minimization.”

Theorem 18.1 (Exponential form of the entropy-balancing weights) Suppose the entropy-balancing problem Equation 18.3 admits a strictly positive feasible point (equivalently, by Theorem 18.2, \(\bar{\mathbf{x}}_{BC}\) lies in the relative interior of the convex hull of \(\{\mathbf{x}_i\}\)). Then Equation 18.3 has a unique solution \(\mathbf{w}^{*}\), and it has the exponential form \[ w_i^{*}=\exp\!\big(\alpha+\boldsymbol{\beta}^\top\boldsymbol{\Delta}_i\big) =\exp\!\big(\alpha+\boldsymbol{\beta}^\top(\mathbf{x}_i-\bar{\mathbf{x}}_{BC})\big), \qquad i=1,\dots,n, \tag{18.5}\] where the multipliers \((\alpha,\boldsymbol{\beta})\in\mathbb{R}\times\mathbb{R}^p\) are determined by the two constraints of Equation 18.3. In particular all weights are strictly positive. Moreover \(\boldsymbol{\beta}\) is any minimizer of the convex log-partition function \[ Q(\boldsymbol{\beta})\;:=\;\log\sum_{i=1}^{n}\exp\!\big(\boldsymbol{\beta}^\top\boldsymbol{\Delta}_i\big), \tag{18.6}\] and given such a \(\boldsymbol{\beta}\) the normalization multiplier is \(e^{\alpha}=n\big/\sum_{i}\exp(\boldsymbol{\beta}^\top\boldsymbol{\Delta}_i)\).

Proof. Sufficiency of KKT. Problem Equation 18.3 minimizes a convex objective (Lemma 18.1) over a set defined by affine equalities and the inequalities \(w_i\ge 0\). For such a convex program a feasible point at which the KKT conditions hold is a global minimizer; uniqueness of the minimizer is the strict convexity of Lemma 18.1. It therefore suffices to exhibit a feasible point and multipliers satisfying KKT.

The Lagrangian. Introduce a multiplier vector \(-\boldsymbol{\beta}\in\mathbb{R}^p\) for the moment constraint (the minus sign is a convention that makes Equation 18.5 come out with a positive exponent) and a multiplier \(\lambda\in\mathbb{R}\) for the normalization constraint. Because we will verify that the nonnegativity constraints are inactive, we may form the Lagrangian for the equality constraints alone: \[ \mathcal{L}(\mathbf{w},\boldsymbol{\beta},\lambda) =\sum_{i=1}^{n} w_i\log w_i -\boldsymbol{\beta}^\top\!\sum_{i=1}^{n} w_i\boldsymbol{\Delta}_i -\lambda\Big(\sum_{i=1}^{n} w_i-n\Big). \tag{18.7}\]

Stationarity. Differentiating Equation 18.7 in \(w_i\) and setting the result to zero, \[ \frac{\partial\mathcal{L}}{\partial w_i} =\log w_i+1-\boldsymbol{\beta}^\top\boldsymbol{\Delta}_i-\lambda=0 \;\Longrightarrow\; w_i=\exp\!\big(\lambda-1+\boldsymbol{\beta}^\top\boldsymbol{\Delta}_i\big). \tag{18.8}\] Absorbing the constant by setting \(\alpha:=\lambda-1\) gives the claimed form \(w_i^{*}=\exp(\alpha+\boldsymbol{\beta}^\top\boldsymbol{\Delta}_i)\). Every such \(w_i^{*}\) is strictly positive, so the inequality constraints \(w_i\ge 0\) are slack, their KKT multipliers are zero, and complementary slackness holds trivially; this confirms that working with the equality-only Lagrangian Equation 18.7 was legitimate.

Primal feasibility determines the multipliers. The exponential point Equation 18.5 is a global minimizer iff \((\alpha,\boldsymbol{\beta})\) can be chosen so that it satisfies the two equality constraints. Substituting Equation 18.5 into the normalization constraint, \[ n=\sum_{i} e^{\alpha}\,e^{\boldsymbol{\beta}^\top\boldsymbol{\Delta}_i} =e^{\alpha}\sum_{i} e^{\boldsymbol{\beta}^\top\boldsymbol{\Delta}_i}, \qquad\text{so}\qquad e^{\alpha}=\frac{n}{\sum_{i} e^{\boldsymbol{\beta}^\top\boldsymbol{\Delta}_i}}, \tag{18.9}\] which solves for \(\alpha\) given any \(\boldsymbol{\beta}\). Substituting Equation 18.5 into the moment constraint and cancelling the positive factor \(e^{\alpha}\) leaves a single vector equation in \(\boldsymbol{\beta}\), \[ \sum_{i=1}^{n}\boldsymbol{\Delta}_i\,\exp\!\big(\boldsymbol{\beta}^\top\boldsymbol{\Delta}_i\big)=\mathbf{0}. \tag{18.10}\]

The dual reduction. Equation Equation 18.10 is precisely the stationarity condition of the log-partition function Equation 18.6. Indeed, writing \(Z(\boldsymbol{\beta})=\sum_i e^{\boldsymbol{\beta}^\top\boldsymbol{\Delta}_i}\) and the tilted probabilities \(p_i(\boldsymbol{\beta})=e^{\boldsymbol{\beta}^\top\boldsymbol{\Delta}_i}/Z(\boldsymbol{\beta})\), the gradient of \(Q=\log Z\) is \[ \nabla Q(\boldsymbol{\beta}) =\frac{1}{Z(\boldsymbol{\beta})}\sum_i\boldsymbol{\Delta}_i e^{\boldsymbol{\beta}^\top\boldsymbol{\Delta}_i} =\sum_i p_i(\boldsymbol{\beta})\,\boldsymbol{\Delta}_i, \tag{18.11}\] so Equation 18.10 is the equation \(\nabla Q(\boldsymbol{\beta})=\mathbf{0}\). The Hessian of \(Q\) is the tilted covariance matrix of the centered covariates, \[ \nabla^2 Q(\boldsymbol{\beta}) =\sum_i p_i(\boldsymbol{\beta})\,\boldsymbol{\Delta}_i\boldsymbol{\Delta}_i^\top -\Big(\sum_i p_i(\boldsymbol{\beta})\,\boldsymbol{\Delta}_i\Big)\Big(\sum_i p_i(\boldsymbol{\beta})\,\boldsymbol{\Delta}_i\Big)^\top =\operatorname{Cov}_{p(\boldsymbol{\beta})}(\boldsymbol{\Delta}), \tag{18.12}\] which is positive semidefinite, being a covariance matrix; hence \(Q\) is convex. (We verify Equation 18.12 directly: with \(Q=\log Z\), \(\partial_k Q=\sum_i p_i\Delta_{ik}\), and since \(\partial_\ell p_i=p_i(\Delta_{i\ell}-\sum_j p_j\Delta_{j\ell})\) (this is the derivative of the softmax weights \(p_i=e^{\boldsymbol{\beta}^\top\boldsymbol{\Delta}_i}/Z\), obtained by the quotient rule), we get \(\partial_\ell\partial_k Q=\sum_i p_i\Delta_{ik}\Delta_{i\ell}-(\sum_i p_i\Delta_{ik})(\sum_j p_j\Delta_{j\ell})\), the \((k,\ell)\) entry of the covariance.) A convex differentiable function attains a global minimum exactly where its gradient vanishes, so the \(\boldsymbol{\beta}\) solving Equation 18.10 are exactly the minimizers of \(Q\). That at least one finite minimizer exists under the stated relative-interior hypothesis is the content of Theorem 18.2, on which we are explicitly conditioning.

Collecting the pieces: with \(\boldsymbol{\beta}\) a minimizer of \(Q\) and \(\alpha\) from Equation 18.9, the point Equation 18.5 is feasible and satisfies KKT, hence is the unique global minimizer \(\mathbf{w}^{*}\). \(\square\)

Remark. Back to the four patients. In Example 18.1 the dual problem is one-dimensional. The centered severities are \(\Delta=+0.5\) for the two severe patients and \(\Delta=-1.5\) for the two mild ones, so \(Q(\beta)=\log\!\big(2e^{0.5\beta}+2e^{-1.5\beta}\big)\), a single smooth convex function of the scalar \(\beta\). Setting \(Q'(\beta)=0\) gives \(e^{0.5\beta}=3e^{-1.5\beta}\), that is \(e^{2\beta}=3\) and \(\beta=\tfrac12\ln 3\), exactly the value we read off by hand. The theorem guarantees this minimizer is unique and the resulting weights strictly positive, which is why we never had to worry about a second solution or a negative weight. \(\blacktriangleleft\)

Remark. Computation. In practice one never iterates over \(\mathbf{w}\); one minimizes the smooth convex function \(Q\) in Equation 18.6 over \(\boldsymbol{\beta}\in\mathbb{R}^p\) by Newton-Raphson, whose gradient Equation 18.11 is the normalized covariate imbalance and whose Hessian Equation 18.12 is the tilted covariance, both already assembled from the data. Convergence is fast because \(Q\) is convex and, near the optimum, well approximated by a quadratic. Software such as the maicChecks and MAIC R packages minimizes the equivalent unnormalized objective \(\widetilde{Q}(\boldsymbol{\beta})=\sum_i e^{\boldsymbol{\beta}^\top\boldsymbol{\Delta}_i}\), whose gradient \(\sum_i\boldsymbol{\Delta}_i e^{\boldsymbol{\beta}^\top\boldsymbol{\Delta}_i}\) is the raw imbalance Equation 18.10; \(Q\) and \(\widetilde{Q}=e^{Q}\) share their minimizer because \(e^{(\cdot)}\) is increasing. \(\blacktriangleleft\)

Remark. The Coq counterpart. The companion development defines the weight function as maic_weight_fn x := expR (linpred x), the exponential of an abstract linear predictor, and proves it strictly positive (Lemma maic_weight_pos); it states moment balance as a separate proposition. The optimization that pins down \((\alpha,\boldsymbol{\beta})\), the KKT and duality content of Theorem 18.1, is a pen-and-paper result and is not formalized there; the algebraic identity it does verify, the effective-sample-size bound, is Theorem 18.6 below. \(\blacktriangleleft\)

This result, the central derivation of MAIC, corresponds to F.3 in the proof catalogue.

18.7 Existence and uniqueness via convex duality

Theorem Theorem 18.1 assumed a strictly positive feasible point. We now characterize exactly when one exists, in terms of where the target mean sits relative to the IPD covariate cloud. The characterization is purely geometric and is the precise mathematical content of “adequate overlap.”

Why prove this at all? A practitioner might just run the optimizer and see whether it converges. The value of the theorem is that it tells us, before fitting, and from covariate summaries alone, whether MAIC is even defined: the method succeeds exactly when the comparator’s average patient could have been produced by some reweighting of the patients we actually observed. That is a statement we can check against the reported covariate ranges of the two trials, and it turns a mysterious “the optimizer did not converge” into a diagnosable overlap failure.

Recall that the convex hull \(\operatorname{conv}\{\mathbf{x}_1,\dots,\mathbf{x}_n\}\) is the set of all convex combinations \(\sum_i p_i\mathbf{x}_i\) with \(p_i\ge0\), \(\sum_i p_i=1\); for a finite point set it is a compact polytope and equals its own closure. Picture the index patients as dots in covariate space; the convex hull is the smallest solid polygon (in two dimensions) or polytope enclosing all of them, the region you could reach by averaging patients. The affine hull is the flat (line, plane, or higher) that the dots span, and the relative interior \(\operatorname{relint}(S)\) of a convex set \(S\) is its interior taken within that flat: for a filled triangle in the plane it is the open interior; for a line segment it is the segment minus its two endpoints. The distinction matters only when the covariates are collinear, so the cloud is genuinely lower-dimensional (a segment inside a plane, say), in which case the ambient interior is empty but the relative interior is not.

The geometry sorts every possible target mean into exactly three cases, which is the whole content of the theorem in a table.

Where the target mean \(\bar{\mathbf{x}}_{BC}\) sits	Feasible?	MAIC weights	What you see numerically
Strictly inside the hull (relative interior)	Yes	unique, finite, all positive	optimizer converges to moderate weights
Exactly on the boundary of the hull	Yes, but only with some weights \(=0\)	no finite exponential solution	some weights driven toward \(0\); extreme weights
Outside the hull	No	none exist	optimizer diverges; no weights can reach the target

Reader translation. “Inside the cloud” means the comparator population is a reachable reweighting of your trial and MAIC works; “on the edge” means the comparator looks like the most extreme patients you enrolled and the method is on the verge of failing; “outside” means the comparator is more extreme than anyone in your trial and no weighting can get there, the hard wall of Theorem 18.8. In the running Example 18.1 the target severity \(0.5\) sits strictly between the observed values \(-1\) and \(+1\), the one-dimensional version of “inside the hull,” so finite positive weights existed.

The formal statement and its proof follow. The proof of Part 2, especially the coercivity argument, is the most technical passage in the chapter; a reader meeting convex analysis for the first time may take the three-case table as the working summary and return to the proof later without loss of continuity.

Theorem 18.2 (Feasibility and finite weights) Let \(\boldsymbol{\Delta}_i=\mathbf{x}_i-\bar{\mathbf{x}}_{BC}\) and \(\mathcal{S}=\operatorname{conv}\{\boldsymbol{\Delta}_1,\dots,\boldsymbol{\Delta}_n\}\).

