17  Population Adjustment Theory

17.1 Introduction

Before any symbols, here is the problem in plain words. Chapter 14’s Bucher method estimates the effect of \(A\) versus \(B\) by subtracting two relative effects that share a common comparator \(C\): the effect of \(A\) versus \(C\) measured in one trial and the effect of \(B\) versus \(C\) measured in another. This subtraction is valid only when each trial’s relative effect is a fixed number that does not depend on the patient mix. If some baseline characteristic changes the size of a treatment effect, an effect modifier, and the two trials enrolled different patients, then each trial reports a relative effect averaged over a different population, and the subtraction silently mixes the two and is biased. This chapter is the systematic repair: instead of treating a trial’s effect as a population-free constant, we model how it depends on the covariates and re-standardize it to a population we actually care about. That is what population adjustment means.

Remark. Running example (anticoagulants). We carry one clinical scenario through the whole chapter. Let \(C\) be warfarin, the old common comparator; let \(A\) be a new oral anticoagulant studied against warfarin in a trial that supplies individual patient data (IPD); and let \(B\) be an established direct oral anticoagulant (DOAC) studied against warfarin in a separate trial that supplies only aggregate data (AgD). The outcome is a major bleed within a year, so a higher risk is worse. The one covariate that matters is old age, \(X=1\) for age \(75\) or over and \(X=0\) otherwise, and age raises the bleeding risk on the new drug \(A\) but not on \(B\) or warfarin, so age is an effect modifier of \(A\) versus \(C\). The \(AC\) trial enrolled mostly younger patients and the \(BC\) trial mostly older patients, and a reimbursement agency wants the \(A\)-versus-\(B\) comparison in a target population that is half older. This is exactly the situation Bucher cannot handle and population adjustment can. We attach numbers to it in Section 17.2 and return to it at every definition and theorem. \(\blacktriangleleft\)

Chapter 14 left us with a clean estimator and a fragile assumption. Bucher’s anchored indirect comparison \(\hat d_{AB}=\hat d_{AC}-\hat d_{BC}\) is unbiased for the cross-study contrast precisely when the relative effects are constant across the trial populations (Theorem 14.1, Definition 14.2), and Chapter 16 showed exactly how that constancy breaks: an effect modifier (Definition 11.3) whose distribution differs between the trials makes each trial report a different marginal contrast, and the Bucher subtraction then mixes populations and is biased (Theorem 16.1). This chapter is the systematic response to that failure. It is the conceptual center of the book.

The idea of population adjustment is to refuse to treat a trial’s reported effect as a population-free constant, and instead to model how the effect depends on the covariates and to re-standardize it to a chosen target population. Doing so requires more than aggregate trial summaries. It requires, for at least one trial, the individual patient data (IPD): the covariate vector, treatment, and outcome of each enrolled subject. The other trials may contribute only aggregate data (AgD): arm-level event counts, means, and covariate summaries. A population-adjusted indirect comparison (PAIC) combines an IPD trial with one or more AgD trials to deliver an effect estimate valid in a stated target population. The methods that do this concretely, matching-adjusted indirect comparison (MAIC, Chapter 18), simulated treatment comparison (STC, Chapter 19), and multilevel network meta-regression (ML-NMR, Chapter 20), all rest on a single body of theory, and that theory is the subject of the present chapter.

The central question is: under what conditions is a population-adjusted indirect comparison consistent for its target estimand? The modern answer, which we follow, has two features that distinguish it from older formulations. First, the conditions are scale-specific. The effect modification, the collapsibility, and the reported effect measure each live on a scale, and whether adjustment succeeds depends on how those scales relate. We must distinguish the model link \(g\) (the scale on which the outcome model is built and on which effect modification is defined, Definition 7.2) from the reporting scale \(h\) (the scale of the effect measure actually quoted to a decision maker). When \(h\circ g^{-1}\) is the identity the scales are aligned (Definition 12.4); when they are not, adjustment that works on \(g\) need not work on \(h\). Second, the classical “if and only if” statements of the early literature are replaced by scale-specific sufficient conditions together with a class-wise necessity result. By class-wise we mean the necessity holds across a whole class of data-generating mechanisms, not that every single mechanism in it is biased; a specific trial might escape bias by numerical coincidence. The “iff” is technically false for non-collapsible marginal estimands such as the odds ratio and hazard ratio, because of non-collapsibility (Definition 7.7, Theorem 11.2): even a perfectly adjusted comparison can fail to transport a marginal odds ratio across populations. We make all of this precise.

The development is organized around the two architectures of indirect comparison inherited from Chapter 14. The anchored setup has a common comparator: an IPD trial of \(A\) versus \(C\) and an AgD trial of \(B\) versus \(C\), with the shared arm \(C\) available to absorb study-level baseline differences. The unanchored setup has no common comparator: a single-arm or otherwise disconnected source of \(A\) and a disconnected source of \(B\), with no shared arm to difference against. The two architectures demand different assumptions. The anchored case needs only that the relative effect transports at the conditional level (Definition 17.4, E.4) and that imbalanced effect modifiers be adjusted (Theorem 17.1, E.6a to E.6e). The unanchored case needs that the absolute conditional outcome transports (Definition 17.5, E.5) and that imbalanced prognostic variables as well as effect modifiers be adjusted (Theorem 17.2, E.7), because there is no common comparator to cancel the prognostic terms. The chapter closes with the shared effect modifier assumption that makes small networks identifiable (Definition 17.6, E.8; Theorem 17.3, E.9), the scale dependence of effect modifier status as an explicit counterexample (Example 17.1, E.10), the question of which parts of these assumptions are testable from data (Proposition 17.4, E.11), and the definition of the target-population estimands themselves via the g-formula (Theorem 17.4, E.12). This chapter owns results E.4 to E.12 of the proof catalogue; it builds on E.0 to E.3 from Chapters 11, 12, and 16, and on the identification machinery of Chapters 9 and 10. A warning threads through all of it and is worth stating at the outset: in a two-study comparison the core constancy assumptions cannot be tested from the data, and must be defended by domain knowledge and sensitivity analysis (Section 17.10).

17.2 Data structures, models, and estimands

We retain the potential-outcome framework of Chapter 9. A population \(\mathcal P\) is a distribution over covariates \(X\in\mathcal X\) with density \(f_{\mathcal P}\), and the conditional mean potential outcome under treatment \(t\) is \(\mu_t^{\mathcal P}(x)=\mathbb E_{\mathcal P}[Y(t)\mid X=x]\). Throughout we assume the within-trial identification conditions of Chapter 9: SUTVA and consistency (Definition 9.3, Theorem 9.2), within-trial conditional exchangeability (Definition 9.7), and within-trial positivity (Definition 9.8), so that each trial’s conditional means are identified from its own data by the g-formula (Theorem 10.4). For a treatment \(t\) and a population \(\mathcal P\) the marginal mean is \[ m_t(\mathcal P)\;=\;\int_{\mathcal X}\mu_t(x)\,f_{\mathcal P}(x)\,dx, \tag{17.1}\] the same quantity written \(\bar\mu_{t(\mathcal P)}\) in Chapter 14; we drop the bar here and write \(m_t(\mathcal P)\) throughout, so the reader need not carry two names for one object.

For orientation, the table below collects the symbols this chapter leans on most heavily. A reader who keeps it in view will find the theorem statements much easier to parse; the two estimands \(\Delta^{\mathrm{Marg}}\) and \(\Delta^{\mathrm{Cond}}\) are defined formally in Section 17.2.3 and consolidated in Section 17.11.

Symbol	Reads as
\(\mathcal P_{AC},\mathcal P_{BC}\)	covariate distributions of the \(AC\) trial (IPD) and the \(BC\) trial (AgD)
\(\mathcal P^{*}\), \(f_{*}\)	the target population and its covariate density
\(g\)	the model link: the scale the outcome model is fitted on (for logistic regression, the logit)
\(h\)	the reporting scale: the scale of the effect measure quoted to a decision maker
\(\mu_t(x)\)	conditional mean outcome (for a binary outcome, the risk) under treatment \(t\) at covariates \(x\)
\(\tau^g_{ab}(x)\)	conditional contrast of \(b\) versus \(a\) at \(x\) on scale \(g\) (sign convention in Section 17.2.2)
\(m_t(\mathcal P)\)	marginal (population-average) mean outcome under \(t\) in population \(\mathcal P\)
\(\Delta^{\mathrm{Marg},h}_{ab}\)	marginal effect: contrast of the transformed marginal means
\(\Delta^{\mathrm{Cond},h}_{ab}\)	population-average conditional effect: average of the stratum contrasts
\(\theta^{\mathrm{anch}}_{AB},\theta^{\mathrm{unanch}}_{AB}\)	the anchored and unanchored PAIC estimands
\(\tilde d_{kC(\mathcal P_j)}\)	trial \(j\)’s reported average conditional contrast of \(k\) versus \(C\)

Remark. Running example, with numbers. Give the anticoagulant scenario the conditional risks that the capstone example Example 17.2 will reuse, with \(X=1\) for age \(75\) or over:

\(x\)	\(\mu_A(x)\) (new drug)	\(\mu_B(x)\) (DOAC)	\(\mu_C(x)\) (warfarin)
\(0\) (younger)	\(0.20\)	\(0.30\)	\(0.40\)
\(1\) (older)	\(0.50\)	\(0.30\)	\(0.40\)

The bleeding risk on the new drug \(A\) jumps from \(0.20\) to \(0.50\) with age, while \(B\) and warfarin stay flat, so age modifies \(A\) versus \(C\) but not \(B\) versus \(C\). The IPD trial enrolled \(\Pr_{AC}(X=1)=0.25\) older patients, the AgD trial \(\Pr_{BC}(X=1)=0.75\), and the target population has \(\Pr_{*}(X=1)=0.50\). Every definition below can be read against this one table. \(\blacktriangleleft\)

Definition 17.1 (Individual patient data and aggregate data) A trial \(j\) provides individual patient data (IPD) if its data set is the empirical distribution of the per-subject triples \((x_i,t_i,y_i)\), so that the conditional mean function \(x\mapsto\mu_t^{(j)}(x)\) is estimable on the trial’s covariate support. A trial provides aggregate data (AgD) if only arm-level summaries are available: for each arm, an outcome summary (for binary outcomes an event count \(r\) out of \(n\), equivalently \(\hat m_t=r/n\)) and covariate summaries (typically the covariate means \(\bar x_j\) and possibly higher moments), but not the per-subject records. \(\blacktriangleleft\)

The asymmetry between IPD and AgD is the entire operational difficulty of population adjustment. From IPD one can recompute the trial’s effect under any target covariate distribution, because one holds the conditional mean function. From AgD one holds only the effect already aggregated over the trial’s own covariate distribution, which one cannot re-standardize without an extra modeling assumption.

17.2.1 The two architectures

The two setups differ only in whether a shared arm connects the treatments, but that one difference drives every assumption in the chapter. The picture drawn in health-technology-assessment courses is a network of treatments joined by trials:

   ANCHORED  (shared comparator C)        UNANCHORED  (no shared arm)

        A ------ C ------ B                   A            B
      IPD: A vs C  B vs C: AgD             IPD or       AgD
                                           single arm   source
      one connected network               two disconnected pieces
      target estimand: A vs B in P*        target estimand: A vs B in P*

In the anchored network a path runs from \(A\) to \(B\) through the shared node \(C\); in the unanchored network \(A\) and \(B\) sit in separate pieces with no path between them.

Definition 17.2 (The anchored and unanchored setups) The anchored two-study setup consists of an IPD trial in population \(\mathcal P_{AC}\) comparing \(A\) with a common comparator \(C\), and an AgD trial in population \(\mathcal P_{BC}\) comparing \(B\) with the same comparator \(C\). The target estimand is a contrast of \(A\) versus \(B\) in a target population \(\mathcal P^{*}\).

The unanchored two-study setup consists of a source of outcome data on \(A\) (typically an IPD single arm or trial) and a separate source of outcome data on \(B\) (typically AgD), with no common comparator: the two evidence sources share no treatment arm. The target estimand is again a contrast of \(A\) versus \(B\) in \(\mathcal P^{*}\). \(\blacktriangleleft\)

The same distinction, laid out as a table, previews the entire chapter. Read it now and refer back as each row is proved.