1. (Feasibility.) The moment-balance system Equation 18.2 has a nonnegative solution with \(\sum_i w_i=n\) if and only if \[ \bar{\mathbf{x}}_{BC}\in\operatorname{conv}\{\mathbf{x}_1,\dots,\mathbf{x}_n\}, \qquad\text{equivalently}\qquad \mathbf{0}\in\mathcal{S}. \tag{18.13}\]

2. (Finite positive weights.) The entropy-balancing problem Equation 18.3 has a (necessarily unique) strictly positive solution of exponential form Equation 18.5 with finite \((\alpha,\boldsymbol{\beta})\) if and only if \[ \bar{\mathbf{x}}_{BC}\in\operatorname{relint}\operatorname{conv}\{\mathbf{x}_1,\dots,\mathbf{x}_n\}, \qquad\text{equivalently}\qquad \mathbf{0}\in\operatorname{relint}\mathcal{S}. \tag{18.14}\] The weight vector \(\mathbf{w}^{*}\) is then unique; the dual parameter \(\boldsymbol{\beta}\) is unique up to addition of any vector orthogonal to every \(\boldsymbol{\Delta}_i\), and is unique outright when the centered covariates \(\{\boldsymbol{\Delta}_i\}\) span \(\mathbb{R}^p\).

Proof. Part 1. The set of achievable weighted means under the normalization \(\sum_i w_i=n\) is exactly the convex hull: putting \(p_i=w_i/n\) gives \(p_i\ge0\), \(\sum_i p_i=1\), and weighted mean \(\sum_i p_i\mathbf{x}_i\), so as \(\mathbf{w}\) ranges over nonnegative vectors summing to \(n\) the weighted mean ranges over \(\{\sum_i p_i\mathbf{x}_i:p_i\ge0,\sum p_i=1\}=\operatorname{conv}\{\mathbf{x}_i\}\). Hence Equation 18.1 is solvable iff \(\bar{\mathbf{x}}_{BC}\in\operatorname{conv}\{\mathbf{x}_i\}\); subtracting \(\bar{\mathbf{x}}_{BC}\) throughout, this is \(\mathbf{0}\in\mathcal{S}\). (That an explicit solution uses at most \(p+1\) of the points is Carathéodory’s theorem, but existence needs only the hull membership.)

Part 2, necessity. Suppose a finite minimizer \(\boldsymbol{\beta}\) of \(Q\) exists, equivalently \(\nabla Q(\boldsymbol{\beta})=\mathbf{0}\). By Equation 18.11 this reads \(\sum_i p_i(\boldsymbol{\beta})\boldsymbol{\Delta}_i=\mathbf{0}\) with all \(p_i(\boldsymbol{\beta})>0\) and \(\sum_i p_i(\boldsymbol{\beta})=1\). Thus \(\mathbf{0}\) is a convex combination of the \(\boldsymbol{\Delta}_i\) with strictly positive coefficients. A strictly positive convex combination of the generators of a polytope lies in its relative interior: indeed for any point \(\mathbf{y}\in\mathcal{S}\), \(\mathbf{y}=\sum_i q_i\boldsymbol{\Delta}_i\) with \(q_i\ge0\), \(\sum_i q_i=1\), and then for small \(\varepsilon>0\) the point \(\mathbf{0}+\varepsilon(\mathbf{0}-\mathbf{y})=\sum_i\big(p_i-\varepsilon(q_i-p_i)\big) \boldsymbol{\Delta}_i\) still has nonnegative coefficients summing to one (because each \(p_i>0\) gives room to subtract), so \(\mathbf{0}\) can be moved a little in the direction opposite any point of \(\mathcal{S}\) while staying in \(\mathcal{S}\), which is the defining property of the relative interior. Hence \(\mathbf{0}\in\operatorname{relint}\mathcal{S}\).

Part 2, sufficiency. Suppose \(\mathbf{0}\in\operatorname{relint}\mathcal{S}\). We show \(Q\) attains a finite minimum. Let \(V=\operatorname{span}\{\boldsymbol{\Delta}_1,\dots,\boldsymbol{\Delta}_n\}\); since \(\mathbf{0}\in\mathcal{S}\) implies \(\mathbf{0}\in\operatorname{aff}\mathcal{S}\), the affine hull is the linear subspace \(V\), and the relative interior of \(\mathcal{S}\) is its interior relative to \(V\). Decompose \(\boldsymbol{\beta}=\boldsymbol{\beta}_V+\boldsymbol{\beta}_\perp\) with \(\boldsymbol{\beta}_V\in V\) and \(\boldsymbol{\beta}_\perp\perp V\). Because each \(\boldsymbol{\Delta}_i\in V\), we have \(\boldsymbol{\beta}^\top\boldsymbol{\Delta}_i=\boldsymbol{\beta}_V^\top\boldsymbol{\Delta}_i\), so \(Q\) does not depend on \(\boldsymbol{\beta}_\perp\); it therefore suffices to minimize \(Q\) over \(V\).

We claim \(Q\) is coercive on \(V\) (a function is coercive when it eventually exceeds any fixed bound as its argument runs off to infinity in every direction), that is \(Q(\boldsymbol{\beta})\to\infty\) as \(\|\boldsymbol{\beta}\|\to\infty\) within \(V\). Since \(\mathbf{0}\in\operatorname{relint}\mathcal{S}=\operatorname{int}_V\mathcal{S}\), there is \(\varepsilon>0\) such that the relative ball \(\{\mathbf{v}\in V:\|\mathbf{v}\|\le\varepsilon\}\subset \mathcal{S}\). The reason for taking a whole ball, rather than one point of \(\mathcal{S}\), is that the same radius \(\varepsilon\) then serves every direction at once, which is exactly what makes the growth bound below hold uniformly in \(\mathbf{u}\). Fix any unit vector \(\mathbf{u}\in V\). Then \(\varepsilon\mathbf{u}\in\mathcal{S}\), so \(\varepsilon\mathbf{u}=\sum_i q_i\boldsymbol{\Delta}_i\) for some convex weights \(q_i\), and \[ \varepsilon=\mathbf{u}^\top(\varepsilon\mathbf{u})=\sum_i q_i\,(\mathbf{u}^\top\boldsymbol{\Delta}_i) \le \max_i\,\mathbf{u}^\top\boldsymbol{\Delta}_i, \] so for every unit \(\mathbf{u}\in V\) there is an index \(i\) with \(\mathbf{u}^\top\boldsymbol{\Delta}_i\ge\varepsilon\). Therefore, for \(t>0\), \[ Q(t\mathbf{u})=\log\sum_i e^{t\,\mathbf{u}^\top\boldsymbol{\Delta}_i} \ge\log\Big(\max_i e^{t\,\mathbf{u}^\top\boldsymbol{\Delta}_i}\Big) =t\,\max_i\mathbf{u}^\top\boldsymbol{\Delta}_i\ge t\varepsilon\xrightarrow[t\to\infty]{}\infty, \] uniformly in the unit vector \(\mathbf{u}\). A convex function that grows at least linearly along every ray from the origin is coercive, so \(Q\), continuous and coercive on the finite-dimensional space \(V\), attains its infimum at some \(\boldsymbol{\beta}^{*}\in V\). There \(\nabla_V Q(\boldsymbol{\beta}^{*})=\mathbf{0}\); and since the full gradient Equation 18.11 is always a combination of the \(\boldsymbol{\Delta}_i\) and hence lies in \(V\), we get \(\nabla Q(\boldsymbol{\beta}^{*})=\mathbf{0}\) in \(\mathbb{R}^p\). By Theorem 18.1 the corresponding Equation 18.5 is the finite positive solution.

Strong duality and uniqueness. Under Equation 18.14 the primal Equation 18.3 has a strictly feasible point (the positive solution just constructed), so Slater’s condition holds and there is no duality gap; the primal optimum \(\mathbf{w}^{*}\) is unique by the strict convexity of \(H\) (Lemma 18.1). The dual objective \(Q\) restricted to \(V\) has Hessian \(\operatorname{Cov}_{p(\boldsymbol{\beta})}(\boldsymbol{\Delta})\), which is positive definite on \(V\) (its null space consists of directions \(\mathbf{v}\) with \(\mathbf{v}^\top\boldsymbol{\Delta}_i\) constant across \(i\); within \(V\), with \(\mathbf{0}\in\operatorname{relint}\mathcal{S}\), the only such direction is \(\mathbf{0}\)), so \(Q\) is strictly convex on \(V\) and its minimizer in \(V\) is unique. Adding any \(\boldsymbol{\beta}_\perp\perp V\) leaves the weights unchanged, which is the stated non-uniqueness; when the \(\boldsymbol{\Delta}_i\) span \(\mathbb{R}^p\) we have \(V=\mathbb{R}^p\) and \(\boldsymbol{\beta}\) is unique outright. \(\square\)

Remark. The two failure modes, geometrically. A target on the boundary of the convex hull, for instance a \(\bar{\mathbf{x}}_{BC}\) equal to the maximum of one covariate over the IPD, is feasible (Part 1) only with some weights driven to zero, and admits no finite positive exponential solution (Part 2): the optimizer sends a component of \(\boldsymbol{\beta}\) to \(\pm\infty\) and the corresponding weights to \(0\) or \(\infty\). A target outside the hull is infeasible: no nonnegative weights can pull the weighted mean out to it, because a weighted average never leaves the convex hull of the averaged points. Numerically, both manifest as failed convergence or extreme weights, and they are the precondition for the no-extrapolation theorem Theorem 18.8. The reduced-dimension subtlety in Part 2 (relative versus absolute interior) is exactly the case of collinear covariates, for instance a covariate that is a fixed affine function of others; entropy balancing then proceeds on the affine subspace the data actually occupy. \(\blacktriangleleft\)

Remark. Overlap diagnostics, in practice. The theorem is not just theory; it is a checklist to run before trusting any MAIC. (1) Compare covariate ranges. For each covariate to be balanced, confirm that the comparator’s reported mean, and if available its range and standard deviation, are compatible with the minimum and maximum seen in the index IPD. A comparator mean near or beyond an index extreme is the boundary or outside case of the table above. (2) Read failed convergence as a message. If the optimizer does not converge or returns enormous weights, do not treat it as a numerical nuisance; it is usually the data telling you the target sits on or outside the hull, so that no finite weighting exists. (3) Inspect the fitted weights. A handful of very large weights, or an effective sample size (Section 18.11) that has collapsed to a small fraction of \(n\), signals a near-boundary target and a fragile analysis, even when the optimizer technically converged. (4) Report balance before and after. Show the standardized covariate differences between index and comparator before weighting and confirm they are essentially zero after weighting, the finite-sample evidence that the balance constraints were actually met. \(\blacktriangleleft\)

This result corresponds to F.4 in the proof catalogue.

18.8 MAIC weights are logistic inverse-probability weights

The exponential form Equation 18.5 is not an accident of the entropy objective; it is the functional form of an inverse probability-of-selection weight. Making the connection precise places MAIC inside the inverse-probability-weighting framework of Chapter 10 and the transportability framework of Chapter 12, and it explains both why MAIC works and what it assumes.

The idea in one sentence, before the algebra. Imagine pooling the \(AC\) and \(BC\) patients and fitting a model that predicts, from covariates alone, which trial a patient came from. A patient who “looks like a \(BC\) patient” gets a high predicted probability of \(BC\) membership. MAIC upweights exactly those \(AC\) patients, the ones who resemble the comparator population, and downweights the \(AC\) patients who look distinctively like the index trial. The theorem below shows that the entropy weights we derived are these inverse-membership-odds weights, so MAIC is quietly doing propensity weighting for trial membership, the same machinery as inverse probability weighting in Chapter 10, but pointed at “which population” instead of “which treatment.” The only twist is that we never see the individual \(BC\) patients, so we cannot fit the membership model by ordinary likelihood; we fit it by matching moments instead.

Introduce a binary selection indicator \(S\) that labels which trial an individual came from: \(S=1\) for the index (\(AC\)) trial and \(S=0\) for the comparator (\(BC\)) trial. Posit a logistic model for trial membership given covariates, \[ \operatorname{logit}P(S=1\mid\mathbf{X}=\mathbf{x}) =\log\frac{P(S=1\mid\mathbf{x})}{P(S=0\mid\mathbf{x})} =\tilde\alpha+\tilde{\boldsymbol{\beta}}^\top\mathbf{x}. \tag{18.15}\]

Theorem 18.3 (Equivalence of MAIC weights and logistic inverse-odds weights) Under the trial-membership model Equation 18.15, the density ratio that transports an index-population average to the comparator population is the inverse selection odds, \[ \frac{f_{BC}(\mathbf{x})}{f_{AC}(\mathbf{x})} =\frac{P(S=0)}{P(S=1)}\cdot\frac{P(S=0\mid\mathbf{x})}{P(S=1\mid\mathbf{x})} =\frac{P(S=0)}{P(S=1)}\,\exp\!\big(-\tilde\alpha-\tilde{\boldsymbol{\beta}}^\top\mathbf{x}\big), \tag{18.16}\] which, as a function of \(\mathbf{x}\), is proportional to the MAIC weight Equation 18.5 under the identification \(\boldsymbol{\beta}=-\tilde{\boldsymbol{\beta}}\). Consequently the MAIC estimator of the comparator-population mean of arm \(A\), \[ \hat\mu_A^{\mathrm{MAIC}} =\frac{\sum_{i}w_i^{*}\,\mathbf{1}\{T_i=A\}\,Y_i}{\sum_{i}w_i^{*}\,\mathbf{1}\{T_i=A\}}, \tag{18.17}\] is the inverse-probability-weighting (Horvitz-Thompson) estimator of Theorem 10.3 applied to the transport problem from \(\mathcal{P}_{AC}\) to \(\mathcal{P}_{BC}\), with selection weights estimated by the moment conditions Equation 18.10 rather than by logistic maximum likelihood.

Proof. The density ratio is the inverse odds. By Bayes’ theorem applied within each trial, the covariate density given membership is \(f_{AC}(\mathbf{x})=P(\mathbf{x}\mid S=1)\) and \(f_{BC}(\mathbf{x})=P(\mathbf{x}\mid S=0)\). (In this line \(P(\mathbf{x})\) and \(P(\mathbf{x}\mid S=s)\) are covariate densities, whereas \(P(S=s)\) and \(P(S=s\mid\mathbf{x})\) are membership probabilities; Bayes’ theorem is what links the two.) Writing \(P(\mathbf{x}\mid S=s)=P(S=s\mid\mathbf{x})\, P(\mathbf{x})/P(S=s)\) and taking the ratio, the marginal covariate density \(P(\mathbf{x})\) cancels: \[ \frac{f_{BC}(\mathbf{x})}{f_{AC}(\mathbf{x})} =\frac{P(S=0\mid\mathbf{x})/P(S=0)}{P(S=1\mid\mathbf{x})/P(S=1)} =\frac{P(S=0)}{P(S=1)}\cdot\frac{P(S=0\mid\mathbf{x})}{P(S=1\mid\mathbf{x})}. \] Under Equation 18.15 the conditional odds is \(P(S=1\mid\mathbf{x})/P(S=0\mid\mathbf{x})= \exp(\tilde\alpha+\tilde{\boldsymbol{\beta}}^\top\mathbf{x})\), so its reciprocal is \(\exp(-\tilde\alpha-\tilde{\boldsymbol{\beta}}^\top\mathbf{x})\), giving Equation 18.16. As a function of \(\mathbf{x}\) this equals a positive constant times \(\exp(-\tilde{\boldsymbol{\beta}}^\top\mathbf{x})\), which matches \(\exp(\alpha+\boldsymbol{\beta}^\top(\mathbf{x}-\bar{\mathbf{x}}_{BC}))\) up to a positive constant exactly when \(\boldsymbol{\beta}=-\tilde{\boldsymbol{\beta}}\). The constant is irrelevant to Equation 18.17 because it cancels between numerator and denominator.

Recognition as inverse-probability weighting. Transporting the index-population conditional mean to \(\mathcal{P}_{BC}\) means computing \(\mathbb{E}_{\mathcal{P}_{BC}}[\mu_A(\mathbf{X})]=\int \mu_A(\mathbf{x})f_{BC}(\mathbf{x})\,d\mathbf{x}\). Writing \(f_{BC}=(f_{BC}/f_{AC})\,f_{AC}\), this is \(\mathbb{E}_{\mathcal{P}_{AC}}[(f_{BC}/f_{AC})(\mathbf{X})\,\mu_A(\mathbf{X})]\), an index-population expectation of the outcome weighted by the density ratio. Estimating it by the self-normalized weighted average over the arm-\(A\) index patients (self-normalized meaning we divide the weighted sum of outcomes by the sum of the weights, so the weights behave like proportions that add to one, and any constant scaling of the weights cancels), with the density ratio Equation 18.16 replaced by the MAIC weight Equation 18.5, is exactly Equation 18.17. This is the inverse-probability-weighting estimator of Theorem 10.3, transposed from “probability of treatment within a population” to “probability of selection into a population,” which is the transportability weighting of Definition 12.3 and Theorem 12.2. The selection weights play the role of the balancing score (Definition 10.1): a function of \(\mathbf{x}\) that, used as a weight, renders the index covariate distribution equal to the comparator’s. \(\square\)

Remark. Method of moments versus maximum likelihood. The only difference between MAIC and a textbook logistic inverse-probability weighting is the estimation criterion for \(\tilde{\boldsymbol{\beta}}\). Ordinary IPW fits the logistic model by maximum likelihood on a sample with individual \(S\)-labels. MAIC has no individual labels from the \(BC\) trial, only its reported moments, so it cannot write a logistic likelihood; instead it chooses \(\boldsymbol{\beta}\) to make the weighted index moments equal the reported comparator moments, which is Equation 18.10. The two criteria target the same population density ratio and coincide asymptotically when the log-odds is genuinely linear in \(\mathbf{x}\); MAIC is thus the method-of-moments implementation of logistic IPW adapted to aggregate comparator data. Its consistency inherits the inverse-probability-weighting requirement that the weight model be correct, here that the density ratio is log-linear in the matched features, which is the same linearity condition that governed the choice of moments in Section 18.4. \(\blacktriangleleft\)

The equivalence, and the one place the two methods part company, is worth a side-by-side.

	Standard logistic IPW	MAIC
Data needed	individual trial labels \(S\) for every patient	index IPD plus comparator summary moments only
Weight form	inverse odds \(\exp(-\tilde\alpha-\tilde{\boldsymbol\beta}^\top\mathbf x)\)	\(\exp(\alpha+\boldsymbol\beta^\top(\mathbf x-\bar{\mathbf x}_{BC}))\), the same shape
How \(\boldsymbol\beta\) is fit	maximum likelihood on labeled data	moment matching (Equation 18.10)
Target it estimates	density ratio \(f_{BC}/f_{AC}\)	the same density ratio
Agree asymptotically when	the membership log-odds is truly linear in \(\mathbf x\)	the same linearity holds

The reason MAIC exists as a separate method is the first row: in an indirect comparison we almost never have the comparator’s patient-level data, only a published table, so maximum likelihood is off the table and moment matching is the best we can do.

This result corresponds to F.5 in the proof catalogue.

18.9 Consistency of anchored MAIC

We now establish that the anchored MAIC estimator targets the right quantity. MAIC is a particular population-adjusted indirect comparison (PAIC), so its consistency is an instance of the general anchored PAIC theorem of Chapter 17 (Theorem 17.1); we give the weighting-specific argument in full and invoke the general theorem for the scale-alignment step that handles non-collapsible measures.

The anchored MAIC estimator combines the reweighted index contrast with the published comparator contrast: \[ \hat d_{AB}^{\mathrm{MAIC,anch}} :=\hat d_{AC}^{\mathrm{MAIC}}-\hat d_{BC}^{\mathrm{AgD}}, \qquad \hat d_{AC}^{\mathrm{MAIC}}=g(\hat\mu_C^{\mathrm{MAIC}})-g(\hat\mu_A^{\mathrm{MAIC}}), \tag{18.18}\] with \(\hat\mu_A^{\mathrm{MAIC}}\) as in Equation 18.17 and \(\hat\mu_C^{\mathrm{MAIC}}\) its arm-\(C\) analogue, on the link scale \(g\).

A word on sign. We write each contrast as comparator-minus-active on the link scale, for example \(\hat d_{AC}^{\mathrm{MAIC}}=g(\hat\mu_C^{\mathrm{MAIC}})-g(\hat\mu_A^{\mathrm{MAIC}})\), so that when the active treatment improves the outcome (a lower severity score, a lower event risk) the contrast comes out positive and reads as “how much better \(A\) is than \(C\).” Orient this to your own outcome: if higher is better, flip the sign. In the running Example 18.1 the benefit \(\mu_C-\mu_A=4+2x\) is positive because lower severity is better and \(A\) lowers it.

Why this matters, and what to watch. This theorem is the payoff: it says the reweight-then-subtract recipe recovers the true \(A\)-versus-\(B\) effect in the comparator population \(\mathcal{P}_{BC}\), the population we could not re-analyze directly. But it does so only under a stack of assumptions, and a health researcher’s real job is to know which of them are checkable and which are acts of faith. The next table is that map; the formal statement and proof follow, and the remark after the proof (Section 18.9, “Where each hypothesis is spent”) revisits each assumption in prose. The one genuinely new idea in the assumption list is shared effect modification (SEMA, Definition 17.6): in plain terms, a covariate such as severity must bend the \(A\)-versus-\(C\) effect in the same way it bends the \(B\)-versus-\(C\) effect, so that the published \(BC\) result can be read as the comparator-population \(B\)-versus-\(C\) effect without our re-estimating it from \(BC\)’s (unavailable) patient data.

Assumption	Plain meaning	Checkable from the two trials?	What goes wrong if it fails
(i) Randomized index trial	the \(A\) and \(C\) arms are comparable without adjustment	Yes, it is a design fact	nothing, if the trial was truly randomized
(ii) Conditional constancy + SEMA	the \(A\)-vs-\(C\) effect is the same function of covariates in both trials, and modifiers act alike on \(A\)-vs-\(C\) and \(B\)-vs-\(C\)	No, untestable from two trials	the carried and published effects do not describe \(\mathcal{P}_{BC}\); residual bias
(iii) All imbalanced effect modifiers measured and balanced	every covariate that changes the \(A\)-vs-\(C\) effect and differs between trials is in \(\mathbf{x}\) and balanced	Partly: you can see imbalance, not prove completeness	leftover confounding-by-population, the Bucher failure MAIC was built to fix
(iv) Overlap	the comparator mean lies inside the index covariate cloud	Yes, compare covariate ranges	weights diverge; estimator undefined or unstable
(v) Right moments for the scale	first moments suffice on a linear scale; higher moments needed on nonlinear ones	Partly, from the link and the reported summaries	mismatch on nonlinear scales; ecological bias

Theorem 18.4 (Consistency of anchored MAIC) Fix a link scale \(g\) and report the \(A\)-versus-\(B\) contrast in the comparator population \(\mathcal{P}_{BC}\). Assume:

1. (Randomization.) The index trial \(AC\) is randomized, so within \(\mathcal{P}_{AC}\), \(\mathbb{E}[Y\mid T=t,\mathbf{X}=\mathbf{x}]=\mu_t(\mathbf{x})\) for \(t\in\{A,C\}\).

2. (Conditional constancy of relative effects, Definition 17.4, with shared effect modification, Definition 17.6.) The conditional \(A\)-versus-\(C\) contrast on scale \(g\) is the same function of \(\mathbf{x}\) in \(\mathcal{P}_{AC}\) and \(\mathcal{P}_{BC}\), and the effect modifiers and their interaction coefficients are shared across \(A\) and \(B\) relative to \(C\), so that \(\hat d_{BC}^{\mathrm{AgD}}\) is consistent for the \(\mathcal{P}_{BC}\) marginal contrast \(d_{BC(\mathcal{P}_{BC})}\).

3. (Complete, balanced effect modifiers.) Every covariate that modifies the \(A\)-versus-\(C\) contrast on scale \(g\) and is imbalanced between \(\mathcal{P}_{AC}\) and \(\mathcal{P}_{BC}\) is included in \(\mathbf{x}\) and balanced by the matched moments.

4. (Overlap.) \(\bar{\mathbf{x}}_{BC}\in\operatorname{relint}\operatorname{conv}\{\mathbf{x}_i\}\), so finite positive weights exist (Theorem 18.2), and the effective sample size grows without bound, \(\mathrm{ESS}\to\infty\), as \(n\to\infty\).

5. (Sufficiency of the matched moments.) The conditional means \(\mu_A(\cdot),\mu_C(\cdot)\) entering the estimand are affine in the matched features (first moments for the identity scale; augmented with the balanced higher moments otherwise), so that weighted index means converge to \(\mathcal{P}_{BC}\) marginal means.

Then the anchored MAIC estimator Equation 18.18 is consistent for the target contrast, \[ \hat d_{AB}^{\mathrm{MAIC,anch}}\;\xrightarrow{\ p\ }\;d_{AB(\mathcal{P}_{BC})}. \] This is consistency, not finite-sample unbiasedness: the weights are estimated from \(\{\mathbf{x}_i\}\), so \(\hat d_{AB}^{\mathrm{MAIC,anch}}\) carries an \(O(1/n)\) finite-sample bias that vanishes in the limit. The reason for the bias is that the same patients are used twice, once to choose the weights and once to average the outcomes, so a chance covariate imbalance in a finite sample nudges the weights and the weighted mean in correlated ways; as \(n\) grows the fitted weights settle to their true values and the correlation washes out, which is why consistency survives even though exact unbiasedness does not.

Proof. Step 1 (exact balance at finite \(n\)). By (iv) and Theorem 18.2 the weights \(w_i^{*}=\exp(\hat\alpha+\hat{\boldsymbol{\beta}}^\top\boldsymbol{\Delta}_i)\) exist, are finite and positive, and by construction satisfy the moment constraint Equation 18.2 exactly, so the weighted index covariate mean equals \(\bar{\mathbf{x}}_{BC}\) for every \(n\).