Question	Anchored: \(AC\) and \(BC\) trials share \(C\)	Unanchored: no shared arm
Data available	IPD on one arm pair, AgD on another, sharing \(C\)	separate sources for \(A\) and for \(B\)
What is transported	the relative effect (a contrast)	the absolute conditional outcome (each arm)
Assumption needed	conditional constancy of relative effects (Definition 17.4)	conditional constancy of absolute effects (Definition 17.5)
Must be adjusted	imbalanced effect modifiers only	imbalanced effect modifiers and prognostic variables
What the comparator cancels	study baseline and purely prognostic terms	nothing: there is no comparator
Why it can fail	an imbalanced effect modifier omitted from \(X\)	any imbalanced predictor of outcome omitted from \(X\)

In the running anticoagulant example the anchored setup is the realistic one: both the new drug \(A\) and the DOAC \(B\) were trialed against warfarin \(C\), so warfarin is the shared arm that absorbs how sick each trial’s patients were at baseline. An unanchored version would arise only if, say, \(A\) had been licensed on a single-arm study with no control group, leaving nothing to difference against; we treat that harder case in Section 17.7.

In the anchored setup the comparator \(C\) plays exactly the role it played in Chapter 14: differencing each trial’s arms against \(C\) removes the study-specific baseline (Proposition 14.4 in Chapter 14). Population adjustment then repairs the residual bias from effect-modifier imbalance that survives that cancellation. In the unanchored setup there is no comparator to difference against, so the baseline itself must be modeled and transported, which is a far stronger demand. The rest of the chapter quantifies that difference.

17.2.2 The conditional outcome model

All of the methods specialize a single regression model, the network meta-regression of Dias et al. (2018), written on the model link \(g\). With \(C\) as the reference treatment, the linear predictor for an individual with covariates \(x\) on arm \(k\) in study \(j\) is \[ \eta_{jk}(x)\;=\;g\!\left(\mu_k^{(j)}(x)\right)\;=\;\mu_j+\boldsymbol\beta_1^\top x+\gamma_k+\boldsymbol\beta_{2,k}^\top x, \qquad \gamma_C=0,\ \boldsymbol\beta_{2,C}=\mathbf 0, \tag{17.2}\] in the notation of Notation: \(\mu_j\) is the study baseline, \(\boldsymbol\beta_1\) the prognostic main effects, \(\gamma_k\) the treatment main effect of \(k\) versus \(C\), and \(\boldsymbol\beta_{2,k}\) the treatment-by-covariate interactions of \(k\) versus \(C\).

To see Equation 17.2 in the flesh, put the anticoagulant trial into it on the logit link \(g=\operatorname{logit}\), with the single covariate \(x=\mathbf{1}\{\mathrm{age}\ge 75\}\). The warfarin arm of the IPD trial has linear predictor \(\eta_{AC,C}(x)=\mu_{AC}+\beta_1 x\); the new-drug arm adds the treatment main effect and its age interaction, \(\eta_{AC,A}(x)=\mu_{AC}+\beta_1 x+\gamma_A+\beta_{2,A}\,x\). Here \(\mu_{AC}\) sets the baseline log-odds of bleeding on warfarin among younger patients, \(\beta_1\) says how age alone shifts that log-odds, \(\gamma_A\) is the log-odds change from switching to the new drug at \(x=0\), and \(\beta_{2,A}\) is the extra log-odds change from switching among older patients, that is, the effect modification by age. A nonzero \(\beta_{2,A}\) is exactly what makes age an effect modifier of \(A\) versus \(C\) on the logit scale.

By Lemma 11.2 the conditional effect of an active treatment \(k\) versus the comparator \(C\) on scale \(g\) is the linear-in-\(x\) contrast \[ \tau^{g}_{kC}(x)\;=\;g\!\left(\mu_C(x)\right)-g\!\left(\mu_k(x)\right)\;=\;\gamma_k+\boldsymbol\beta_{2,k}^\top x, \tag{17.3}\] using the reference-second index convention of Chapter 14, so that \(\tau^{g}_{kC}(x)\) is the conditional analogue of the trial contrast \(d_{kC}=\eta_C-\eta_k\) and a positive value means \(k\) improves the (worse is larger) outcome relative to \(C\). The covariate component \(X_\ell\) is an effect modifier of \(k\) versus \(C\) on scale \(g\) if and only if its interaction coefficient \(\beta_{2,k,\ell}\) is nonzero (Lemma 11.2).

Remark. Sign convention, read this once. We follow Chapter 14’s reference-second convention: the second subscript names the reference treatment. So \(\tau^g_{kC}(x)=g(\mu_C(x))-g(\mu_k(x))\) subtracts the active-treatment term from the comparator term, and because the outcome is “worse is larger” (a bleed, a death), a positive \(\tau^g_{kC}\) means treatment \(k\) is better than \(C\): it lowers the bad outcome. This is the opposite of the naive “\(k\) minus \(C\)” reading, and every number in the worked examples depends on it. One line to memorize: a positive contrast means the first-named treatment wins. \(\blacktriangleleft\)

We do not require the linear form Equation 17.2 for the abstract theorems below; they hold for an arbitrary conditional mean model, with Equation 17.2 used only when we need explicit parameters (for the shared effect modifier assumption and for the worked example).

17.2.3 The two estimand scales

Chapter 11 introduced the two population-level effects that a single scale admits, and we will need both.

The difference between these two is the subtlest idea in the chapter, so meet it first with numbers. Take two equally sized strata, younger and older, with bleeding risk under drug \(b\) of \(0.30\) in both and under drug \(a\) of \(0.20\) (young) and \(0.50\) (old). The marginal way pools first: the population risk under \(b\) is \(0.30\) and under \(a\) is \(0.35\), and it contrasts those two pooled numbers. The conditional way contrasts first: the effect is \(0.30-0.20=0.10\) among the young and \(0.30-0.50=-0.20\) among the old, and it averages those two effects to \(-0.05\). On the risk-difference scale these happen to agree (\(-0.05\) either way); on the odds-ratio scale they will not, and that gap is the whole story of non-collapsibility below. Keep the slogan: marginal pools then contrasts; conditional contrasts then pools.

On a reporting scale \(h\), in population \(\mathcal P\), the marginal effect (Definition 11.2) is the contrast of the transformed marginal means, \[ \Delta^{\mathrm{Marg},h}_{ab}(\mathcal P)\;=\;h\!\left(m_b(\mathcal P)\right)-h\!\left(m_a(\mathcal P)\right), \tag{17.4}\] and the population-average conditional effect (Definition 11.1) is the average of the stratum-specific contrasts, \[ \Delta^{\mathrm{Cond},h}_{ab}(\mathcal P)\;=\;\int_{\mathcal X}\Bigl[h\!\left(\mu_b(x)\right)-h\!\left(\mu_a(x)\right)\Bigr]f_{\mathcal P}(x)\,dx \;=\;\int_{\mathcal X}\tau^{h}_{ab}(x)\,f_{\mathcal P}(x)\,dx . \tag{17.5}\] When \(h\) is the identity the two coincide for every population, because the link passes through the integral; for a nonlinear \(h\) they differ in general by Jensen’s inequality, the collapsibility gap of Definition 11.6 (this is catalogue E.0). The marginal effect answers “what happens to the population mean outcome if everyone switches from \(a\) to \(b\)”; the conditional effect answers “what is the average within-stratum effect.” Both are legitimate, they answer different questions, and the theorems below are careful to say which one a given method recovers.

The two scales that recur, the model link \(g\) and the reporting scale \(h\), are easy to confuse because a health researcher meets them as named effect measures, not as functions. The cheat sheet below maps the common measures to their roles and flags where the chapter’s caveats bite.

Named measure	Model link \(g\)	Reporting scale \(h\)	Collapsible?	Caveat that bites
Risk difference	identity	identity	yes, directly	none: adjustment recovers the marginal effect
Risk ratio	log	log	yes, if contrast constant	mild: needs a constant conditional contrast
Odds ratio	logit	logit	no	part (c): only the conditional odds ratio is recovered
Hazard ratio	log-hazard	log-hazard	no	part (c): only the conditional hazard ratio is recovered
Mean survival (RMST)	log-hazard	time scale	scale misaligned	part (d): balancing on \(g\) need not deliver \(h\)

Two grades of collapsibility appear in the table. A measure is directly collapsible if the marginal and conditional effects coincide for every covariate distribution (the risk difference), and merely collapsible if they coincide only when the conditional contrast is constant across strata (the risk ratio). The non-collapsible rows, the odds ratio and hazard ratio, are where “adjust perfectly and still read off a population-specific number” happens; the last row, where \(h\) is not even \(g\), is scale misalignment.

We abbreviate the trial’s reported average conditional contrast, following Chapter 16, by \[ \tilde d_{kC(\mathcal P_j)}\;=\;\Delta^{\mathrm{Cond},g}_{kC}(\mathcal P_j)\;=\;\int\tau^{g}_{kC}(x)\,f_j(x)\,dx . \tag{17.6}\]

Definition 17.3 (Balance and the adjustment set) A covariate (or covariate component) is balanced across two populations if it has the same distribution in both, and imbalanced otherwise. The adjustment set \(X\) of a PAIC is the collection of covariates that the method measures and balances or regresses on; covariates outside \(X\) are unadjusted. We write the full covariate as \((X,W)\) when we wish to single out an unadjusted component \(W\). \(\blacktriangleleft\)

17.3 Conditional constancy of relative effects

The assumption that licenses an anchored comparison is that the conditional relative effect is the same quantity in the trial population and in the target. This is weaker than the marginal constancy of Chapter 14, and it is exactly the structural invariance that Chapter 12 required for transportability.

Definition 17.4 (Conditional constancy of relative effects (anchored)) Two populations \(\mathcal P_1,\mathcal P_2\) satisfy conditional constancy of relative effects on scale \(g\) with respect to \(X\), for the treatment pair \((a,b)\), if the conditional effect Equation 11.2 is the same function of \(x\) in both populations on the common covariate support: \[ \tau^{\mathcal P_1,g}_{ab}(x)\;=\;\tau^{\mathcal P_2,g}_{ab}(x)\qquad\text{for all }x . \tag{17.7}\] Equivalently, the conditional contrast \(\tau^{g}_{ab}(\cdot)\) is a population-free function. (Corresponds to E.4 in the proof catalogue.) \(\blacktriangleleft\)

Remark. Reader translation. Conditional constancy says the treatment effect at a fixed patient profile is a property of the drug, not of who happened to enroll. In the anticoagulant example it asks that the age-specific risk difference of the new drug versus warfarin, \(0.20\) among the young and \(-0.10\) among the old, be the same numbers in the \(AC\) trial, in the \(BC\) trial, and in the target. What is allowed to differ between populations is the mix of young and old; what must not differ is the effect within each age group. This is weaker than Chapter 14’s marginal constancy, which demands the population-averaged effect itself be identical, and it is exactly what lets us re-standardize a trial to a new patient mix. \(\blacktriangleleft\)

Remark. Conditional versus marginal constancy. Constancy of marginal relative effects (Definition 14.2) asserts that the population-level number \(d_{ab(\mathcal P)}\) is the same in every population. Conditional constancy Equation 17.7 asserts only that the per-stratum contrast \(\tau^g_{ab}(x)\) is the same; the marginal effect may still differ across populations because it averages the same \(\tau^g_{ab}(\cdot)\) against different covariate densities. Marginal constancy is therefore the stronger statement: by Equation 17.5, if \(\tau^g_{ab}(\cdot)\) is population-free and either the covariate distributions agree or the conditional contrast is constant in \(x\), then the population-average conditional effects agree, but in general they do not. Conditional constancy is the natural assumption for population adjustment precisely because it permits the marginal effect to vary with the population while pinning down the mechanism that generates that variation. \(\blacktriangleleft\)

Conditional constancy is the hypothesis that makes the trial’s conditional model portable. The next proposition records the consequence we will use repeatedly: under Equation 17.7 the population-average conditional effect in any target is identified from the source conditional contrast and the target covariate distribution, by the transport g-formula of Chapter 12.