Step 2 (the weighted arm mean targets the comparator marginal mean). Consider arm \(A\). By (i), for \(T_i=A\) write \(Y_i=\mu_A(\mathbf{x}_i)+\varepsilon_i\) with \(\mathbb{E}[\varepsilon_i\mid\mathbf{x}_i, T_i=A]=0\) and \(\operatorname{Var}(\varepsilon_i\mid\mathbf{x}_i,T_i=A)\le\sigma^2<\infty\). Then \[ \hat\mu_A^{\mathrm{MAIC}} =\underbrace{\frac{\sum_{i:T_i=A}w_i^{*}\,\mu_A(\mathbf{x}_i)}{\sum_{i:T_i=A}w_i^{*}}}_{\text{(signal)}} +\underbrace{\frac{\sum_{i:T_i=A}w_i^{*}\,\varepsilon_i}{\sum_{i:T_i=A}w_i^{*}}}_{\text{(noise)}}. \] Under (v), \(\mu_A(\mathbf{x})=a_0+\mathbf{a}^\top\mathbf{x}\) is affine, so the signal term equals \(a_0+\mathbf{a}^\top\bar{\mathbf{x}}^{A}_{w}\), where \(\bar{\mathbf{x}}^{A}_{w}\) is the weighted covariate mean over arm \(A\). Randomization (i) makes the covariate distribution in arm \(A\) identical to the overall index covariate distribution, and the weights depend only on \(\mathbf{x}\), so the within-arm weighted covariate mean converges to the overall weighted covariate mean, which by Step 1 equals \(\bar{\mathbf{x}}_{BC}\); hence the signal term converges to \(a_0+\mathbf{a}^\top\bar{\mathbf{x}}_{BC}\). By conditional constancy (ii) the function \(\mu_A\) is the same in \(\mathcal{P}_{BC}\), so \(a_0+\mathbf{a}^\top\bar{\mathbf{x}}_{BC}=\mathbb{E}_{\mathcal{P}_{BC}}[\mu_A(\mathbf{X})]= \bar\mu_{A(\mathcal{P}_{BC})}\). The noise term has conditional mean zero and conditional variance bounded by \(\sigma^2\sum_{i:T_i=A}(w_i^{*})^2/(\sum_{i:T_i=A}w_i^{*})^2=\sigma^2/\mathrm{ESS}_A\), which tends to zero by (iv); by Chebyshev’s inequality the noise term converges to zero in probability. Therefore \(\hat\mu_A^{\mathrm{MAIC}}\xrightarrow{p}\bar\mu_{A(\mathcal{P}_{BC})}\). The same argument gives \(\hat\mu_C^{\mathrm{MAIC}}\xrightarrow{p}\bar\mu_{C(\mathcal{P}_{BC})}\).

Step 3 (the reweighted contrast). The link \(g\) is continuous, so by the continuous mapping theorem \[ \hat d_{AC}^{\mathrm{MAIC}}=g(\hat\mu_C^{\mathrm{MAIC}})-g(\hat\mu_A^{\mathrm{MAIC}}) \xrightarrow{p} g(\bar\mu_{C(\mathcal{P}_{BC})})-g(\bar\mu_{A(\mathcal{P}_{BC})}) =d_{AC(\mathcal{P}_{BC})}, \] the marginal \(A\)-versus-\(C\) effect in the comparator population.

Step 4 (anchored subtraction in a single population). By (ii) the comparator estimate satisfies \(\hat d_{BC}^{\mathrm{AgD}}\xrightarrow{p}d_{BC(\mathcal{P}_{BC})}\). Both surviving contrasts now live in the same population \(\mathcal{P}_{BC}\), so the within-population additivity of relative effects (Lemma 14.1, evaluated in \(\mathcal{P}_{BC}\)) gives \(d_{AC(\mathcal{P}_{BC})}-d_{BC(\mathcal{P}_{BC})}=d_{AB(\mathcal{P}_{BC})}\). Combining Steps 3 and 4 by Slutsky’s theorem yields \(\hat d_{AB}^{\mathrm{MAIC,anch}}=\hat d_{AC}^{\mathrm{MAIC}}-\hat d_{BC}^{\mathrm{AgD}} \xrightarrow{p}d_{AB(\mathcal{P}_{BC})}\).

Non-collapsible scales. For collapsible measures (the identity and log scales, Definition 11.6) Step 4 is immediate. For non-collapsible measures (the log odds ratio, the log hazard ratio, Theorem 11.2) the marginal contrast is not the average of conditional contrasts, and the subtraction in Step 4 is licensed only under the scale-alignment conditions catalogued by the general anchored PAIC theorem (Theorem 17.1), which (ii) and (iii) supply; we invoke that theorem for the marginalization-then-subtraction step on such scales rather than reproving the collapsibility bookkeeping here. \(\square\)

Remark. What non-collapsibility demands, concretely. Because most health-technology submissions report odds ratios or hazard ratios, the last step of the proof deserves to be spelled out, not just cited. On a collapsible scale (identity, log-risk) the marginal effect is the covariate-average of the conditional effects, so once MAIC has matched the covariate distribution the marginal \(A\)-versus-\(C\) effect transports directly and the Bucher subtraction goes through. On a non-collapsible scale (log odds ratio, log hazard ratio) this is false: as Example 18.2 showed, the marginal contrast depends on the covariate spread and on the baseline risk, not only on what conditional effect the patients carry. Two concrete consequences follow. First, matching the mean covariate does not by itself make two marginal odds ratios comparable; you must match enough of the covariate distribution that the two populations marginalize the same way, which is why higher moments matter here (Section 18.4). Second, even after matching, the two marginal contrasts must be brought to a common baseline risk before they are subtracted; this is the “scale-alignment” content of Theorem 17.1, and in an anchored comparison it is the shared comparator \(C\) together with SEMA (ii) that supplies the common baseline. The systematic fix that never needs this bookkeeping is to integrate an individual-level model over the target covariate distribution, which is multilevel network meta-regression (Chapter 20); MAIC instead leans on the anchor and the shared-modifier structure to keep the two marginal effects on speaking terms. The practical warning for a reader is blunt: a MAIC that balances covariate means and then differences two published odds ratios has not automatically produced a valid comparison unless the scale-alignment conditions hold. \(\blacktriangleleft\)

Remark. Where each hypothesis is spent, and what fails without it. Overlap (iv) is what makes the weights finite and the noise average out; without it MAIC is undefined or unstable (Theorem 18.8, Theorem 18.6). Balance of every imbalanced effect modifier (iii) is what makes the reweighted contrast equal the comparator contrast; omitting an imbalanced modifier leaves residual confounding-by- population and asymptotic bias, exactly the Bucher failure of Theorem 16.1 that MAIC was built to repair. Conditional constancy with shared modification (ii) is what licenses transporting \(\mu_A\) to \(\mathcal{P}_{BC}\) and identifying \(\hat d_{BC}^{\mathrm{AgD}}\) with the \(\mathcal{P}_{BC}\) contrast; it is the population-adjustment analogue of transitivity and the only assumption that is genuinely untestable from these two trials (Proposition 17.4). Sufficiency of the matched moments (v) is the linearity condition that makes first-moment matching enough; on nonlinear scales one matches more moments, and Theorem 17.3 records what is then identified. Randomization (i) supplies the within-trial unconfoundedness for free. \(\blacktriangleleft\)

This result corresponds to F.6 in the proof catalogue.

18.10 Consistency of unanchored MAIC

When the index and comparator trials share no common comparator, there is no anchor to difference, and MAIC must compare the absolute outcomes of \(A\) and \(B\) across populations directly. The estimator becomes \[ \hat d_{AB}^{\mathrm{MAIC,unanch}} :=g(\hat\mu_A^{\mathrm{MAIC}})-g(\hat\mu_B^{\mathrm{MAIC}}), \tag{18.19}\] where each weighted arm mean is transported to a common target population \(\mathcal{P}^{*}\). The price of discarding the anchor is a far stronger and largely untestable assumption set; this is the MAIC instance of Theorem 17.2.

Why this matters, and why it is dangerous. Without a shared comparator \(C\) there is no subtraction to cancel anything, so we must estimate the absolute outcome under \(A\) and under \(B\) separately and difference them. The single most important practical consequence is that we can no longer ignore purely prognostic variables: in the anchored case a covariate that shifts both arms equally cancels out of the contrast (Section 18.9), but here it shifts \(\mu_A\) and \(\mu_B\) and does not cancel, so every imbalanced prognostic variable, not only every effect modifier, must be measured and balanced. That is usually impossible to guarantee, which is why unanchored MAIC is a last resort.

Why the key assumption is untestable. Assumption (i) below says the conditional mean \(\mu_t(\mathbf{x})\) is the same in the source trial and in the target population \(\mathcal{P}^{*}\). We observe \(\mu_t\) only in the source trial; we never observe it in the target, because the whole reason we are doing an indirect comparison is that the target has no arm on treatment \(t\). There is therefore nothing in the data to check the assumption against: it is an act of faith about unmeasured differences between the two populations. Concretely, suppose \(A\) and \(B\) are two oncology drugs studied in separate single-arm or non-overlapping trials with no common comparator. If age is prognostic (older patients fare worse on either drug) and the \(A\) trial enrolled younger patients than the \(B\) trial, then the estimated absolute outcome under \(A\) is computed in a younger mix and under \(B\) in an older mix; unless age is fully measured and balanced, the head-to-head difference confounds “drug” with “age.” An anchored design would have let the shared comparator absorb the age effect by cancellation; the unanchored design has no such protection.

	Anchored MAIC	Unanchored MAIC
Needs a common comparator \(C\)?	Yes	No
Estimand built from	the contrast \(\mu_A-\mu_C\)	the absolute means \(\mu_A\) and \(\mu_B\)
Must balance	imbalanced effect modifiers	imbalanced effect modifiers and prognostic variables
Key assumption	conditional constancy of relative effects, plus SEMA	conditional constancy of absolute effects
Checkable from the data?	partly (see the anchored table)	essentially not
Verdict	preferred whenever an anchor exists	last resort, with heavy sensitivity analysis

Theorem 18.5 (Consistency of unanchored MAIC) Fix a link scale \(g\) and a target population \(\mathcal{P}^{*}\). Assume:

1. (Conditional constancy of absolute effects, Definition 17.5.) For each treatment \(t\in\{A,B\}\) the conditional mean function \(\mu_t(\mathbf{x})\) is the same in its source population and in \(\mathcal{P}^{*}\).

2. (Complete, balanced effect modifiers and prognostic variables.) Every covariate that is an effect modifier on scale \(g\) or a prognostic variable (Definition 11.5) for either treatment, and is imbalanced between any of the source populations and \(\mathcal{P}^{*}\), is included in \(\mathbf{x}\) and balanced.

3. (Overlap.) The target moments lie in the relative interior of the supported features of each source, so finite positive weights exist and \(\mathrm{ESS}\to\infty\).

4. (Sufficiency of the matched moments.) Each \(\mu_t(\cdot)\) is affine in the matched features.

Then the unanchored MAIC estimator Equation 18.19 is consistent for the target contrast, \[ \hat d_{AB}^{\mathrm{MAIC,unanch}}\;\xrightarrow{\ p\ }\;d_{AB(\mathcal{P}^{*})} =g(\bar\mu_{A(\mathcal{P}^{*})})-g(\bar\mu_{B(\mathcal{P}^{*})}). \]

Proof. We give the argument in full, because its structure differs from the anchored case: there is no comparator subtraction to cancel a prognostic variable, so each absolute conditional mean must be identified outright.

Step 1 (identify the source conditional mean). The arm-\(A\) source IPD identifies the source-conditional mean \(\mu_A^{\mathrm{src}}(\mathbf{x})=\mathbb{E}_{\mathrm{src}}[Y\mid T=A,\mathbf{X}=\mathbf{x}]\). Under (i) this equals the target-conditional mean \(\mu_A(\mathbf{x})=\mathbb{E}_{\mathcal{P}^{*}}[Y(A)\mid \mathbf{X}=\mathbf{x}]\). Assumption (i) is strong: it requires that, conditional on \(\mathbf{x}\), no variable left out of \(\mathbf{x}\) shifts the absolute mean of \(Y(A)\) between source and target; by (ii) every imbalanced prognostic variable is in \(\mathbf{x}\), which is what reduces (i) to within-population unconfoundedness plus the absence of differentially distributed unmeasured prognostic variables.

Step 2 (identify the target marginal mean). By the g-formula (Theorem 10.4), \[ \bar\mu_{A(\mathcal{P}^{*})}=\mathbb{E}_{\mathbf{X}\sim f_{*}}[\mu_A(\mathbf{X})] =\int\mu_A(\mathbf{x})\,f_{*}(\mathbf{x})\,d\mathbf{x}. \]

Step 3 (the weighted estimator hits it). MAIC reweights the source-\(A\) IPD to match the target moments, so by the affine-mean argument of Step 2 of Theorem 18.4, applied now to the absolute mean rather than a contrast, \[ \hat\mu_A^{\mathrm{MAIC}} =\frac{\sum_i w_i^{*}\,\mathbf{1}\{T_i=A\}\,Y_i}{\sum_i w_i^{*}\,\mathbf{1}\{T_i=A\}} \xrightarrow{p}\int\mu_A(\mathbf{x})\,f_{*}(\mathbf{x})\,d\mathbf{x}=\bar\mu_{A(\mathcal{P}^{*})}. \] Here is the decisive difference from the anchored case: the weighted mean must converge to the absolute target mean \(\bar\mu_{A(\mathcal{P}^{*})}\), not to a difference of conditional means against a shared comparator. There is no subtraction to cancel a prognostic variable, so the full conditional mean function \(\mu_A(\cdot)\), not merely the CATE \(\tau(\cdot)\), must be transported, which is why (ii) demands prognostic variables and not only effect modifiers.

Step 4 (the other arm and the contrast). The symmetric argument gives \(\hat\mu_B^{\mathrm{MAIC}}\xrightarrow{p}\bar\mu_{B(\mathcal{P}^{*})}\). By the continuous mapping theorem, \[ \hat d_{AB}^{\mathrm{MAIC,unanch}}=g(\hat\mu_A^{\mathrm{MAIC}})-g(\hat\mu_B^{\mathrm{MAIC}}) \xrightarrow{p}g(\bar\mu_{A(\mathcal{P}^{*})})-g(\bar\mu_{B(\mathcal{P}^{*})})=d_{AB(\mathcal{P}^{*})}. \qquad\square \]

Remark. Why prognostic variables matter here but not in the anchored case. In the anchored estimator the quantity that enters is the CATE \(\tau(\mathbf{x})=\mu_A(\mathbf{x})-\mu_C(\mathbf{x})\). If \(w\) is a prognostic variable for both \(A\) and \(C\) that is not an effect modifier on the chosen scale, then it shifts \(\mu_A(\mathbf{x},w)\) and \(\mu_C(\mathbf{x},w)\) by the same amount, so it cancels in the difference: \(\mu_A(\mathbf{x},w)-\mu_C(\mathbf{x},w)=\mu_A(\mathbf{x})-\mu_C(\mathbf{x})\), and omitting \(w\) from \(\mathbf{x}\) does not bias the anchored estimate. In the unanchored estimator the quantity that enters is \(\mu_A(\mathbf{x})\) alone. If \(w\) is an omitted prognostic variable for \(A\), then \(\mu_A(\mathbf{x},w)\) depends on \(w\), and integrating it against the source’s \(f(w\mid\mathbf{x})\) rather than the target’s \(f^{*}(w\mid\mathbf{x})\) gives a different answer; the absolute mean is biased. This is the formal reason that unanchored population adjustment requires strictly more covariate coverage than anchored, and it is identical in form to the baseline-cancellation argument of Proposition 14.4 in Chapter 14: the anchor cancels the baseline; without it the baseline must be modeled. \(\blacktriangleleft\)

Remark. Practical defensibility. Conditional constancy of absolute effects (i) is essentially untestable in the two-study setting (Proposition 17.4), and the adjustment set demanded by (ii), every imbalanced effect modifier and prognostic variable, is usually prohibitive. For these reasons Phillippo et al. (2016) treats unanchored comparisons as a last resort, to be accompanied by extensive sensitivity analysis over the adjustment set. On non-collapsible scales the target marginal contrast also depends on the target baseline risk, a further complication (Theorem 12.6); the systematic remedy is multilevel unanchored meta-regression (Chapter 26). \(\blacktriangleleft\)

This result corresponds to F.7 in the proof catalogue.

18.11 The effective sample size and its bound

Reweighting is not free. A weighted estimator behaves, for the purpose of variance, like an unweighted estimator on a smaller sample, and the size of that smaller sample is the effective sample size.

Definition 18.3 (Effective sample size) For nonnegative weights \(w_1,\dots,w_n\), not all zero, the effective sample size is \[ \mathrm{ESS}\;:=\;\frac{\big(\sum_{i=1}^{n} w_i\big)^2}{\sum_{i=1}^{n} w_i^2}. \tag{18.20}\] It is invariant to rescaling all weights by a common positive constant. \(\blacktriangleleft\)

Reader translation. Read the ESS as “how many equally weighted patients this weighted sample is worth.” If you threw away your weighted sample of \(n\) patients and replaced it with \(\mathrm{ESS}\) fresh patients, each counted once, you would get an estimate of the same precision. When the weights are equal the two are the same and \(\mathrm{ESS}=n\); when a few patients carry most of the weight the sample is effectively much smaller, because the estimate leans on those few. The following table makes the collapse concrete for the four-patient running example; notice how emphasis concentrating onto one patient drives the ESS toward \(1\).

Weights (\(n=4\))	Situation	\(\mathrm{ESS}\)	\(\mathrm{ESS}/n\)
\((1,1,1,1)\)	uniform (no adjustment)	\(4.00\)	\(1.00\)
\((1.5,1.5,0.5,0.5)\)	running Example 18.1 (one covariate matched)	\(3.20\)	\(0.80\)
\((2.25,0.75,0.75,0.25)\)	Section 18.16 (two covariates matched)	\(2.56\)	\(0.64\)
\((2.5,0.5,0.5,0.5)\)	one patient dominant	\(2.29\)	\(0.57\)
\((4,0,0,0)\)	all weight on one patient	\(1.00\)	\(0.25\)

Theorem 18.6 (The effective-sample-size bound) For any nonnegative weights \(w_1,\dots,w_n\), not all zero, \[ \mathrm{ESS}=\frac{\big(\sum_i w_i\big)^2}{\sum_i w_i^2}\;\le\;n, \] with equality if and only if all the weights are equal.

[machine-checked: theories/MAIC.v, Theorem ESS_upper_bound]

Proof. This is the Cauchy-Schwarz inequality (Chapter 2) applied to the vectors \(\mathbf{w}=(w_1,\dots,w_n)\) and \(\mathbf{1}_n=(1,\dots,1)\) in \(\mathbb{R}^n\): \[ \Big(\sum_i w_i\Big)^2=\langle\mathbf{w},\mathbf{1}_n\rangle^2 \le\|\mathbf{w}\|_2^2\,\|\mathbf{1}_n\|_2^2=\Big(\sum_i w_i^2\Big)\cdot n. \] Dividing by \(\sum_i w_i^2>0\) gives \(\mathrm{ESS}\le n\). We exhibit the bound and its equality case self-containedly through Lagrange’s identity, which makes the slack explicit: \[ n\sum_i w_i^2-\Big(\sum_i w_i\Big)^2 =\sum_{i}\sum_{j}w_i^2-\sum_i\sum_j w_iw_j =\tfrac12\sum_{i}\sum_{j}\big(w_i^2-2w_iw_j+w_j^2\big) =\sum_{i<j}(w_i-w_j)^2\ge 0, \] where the second equality writes \(n\sum_i w_i^2=\sum_j\sum_i w_i^2\) and uses the symmetry of the double sum under interchange of \(i\) and \(j\) to symmetrize. The right-hand side is a sum of squares, hence nonnegative, which gives \((\sum_i w_i)^2\le n\sum_i w_i^2\) and therefore \(\mathrm{ESS}\le n\); it vanishes if and only if \(w_i=w_j\) for all pairs \(i<j\), that is, if and only if all weights are equal. \(\square\)

Remark. The Coq proof. The companion development proves this bound without admissions. The elementary inequality \(2xy\le x^2+y^2\) is Lemma two_mul_le_sqr (a rearrangement of \(0\le(x-y)^2\)); an induction on \(n\) assembles it into the Cauchy-Schwarz statement \((\sum_i w_i)^2\le n\sum_i w_i^2\), named cauchy_schwarz_ones in its general form and cauchy_schwarz_n_weights specialized to the weight vector; dividing by the positive \(\sum_i w_i^2\) yields Theorem ESS_upper_bound. The inductive step is exactly the symmetrization used in Lagrange’s identity above. \(\blacktriangleleft\)

The bound is what gives the ESS its meaning as a sample size: the most informative weighting, uniform weighting, retains all \(n\) observations, and any departure strictly reduces the effective count.

This result corresponds to F.8 in the proof catalogue.

18.12 Variance inflation

The effective sample size is not a metaphor; it is the literal denominator in the variance of a weighted mean.

Theorem 18.7 (Variance of a weighted mean and the inflation factor) Let \(Y_1,\dots,Y_n\) be uncorrelated with common mean \(\theta\) and common variance \(\sigma^2\), and let \(w_1,\dots,w_n\ge 0\) be fixed weights, not all zero. The weighted mean \(\hat\theta_w=\sum_i w_iY_i/\sum_i w_i\) is unbiased for \(\theta\) and has variance \[ \operatorname{Var}(\hat\theta_w)=\sigma^2\,\frac{\sum_i w_i^2}{(\sum_i w_i)^2} =\frac{\sigma^2}{\mathrm{ESS}}. \tag{18.21}\] Consequently its variance is the unweighted variance \(\sigma^2/n\) inflated by the factor \[ \frac{\operatorname{Var}(\hat\theta_w)}{\sigma^2/n}=\frac{n}{\mathrm{ESS}}\;\ge\;1, \tag{18.22}\] with equality if and only if the weights are uniform (Theorem 18.6).

Proof. Unbiasedness is linearity of expectation: \(\mathbb{E}[\hat\theta_w]=\sum_i w_i\theta/\sum_i w_i=\theta\). For the variance, the \(Y_i\) are uncorrelated with variance \(\sigma^2\), so \[ \operatorname{Var}(\hat\theta_w) =\frac{1}{(\sum_i w_i)^2}\sum_i w_i^2\operatorname{Var}(Y_i) =\frac{\sigma^2\sum_i w_i^2}{(\sum_i w_i)^2} =\frac{\sigma^2}{\mathrm{ESS}}, \] which is Equation 18.21. Dividing by \(\sigma^2/n\) gives the inflation factor \(n/\mathrm{ESS}\), and \(n/\mathrm{ESS}\ge 1\) is the bound Theorem 18.6, with the same equality condition. \(\square\)

Remark. The fundamental MAIC trade-off. Equation Equation 18.22 is the quantitative form of the central tension in population adjustment. Balancing more covariates, or balancing a fixed set more tightly against a distant target, reduces bias but spreads the weights and lowers the ESS, inflating the variance: \(\mathrm{ESS}=n/2\) doubles the variance, \(\mathrm{ESS}=n/10\) tenfolds it, equivalent to discarding ninety percent of the data. A common rule of thumb flags \(\mathrm{ESS}/n<0.5\) as substantial information loss and \(\mathrm{ESS}/n<0.1\) as an analysis dominated by a handful of patients. Because the weights are in fact estimated, Equation 18.21 is the leading term of the true variance; the honest variance that accounts for weight estimation is the sandwich of Theorem 18.9, and the bias-variance trade-off resurfaces in the simulation theory of Chapter 29. \(\blacktriangleleft\)

This result corresponds to F.9 in the proof catalogue.

18.13 MAIC cannot extrapolate

The punchline first. Suppose the comparator \(BC\) trial enrolled patients aged \(75\) to \(85\), but your index \(AC\) trial enrolled no one over \(70\). Then MAIC cannot tell you what treatment \(A\) does at age \(80\), no matter how cleverly the weights are chosen, because you have no patients there to reweight. Reweighting can only shift emphasis among patients you already observed; it cannot conjure a treatment effect in a region of covariate space your trial never entered. This section proves that this is not a shortcoming of a particular estimator but a hard identification barrier: the target quantity is simply not determined by the data MAIC uses. The remedy is to model the effect across covariate space, which is what simulated treatment comparison (Chapter 19) and multilevel network meta-regression (Chapter 20) do, at the cost of a parametric assumption MAIC avoids.

Overlap is not merely a regularity condition for finite weights; it is a hard limit on what reweighting can ever learn. If the target population places mass where the IPD never looked, no choice of finite weights can recover the missing information, and the target estimand is not identified from the data MAIC uses.

Theorem 18.8 (Non-identification beyond the covariate support) Suppose there is a measurable region \(\mathcal{X}_0\subseteq\mathcal{X}\) with positive target probability but zero index support, \[ P_{\mathcal{P}_{BC}}(\mathbf{X}\in\mathcal{X}_0)>0 \qquad\text{and}\qquad P_{\mathcal{P}_{AC}}(\mathbf{X}\in\mathcal{X}_0)=0. \] Then the comparator-population contrast \(d_{AC(\mathcal{P}_{BC})}\) is not identified from the index IPD law of \((\mathbf{X},T,Y)\) together with the comparator density \(f_{BC}\): there exist two data-generating processes with identical index laws and identical \(f_{BC}\) but different \(d_{AC(\mathcal{P}_{BC})}\). In particular, no estimator that is a functional of the index empirical distribution and the comparator moments, and so no finite-weight MAIC estimator Equation 18.18, can be consistent for \(d_{AC(\mathcal{P}_{BC})}\) across all such processes.

Proof. The target contrast decomposes over the support gap. On the identity scale (the argument is identical on any scale after applying \(g\)), with CATE \(\tau(\mathbf{x})=\mu_A(\mathbf{x})-\mu_C(\mathbf{x})\), \[ d_{AC(\mathcal{P}_{BC})}=\int_{\mathcal{X}\setminus\mathcal{X}_0}\tau(\mathbf{x})\,f_{BC}(\mathbf{x})\,d\mathbf{x} +\int_{\mathcal{X}_0}\tau(\mathbf{x})\,f_{BC}(\mathbf{x})\,d\mathbf{x}. \tag{18.23}\] The first integral is over the region the index trial observes and is a functional of the index law and \(f_{BC}\); the second integral is over \(\mathcal{X}_0\), where the index trial contributes no observations.