Proposition 17.1 (Conditional effects transport under conditional constancy) Suppose conditional constancy of relative effects on scale \(g\) (Definition 17.4) holds from a source population \(\mathcal P\) to a target \(\mathcal P^{*}\), with target support contained in the source support (overlap, equivalently the positivity condition of Definition 9.8). Then the source conditional contrast \(\tau^g_{ab}(\cdot)\), identified within the source by the g-formula (Theorem 10.4), determines the target population-average conditional effect by the transport g-formula (Theorem 12.2): \[ \Delta^{\mathrm{Cond},g}_{ab}(\mathcal P^{*})\;=\;\int_{\mathcal X}\tau^g_{ab}(x)\,f_{\mathcal P^{*}}(x)\,dx \;=\;\mathbb E_{X\sim\mathcal P}\!\left[\,w(X)\,\tau^g_{ab}(X)\,\right],\qquad w(x)=\frac{f_{\mathcal P^{*}}(x)}{f_{\mathcal P}(x)} . \tag{17.8}\]

Proof. By conditional constancy the function \(\tau^g_{ab}(\cdot)\) identified in the source equals the corresponding function in the target on the common support. The target population-average conditional effect is by definition Equation 17.5 the expectation of this function under \(f_{\mathcal P^{*}}\), which is the first equality. For the second, change measure from the target density to the source density: wherever \(f_{\mathcal P}(x)>0\) we have \(f_{\mathcal P^{*}}(x)=w(x)f_{\mathcal P}(x)\), and overlap guarantees the target support is covered, so \(\int\tau^g_{ab}(x)f_{\mathcal P^{*}}(x)\,dx=\int\tau^g_{ab}(x)w(x)f_{\mathcal P}(x)\,dx =\mathbb E_{X\sim\mathcal P}[w(X)\tau^g_{ab}(X)]\). This is the reweighting form Equation 12.6 of the transport g-formula applied to the integrand \(\tau^g_{ab}\), which is itself a fixed function of \(x\) constructed from the source conditional means. \(\square\)

Remark. Why this matters, and why the density ratio is legitimate. This proposition is the workhorse of the chapter: it says an effect measured in one population can be recomputed for another just by reweighting. The reweighting step rewards a slow reading. In the discrete case it is elementary. Suppose the covariate takes finitely many values; then \(\int\tau^g_{ab}(x)f_{\mathcal P^{*}}(x)\,dx\) is the sum \(\sum_x\tau^g_{ab}(x)f_{\mathcal P^{*}}(x)\), and multiplying and dividing each term by \(f_{\mathcal P}(x)\) rewrites it as \(\sum_x[f_{\mathcal P^{*}}(x)/f_{\mathcal P}(x)]\,\tau^g_{ab}(x)f_{\mathcal P}(x)\), a source-weighted average of the same contrast with each stratum reweighted by \(w(x)=f_{\mathcal P^{*}}(x)/f_{\mathcal P}(x)\). Strata over-represented in the target get weight above \(1\); under-represented strata get weight below \(1\). This is the importance-sampling (Radon-Nikodym) identity, and it is the theoretical target that MAIC will estimate. The overlap assumption is what keeps the weight finite: if the target puts mass where the source has none, \(f_{\mathcal P}(x)=0\) makes \(w(x)\) undefined, and no reweighting of source data can conjure an effect for a patient profile the source never observed. That is the precise sense in which population adjustment transports but does not extrapolate. \(\blacktriangleleft\)

The two faces of Equation 17.8 are the two faces of population adjustment. The integral form is outcome regression: model \(\tau^g_{ab}(\cdot)\) from IPD, then average it over the target covariate distribution. This is STC and the regression core of ML-NMR. The reweighting form is weighting: reweight the IPD by \(w\) to the target and average the raw contrast. This is MAIC. The remainder of the chapter establishes when each delivers the target estimand.

17.4 Conditional constancy of absolute effects

The unanchored setup has no comparator to difference against, so it must transport the absolute conditional outcome, not merely the contrast. The corresponding assumption is strictly stronger.

Definition 17.5 (Conditional constancy of absolute effects (unanchored)) Two populations \(\mathcal P_1,\mathcal P_2\) satisfy conditional constancy of absolute effects with respect to \(X\) if for every treatment \(t\) and every \(x\) on the common support, \[ \mu_t^{\mathcal P_1}(x)\;=\;\mu_t^{\mathcal P_2}(x) . \tag{17.9}\] That is, the entire conditional mean function, arm by arm, is population-free. (Corresponds to E.5 in the proof catalogue.) \(\blacktriangleleft\)

Remark. Reader translation. Absolute constancy is a far heavier promise than relative constancy: it asks that the actual risk on each drug, arm by arm, at a fixed patient profile, be identical across populations, not merely that the gap between drugs be identical. In the anticoagulant example it would require the older-patient bleeding risk on warfarin to be exactly \(0.40\) in every population, and likewise for every arm. Relative constancy lets the whole risk profile float up or down between populations as long as the treatment contrast holds; absolute constancy forbids even that. The unanchored setup needs the heavier promise because, with no comparator arm to difference against, it must reconstruct each arm’s risk from scratch. \(\blacktriangleleft\)

Proposition 17.2 (Absolute constancy is strictly stronger than relative constancy) Conditional constancy of absolute effects (Definition 17.5) implies conditional constancy of relative effects on every scale \(g\) (Definition 17.4), for every treatment pair. The converse is false: there exist conditional mean functions that satisfy relative constancy on a scale \(g\) but not absolute constancy.

Proof. If \(\mu_t^{\mathcal P_1}(x)=\mu_t^{\mathcal P_2}(x)\) for every \(t\) and \(x\), then for any scale \(g\) and any pair \((a,b)\), \(\tau^{\mathcal P_1,g}_{ab}(x)=g(\mu_b^{\mathcal P_1}(x))-g(\mu_a^{\mathcal P_1}(x)) =g(\mu_b^{\mathcal P_2}(x))-g(\mu_a^{\mathcal P_2}(x))=\tau^{\mathcal P_2,g}_{ab}(x)\), which is Equation 17.7. For the converse, let the covariate be \((X,W)\) and let the absolute means depend on a component \(W\) that is not adjusted; take, for a single covariate value and on the identity scale, \(\mu_a^{\mathcal P_1}(x,w)=\mu_b^{\mathcal P_1}(x,w)=w\) and \(\mu_a^{\mathcal P_2}(x,w)=\mu_b^{\mathcal P_2}(x,w)=w+1\). Then the contrast is identically zero in both populations, so relative constancy holds on the identity scale, while the absolute means differ by \(1\), so absolute constancy fails. \(\square\)

The witness in the proof is the formal seed of the difference between anchored and unanchored adjustment: a covariate can shift both arms equally, raising the baseline risk while leaving the treatment contrast untouched, so the absolute level changes but the contrast does not. Such a covariate is invisible to an anchored comparison and fatal to an unanchored one. We now isolate that phenomenon as the effect-modifier-versus-prognostic distinction, which is the algebraic heart of the chapter.

17.5 The effect modifier versus prognostic distinction

Recall from Chapter 11 the four-cell classification of a covariate by whether it is prognostic (affects the conditional means, Definition 11.5) and whether it is an effect modifier (affects the conditional contrast, Definition 11.3). The single cell that powers anchored comparison is the purely prognostic one: a covariate that moves both arms but not their contrast. We prove the two cancellation facts that separate the anchored and unanchored architectures. Both have machine-checked counterparts in the companion Coq development.

For a reader arriving directly at Part IV, here are the two words in one line each. A prognostic variable moves the outcome under both arms: sicker patients bleed more whatever drug they take. An effect modifier moves the difference between arms: the new drug’s advantage over warfarin shrinks or reverses with age. A covariate can be one, the other, both, or neither, the four cells of Chapter 11:

	Not an effect modifier (contrast flat in it)	Effect modifier (contrast varies in it)
Not prognostic (outcome flat in it)	irrelevant: ignore it	uncommon, but must be adjusted
Prognostic (outcome varies in it)	purely prognostic: cancels when anchored	prognostic and modifying: always adjust

The load-bearing cell is the lower left, purely prognostic. In the anticoagulant example, age sits in the lower right for the \(A\)-versus-\(C\) contrast (prognostic and a modifier), whereas a shared frailty score that raised every arm’s bleeding risk equally would sit in the lower left. The two lemmas below prove exactly why the lower-left cell is free in an anchored comparison and fatal in an unanchored one.

Lemma 17.1 (Anchored cancellation of purely prognostic covariates) Let the full covariate be \((X,W)\) and fix a scale \(g\) and a treatment pair \((k,C)\). Suppose \(W\) is not an effect modifier of \(k\) versus \(C\) on scale \(g\), in the sense that the conditional contrast does not depend on \(w\): \[ \tau^{g}_{kC}(x,w)\;=\;\tau^{g}_{kC}(x,w')\qquad\text{for all }x,w,w' . \tag{17.10}\] Then, writing \(\tau^g_{kC}(x)\) for the common value, the population-average conditional contrast over any population \(\mathcal P\) depends on \(\mathcal P\) only through the marginal distribution of \(X\): \[ \int\tau^g_{kC}(x,w)\,f_{\mathcal P}(x,w)\,dx\,dw\;=\;\int\tau^g_{kC}(x)\,f_{\mathcal P}(x)\,dx . \tag{17.11}\] In particular, omitting \(W\) from the adjustment set does not change the anchored estimand, whatever the distribution of \(W\) given \(X\) and however imbalanced \(W\) may be.

[machine-checked: theories/EffectModifiers.v, Theorem anchored_PV_cancellation]

Proof. By hypothesis Equation 17.10 the integrand \(\tau^g_{kC}(x,w)\) equals the constant-in-\(w\) value \(\tau^g_{kC}(x)\) for every \(w\). Hence \[ \int\tau^g_{kC}(x,w)\,f_{\mathcal P}(x,w)\,dx\,dw =\int\tau^g_{kC}(x)\left(\int f_{\mathcal P}(x,w)\,dw\right)dx =\int\tau^g_{kC}(x)\,f_{\mathcal P}(x)\,dx, \] the inner integral being the marginal density of \(X\). The right-hand side does not involve the conditional distribution of \(W\) given \(X\), so it is unchanged if \(W\) is left unadjusted and unchanged across populations with different \(W\)-distributions. \(\square\)

Lemma 17.2 (A prognostic variable need not be an effect modifier) “Not an effect modifier of \(k\) versus \(C\)” does not imply “not prognostic for \(k\).” Concretely, there exist conditional mean functions \(\mu_k,\mu_C\) such that the contrast \(\mu_k(x,w)-\mu_C(x,w)\) is constant in \(w\) (so \(W\) is not an effect modifier on the identity scale) while \(\mu_k(x,w)\) depends on \(w\) (so \(W\) is prognostic for \(k\)).

[machine-checked: theories/EffectModifiers.v, Theorem not_EM_does_not_imply_not_PV]

Proof. Take \(\mu_k(x,w)=\mu_C(x,w)=w\). Then \(\mu_k(x,w)-\mu_C(x,w)=0\) for all \(w\), so \(W\) is not an effect modifier; but \(\mu_k(x,w)=w\) is nonconstant in \(w\), so \(W\) is prognostic for \(k\). In words, this \(W\) is the shared-frailty kind of covariate that raises every arm’s outcome by the same amount, changing who bleeds without changing which drug is better. The same two functions witness that absolute constancy is strictly stronger than relative constancy (Proposition 17.2). \(\square\)

The contrast between Lemma 17.1 and Lemma 17.2 is the whole story in miniature. In an anchored comparison the estimand depends on \(W\) only through the contrast \(\tau^g_{kC}\), so a purely prognostic \(W\) cancels and may be ignored (Lemma 17.1). In an unanchored comparison the estimand depends on the absolute means \(\mu_k\) themselves, which a prognostic \(W\) moves even when it modifies no contrast (Lemma 17.2), so it cannot be ignored. This is why the anchored case adjusts for effect modifiers while the unanchored case must adjust for prognostic variables as well.

17.6 The anchored consistency theorem

We now assemble the central anchored result. Define the anchored PAIC estimand at a target \(\mathcal P^{*}\) as the difference of the two conditional contrasts, each standardized to the target: \[ \theta^{\mathrm{anch}}_{AB}(\mathcal P^{*}) \;:=\;\Delta^{\mathrm{Cond},g}_{AC}(\mathcal P^{*})-\Delta^{\mathrm{Cond},g}_{BC}(\mathcal P^{*}) \;=\;\int\tau^g_{AC}(x)\,f_{*}(x)\,dx-\int\tau^g_{BC}(x)\,f_{*}(x)\,dx . \tag{17.12}\] This is the population-level object that the marginal-form STC, ML-NMR, and moment-balanced MAIC estimators target; in the two-study MAIC special case one takes \(\mathcal P^{*}=\mathcal P_{BC}\), so that the \(BC\) contrast is reported directly and only the \(AC\) contrast is re-standardized. The first lemma is pure algebra and is the conditional analogue of Bucher’s additivity (Lemma 14.1). It looks trivial, but it is the linchpin of the anchored architecture: it is the reason that subtracting two separately re-standardized contrasts yields the correct \(A\)-versus-\(B\) answer, and it holds pointwise on every scale.