To prove non-identification, construct two processes that agree on everything observable yet differ in Equation 18.23. Let the index law of \((\mathbf{X},T,Y)\) and the comparator density \(f_{BC}\) be fixed. Choose two conditional-contrast functions \(\tau_1,\tau_2\) that coincide on the index support \(\operatorname{supp}(f_{AC})\subseteq\mathcal{X}\setminus\mathcal{X}_0\) but differ on \(\mathcal{X}_0\) in a way that changes the gap integral, that is \[ \int_{\mathcal{X}_0}\tau_1(\mathbf{x})\,f_{BC}(\mathbf{x})\,d\mathbf{x} \neq\int_{\mathcal{X}_0}\tau_2(\mathbf{x})\,f_{BC}(\mathbf{x})\,d\mathbf{x}; \] such functions exist because \(f_{BC}\) has positive mass on \(\mathcal{X}_0\), so the linear functional \(\tau\mapsto\int_{\mathcal{X}_0}\tau f_{BC}\) is not identically zero on functions supported in \(\mathcal{X}_0\). Because \(\tau_1\) and \(\tau_2\) agree on \(\operatorname{supp}(f_{AC})\) and the index trial never visits \(\mathcal{X}_0\), the two processes induce the same index law of \((\mathbf{X},T,Y)\) and the same \(f_{BC}\); yet by Equation 18.23 their values of \(d_{AC(\mathcal{P}_{BC})}\) differ by the nonzero amount \(\int_{\mathcal{X}_0}(\tau_1-\tau_2)f_{BC}\). Hence the contrast is not a function of the observable law, which is non-identification. Any estimator built from the index empirical distribution and the comparator moments, MAIC included, takes the same value under the two processes and so is inconsistent for at least one of them. \(\square\)

Remark. Reweighting interpolates, it does not extrapolate. A finite-weight estimator outputs a weighted combination of the observed conditional contrasts \(\{\tau(\mathbf{x}_i)\}\), which can only land in their convex hull; it has no mechanism to report a value of \(\tau\) at an unobserved \(\mathbf{x}\in\mathcal{X}_0\). This is the population-level shadow of the convex-hull feasibility condition Equation 18.13: when \(\bar{\mathbf{x}}_{BC}\) escapes the hull, the weights diverge (Theorem 18.2), and the missing-mass integral in Equation 18.23 is exactly what the divergence is failing to capture. The remedy is not a better weighting but a model: an outcome model that interpolates \(\tau\) across covariate space, which is what simulated treatment comparison (Chapter 19) and multilevel network meta-regression (Chapter 20) provide, at the cost of a parametric assumption that MAIC avoids. Diagnostics should compare the index and comparator covariate ranges before any MAIC is attempted. \(\blacktriangleleft\)

This result corresponds to F.10 in the proof catalogue.

18.14 Standard errors: the sandwich

The one-line rule for practice. Do not compute your standard errors as if the weights were known numbers. You chose the weights by fitting them to the very same patients whose outcomes you are now averaging, and pretending otherwise generally understates the uncertainty. The two honest options are the sandwich variance developed here and the bootstrap of Section 18.15; the naive fixed-weight standard error is not one of them.

The estimated weights make the naive variance Equation 18.21 wrong: it treats the weights as fixed when they were in fact fitted to the same data. The honest variance treats the entire procedure, weight fitting and outcome averaging, as a single system of estimating equations and applies the M-estimation sandwich of Chapter 5 (Definition 5.11).

Stack the parameters as \(\boldsymbol{\theta}=(\boldsymbol{\beta},\mu_A,\mu_C)\) and define the per-patient estimating function \(\boldsymbol{\psi}(\,\cdot\,;\boldsymbol{\theta})\) with three blocks: the balance block, which fits \(\boldsymbol{\beta}\), and the two outcome blocks, which fit the weighted arm means, \[ \boldsymbol{\psi}(\mathbf{x}_i,T_i,Y_i;\boldsymbol{\theta}) =\begin{pmatrix} e^{\boldsymbol{\beta}^\top\boldsymbol{\Delta}_i}\,\boldsymbol{\Delta}_i\\[2pt] \mathbf{1}\{T_i=A\}\,e^{\boldsymbol{\beta}^\top\boldsymbol{\Delta}_i}\,(Y_i-\mu_A)\\[2pt] \mathbf{1}\{T_i=C\}\,e^{\boldsymbol{\beta}^\top\boldsymbol{\Delta}_i}\,(Y_i-\mu_C) \end{pmatrix}, \qquad \sum_{i=1}^{n}\boldsymbol{\psi}(\mathbf{x}_i,T_i,Y_i;\hat{\boldsymbol{\theta}})=\mathbf{0}. \tag{18.24}\] The first block is the balance condition Equation 18.10; the second and third define \(\hat\mu_A^{\mathrm{MAIC}}\) and \(\hat\mu_C^{\mathrm{MAIC}}\) of Equation 18.17. The normalization constant \(\alpha\) does not appear in the balance block because it multiplies the whole balance equation by the positive factor \(e^{\alpha}\), which cannot change whether that equation equals zero; only \(\boldsymbol{\beta}\) is a free parameter to be fit here, and \(\alpha\) is recovered afterward from the normalization (Equation 18.9).

Theorem 18.9 (Sandwich variance for MAIC) Suppose the M-estimator \(\hat{\boldsymbol{\theta}}\) solving Equation 18.24 is consistent for \(\boldsymbol{\theta}_0\) with the standard regularity conditions of Chapter 5 (the map \(\boldsymbol{\theta}\mapsto\mathbb{E}[\boldsymbol{\psi}]\) is differentiable with nonsingular Jacobian \(A=\mathbb{E}[\nabla_{\boldsymbol{\theta}}\boldsymbol{\psi}(\,\cdot\,;\boldsymbol{\theta}_0)]\), and \(B=\mathbb{E}[\boldsymbol{\psi}\boldsymbol{\psi}^\top]\) is finite). Then \[ \sqrt{n}\,(\hat{\boldsymbol{\theta}}-\boldsymbol{\theta}_0)\xrightarrow{\ d\ }\mathcal{N}(\mathbf{0},\, \mathbb{V}),\qquad \mathbb{V}=A^{-1}B\,A^{-\top}, \tag{18.25}\] the sandwich variance of Definition 5.11, consistently estimated by \(\hat{\mathbb{V}}=\hat A^{-1}\hat B\,\hat A^{-\top}\) with \(\hat A=\tfrac1n\sum_i\nabla_{\boldsymbol{\theta}}\boldsymbol{\psi}_i\) and \(\hat B=\tfrac1n\sum_i\boldsymbol{\psi}_i\boldsymbol{\psi}_i^\top\) evaluated at \(\hat{\boldsymbol{\theta}}\). The asymptotic variance of the adjusted contrast \(\hat d_{AC}^{\mathrm{MAIC}}=g(\hat\mu_C)-g(\hat\mu_A)\) follows by the delta method, and that of the anchored estimator Equation 18.18 by adding the independent variance of \(\hat d_{BC}^{\mathrm{AgD}}\).

Proof. Write \(M_n(\boldsymbol{\theta})=\tfrac1n\sum_i\boldsymbol{\psi}(\,\cdot\,;\boldsymbol{\theta})\), so that \(M_n(\hat{\boldsymbol{\theta}})=\mathbf{0}\) and \(M_n(\boldsymbol{\theta})\to M(\boldsymbol{\theta})=\mathbb{E}[\boldsymbol{\psi}]\) with \(M(\boldsymbol{\theta}_0)=\mathbf{0}\). By a first-order Taylor expansion of \(M_n\) about \(\boldsymbol{\theta}_0\), with \(\tilde{\boldsymbol{\theta}}\) on the segment between \(\hat{\boldsymbol{\theta}}\) and \(\boldsymbol{\theta}_0\), \[ \mathbf{0}=M_n(\hat{\boldsymbol{\theta}}) =M_n(\boldsymbol{\theta}_0)+\nabla M_n(\tilde{\boldsymbol{\theta}})\,(\hat{\boldsymbol{\theta}}-\boldsymbol{\theta}_0). \] By the law of large numbers \(\nabla M_n(\tilde{\boldsymbol{\theta}})\xrightarrow{p}A\) (using consistency of \(\hat{\boldsymbol{\theta}}\) and continuity of the Jacobian), and \(A\) is nonsingular by hypothesis, so for large \(n\) it may be inverted: \[ \sqrt{n}\,(\hat{\boldsymbol{\theta}}-\boldsymbol{\theta}_0) =-\big(\nabla M_n(\tilde{\boldsymbol{\theta}})\big)^{-1}\sqrt{n}\,M_n(\boldsymbol{\theta}_0) =-A^{-1}\sqrt{n}\,M_n(\boldsymbol{\theta}_0)+o_p(1). \] By the central limit theorem, \(\sqrt{n}\,M_n(\boldsymbol{\theta}_0)=\tfrac1{\sqrt n}\sum_i \boldsymbol{\psi}(\,\cdot\,;\boldsymbol{\theta}_0)\xrightarrow{d}\mathcal{N}(\mathbf{0},B)\) since the summands are independent and identically distributed with mean \(M(\boldsymbol{\theta}_0)=\mathbf{0}\) and covariance \(B\). Slutsky’s theorem then gives \(\sqrt{n}(\hat{\boldsymbol{\theta}}-\boldsymbol{\theta}_0)\xrightarrow{d}-A^{-1}\mathcal{N}(\mathbf{0},B) =\mathcal{N}(\mathbf{0},A^{-1}BA^{-\top})\), which is Equation 18.25. Consistency of \(\hat{\mathbb{V}}=\hat A^{-1}\hat B\hat A^{-\top}\) is the law of large numbers applied to \(\hat A\) and \(\hat B\) together with continuity of matrix inversion. The delta method (the smooth-map asymptotics of Chapter 5, the same tool behind Theorem 5.6) propagates Equation 18.25 through \(g\) to the contrast, and the comparator term, being from a disjoint trial, contributes an additive independent variance as in the Bucher variance (Theorem 14.2). \(\square\)

Remark. Why naive standard errors are wrong, structurally. The Jacobian \(A\) of Equation 18.24 is block lower-triangular: the balance block depends only on \(\boldsymbol{\beta}\), but the outcome blocks depend on both \(\boldsymbol{\beta}\) and the means, since the weights \(e^{\boldsymbol{\beta}^\top \boldsymbol{\Delta}_i}\) carry \(\boldsymbol{\beta}\) into the outcome equations. Inverting a block-triangular \(A\) propagates the sampling uncertainty of \(\hat{\boldsymbol{\beta}}\) into the variance of \((\hat\mu_A,\hat\mu_C)\) through the off-diagonal block. A naive analysis that treats the weights as fixed drops that off-diagonal block, that is, it pretends the balance block and the outcome blocks are uncoupled; it therefore generally misstates the variance. In typical MAIC applications the weight estimation adds positively correlated noise and the naive estimator is anticonservative; the sign is not universal, which is exactly why the full sandwich, not a fixed-weight shortcut, is needed. \(\blacktriangleleft\)

This result corresponds to F.11 in the proof catalogue.

18.15 Standard errors: the bootstrap

The one-line rule for practice. To bootstrap a MAIC, resample the index patients with replacement and refit the weights from scratch on every resample, then recompute the adjusted contrast; the spread of those recomputed contrasts is your standard error. The common mistake is to freeze the fitted weights and resample only the outcomes, which quietly reproduces the wrong fixed-weight variance of Section 18.14. Refitting the weights each time is what lets the bootstrap feel the uncertainty in the weights themselves.

One technical term recurs below. A map \(\Psi\) of the data distribution is Hadamard-differentiable when it is smooth enough, in the functional sense used for the delta method in Chapter 5, that a small perturbation of the input distribution produces an approximately linear change in the output; it is the infinite-dimensional analogue of “differentiable,” and it is exactly the condition under which resampling the input (the bootstrap) correctly reproduces the sampling distribution of the output. (Fréchet differentiability, mentioned in the proof, is a slightly stronger version of the same idea.)

The sandwich variance requires differentiating the estimating equations; the nonparametric bootstrap sidesteps the algebra by resampling, and it is valid here because the whole MAIC map is smooth in the empirical distribution.

Theorem 18.10 (Validity of the nonparametric bootstrap for MAIC) Let \(\hat d_{AC}^{\mathrm{MAIC}}=\Psi(\mathbb{P}_n)\), where \(\mathbb{P}_n\) is the empirical distribution of the index IPD and \(\Psi\) is the map that fits the entropy-balancing weights, forms the weighted arm means, and returns the contrast. Under the regularity conditions of Theorem 18.9 (in particular overlap, so that the balance Jacobian \(\partial_{\boldsymbol{\beta}}\mathbb{E}[e^{\boldsymbol{\beta}^\top\boldsymbol{\Delta}} \boldsymbol{\Delta}]\) is nonsingular at the solution), \(\Psi\) is Hadamard-differentiable at the index law, and therefore the nonparametric bootstrap is consistent: the bootstrap distribution of \(\sqrt{n}\big(\Psi(\mathbb{P}_n^{*})-\Psi(\mathbb{P}_n)\big)\) converges to the same limiting law as \(\sqrt{n}\big(\Psi(\mathbb{P}_n)-\Psi(\mathbb{P})\big)\), conditionally on the data and in probability. Bootstrap percentile and studentized intervals are thus asymptotically valid.

Proof. We verify the hypotheses of the functional delta method for the bootstrap (the named theorem we invoke is Theorem 23.9 of Vaart (1998)), namely that \(\Psi\) is a composition of Hadamard-differentiable maps of the empirical distribution.

First, the weight parameter is a smooth functional of the law. Define \(\mathbf{G}(\boldsymbol{\beta},F)=\int e^{\boldsymbol{\beta}^\top(\mathbf{x}-\bar{\mathbf{x}}_{BC})} (\mathbf{x}-\bar{\mathbf{x}}_{BC})\,dF(\mathbf{x})\); the fitted \(\boldsymbol{\beta}(F)\) solves \(\mathbf{G}(\boldsymbol{\beta},F)=\mathbf{0}\), which is Equation 18.10 for \(F=\mathbb{P}_n\). The partial derivative \(\partial_{\boldsymbol{\beta}}\mathbf{G}=\int e^{\boldsymbol{\beta}^\top(\mathbf{x}-\bar{\mathbf{x}}_{BC})} (\mathbf{x}-\bar{\mathbf{x}}_{BC})(\mathbf{x}-\bar{\mathbf{x}}_{BC})^\top\,dF\) is a positive multiple of a covariance-type matrix and is nonsingular at the solution under overlap (it is \(Z\cdot\nabla^2 Q\) in the notation of Equation 18.12, positive definite when the centered covariates span). The implicit function theorem then makes \(\boldsymbol{\beta}(F)\) Fréchet-, hence Hadamard-, differentiable in \(F\) along the directions generated by empirical processes.