Lemma 17.3 (Pointwise additivity of conditional contrasts) For every \(x\) and every scale \(g\), the conditional contrasts obey \[ \tau^g_{AB}(x)\;=\;\tau^g_{AC}(x)-\tau^g_{BC}(x). \tag{17.13}\] Consequently, for any population \(\mathcal P\), the population-average conditional effects obey \(\Delta^{\mathrm{Cond},g}_{AB}(\mathcal P)=\Delta^{\mathrm{Cond},g}_{AC}(\mathcal P)-\Delta^{\mathrm{Cond},g}_{BC}(\mathcal P)\).

Proof. Using the reference-second convention \(\tau^g_{ab}(x)=g(\mu_b(x))-g(\mu_a(x))\) of Equation 11.2, \[ \tau^g_{AC}(x)-\tau^g_{BC}(x)=\bigl[g(\mu_C(x))-g(\mu_A(x))\bigr]-\bigl[g(\mu_C(x))-g(\mu_B(x))\bigr] =g(\mu_B(x))-g(\mu_A(x))=\tau^g_{AB}(x), \] the comparator term \(g(\mu_C(x))\) canceling pointwise. Integrating both sides against \(f_{\mathcal P}\) and using linearity of the integral (Equation 17.5) gives the population-average statement. \(\square\)

Before the theorem, here is the plan in words, because the statement below has five parts that are easy to conflate. Part (a) is the good news and does the real work: on the scale the model is fitted, adjusting for imbalanced effect modifiers recovers the target’s average conditional effect. Parts (b) to (e) are all about which effect measure that recovered number equals. If the reporting scale is collapsible and aligned with the model (a risk difference), the conditional number is also the marginal policy number (b). If it is non-collapsible (an odds ratio or hazard ratio), the recovered number is the conditional one, which is a genuinely different quantity from the marginal one, so it must not be read as a marginal odds ratio (c). If the reporting scale is not even the model scale (a hazard model but an RMST target), balancing on the model scale need not deliver the target at all (d). And (e) is the one step you cannot skip: leave an imbalanced effect modifier out of the adjustment set and the estimate is biased. The roadmap table collects this.

Part	Condition	What the anchored estimand equals	Plain reading
(a)	(i)-(iv), any scale \(g\)	target population-average conditional effect \(\Delta^{\mathrm{Cond},g}_{AB}(\mathcal P^{*})\)	adjust imbalanced effect modifiers and you recover the conditional effect
(b)	aligned \(h=g\), collapsible	also the target marginal effect	on a risk-difference scale, conditional equals marginal, so you also get the policy number
(c)	aligned, non-collapsible	the conditional effect, not the marginal one	on an odds/hazard-ratio scale you get the conditional measure, not the marginal one
(d)	misaligned \(h\circ g^{-1}\neq\mathrm{id}\)	in general neither, on scale \(h\)	a hazard model does not hand you an RMST target for free
(e)	an imbalanced effect modifier omitted from \(X\)	a biased number	skipping a relevant effect modifier reintroduces the Bucher bias

Remark. Reader translation of the theorem. If this is an anchored comparison, the operational instruction is short: adjust for every effect modifier that is imbalanced across the trials and the target, on the scale you plan to analyze, then read the result as the conditional effect on that scale. Do not expect it to be a marginal odds ratio or hazard ratio unless the estimand and scale conditions of parts (b) to (d) also line up. Everything else in the theorem is the fine print behind those two sentences. \(\blacktriangleleft\)

Theorem 17.1 (Anchored PAIC consistency (scale-specific)) Consider the anchored two-study setup (Definition 17.2) targeting population \(\mathcal P^{*}\) on model link \(g\). Assume

1. conditional constancy of relative effects on scale \(g\) from each trial population to \(\mathcal P^{*}\) (Definition 17.4);

2. within-trial identification of the conditional contrasts \(\tau^g_{AC}(\cdot),\tau^g_{BC}(\cdot)\) (Theorem 10.4);

3. every covariate that is an effect modifier on scale \(g\) of \(A\) versus \(C\) or of \(B\) versus \(C\), and that is imbalanced between some pair of \(\{\mathcal P_{AC},\mathcal P_{BC},\mathcal P^{*}\}\), lies in the adjustment set \(X\) (Definition 17.3);

4. overlap: the support of \(f_{*}\) on \(X\) is contained in the supports of \(f_{AC}\) and \(f_{BC}\) (Definition 9.8).

Then the following hold.

(a) Conditional recovery, any scale. The anchored PAIC estimand Equation 17.12 equals the target population-average conditional effect, \[ \theta^{\mathrm{anch}}_{AB}(\mathcal P^{*})\;=\;\Delta^{\mathrm{Cond},g}_{AB}(\mathcal P^{*}). \tag{17.14}\]

(b) Marginal recovery on an aligned collapsible scale. If in addition the reporting scale is aligned, \(h=g\) (Definition 12.4), and the effect measure is collapsible on scale \(g\) (Definition 11.6), and either

• the measure is directly collapsible on \(g\) (for instance the identity scale, Proposition 11.3), or
• the conditional \(A\)-versus-\(B\) contrast \(\tau^g_{AB}(\cdot)\) is constant on the support (for instance under the shared effect modifier assumption Definition 17.6),

then the estimand also equals the target marginal effect, \(\theta^{\mathrm{anch}}_{AB}(\mathcal P^{*})=\Delta^{\mathrm{Marg},h}_{AB}(\mathcal P^{*})\).

(c) The recovered estimand is the conditional effect, not a non-collapsible marginal one. If the effect measure is non-collapsible on scale \(g\) (the odds ratio on the logit link, the hazard ratio on the log-hazard link, Theorem 11.2), then in general \(\theta^{\mathrm{anch}}_{AB}(\mathcal P^{*})\neq\Delta^{\mathrm{Marg},g}_{AB}(\mathcal P^{*})\), even under SEMA and scale alignment (Theorem 12.6); only the conditional effect of part (a) is recovered.

(d) Scale misalignment. If the reporting scale is misaligned, \(h\circ g^{-1}\neq\mathrm{id}\) (for instance a hazard model, \(g=\log\)-hazard, with a restricted mean survival time target, \(h\) on the time scale), then balancing on \(g\) does not in general recover the target contrast on \(h\), conditional or marginal.

(e) Class-wise necessity. Over the class of data-generating mechanisms in which some covariate is an effect modifier on scale \(g\) and is imbalanced between two of the three populations but is omitted from \(X\), the estimand is asymptotically biased. No claim is made for any specific mechanism: a particular mechanism may have zero bias by numerical coincidence.

(Corresponds to E.6a to E.6e in the proof catalogue.)

Proof. (a). By assumption (ii) the contrasts \(\tau^g_{AC}(\cdot)\) and \(\tau^g_{BC}(\cdot)\) are identified within their trials, and by assumption (i) they are the same functions in \(\mathcal P^{*}\). By assumption (iii) every effect modifier that is imbalanced lies in \(X\), so each contrast is a function of the adjusted covariates \(X\) alone on the support that matters: any unadjusted covariate \(W\) is, by (iii), not an imbalanced effect modifier, hence either balanced (its conditional distribution given \(X\) agrees across the populations) or not an effect modifier (so Lemma 17.1 applies and \(W\) drops out of the contrast). In either case the standardization to \(f_{*}\) is governed by the distribution of \(X\), and Proposition 17.1 gives, for \(t\in\{A,B\}\), \(\Delta^{\mathrm{Cond},g}_{tC}(\mathcal P^{*})=\int\tau^g_{tC}(x)f_{*}(x)\,dx\), the quantity appearing in Equation 17.12, with overlap (iv) ensuring the integral is well defined. Subtracting and applying the pointwise additivity Lemma 17.3, \[ \theta^{\mathrm{anch}}_{AB}(\mathcal P^{*}) =\int\bigl[\tau^g_{AC}(x)-\tau^g_{BC}(x)\bigr]f_{*}(x)\,dx =\int\tau^g_{AB}(x)\,f_{*}(x)\,dx =\Delta^{\mathrm{Cond},g}_{AB}(\mathcal P^{*}), \] which is Equation 17.14.

(b). By part (a) the estimand equals \(\Delta^{\mathrm{Cond},g}_{AB}(\mathcal P^{*})\). It remains to show this conditional effect equals the marginal effect \(\Delta^{\mathrm{Marg},h}_{AB}(\mathcal P^{*})\) under the stated conditions, on the aligned scale \(h=g\). In the first case the measure is directly collapsible on \(g\): by Definition 11.6 the marginal effect equals the population-average conditional effect for every covariate distribution, so \(\Delta^{\mathrm{Marg},g}_{AB}(\mathcal P^{*})=\Delta^{\mathrm{Cond},g}_{AB}(\mathcal P^{*})\) identically, and the conclusion follows; the identity scale is the canonical instance (Proposition 11.3), giving E.6a. In the second case the conditional contrast \(\tau^g_{AB}(\cdot)\) is constant on the support, say \(\tau^g_{AB}(x)\equiv c\); then \(\Delta^{\mathrm{Cond},g}_{AB}(\mathcal P^{*})=\int c\,f_{*}=c\), and collapsibility of the measure (Definition 11.6, first bullet) gives that a constant conditional effect equals the marginal effect, \(\Delta^{\mathrm{Marg},g}_{AB}(\mathcal P^{*})=c\); this is the situation of Theorem 12.5, and with \(g=\log\) for the risk ratio it gives E.6b.

(c). Suppose the measure is non-collapsible on \(g\), and suppose, for contradiction with the generic claim, that recovery of the marginal effect held for all mechanisms satisfying (i) to (iv) plus SEMA and alignment. Under SEMA the conditional contrast \(\tau^g_{AB}(\cdot)\equiv c\) is constant (Definition 17.6 below), so part (a) gives \(\theta^{\mathrm{anch}}_{AB}(\mathcal P^{*})=c\). But non-collapsibility means precisely that there is a covariate distribution and a pair of stratum means with constant conditional effect \(c\) whose marginal effect differs from \(c\) (Definition 7.7, Theorem 11.2): for the odds ratio the marginal effect is strictly attenuated toward the null. Hence \(\Delta^{\mathrm{Marg},g}_{AB}(\mathcal P^{*})\neq c=\theta^{\mathrm{anch}}_{AB}(\mathcal P^{*})\) for such a mechanism, contradicting universal recovery. The “if and only if” of Theorem 12.6 makes this sharp: under SEMA and alignment, direct transportability of the marginal effect holds for all populations if and only if the measure is collapsible. Thus only the conditional effect is recovered in general.

(d). Misalignment means \(h=H\circ g\) with \(H=h\circ g^{-1}\) a nonidentity, nonlinear map. Part (a) recovers the \(g\)-scale conditional contrast \(\Delta^{\mathrm{Cond},g}_{AB}(\mathcal P^{*})\). The target on the reporting scale is built from the \(h\)-scale conditional means \(h(\mu_t(x))=H(g(\mu_t(x)))\), and because \(H\) is nonlinear the \(h\)-scale conditional contrast \(h(\mu_B(x))-h(\mu_A(x))\) is not a function of the \(g\)-scale contrast \(g(\mu_B(x))-g(\mu_A(x))\) alone; it depends separately on the baseline \(g(\mu_A(x))\). Transporting the \(g\)-scale contrast therefore does not determine the \(h\)-scale contrast, conditional or marginal. The survival instance, where \(g\) is the log-hazard and \(h\) encodes the restricted mean survival time through the nonlinear functional \(\mathrm{RMST}_t(\tau\mid x)=\int_0^\tau\exp\!\bigl(-\int_0^u h_t(s\mid x)\,ds\bigr)du\) of the hazard, is E.6e; we defer its survival-specific development to the later chapters on general likelihoods and ML-UMR, and record here only that the obstruction is exactly the nonlinearity of \(H\).

(e). Suppose a covariate \(W\) is an effect modifier on \(g\) of, say, \(A\) versus \(C\), and is imbalanced between \(\mathcal P_{AC}\) and \(\mathcal P^{*}\), but is omitted from \(X\). Then the standardization in part (a) computes, for the \(AC\) contrast, \(\int\tau^g_{AC}(x,w)\,f^{AC}(w\mid x)\,f_{*}(x)\,dx\,dw\) rather than the target \(\int\tau^g_{AC}(x,w)\,f^{*}(w\mid x)\,f_{*}(x)\,dx\,dw\), and the difference is the failure-of-Bucher residual of Theorem 16.1, \[ \theta^{\mathrm{anch}}_{AB}(\mathcal P^{*})-\Delta^{\mathrm{Cond},g}_{AB}(\mathcal P^{*}) =\int\tau^g_{AC}(x,w)\,[\,f^{AC}(w\mid x)-f^{*}(w\mid x)\,]\,f_{*}(x)\,dx\,dw, \] which by Theorem 16.1(2) is nonzero for some mechanism in the stated class, since \(\tau^g_{AC}\) depends nontrivially on \(w\) (effect modification) and the conditional distributions of \(W\) differ (imbalance). The statement is class-wise: by Theorem 16.1(1) the residual vanishes identically only when the modifier is balanced or absent, but a specific mechanism may zero the integral by cancellation even with an unadjusted imbalanced modifier. \(\square\)

Remark. Where the comparator earns its keep. Part (a) is the engine; parts (b) to (e) are about which scale the engine’s output lives on. The reason the anchored estimand needs only effect modifiers in \(X\), and not all prognostic variables, is Lemma 17.1: a purely prognostic \(W\) enters \(\tau^g_{AC}\) and \(\tau^g_{BC}\) only through the comparator term \(g(\mu_C(x,w))\), which is the same in both and cancels in the contrast. This is the conditional-level descendant of the baseline cancellation (Proposition 14.4) that made anchored comparison robust in Chapter 14. The unanchored case forfeits exactly this cancellation. \(\blacktriangleleft\)

Remark. Why this matters. Part (a) is population adjustment stated as a regression fact: it is standardization, the same operation that turns a fitted outcome model into a marginal prediction in Chapter 10, now aimed at a different population than the one that supplied the data. The anchored theorem is the formal license for the whole of Part IV. It says the effect-modifier bookkeeping that broke Bucher in Chapter 16 is fully repaired by adjusting the imbalanced modifiers, and it pins down the estimand each downstream method (MAIC, STC, ML-NMR) may claim. In the anticoagulant example, part (a) promises that adjusting for age turns the biased Bucher answer of Step 3 in Example 17.2 into the exact target risk difference of Step 4, while part (c) warns that the same adjustment on the odds-ratio scale returns only the conditional odds ratio, not the marginal one a reimbursement dossier may quote. \(\blacktriangleleft\)

For completeness we record the finite, exact algebraic identity underlying part (a) in the form proved in the Coq development: once the adjusted \(AC\) contrast has been correctly identified in the target and the AgD \(BC\) contrast is unbiased, the Bucher subtraction returns the target contrast.

Corollary 17.1 (The adjusted Bucher identity) Let \(\hat d_{AC}^{\mathrm{adj}}\) be an estimator of the target \(A\)-versus-\(C\) contrast with \(\hat d_{AC}^{\mathrm{adj}}=d_{AC(\mathcal P^{*})}\), let \(\hat d_{BC}\) satisfy \(\hat d_{BC}=d_{BC(\mathcal P^{*})}\), and let transitivity hold in \(\mathcal P^{*}\) in the form \(d_{AB(\mathcal P^{*})}=d_{AC(\mathcal P^{*})}-d_{BC(\mathcal P^{*})}\). Then \(\hat d_{AC}^{\mathrm{adj}}-\hat d_{BC}=d_{AB(\mathcal P^{*})}\).

[machine-checked: theories/EffectModifiers.v, Theorem anchored_unbiased]

Proof. Substitute the two identifications and the transitivity relation: \(\hat d_{AC}^{\mathrm{adj}}-\hat d_{BC}=d_{AC(\mathcal P^{*})}-d_{BC(\mathcal P^{*})}=d_{AB(\mathcal P^{*})}\). \(\square\)

17.7 The unanchored consistency theorem

Without a comparator, the estimator must transport each absolute marginal mean separately, and there is no cancellation to relieve it of the prognostic variables.

The unanchored setup is a method of last resort, and it helps to know when a health researcher is forced into it. It arises when the two treatments were never controlled against a common arm: perhaps \(A\) was licensed on a single-arm study, common for accelerated approvals in oncology and rare disease, while \(B\) comes from a placebo-controlled trial, so there is no shared placebo or active comparator to difference against. Stripped of the comparator, the estimator can no longer content itself with contrasts; it must rebuild each arm’s absolute outcome in the target population and only then take a difference. That is why the assumptions jump from relative to absolute constancy and from effect modifiers to all prognostic variables.

Define the unanchored PAIC estimand at target \(\mathcal P^{*}\) on reporting scale \(h\) as \[ \theta^{\mathrm{unanch}}_{AB}(\mathcal P^{*})\;:=\;h\!\left(m_B(\mathcal P^{*})\right)-h\!\left(m_A(\mathcal P^{*})\right) \;=\;\Delta^{\mathrm{Marg},h}_{AB}(\mathcal P^{*}), \tag{17.15}\] with \(m_t(\mathcal P^{*})=\int\mu_t(x)f_{*}(x)\,dx\) as in Equation 17.1.

Theorem 17.2 (Unanchored PAIC consistency) Consider the unanchored two-study setup (Definition 17.2) targeting \(\mathcal P^{*}\) on reporting scale \(h\). Assume

1. conditional constancy of absolute effects (Definition 17.5): the conditional mean function \(\mu_t(\cdot)\) is the same in each source and in \(\mathcal P^{*}\);

2. within-trial identification of \(\mu_A(\cdot)\) and \(\mu_B(\cdot)\) (Theorem 10.4);

3. every covariate that is either an effect modifier on the modeled scale or a prognostic variable for \(A\) or \(B\), and that is imbalanced between some pair of the source and target populations, lies in the adjustment set \(X\);

4. overlap (Definition 9.8).

Then the unanchored PAIC estimand Equation 17.15 is recovered: each marginal mean is identified by the transport g-formula, \[ m_t(\mathcal P^{*})\;=\;\int\mu_t(x)\,f_{*}(x)\,dx \;=\;\mathbb E_{X\sim\mathcal P_{\mathrm{src},t}}\!\left[\,w_t(X)\,\mu_t(X)\,\right],\qquad w_t(x)=\frac{f_{*}(x)}{f_{\mathrm{src},t}(x)}, \tag{17.16}\] and applying \(h\) and differencing gives \(\theta^{\mathrm{unanch}}_{AB}(\mathcal P^{*})=\Delta^{\mathrm{Marg},h}_{AB}(\mathcal P^{*})\).

Necessity (over a class). Over the class of mechanisms with an imbalanced prognostic variable for \(A\) or \(B\) omitted from \(X\), the estimand is asymptotically biased, with bias for the \(A\) arm \[ \int\!\!\int\mu_A(x,w)\,[\,f_{\mathrm{src},A}(w\mid x)-f^{*}(w\mid x)\,]\,f_{*}(x)\,dw\,dx, \tag{17.17}\] which is nonzero whenever \(\mu_A\) depends on the omitted \(w\) (prognostic) and its conditional distribution differs between source and target (imbalance), with no comparator term to cancel it.

(Corresponds to E.7 in the proof catalogue.)

Proof. By assumption (ii) the function \(\mu_t(\cdot)\) is identified within its source, and by (i) it is the same function in \(\mathcal P^{*}\). By (iii) every imbalanced prognostic variable lies in \(X\), so on the relevant support \(\mu_t\) is a function of \(X\) alone: any omitted covariate is either balanced, with the same conditional distribution given \(X\) in source and target, or nonprognostic, in which case it does not enter \(\mu_t\). Applying the transport g-formula Theorem 12.2 to the function \(\mu_t\) gives Equation 17.16, with overlap (iv) guaranteeing the integral and the reweighting are well defined. Both arms are handled this way; there is no subtraction step, so the two transports are independent. Applying \(h\) to the two recovered marginal means and differencing yields \(\theta^{\mathrm{unanch}}_{AB}(\mathcal P^{*})=h(m_B(\mathcal P^{*}))-h(m_A(\mathcal P^{*}))\), which is the target marginal effect.

For the necessity claim, let \(W\) be prognostic for \(A\) and imbalanced, but omitted from \(X\). The estimator that adjusts only on \(X\) computes the source within-stratum average \(\bar\mu_A^{\mathrm{src}}(x)=\int\mu_A(x,w)f_{\mathrm{src},A}(w\mid x)\,dw\) and standardizes it to \(f_{*}\), producing \(\int\bar\mu_A^{\mathrm{src}}(x)f_{*}(x)\,dx\). The target requires the target within-stratum average \(\bar\mu_A^{*}(x)=\int\mu_A(x,w)f^{*}(w\mid x)\,dw\), so the bias of the \(A\) marginal mean is \(\int[\bar\mu_A^{\mathrm{src}}(x)-\bar\mu_A^{*}(x)]f_{*}(x)\,dx\), which is Equation 17.17. Because \(\mu_A\) depends on \(w\) and the conditional laws of \(W\) differ, the inner difference is generically nonzero, and there is no comparator arm whose identical bias would cancel it in a contrast. By contrast, in the anchored case the analogous term enters through the comparator and cancels (Lemma 17.1), which is precisely why a purely prognostic imbalance is harmless there and harmful here. \(\square\)

Remark. Reader translation, and the shared prognostic factor assumption. The instruction for an unanchored comparison is the anchored one with the safety catch removed: adjust for every imbalanced covariate that predicts the outcome, whether or not it modifies any treatment effect. In the model Equation 17.2 the matching structural hypothesis, the unanchored analogue of SEMA, is that the prognostic main effects are shared across the compared treatments’ evidence sources; this is the shared prognostic factor assumption (SPFA) of Notation. Because it constrains the baseline and not just the interactions, SPFA is even harder to defend than SEMA. The practical reading of overlap here is the one that should worry a health researcher: the target population must not contain patient profiles the source never enrolled, because the weight \(w_t(x)=f_{*}(x)/f_{\mathrm{src},t}(x)\) is undefined where the source density is zero. If the target has sicker patients than any trial subject, the method is extrapolating, not transporting, and the guarantee lapses. \(\blacktriangleleft\)

Remark. The price of dropping the anchor. The anchored theorem asked for effect modifiers in \(X\); the unanchored theorem asks for effect modifiers and prognostic variables. By Lemma 17.2 the second demand is strictly stronger: a covariate can be prognostic without modifying any contrast, and such a covariate is invisible to the anchored adjustment but must be measured and balanced for the unanchored one. In practice this is close to prohibitive, because it requires having measured essentially every driver of the outcome. This is the formal reason that unanchored comparison is a method of last resort, reserved for disconnected networks (Chapter 26), and that Phillippo et al. (2016) demands extensive sensitivity analysis whenever it is used. \(\blacktriangleleft\)

Remark. Non-collapsibility, again. Equation Equation 17.16 recovers the marginal means \(m_t\) themselves, so for a non-collapsible reporting scale the unanchored estimand is the genuine marginal effect in \(\mathcal P^{*}\), not merely a conditional one. The subtlety of part (c) of the anchored theorem does not arise here, because nothing was differenced before applying \(h\). What does carry over is that the recovered marginal effect is specific to \(\mathcal P^{*}\): a non-collapsible marginal effect computed for one target does not transport unchanged to another, exactly as in Theorem 11.2. The untestability of the absolute-constancy assumption (i) is taken up in Proposition 17.4. \(\blacktriangleleft\)

17.8 The shared effect modifier assumption

The anchored theorem assumed the conditional contrasts \(\tau^g_{AC}\) and \(\tau^g_{BC}\) were both available. In the smallest network, one IPD trial of \(A\) versus \(C\) and one AgD trial of \(B\) versus \(C\), the IPD identifies the interactions for \(A\) but the AgD cannot identify the interactions for \(B\) separately from its main effect. The assumption that restores identifiability is that the active treatments share their effect-modifier structure.