Second, given \(\boldsymbol{\beta}\), the weighted arm means \(\mu_t(F,\boldsymbol{\beta})=\int\mathbf{1}\{T=t\}e^{\boldsymbol{\beta}^\top(\mathbf{x}-\bar{\mathbf{x}}_{BC})} y\,dF\big/\int\mathbf{1}\{T=t\}e^{\boldsymbol{\beta}^\top(\mathbf{x}-\bar{\mathbf{x}}_{BC})}\,dF\) are ratios of integrals of bounded continuous transformations against \(F\), hence Hadamard-differentiable in \((F,\boldsymbol{\beta})\) wherever the denominator is bounded away from zero (guaranteed by positivity of the weights, Theorem 18.1). Third, the link \(g\) and the subtraction are smooth maps of the means. The composition \(\Psi\) of Hadamard-differentiable maps is Hadamard-differentiable by the chain rule for such maps (Vaart (1998), Theorem 20.9).

By the empirical-process central limit theorem \(\sqrt{n}(\mathbb{P}_n-\mathbb{P})\) converges to a tight Gaussian process, and that same process governs the bootstrap empirical measure. The functional delta method for the bootstrap (Vaart (1998), Theorem 23.9) then transfers the convergence through the Hadamard-differentiable \(\Psi\): the bootstrapped statistic and the original statistic share the limiting Gaussian law, which is precisely the sandwich law of Theorem 18.9. \(\square\)

Remark. Resample the whole procedure. Validity requires bootstrapping the entire map \(\Psi\), refitting the weights on each resample, not freezing the weights and resampling only the outcomes; freezing the weights reproduces the naive fixed-weight variance and inherits its bias (Theorem 18.9 remark). The bootstrap and the sandwich agree to first order and typically differ only by small-sample fluctuation; the bootstrap is often preferred in practice because it sidesteps assembling the Jacobian blocks of Equation 18.24 by hand, while the sandwich is cheaper when many bootstrap replicates would be costly. \(\blacktriangleleft\)

This result corresponds to F.12 in the proof catalogue.

18.16 A fully worked two-covariate example

We carry a complete anchored MAIC from raw covariates to an adjusted indirect comparison, choosing the numbers so that every weight, the effective sample size, and the adjusted contrast are exact and checkable by hand. The example is genuinely two-dimensional, with one effect modifier and one purely prognostic covariate, so it also exhibits the dividing line between covariates that must be balanced and covariates whose balance is harmless but costly.

Example 18.3 (Anchored MAIC with one effect modifier and one prognostic covariate) The index trial \(AC\) has, in each arm, four patients sitting at the corners of a centered square in standardized covariate space, \[ \mathbf{x}\in\big\{(1,1),\,(1,-1),\,(-1,1),\,(-1,-1)\big\}, \] where \(x_1\) is an effect modifier of the \(A\)-versus-\(C\) benefit and \(x_2\) is purely prognostic. The index covariate mean is the center, \(\bar{\mathbf{x}}_{AC}=(0,0)\). The comparator trial \(BC\) reports the covariate mean \[ \bar{\mathbf{x}}_{BC}=(0.5,\,0.5), \] so the index population is imbalanced against the comparator on both covariates. The outcome \(Y\) is a continuous severity score (lower is better) with conditional means, on the identity scale, given by \[ \mu_C(\mathbf{x})=20+2x_1+2x_2, \qquad \mu_A(\mathbf{x})=\mu_C(\mathbf{x})-\big(4+x_1\big)=16+x_1+2x_2. \] Thus the conditional benefit of \(A\) over \(C\) is \(\mu_C(\mathbf{x})-\mu_A(\mathbf{x})=4+x_1\), which depends on the effect modifier \(x_1\) but not on \(x_2\); the prognostic covariate \(x_2\) shifts both arms equally and cancels in the contrast.

Step 1: set up the balance equation. The centered covariates are \(\boldsymbol{\Delta}_i=\mathbf{x}_i-(0.5,0.5)\), which span \(\mathbb{R}^2\), and the target \((0.5,0.5)\) lies strictly inside the square \(\operatorname{conv}\{(\pm1,\pm1)\}=[-1,1]^2\), so by Theorem 18.2 a unique finite positive weight vector exists. The weights have the form \(w(\mathbf{x})=\exp(\alpha+\beta_1(x_1-0.5)+\beta_2(x_2-0.5))\), and we must solve the balance condition Equation 18.10.

Step 2: exploit symmetry. The point set and the target are both invariant under swapping the two coordinates, so the unique solution must satisfy \(\beta_1=\beta_2=:\beta\). Then, up to the constant \(\alpha\), \(w(\mathbf{x})\propto e^{\beta(x_1+x_2)}\), and the four corners take values \(x_1+x_2\in \{2,0,0,-2\}\). The weighted mean of \(x_1\) is \[ \frac{\sum w\,x_1}{\sum w} =\frac{e^{2\beta}(1)+1(1)+1(-1)+e^{-2\beta}(-1)}{e^{2\beta}+1+1+e^{-2\beta}} =\frac{e^{2\beta}-e^{-2\beta}}{e^{2\beta}+e^{-2\beta}+2} =\frac{2\sinh 2\beta}{2\cosh 2\beta+2}=\tanh\beta, \] using \(\sinh 2\beta=2\sinh\beta\cosh\beta\) and \(\cosh 2\beta+1=2\cosh^2\beta\). Setting this equal to the target \(0.5\) gives \(\tanh\beta=\tfrac12\), that is \[ \beta=\operatorname{artanh}(0.5)=\tfrac12\ln\frac{1+0.5}{1-0.5}=\tfrac12\ln 3\approx 0.5493, \qquad e^{2\beta}=3,\quad e^{-2\beta}=\tfrac13. \] By symmetry the weighted mean of \(x_2\) is also \(\tanh\beta=0.5\), so both moments are matched.

Step 3: the explicit weights. Up to scale the corner weights are proportional to \(e^{\beta(x_1+x_2)}\in\{3,\,1,\,1,\,\tfrac13\}\), summing to \(3+1+1+\tfrac13=\tfrac{16}{3}\). Normalizing the four pattern weights to sum to \(4\) multiplies by \(4/(16/3)=\tfrac34\): \[ w_{(1,1)}=\tfrac94=2.25,\quad w_{(1,-1)}=\tfrac34=0.75,\quad w_{(-1,1)}=\tfrac34=0.75,\quad w_{(-1,-1)}=\tfrac14=0.25. \tag{18.26}\] These sum to \(4\), and one checks the balance directly: the weighted mean of \(x_1\) is \((2.25\cdot 1+0.75\cdot 1+0.75\cdot(-1)+0.25\cdot(-1))/4=2/4=0.5\), and likewise the weighted mean of \(x_2\) is \((2.25-0.75+0.75-0.25)/4=0.5\). The normalization multiplier follows from \(w_{(1,1)}=e^{\alpha}e^{\beta}=e^{\alpha}\sqrt3=2.25\), giving \(e^{\alpha}=2.25/\sqrt3=\tfrac{3\sqrt3}{4}\) and \(\alpha=\tfrac32\ln 3-\ln 4\approx 0.2616\).

Step 4: the effective sample size. With the four pattern weights Equation 18.26, \[ \sum_i w_i=4,\qquad \sum_i w_i^2=2.25^2+0.75^2+0.75^2+0.25^2=5.0625+0.5625+0.5625+0.0625=6.25, \] so \[ \mathrm{ESS}=\frac{4^2}{6.25}=\frac{16}{6.25}=2.56,\qquad \frac{\mathrm{ESS}}{n}=\frac{2.56}{4}=0.64. \] Matching both covariate means cost \(36\%\) of the effective sample. The variance inflation factor is \(n/\mathrm{ESS}=4/2.56=1.5625\) (Theorem 18.7), so a weighted arm mean has standard error \(\sqrt{1.5625}=1.25\) times the unweighted standard error, a \(25\%\) increase. (The bound \(\mathrm{ESS}\le n\) of Theorem 18.6 is comfortably respected.)

Step 5: the weighted arm means. The arm-\(A\) outcomes at the four patterns are \(\mu_A(1,1)=19\), \(\mu_A(1,-1)=15\), \(\mu_A(-1,1)=17\), \(\mu_A(-1,-1)=13\); the arm-\(C\) outcomes are \(\mu_C(1,1)=24\), \(\mu_C(1,-1)=20\), \(\mu_C(-1,1)=20\), \(\mu_C(-1,-1)=16\) (we set \(Y\) equal to its conditional mean to keep the arithmetic exact; with noise the estimator is consistent rather than exact, Theorem 18.4). The weighted arm means are \[ \hat\mu_A^{\mathrm{MAIC}}=\frac{2.25(19)+0.75(15)+0.75(17)+0.25(13)}{4}=\frac{70}{4}=17.5, \] \[ \hat\mu_C^{\mathrm{MAIC}}=\frac{2.25(24)+0.75(20)+0.75(20)+0.25(16)}{4}=\frac{88}{4}=22. \] Both equal the comparator-population truth: \(\bar\mu_{A(\mathcal{P}_{BC})}=16+0.5+2(0.5)=17.5\) and \(\bar\mu_{C(\mathcal{P}_{BC})}=20+2(0.5)+2(0.5)=22\), confirming Step 2 of Theorem 18.4.

Step 6: the adjusted index contrast. On the identity scale the benefit of \(A\) over \(C\) in the comparator population is \[ \hat d_{AC}^{\mathrm{MAIC}}=\hat\mu_C^{\mathrm{MAIC}}-\hat\mu_A^{\mathrm{MAIC}}=22-17.5=4.5, \] matching the true comparator-population benefit \(4+\bar x_{1,BC}=4+0.5=4.5\). The unadjusted index benefit, from the unweighted arm means \(\bar\mu_A^{\mathrm{unw}}=(19+15+17+13)/4=16\) and \(\bar\mu_C^{\mathrm{unw}}=(24+20+20+16)/4=20\), is \(20-16=4\), the benefit in the index population (\(4+\bar x_{1,AC}=4+0=4\)). MAIC has moved the estimate from \(4\) to \(4.5\), correcting for the higher value of the effect modifier \(x_1\) in the comparator population.

Step 7: the anchored indirect comparison. Suppose the comparator trial reports a \(B\)-versus-\(C\) benefit of \(\hat d_{BC}^{\mathrm{AgD}}=3.0\) in its own population. The anchored MAIC estimate of the \(A\)-versus-\(B\) benefit in \(\mathcal{P}_{BC}\) is \[ \hat d_{AB}^{\mathrm{MAIC,anch}}=\hat d_{AC}^{\mathrm{MAIC}}-\hat d_{BC}^{\mathrm{AgD}}=4.5-3.0=1.5, \] the target estimand \(d_{AB(\mathcal{P}_{BC})}=d_{AC(\mathcal{P}_{BC})}-d_{BC(\mathcal{P}_{BC})}=4.5-3.0= 1.5\). The unadjusted Bucher comparison would have used the index-population benefit \(4.0\) and produced \(4.0-3.0=1.0\), biased by \(-0.5\). The bias is exactly the effect-modification discrepancy \(d_{AC(\mathcal{P}_{BC})}-d_{AC(\mathcal{P}_{AC})}=4.5-4.0=0.5\), caused by the imbalance in the effect modifier \(x_1\) (\(\bar x_1=0.5\) in \(\mathcal{P}_{BC}\) versus \(0\) in \(\mathcal{P}_{AC}\)); this is precisely the transitivity failure of Theorem 16.1 that population adjustment removes. \(\blacktriangleleft\)

Remark. What the prognostic covariate cost. The effect modifier \(x_1\) had to be balanced: it drives the \(0.5\) bias correction. The prognostic covariate \(x_2\) did not need balancing for the contrast, because it cancels between the arms (\(\mu_A\) and \(\mu_C\) both carry \(+2x_2\)), yet we balanced it anyway, and doing so contributed to the drop in effective sample size from \(4\) to \(2.56\). Had we balanced only \(x_1\), the weights would have been less variable and the ESS higher, at no cost in bias on the identity scale. This is the anchored-case phenomenon of Theorem 18.4: only imbalanced effect modifiers are mandatory; balancing pure prognostic variables is harmless for bias but spends precision. In the unanchored case (Theorem 18.5) the prognostic \(x_2\) would not cancel, and balancing it would be mandatory, which is one more reason the unanchored estimator is so much more expensive. \(\blacktriangleleft\)

18.17 Practitioner checklist

The theory of this chapter condenses into a short list of things to do, and to check in someone else’s analysis, when performing or reviewing a MAIC. Each item names the result it rests on.