Definition 17.6 (Shared effect modifier assumption (SEMA)) In the model Equation 17.2, the shared effect modifier assumption asserts that the treatment-by-covariate interactions are the same for the active treatments being compared: \[ \boldsymbol\beta_{2,A}\;=\;\boldsymbol\beta_{2,B}\;=:\;\boldsymbol\beta_{2}. \tag{17.18}\] (Corresponds to E.8 in the proof catalogue.) \(\blacktriangleleft\)

Remark. Reader translation: what SEMA says, and how it differs from conditional constancy. SEMA says the covariates bend the two active treatments’ effects in the same way. In the anticoagulant example it asks that age change the benefit of the new drug \(A\) versus warfarin by exactly the amount it changes the benefit of the DOAC \(B\) versus warfarin, so that when we difference \(A\) against \(B\) the age term cancels and the \(A\)-versus-\(B\) effect stops depending on age. Contrast this with conditional constancy (Definition 17.4), which is an assumption across populations, the same effect function in every trial and the target; SEMA is an assumption across treatments within one model, the same interaction vector for \(A\) and \(B\). One is portability of a fixed effect to a new patient mix; the other is a shared shape of effect modification between two drugs.

A plausible instance: \(A\) and \(B\) are two drugs of the same pharmacological class and the only strong modifier is disease severity, which blunts any active treatment’s advantage over the comparator by a similar amount, so severity modifies \(A\)-versus-\(C\) and \(B\)-versus-\(C\) alike and cancels. An implausible instance: \(A\) is a targeted therapy whose benefit is concentrated in a biomarker-positive subgroup while \(B\) is a broad-acting comparator whose benefit is flat across that biomarker; then the biomarker modifies \(A\) but not \(B\), SEMA fails, and the active-versus-active contrast genuinely depends on the biomarker mix. \(\blacktriangleleft\)

Proposition 17.3 (SEMA makes the active-versus-active conditional contrast constant) Under the model Equation 17.2 and SEMA (Definition 17.6), the conditional effect of \(A\) versus \(B\) on scale \(g\) is constant in \(x\): \[ \tau^g_{AB}(x)\;=\;\gamma_A-\gamma_B\qquad\text{for all }x . \tag{17.19}\]

Proof. By Equation 17.3, \(\tau^g_{AC}(x)=\gamma_A+\boldsymbol\beta_{2,A}^\top x\) and \(\tau^g_{BC}(x)=\gamma_B+\boldsymbol\beta_{2,B}^\top x\). By Lemma 17.3, \[ \tau^g_{AB}(x)=\tau^g_{AC}(x)-\tau^g_{BC}(x) =(\gamma_A-\gamma_B)+(\boldsymbol\beta_{2,A}-\boldsymbol\beta_{2,B})^\top x . \] SEMA Equation 17.18 sets the interaction difference to zero, leaving the constant \(\gamma_A-\gamma_B\). \(\square\)

Proposition Proposition 17.3 is what links SEMA to part (b) of the anchored theorem: with the active-versus-active conditional contrast constant, a collapsible aligned reporting scale recovers the marginal effect (Theorem 12.5), whereas a non-collapsible one still does not (Theorem 12.6). SEMA buys identifiability on the linear-predictor scale; it does not by itself buy transportability of a non-collapsible marginal effect.

Theorem 17.3 (Identifiability under SEMA in the smallest network) Consider one IPD trial on \(\{A,C\}\) in \(\mathcal P_{AC}\) and one AgD trial on \(\{B,C\}\) in \(\mathcal P_{BC}\), modeled by Equation 17.2 under SEMA (Definition 17.6). Suppose the IPD design has full column rank (both arms present and the covariates have nondegenerate within-arm variation) and the AgD reports its \(B\)-versus-\(C\) average conditional contrast on scale \(g\) together with the covariate mean \(\bar x_{BC}\). Then the target-population linear-predictor contrast \(\gamma_A-\gamma_B\) is identified from the combined IPD-plus-AgD likelihood. (Corresponds to E.9 in the proof catalogue.)

Proof. From the IPD trial, the conditional mean model Equation 17.2 restricted to arms \(A\) and \(C\) is a generalized linear model with linear predictor \(\mu_{AC}+\boldsymbol\beta_1^\top x\) on the \(C\) arm and \(\mu_{AC}+\gamma_A+(\boldsymbol\beta_1+\boldsymbol\beta_{2})^\top x\) on the \(A\) arm. With full column rank of the design and a strictly monotone link, the coefficients of a generalized linear model are identified (Chapter 7), meaning the map from coefficients to the fitted conditional means is injective, so that no two distinct coefficient vectors produce the same fit: the \(C\) arm identifies \((\mu_{AC},\boldsymbol\beta_1)\), and the \(A\) arm then identifies \((\gamma_A,\boldsymbol\beta_{2})\). In particular the shared interaction vector \(\boldsymbol\beta_{2}\) is known from the IPD.

From the AgD trial, the reported \(B\)-versus-\(C\) average conditional contrast on scale \(g\) is, by Equation 17.6 and the linear form Equation 17.3 under SEMA, \[ \tilde d_{BC(\mathcal P_{BC})}=\int\bigl(\gamma_B+\boldsymbol\beta_{2}^\top x\bigr)f_{BC}(x)\,dx =\gamma_B+\boldsymbol\beta_{2}^\top\bar x_{BC}, \] a single linear equation in the one unknown \(\gamma_B\), since \(\boldsymbol\beta_{2}\) is known from the IPD and \(\bar x_{BC}\) is reported. Solving, \(\gamma_B=\tilde d_{BC(\mathcal P_{BC})}-\boldsymbol\beta_{2}^\top\bar x_{BC}\), so \(\gamma_B\) is identified. Hence \(\gamma_A-\gamma_B\) is identified from the joint data. The network is connected through the shared comparator \(C\), so the rank condition for network identifiability (Theorem 15.3) is met trivially. \(\square\)

Remark. What identifiability does and does not give. Theorem Theorem 17.3 identifies the contrast on the linear-predictor scale \(g\). It does not identify a marginal effect on a non-collapsible reporting scale: that requires the joint covariate distribution and the baseline risk of the target, through the non-collapsibility of Theorem 11.2, and is governed by part (c) of Theorem 17.1 and by Theorem 12.6. This is the same caveat that recurs whenever the odds ratio or hazard ratio is the quoted measure. \(\blacktriangleleft\)

17.9 Scale dependence of effect modifier status

The conditions of the consistency theorems are stated “on scale \(g\),” and the qualifier is not decorative: whether a covariate is an effect modifier can change with the scale. The following worked counterexample, with arithmetic the reader can check by hand, exhibits a covariate that modifies the effect on the logit scale but not on the risk-difference scale.

Example 17.1 (A covariate that is an effect modifier on the logit scale but not the identity scale) Let \(X\in\{0,1\}\) be binary, let treatment be \(t\in\{0,1\}\), and let the outcome be binary with conditional risks \[ \mu_0(0)=0.1,\quad \mu_1(0)=0.2,\quad \mu_0(1)=0.5,\quad \mu_1(1)=0.6 . \]

Risk-difference (identity) scale. The conditional risk differences are \[ \tau^{\mathrm{id}}(0)=\mu_1(0)-\mu_0(0)=0.2-0.1=0.1,\qquad \tau^{\mathrm{id}}(1)=\mu_1(1)-\mu_0(1)=0.6-0.5=0.1 . \] They are equal, so the conditional risk difference does not depend on \(x\): by Definition 11.3, \(X\) is not an effect modifier on the identity scale.

Log-odds-ratio (logit) scale. The conditional log odds ratios are \[ \tau^{\mathrm{logit}}(0)=\operatorname{logit}(0.2)-\operatorname{logit}(0.1) =\log\tfrac{0.2/0.8}{0.1/0.9}=\log 2.25\approx 0.8109, \] \[ \tau^{\mathrm{logit}}(1)=\operatorname{logit}(0.6)-\operatorname{logit}(0.5) =\log\tfrac{0.6/0.4}{0.5/0.5}=\log 1.5\approx 0.4055 . \] They differ, so by Definition 11.3, \(X\) is an effect modifier on the logit scale.

(Corresponds to E.10 in the proof catalogue.) \(\blacktriangleleft\)

Remark. Consequences for practice. The example shows that “\(X\) is not an effect modifier,” and therefore the entire question “which covariates must be in the adjustment set,” is meaningless until the scale is named. A SEMA that holds on the identity scale need not hold on the logit scale, and the adjustment that licenses an anchored mean-difference comparison need not license an anchored odds-ratio comparison. This is the finite, concrete face of the scale specificity that makes the older scale-free “iff” formulations untenable, and it is why Theorem 17.1 is stated one scale at a time. \(\blacktriangleleft\)

17.10 Testable and untestable parts of constancy

The constancy assumptions E.4, E.5, and SEMA are the load-bearing hypotheses of population adjustment, and it is natural to ask whether the data can check them. The answer divides sharply by network size.

The short answer, and the single most important practical message of the chapter, is that in a two-study comparison these assumptions cannot be checked from the data at all; they must be argued from external knowledge. The reason is a counting mismatch. The AgD trial hands us exactly one number about treatment \(B\)’s effect modification, a single average contrast at a single covariate mean, but the interactions we would need to verify are a whole vector, one coefficient per covariate. One number cannot pin down a vector. Geometrically, that one equation carves out a flat hyperplane of \((\gamma_B,\boldsymbol\beta_{2,B})\) pairs that all reproduce the observed summary; SEMA picks one point on that hyperplane, the point where \(B\)’s interactions match \(A\)’s, and nothing in the data says whether that point is the right one. The proposition makes this precise.

Proposition 17.4 (Testability of the shared effect modifier assumption) In the smallest anchored network of Theorem 17.3 (one IPD trial on \(\{A,C\}\), one AgD trial on \(\{B,C\}\)), SEMA (Definition 17.6) imposes no restriction on the observable data distribution: the SEMA model and the unrestricted model with \(\boldsymbol\beta_{2,B}\neq\boldsymbol\beta_{2,A}\) fit the observed summaries identically. Hence SEMA is untestable from these data. If instead a second AgD trial on \(\{B,C\}\) is available at a covariate mean \(\bar x_{BC}'\neq\bar x_{BC}\), then SEMA implies the over-identifying restriction \[ \tilde d_{BC(\mathcal P_{BC})}-\tilde d_{BC(\mathcal P_{BC}')}\;=\;\boldsymbol\beta_{2}^\top\bigl(\bar x_{BC}-\bar x_{BC}'\bigr), \tag{17.20}\] and is therefore testable on the covariate directions spanned by \(\bar x_{BC}-\bar x_{BC}'\). (Corresponds to E.11 in the proof catalogue.)

Proof. From the AgD trial the only functional of the \(B\)-versus-\(C\) interactions that enters the likelihood is the scalar \(\tilde d_{BC(\mathcal P_{BC})}=\gamma_B+\boldsymbol\beta_{2,B}^\top\bar x_{BC}\) (the average conditional contrast at the reported covariate mean). Given the IPD, which fixes \(\boldsymbol\beta_{2}\) and the remaining parameters, the set of \((\gamma_B,\boldsymbol\beta_{2,B})\) consistent with the AgD is the entire hyperplane \(\{(\gamma_B,\boldsymbol\beta_{2,B}):\gamma_B+\boldsymbol\beta_{2,B}^\top\bar x_{BC}=\tilde d_{BC(\mathcal P_{BC})}\}\). The SEMA point \((\,\tilde d_{BC(\mathcal P_{BC})}-\boldsymbol\beta_{2}^\top\bar x_{BC},\ \boldsymbol\beta_{2}\,)\) lies on this hyperplane, but so does every other point; all give the same observable summary. Therefore no function of the data distinguishes the SEMA model from an unrestricted one, and SEMA is untestable. The target contrast \(\gamma_A-\gamma_B\) does depend on which point of the hyperplane is selected, which is why SEMA is simultaneously necessary for identification and beyond the reach of the data.

With a second AgD trial at \(\bar x_{BC}'\) the data deliver a second scalar \(\tilde d_{BC(\mathcal P_{BC}')}=\gamma_B+\boldsymbol\beta_{2,B}^\top\bar x_{BC}'\). Under SEMA both must be explained by the single, IPD-fixed \(\boldsymbol\beta_{2}\), so their difference must equal \(\boldsymbol\beta_{2}^\top(\bar x_{BC}-\bar x_{BC}')\); any departure of the observed difference from this predicted value falsifies SEMA on the spanned direction. This is the over-identifying restriction Equation 17.20, and it is the node-splitting logic of Definition 15.4 specialized to the regression coefficients. \(\square\)

Remark. The general principle. Conditional constancy of relative effects (E.4) and of absolute effects (E.5) are likewise testable only when the network carries direct and indirect evidence for the same contrast, so that the consistency equation provides an over-identifying restriction (node-splitting, Definition 15.4). In the two-study PAIC setting the model is just-identified for the target contrast, no restriction remains, and the assumptions must be defended by domain knowledge: pharmacological similarity of \(A\) and \(B\), mechanistic argument, and sensitivity analysis over plausible violations. This is the methodological warning of Phillippo et al. (2016). \(\blacktriangleleft\)

17.11 Target-population estimands via the g-formula

We close the abstract development by pinning down the estimands the methods target, and the precise sense in which they are identified. Both are g-formula functionals of the conditional means and the target covariate distribution; they connect this chapter to the transportability theory of Chapter 12. Much of this theorem’s content has appeared piecemeal above; the value of stating it once is that it is the clean summary of what is identified and when the two estimands coincide, and it is the statement the method chapters cite.

Theorem 17.4 (Target-population estimands and their identification) Fix a target population \(\mathcal P^{*}\) and a reporting scale \(h\). Define the target marginal effect \[ \Delta^{\mathrm{Marg},h}_{ab}(\mathcal P^{*})=h\!\left(\textstyle\int\mu_b(x)f_{*}(x)\,dx\right)-h\!\left(\textstyle\int\mu_a(x)f_{*}(x)\,dx\right) \tag{17.21}\] and the target population-average conditional effect \[ \Delta^{\mathrm{Cond},h}_{ab}(\mathcal P^{*})=\int\bigl[h(\mu_b(x))-h(\mu_a(x))\bigr]f_{*}(x)\,dx . \tag{17.22}\] Assume within-trial identification (Theorem 10.4), the relevant constancy assumption (Definition 17.4 for a contrast, or Definition 17.5 for the separate arms), and overlap (Definition 9.8). Then:

(a) Identification. Both estimands are identified from the source conditional means and the target covariate distribution, by the transport g-formula (Theorem 12.2): Equation 17.21 applies the formula to each \(\mu_t\), and Equation 17.22 applies it to the function \(x\mapsto h(\mu_b(x))-h(\mu_a(x))\).

(b) Coincidence under collapsibility. If \(h\) is directly collapsible on its scale (the identity scale, Proposition 11.3), or if \(h\) is collapsible and the conditional contrast is constant on the support (the log risk ratio on the log scale, Theorem 12.5), then \(\Delta^{\mathrm{Marg},h}_{ab}(\mathcal P^{*})=\Delta^{\mathrm{Cond},h}_{ab}(\mathcal P^{*})\).

(c) Divergence under non-collapsibility. If \(h\) is non-collapsible (logit, log-hazard), the two estimands differ in general (Theorem 11.2), and the marginal estimand depends on the target baseline risk, not only on the transported contrast.

(Corresponds to E.12 in the proof catalogue.)

Proof. (a). By within-trial identification and the relevant constancy assumption, the conditional means \(\mu_t(\cdot)\) (for the unanchored or marginal target) or the conditional contrasts \(\tau^h_{ab}(\cdot)\) (for the conditional target) are population-free functions identified from the source. The transport g-formula Theorem 12.2 identifies the expectation under \(f_{*}\) of any such fixed function of \(x\): applied to \(\mu_b\) and \(\mu_a\) it yields the two marginal means \(m_b(\mathcal P^{*})\) and \(m_a(\mathcal P^{*})\), and applying \(h\) and differencing gives Equation 17.21; applied to the function \(h(\mu_b(\cdot))-h(\mu_a(\cdot))\) it yields Equation 17.22 directly. Overlap guarantees the integrals are well defined.

(b). Direct collapsibility means, by Definition 11.6, that the marginal effect equals the population-average conditional effect for every covariate distribution. On the identity scale this is Proposition 11.3; for the log risk ratio with a constant conditional contrast it is the collapsibility of Theorem 12.5. In either case Equation 17.21 equals Equation 17.22.

(c). For a non-collapsible measure there exist, by Theorem 11.2, a covariate distribution and stratum means with constant conditional effect whose marginal effect differs from it; a fortiori Equation 17.21 and Equation 17.22 differ. The marginal odds ratio Equation 17.21 is built from the marginal risks \(m_t(\mathcal P^{*})\), which depend on the full target covariate distribution and hence on the target baseline risk, whereas Equation 17.22 averages stratum contrasts; the two coincide only in the collapsible case. \(\square\)

Remark. Which estimand to target. The marginal estimand answers the population-policy question that a health technology assessment usually asks: what is the average effect in the target population if everyone receives \(b\) rather than \(a\). The conditional estimand answers the within-stratum question that clinical reasoning about a specific patient often uses. The two are different scientific quantities, and on a non-collapsible scale they are different numbers; the analyst must state which one is intended before any claim of consistency is meaningful. The recurring lesson of this chapter is that a population-adjusted method is consistent for a named estimand on a named scale, and that naming both is part of the specification, not an afterthought. \(\blacktriangleleft\)

17.12 A fully worked anchored comparison

We carry one example from conditional risks to a corrected indirect effect, with every number checkable by hand, and use it to show the unadjusted bias, its removal by adjustment, and the scale dependence of the conclusion. The outcome is binary and unfavorable, so a larger risk is worse, and we use the reference-second convention of Chapter 14 in which a positive contrast means the active treatment lowers the risk relative to the comparator.

Example 17.2 (Anchored population adjustment with a binary effect modifier) This is the anticoagulant scenario of the running example, now with every number. Read \(A\) as the new oral anticoagulant, \(B\) as the standard DOAC, \(C\) as warfarin, \(X=1\) as age \(75\) or over, and the outcome as a major bleed, so a larger risk is worse. A single binary covariate \(X\in\{0,1\}\) is present. The conditional risks, taken to be the same function in every population (which makes both constancy assumptions hold and the numbers concrete), are

\(x\)	\(\mu_A(x)\)	\(\mu_B(x)\)	\(\mu_C(x)\)
\(0\)	\(0.20\)	\(0.30\)	\(0.40\)
\(1\)	\(0.50\)	\(0.30\)	\(0.40\)

The IPD trial \(AC\) has \(\Pr_{AC}(X=1)=0.25\), the AgD trial \(BC\) has \(\Pr_{BC}(X=1)=0.75\), and the target population \(\mathcal P^{*}\) has \(\Pr_{*}(X=1)=0.50\).

Step 1: conditional contrasts on the identity scale. With \(\tau^{\mathrm{id}}_{AC}(x)=\mu_C(x)-\mu_A(x)\), \(\tau^{\mathrm{id}}_{BC}(x)=\mu_C(x)-\mu_B(x)\), and \(\tau^{\mathrm{id}}_{AB}(x)=\mu_B(x)-\mu_A(x)\), \[ \tau^{\mathrm{id}}_{AC}(0)=0.40-0.20=0.20,\qquad \tau^{\mathrm{id}}_{AC}(1)=0.40-0.50=-0.10, \] \[ \tau^{\mathrm{id}}_{BC}(0)=0.40-0.30=0.10,\qquad \tau^{\mathrm{id}}_{BC}(1)=0.40-0.30=0.10 . \] So \(X\) is an effect modifier of \(A\) versus \(C\) (the contrast moves from \(0.20\) to \(-0.10\)) but not of \(B\) versus \(C\) (constant at \(0.10\)). The target contrast \(\tau^{\mathrm{id}}_{AB}(x)=\mu_B(x)-\mu_A(x)\) is \(0.10\) at \(x=0\) and \(-0.20\) at \(x=1\), and indeed equals \(\tau^{\mathrm{id}}_{AC}(x)-\tau^{\mathrm{id}}_{BC}(x)\) as Lemma 17.3 requires.

Step 2: the target. Standardizing to \(f_{*}\) (equal weights \(0.5,0.5\)), \[ \Delta^{\mathrm{Cond,id}}_{AC}(\mathcal P^{*})=0.5(0.20)+0.5(-0.10)=0.05,\qquad \Delta^{\mathrm{Cond,id}}_{BC}(\mathcal P^{*})=0.10, \] so the target effect of \(A\) versus \(B\) is \(\Delta^{\mathrm{Cond,id}}_{AB}(\mathcal P^{*})=0.05-0.10=-0.05\), a risk difference of \(-0.05\) (\(A\) raises the risk by \(0.05\) relative to \(B\) in the target).

Step 3: the unadjusted Bucher comparison. Each trial reports its own average conditional contrast Equation 17.6: \[ \tilde d_{AC(\mathcal P_{AC})}=0.75(0.20)+0.25(-0.10)=0.125,\qquad \tilde d_{BC(\mathcal P_{BC})}=0.10 . \] The naive Bucher estimand is \(\tilde d_{AC(\mathcal P_{AC})}-\tilde d_{BC(\mathcal P_{BC})}=0.125-0.10=0.025\), against a target of \(-0.05\). The bias is \(0.025-(-0.05)=0.075\). By the decomposition of Theorem 16.1 it is \(b_{AC}-b_{BC}\) with \[ b_{AC}=\int\tau^{\mathrm{id}}_{AC}(x)[f_{AC}(x)-f_{*}(x)]\,dx=0.20(0.75-0.50)+(-0.10)(0.25-0.50)=0.05+0.025=0.075, \] and \(b_{BC}=0.10\bigl[(0.25-0.50)+(0.75-0.50)\bigr]=0\) because \(\tau^{\mathrm{id}}_{BC}\) is constant. The entire bias comes from the imbalanced effect modifier in the \(A\)-versus-\(C\) contrast.

Step 4: the adjusted comparison. Re-standardize the \(AC\) contrast to the target, which is exactly Step 2: \(\Delta^{\mathrm{Cond,id}}_{AC}(\mathcal P^{*})=0.05\). The adjusted anchored estimand is \[ \theta^{\mathrm{anch}}_{AB}(\mathcal P^{*})=0.05-0.10=-0.05=\Delta^{\mathrm{Cond,id}}_{AB}(\mathcal P^{*}), \] zero bias, as Theorem 17.1(a) guarantees, and since the identity scale is directly collapsible (Proposition 11.3) this is also the target marginal risk difference, by part (b).

Step 5: weighting realization and effective sample size. A MAIC reweights the IPD to the target covariate distribution. With binary \(X\), matching \(\Pr(X=1)\) from \(0.25\) to \(0.50\) uses weights proportional to \(w_1=0.50/0.25=2\) for the \(X=1\) subjects and \(w_0=0.50/0.75=2/3\) for the \(X=0\) subjects. In an IPD sample of \(n=200\) with \(50\) subjects at \(X=1\) and \(150\) at \(X=0\), the weighted contrast is \[ \frac{w_0\,(150)\,(0.20)+w_1\,(50)\,(-0.10)}{w_0\,(150)+w_1\,(50)} =\frac{(2/3)(150)(0.20)+2(50)(-0.10)}{(2/3)(150)+2(50)} =\frac{20-10}{100+100}=\frac{10}{200}=0.05, \] recovering Step 2. The effective sample size (Definition 10.1 and the notation of Notation) is \[ \mathrm{ESS}=\frac{(\sum_i w_i)^2}{\sum_i w_i^2}=\frac{200^2}{50\cdot 2^2+150\cdot(2/3)^2} =\frac{40000}{200+200/3}=\frac{40000}{800/3}=150, \] Intuitively, we are upweighting the \(50\) older (\(X=1\)) subjects by a factor of \(2\) so that they stand in for the larger older share of the target, and downweighting the \(150\) younger subjects; leaning the estimate on fewer, heavily weighted subjects is what shrinks the effective sample size. So balancing one covariate has cost a quarter of the nominal sample. The ESS is a preview of the variance inflation that Chapter 18 studies in detail.