1. Confirm overlap first. Compare the comparator’s reported covariate means (and ranges and standard deviations, when given) with the minimum and maximum in the index IPD. A comparator mean near or beyond an index extreme, or an optimizer that fails to converge or returns enormous weights, is an overlap failure, not a software glitch (Theorem 18.2, Theorem 18.8).
2. Balance every imbalanced effect modifier. These are mandatory in both anchored and unanchored MAIC. On a nonlinear link (log odds ratio, log hazard ratio) balance higher moments as well, because first-moment matching is insufficient there (Example 18.2, Section 18.4).
3. Prefer anchored whenever a common comparator exists. Anchored MAIC needs only effect modifiers; unanchored MAIC also needs every prognostic variable and rests on an untestable absolute-constancy assumption. Use unanchored only as a last resort, with sensitivity analysis over the adjustment set (Section 18.9, Section 18.10).
4. Report the effective sample size and flag low values. State \(\mathrm{ESS}\) and \(\mathrm{ESS}/n\); treat \(\mathrm{ESS}/n<0.5\) as substantial information loss and \(\mathrm{ESS}/n<0.1\) as an analysis driven by a handful of patients. Show standardized covariate differences before and after weighting to confirm balance was achieved (Theorem 18.6, Theorem 18.7).
5. Use honest standard errors. Compute uncertainty with the sandwich variance or a full bootstrap that refits the weights on each resample, never as if the weights were fixed; combine it with the published comparator effect’s own standard error, which is independent (Theorem 18.9, Theorem 18.10).
6. Mind the scale. On non-collapsible scales, matching covariate means does not by itself make two marginal odds ratios or hazard ratios comparable; confirm that the scale-alignment conditions hold or move to a model-based method (Theorem 11.2, Chapter 20).

18.18 Notes and references

Matching-adjusted indirect comparison was introduced by Signorovitch et al. (2010), who proposed reweighting the individual patient data of one trial so that its weighted baseline moments match those reported by a comparator trial, and recognized the weights as inverse-odds weights from a logistic model of trial membership. The entropy-balancing formulation, the derivation of the exponential weight form, the convex-duality analysis of existence, the effective-sample-size diagnostic, and the placement of MAIC within the anchored and unanchored population-adjustment framework follow the thesis of Phillippo (2019) and the NICE Decision Support Unit guidance Phillippo et al. (2016); the latter is the standard practical reference for which covariates to balance, the anchored-versus-unanchored decision, and the treatment of unanchored comparisons as a last resort. The maximum-entropy reading of covariate balancing, with its exponential-family solution and its convex dual, is the entropy-balancing method familiar from the causal-inference literature; the equivalence between entropy balancing and a calibrated logistic propensity model is a recurring theme there, and Theorem 18.3 is its MAIC instance, connected to the inverse-probability-weighting theory of Chapter 10 (Theorem 10.3, Definition 10.1) and the transportability theory of Chapter 12 (Definition 12.3, Theorem 12.2).

The convex-analytic facts underlying Theorem 18.2, that the gradient of a log-partition function maps onto the relative interior of the convex hull of the sufficient statistics and that the infimum is attained exactly when the origin lies in that relative interior, are standard in the theory of exponential families and convex duality; the coercivity argument given here is self-contained and avoids appealing to the general machinery of Rockafellar’s Convex Analysis. The effective-sample-size bound Theorem 18.6 is the Cauchy-Schwarz inequality, and the variance interpretation Equation 18.21 is the classical effective-sample-size formula for weighted means; both appear as Eq. (6.1) of Phillippo (2019). The non-extrapolation theorem Theorem 18.8 is the population-adjustment form of a positivity or overlap requirement, and its failure under poor covariate overlap is documented in the simulation studies of Phillippo (2019); the contrast with model-based methods that can interpolate motivates simulated treatment comparison (Chapter 19) and multilevel network meta-regression (Chapter 20). The sandwich variance Theorem 18.9 is M-estimation theory applied to the stacked weight-and-outcome equations (Definition 5.11), and the bootstrap validity Theorem 18.10 is the functional delta method for the bootstrap (Vaart (1998), Theorems 20.9, 23.9). The distinction between consistency and finite-sample unbiasedness, emphasized throughout Section 18.9 and Section 18.10, corrects a looseness in some of the early MAIC literature; it matters because the weights are estimated.

The companion Coq development formalizes the algebraic core of MAIC in theories/MAIC.v. The weight function \(w(\mathbf{x})=\exp(\text{linear predictor})\) is maic_weight_fn, proved positive in maic_weight_pos; the effective sample size is ESS, and the bound \(\mathrm{ESS}\le n\) is Theorem ESS_upper_bound, the machine-checked counterpart of Theorem 18.6. Its proof is built from the elementary two_mul_le_sqr (\(2xy\le x^2+y^2\)), assembled by induction into cauchy_schwarz_ones and its weight specialization cauchy_schwarz_n_weights, and the development records no admissions for these results. The anchored and unanchored estimators appear as effect_AC_MAIC and effect_AB_anchored, with the substantive consistency content (Theorem 18.4, Theorem 18.5) carried by named section hypotheses standing for conditional constancy, complete effect-modifier coverage, shared effect modification, and overlap, rather than as formalized probabilistic theorems; the optimization that fixes \((\alpha,\boldsymbol{\beta})\) in Theorem 18.1 is likewise pen-and-paper. Chapter 30 walks through MAIC.v in detail.

This chapter corresponds to results F.1 (method-of-moments balance), F.2 (the entropy objective), F.3 (the exponential weights), F.4 (existence and uniqueness), F.5 (the logistic-IPW equivalence), F.6 (anchored consistency), F.7 (unanchored consistency), F.8 (the ESS bound), F.9 (variance inflation), F.10 (the no-extrapolation theorem), F.11 (the sandwich variance), and F.12 (bootstrap validity) of the proof catalogue in proofs/F_maic.md. The general population-adjustment theory that MAIC specializes, conditional constancy in its relative and absolute forms (Definition 17.4, Definition 17.5), the shared-effect-modifier assumption (Definition 17.6), and the anchored and unanchored PAIC consistency theorems (Theorem 17.1, Theorem 17.2), is developed in Chapter 17. The regression-based alternative to reweighting, simulated treatment comparison, and its asymptotic equivalence to MAIC under a correctly specified linear outcome model (Theorem 19.7), is the subject of Chapter 19.

18.19 Exercises

Throughout, a starred exercise, marked \(\star\), has a full worked solution in Appendix F. The first three exercises are warm-ups and interpretation checks that need only the ideas of this chapter; the later ones build toward the proofs, and the two starred ones are the most demanding.

Exercise 18.1 Reading the weights. (Warm-up.) In the running Example 18.1 the two severe patients received weight \(1.5\) each and the two mild patients weight \(0.5\) each. (a) In one sentence, say what a weight of \(1.5\) means for a severe patient, in terms of how many target-population patients they stand in for. (b) Why are the severe patients upweighted rather than the mild ones, given that the comparator trial was sicker? (c) Recompute the effective sample size from these weights and state, in plain words, how many equally weighted patients this reweighted sample is worth. (d) Without recomputing anything, would balancing a second imbalanced covariate raise or lower the ESS, and which result of this chapter guarantees the direction?

Exercise 18.2 Effect modifier or prognostic variable? (Interpretation.) For each covariate below, state whether it is most naturally an effect modifier, a prognostic variable, or both, and then say whether an anchored MAIC and an unanchored MAIC would each be obliged to balance it, assuming it is imbalanced between the trials. (a) A biomarker that predicts worse outcomes on every treatment but does not change the size of \(A\)’s benefit over \(C\). (b) Disease severity, which both worsens outcomes and enlarges the benefit of \(A\). (c) A covariate perfectly balanced between the two trials. Justify each answer in one sentence using the cancellation argument of Section 18.9.

Exercise 18.3 Judging a published MAIC. (Interpretation.) A submission reports an unanchored MAIC comparing drug \(A\) (from an IPD trial of \(n=400\) patients) against drug \(B\), balancing three covariates, with an effective sample size \(\mathrm{ESS}=28\) and a \(95\%\) confidence interval computed as though the weights were fixed. The comparator’s reported mean for one of the three covariates lies just above the largest value observed in the IPD. List, in order of severity, the credibility concerns you would raise, and name the result of this chapter behind each. (Consider at least: the anchored-versus-unanchored choice, the overlap of that one covariate, the \(\mathrm{ESS}/n\) ratio, and the variance method.)

Exercise 18.4 A one-covariate MAIC by hand. (Computation.) An index trial has four patients with a single covariate \(x\in\{-2,-1,1,2\}\) (one per value) and a comparator that reports mean \(\bar x_{BC}=0.5\). Write the balance equation \(\sum_i(x_i-0.5)e^{\beta(x_i-0.5)}=0\), solve for \(\beta\) numerically to three decimal places, report the normalized weights summing to \(4\), verify that the weighted mean is \(0.5\), and compute the effective sample size and the variance inflation factor. Comment on how the ESS would change if the target were \(\bar x_{BC}=1.5\) instead.

Exercise 18.5 Balancing a variance. (Computation.) Continue Example 18.3 but now require, in addition to matching both means at \(0.5\), that the weighted variance of \(x_1\) equal a reported comparator value \(s^2_{1,BC}=0.5\). Write the augmented feature map \(\boldsymbol{\phi}(\mathbf{x})=(x_1,x_2,x_1^2)\) and the corresponding three balance equations, explain why the solution generally has \(\beta_1\ne\beta_2\) once the squared term is added, and argue from Theorem 18.6 why adding the second-moment constraint can only decrease (never increase) the effective sample size.

Exercise 18.6 Recovering the logistic weights. (Computation.) Using Theorem 18.3, suppose a logistic trial-membership model has fitted coefficients \(\tilde\alpha=0.3\) and \(\tilde{\boldsymbol{\beta}}=(0.8,-0.4)\). Write the inverse-odds weight as a function of \(\mathbf{x}\), identify the MAIC parameter \(\boldsymbol{\beta}\), and explain in one paragraph why the MAIC method of moments and logistic maximum likelihood give the same weight form but generally different fitted values when the comparator supplies only covariate means.

Exercise 18.7 The ESS and majorization. (Challenge.) Show that among all nonnegative weight vectors with a fixed sum, the effective sample size Equation 18.20 is maximized uniquely by the uniform vector, and more generally that if \(\mathbf{w}'\) majorizes \(\mathbf{w}\) (the sorted partial sums of \(\mathbf{w}'\) dominate those of \(\mathbf{w}\), with equal total, so \(\mathbf{w}'\) is the more concentrated or lopsided vector) then \(\mathrm{ESS}(\mathbf{w}')\le\mathrm{ESS}(\mathbf{w})\). Deduce Theorem 18.6 as the special case where \(\mathbf{w}'\) concentrates all weight on one coordinate. (Hint: \(\mathbf{w}\mapsto\sum_i w_i^2\) is Schur-convex, meaning it never decreases when weight is shifted from a smaller coordinate to a larger one; a Schur-convex function is therefore smallest at the most equal vector and largest at the most concentrated, which is exactly what the ESS bound records.)

Exercise 18.8 (\(\star\)) Decomposing the adjustment. (Proof.) In Example 18.3, replace the conditional benefit \(4+x_1\) by the general form \(b_0+b_1x_1+b_2x_2\) and the comparator mean by a general \((\bar x_{1},\bar x_{2})\) inside the square. Derive a closed-form expression for the MAIC-adjusted benefit \(\hat d_{AC}^{\mathrm{MAIC}}\) in terms of \(b_0,b_1,b_2\) and the target means, prove that it equals the true comparator-population benefit \(b_0+b_1\bar x_1+b_2\bar x_2\) for any weights achieving moment balance (not only the entropy weights), and conclude that on the identity scale with a linear benefit the choice of balancing objective affects the variance (through the ESS) but not the point estimate. Explain how this exact-on-the-mean property fails on the logit scale, connecting to the higher-moment discussion of Section 18.4 and Definition 7.7.

Exercise 18.9 (\(\star\)) Naive versus sandwich variance. (Challenge.) Consider the stacked estimating equations Equation 18.24 for a single-arm weighted mean (drop the \(C\) block). Write the \(2\times 2\) block-triangular Jacobian \(A\) in the scalar-covariate case, invert it explicitly, and show that the asymptotic variance of \(\hat\mu_A\) has the fixed-weight term \(\sigma^2/\mathrm{ESS}\) plus a correction attributable to estimating \(\boldsymbol{\beta}\). Identify the data configuration in which the correction vanishes (so that the naive and sandwich variances agree) and interpret it: it is the case where the outcome is uncorrelated, within the weighting, with the balancing covariate. Relate your finding to the remark following Theorem 18.9 on why fixed-weight standard errors are generally anticonservative.

Bucher, Heiner C., Gordon H. Guyatt, Lauren E. Griffith, and Stephen D. Walter. 1997. “The Results of Direct and Indirect Treatment Comparisons in Meta-Analysis of Randomized Controlled Trials.” Journal of Clinical Epidemiology 50 (6): 683–91.

Phillippo, David M., A. E. Ades, Sofia Dias, Stephen Palmer, Keith R. Abrams, and Nicky J. Welton. 2016. NICE DSU Technical Support Document 18: Methods for Population-Adjusted Indirect Comparisons in Submissions to NICE. Decision Support Unit, ScHARR, University of Sheffield.

Phillippo, David Mark. 2019. “Calibration of Treatment Effects in Network Meta-Analysis Using Individual Patient Data.” PhD thesis, University of Bristol.

Signorovitch, James E., Eric Q. Wu, Andrew P. Yu, et al. 2010. “Comparative Effectiveness Without Head-to-Head Trials: A Method for Matching-Adjusted Indirect Comparisons.” PharmacoEconomics 28 (10): 935–45.

Vaart, A. W. van der. 1998. Asymptotic Statistics. Cambridge University Press.