Step 6: the mismatched scale. Suppose instead the decision maker wants the marginal odds ratio of \(A\) versus \(B\) in \(\mathcal P^{*}\). The target marginal risks are \(m_A(\mathcal P^{*})=0.5(0.20)+0.5(0.50)=0.35\), \(m_B(\mathcal P^{*})=0.30\), so the target marginal log odds ratio is \[ \Delta^{\mathrm{Marg,logit}}_{AB}(\mathcal P^{*})=\operatorname{logit}(0.30)-\operatorname{logit}(0.35) =-0.8473-(-0.6190)=-0.2283,\qquad \mathrm{OR}\approx 0.796 . \] The population-average conditional log odds ratio is \[ \Delta^{\mathrm{Cond,logit}}_{AB}(\mathcal P^{*})=0.5\bigl[\operatorname{logit}(0.30)-\operatorname{logit}(0.20)\bigr] +0.5\bigl[\operatorname{logit}(0.30)-\operatorname{logit}(0.50)\bigr]=0.5(0.5390)+0.5(-0.8473)=-0.1542, \] with \(\mathrm{OR}\approx 0.857\). The reason the two disagree is the nonlinearity of the logit: the marginal odds ratio forms the population risks first and then takes a single ratio, while the conditional odds ratio takes a ratio within each age group and then averages, and averaging risks is not the same operation as averaging log-odds, so the two odds ratios part company even though not one conditional risk has changed. An anchored PAIC fit on the logit link recovers the conditional value \(-0.1542\) (part (a)), but it does not recover the marginal value \(-0.2283\): the two differ by the collapsibility gap of the odds ratio (Theorem 11.2), and no amount of covariate balancing on the logit scale closes it (part (c), Theorem 12.6). The identity-scale adjustment of Steps 2 to 5 is exact for the risk difference because that scale is directly collapsible; the same data adjusted on the logit scale identify only the conditional odds ratio. The number you can trust depends on the estimand and the scale you asked for. \(\blacktriangleleft\)

17.13 How the methods instantiate the theory

The three method chapters that follow are three ways to compute the standardized contrasts Equation 17.12 and Equation 17.8, differing in how they model the conditional means and how they average to the target.

• MAIC (Chapter 18) uses the reweighting form of Equation 17.8: it estimates weights \(w(x)=f_{*}(x)/f_{AC}(x)\) by entropy balancing on the covariate moments, then forms the weighted IPD contrast. It is nonparametric in the outcome model and pays for it in effective sample size, as Step 5 foreshadows. The density ratio \(w(x)=f_{*}(x)/f_{AC}(x)\) is only the theoretical target; because the analyst knows neither density, MAIC does not estimate them, it instead finds weights of the form \(w_i=\exp(\alpha+\boldsymbol\beta^\top x_i)\) that force the reweighted IPD covariate moments to match the reported target moments, which is the entropy-balancing solution of Chapter 18. That is how the abstract \(f_{*}/f\) of this chapter becomes a computable procedure.
• STC (Chapter 19) uses the integral form: it fits an outcome regression for \(\tau^g_{AC}(\cdot)\) on the IPD and averages it over the target covariate distribution. It is parametric in the outcome model and must be marginalized correctly to avoid the conditional-versus-marginal confusion of part (c).
• ML-NMR (Chapter 20) fits the full model Equation 17.2 jointly to IPD and AgD by integrating the individual model up to the aggregate likelihood, and standardizes to any target by the g-formula. It is the most general and reduces to the others in special cases.

In one sentence each: MAIC reweights the IPD patients so their covariate profile matches the target, then contrasts the weighted outcomes; STC fits an outcome regression on the IPD and reads off the contrast predicted at the target covariate profile; ML-NMR writes a single model for both the IPD and the AgD and integrates the individual model up to the aggregate level, so it can standardize to any target at once. All three compute the same standardized contrast Equation 17.12; they differ in what they must assume about the outcome model and in how much precision they spend.

Each is consistent for its named estimand under the hypotheses of Theorem 17.1 or Theorem 17.2; the differences are in efficiency, in robustness to model misspecification, and in which estimand they most naturally produce.

17.14 Notes and references

The modern theory of population adjustment is due to Phillippo (2019), whose thesis introduced the effect-modifier framing of the anchored case and the prognostic-variable framing of the unanchored case, and to the NICE Technical Support Document of Phillippo et al. (2016), which set the practical guidance on when each comparison is admissible. The scale-aware refinement followed here, distinguishing the model link \(g\) from the reporting scale \(h\), separating direct from ordinary collapsibility, and replacing the classical scale-free “if and only if” with scale-specific sufficiency plus class-wise necessity, is the contribution of Chandler and Ishak (2026); the present chapter adopts their estimand-based view throughout and is in harmony with the transportability development of Chapter 12. The original methods papers, Signorovitch et al. (2010) for MAIC and Caro and Ishak (2010) for STC, predate this theory and motivate it; the multilevel synthesis is Phillippo et al. (2020). The conditional constancy assumptions E.4 and E.5 are the population-adjustment analogues of the structural invariance of Chapter 12, and the anchored cancellation of purely prognostic covariates is the conditional-level descendant of the baseline cancellation that made Bucher’s anchored comparison robust in Chapter 14; the connection between balancing scores and reweighting traces to Rosenbaum and Rubin (1983).

This chapter corresponds to results E.4 (conditional constancy of relative effects, Definition 17.4), E.5 (conditional constancy of absolute effects, Definition 17.5), E.6a to E.6e (the scale-specific anchored consistency theorem, Theorem 17.1), E.7 (unanchored consistency, Theorem 17.2), E.8 (the shared effect modifier assumption, Definition 17.6), E.9 (identifiability under SEMA, Theorem 17.3), E.10 (scale dependence of effect modifier status, Example 17.1), E.11 (testable and untestable constancy, Proposition 17.4), and E.12 (target-population estimands, Theorem 17.4) of the proof catalogue in proofs/E_population_adjustment_theory.md. The base results E.0 to E.3 it builds on are developed in Chapters 11 (collapsibility gap and the effect-modifier and prognostic definitions, Definition 11.6, Definition 11.3, Definition 11.5), 12 (transportability and scale alignment, Theorem 12.2, Definition 12.4, Theorem 12.6), and 16 (the failure-of-Bucher decomposition, Theorem 16.1).

Three algebraic cores have machine-checked counterparts in the companion Coq development, in theories/EffectModifiers.v. The anchored cancellation of a purely prognostic covariate (Lemma 17.1) is Theorem anchored_PV_cancellation; the strict-stronger relation between prognostic and effect-modifier status (Lemma 17.2, the same witness that absolute constancy is strictly stronger than relative constancy) is Theorem not_EM_does_not_imply_not_PV; and the adjusted Bucher identity (Corollary 17.1) is Theorem anchored_unbiased. The Coq file deliberately does not formalize the scale-specific consistency theorems E.6a to E.6e or E.7, which depend on estimand scale, collapsibility, target population, imbalance, overlap, and model correctness; it captures the algebraic distinction between the anchored and unanchored architectures, which is the invariant that survives all the scale-specific case analysis.

17.15 Exercises

A starred exercise (\(\star\)) has a full worked solution in Appendix F. The first two exercises are warm-ups that only ask you to read the running example and the tables; the computational and proof-heavy exercises follow.

Exercise 17.1 (Warm-up, interpretation.) In the anticoagulant running example, age raises the bleeding risk on the new drug \(A\) but not on the DOAC \(B\) or warfarin \(C\). Using the one-line definitions of Section 17.5, state whether age is prognostic for \(A\), whether it is prognostic for \(B\), and whether it is an effect modifier of \(A\) versus \(C\) and of \(B\) versus \(C\). Then say in one sentence why, in an anchored comparison, only its effect-modifier status forces it into the adjustment set.

Exercise 17.2 (Warm-up, interpretation.) Suppose instead the new drug \(A\) had been approved on a single-arm study with no control group, while \(B\) remains a warfarin-controlled trial. Which architecture, anchored or unanchored, are you now in; which constancy assumption do you need (Definition 17.4 or Definition 17.5); and which covariates must enter the adjustment set? Answer from the “what changes by architecture” table in Section 17.2.1, in three short sentences.

Exercise 17.3 Conditional additivity on three scales. Verify Lemma 17.3, \(\tau^g_{AB}(x)=\tau^g_{AC}(x)-\tau^g_{BC}(x)\), directly for the identity, log, and logit links, using the conditional risks of Example 17.2 at \(x=1\). Then explain in one sentence why the identity is purely algebraic and holds on every scale, even though the resulting marginal effects do not subtract the same way on a non-collapsible scale.

Exercise 17.4 A different imbalance. In the setting of Example 17.2, change the target to \(\Pr_{*}(X=1)=0.75\) (so \(\mathcal P^{*}=\mathcal P_{BC}\), the two-study MAIC case). Recompute the target conditional risk difference \(\Delta^{\mathrm{Cond,id}}_{AB}(\mathcal P^{*})\), the unadjusted Bucher estimand, and the adjusted estimand, and confirm that the adjusted estimand hits the target while the unadjusted one does not. Identify the value of \(\Pr_{*}(X=1)\) at which the unadjusted Bucher estimand would happen to be unbiased, and explain why this is a numerical coincidence rather than a vindication of the unadjusted method.

Exercise 17.5 Purely prognostic covariate. Augment Example 17.2 with a second binary covariate \(W\) that is purely prognostic: set \(\mu_t(x,w)=\mu_t(x)+0.1\,w\) for all three treatments. Show that \(W\) is prognostic for every treatment but is not an effect modifier of any contrast on the identity scale, and verify numerically that omitting \(W\) from the adjustment set leaves the anchored estimand unchanged even when \(W\) is imbalanced (\(\Pr_{AC}(W=1)=0.2\), \(\Pr_{*}(W=1)=0.8\)). Then show that the unanchored estimand of \(\mu_A\) alone does change, illustrating Theorem 17.2 and Lemma 17.2.

Exercise 17.6 Identification arithmetic under SEMA. Suppose an IPD trial on \(\{A,C\}\) yields \(\gamma_A=-0.6\) and a shared interaction \(\beta_2=0.5\) on a single covariate, and an AgD trial on \(\{B,C\}\) reports an average conditional contrast \(\tilde d_{BC}=-0.1\) at covariate mean \(\bar x_{BC}=0.4\). Using Theorem 17.3, find \(\gamma_B\) and the target linear-predictor contrast \(\gamma_A-\gamma_B\). Then show that if the true interaction for \(B\) were \(\beta_{2,B}=0.2\neq\beta_2\) (SEMA false) but unknown, the same AgD summary would be produced by a different \(\gamma_B\), and compute the resulting error in \(\gamma_A-\gamma_B\).

Exercise 17.7 Scale dependence both ways. Construct conditional risks \(\mu_0(x),\mu_1(x)\) for binary \(x\) such that the covariate is an effect modifier on the identity scale but not on the logit scale (the reverse of Example 17.1). Prove that such a configuration requires the conditional risks to satisfy a specific algebraic constraint, and exhibit one explicit numerical solution.

Exercise 17.8 (\(\star\)) The unanchored bias formula. Prove Equation 17.17 from first principles, and then specialize it to a single binary unadjusted prognostic variable \(W\) with \(\mu_A(x,w)=\alpha(x)+\rho\,w\), \(\Pr_{\mathrm{src}}(W=1)=p\), \(\Pr_{*}(W=1)=q\), showing the bias of the \(A\) marginal mean equals \(\rho\,(p-q)\). Explain why the corresponding term in an anchored comparison would be multiplied by zero, and connect this to Lemma 17.1. Discuss why this makes the unanchored adjustment set requirement of Theorem 17.2 practically far harder to satisfy than the anchored one.

Exercise 17.9 (\(\star\)) Non-collapsibility obstructs marginal transport. Using the conditional risks of Example 17.2, compute the marginal odds ratio of \(A\) versus \(B\) in three populations: \(\Pr(X=1)\in\{0.25,0.50,0.75\}\). Show that the marginal log odds ratio changes across the three even though the conditional risks (hence the conditional odds ratios and the SEMA structure) are identical, and that the change is not a difference in covariate balance of any effect modifier but a pure non-collapsibility effect. Conclude that, on the odds-ratio scale, even a fully adjusted anchored comparison delivers a population-specific number, and relate this to part (c) of Theorem 17.1 and to Theorem 12.6.

Caro, J. Jaime, and K. Jack Ishak. 2010. “No Head-to-Head Trial? Simulate the Missing Arms.” PharmacoEconomics 28 (10): 957–67.

Chandler, Conor, and K. Jack Ishak. 2026. “Reframing Population-Adjusted Indirect Comparisons as a Transportability Problem: An Estimand-Based Perspective and Implications for Health Technology Assessment.” arXiv Preprint arXiv:2602.17041.

Dias, Sofia, A. E. Ades, Nicky J. Welton, Jeroen P. Jansen, and Alexander J. Sutton. 2018. Network Meta-Analysis for Decision-Making. Statistics in Practice. Wiley.

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.

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

Rosenbaum, Paul R., and Donald B. Rubin. 1983. “The Central Role of the Propensity Score in Observational Studies for Causal Effects.” Biometrika 70 (1): 41–55.

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.