12  Transportability: The Estimand-Based Theory

12.1 Introduction

A randomized trial answers a question about the patients it enrolled. A reimbursement decision asks a question about a different group of patients: the population that will receive the drug if it is approved. The two groups are almost never the same. Trials recruit through eligibility criteria, run at specialized centers, and finish years before the decision is made; the decision population is broader, older, sicker, or simply distributed differently across the prognostic and effect-modifying covariates that govern how the treatment works. The central practical problem of health technology assessment (HTA, Notation) is therefore not estimation but transport: given an effect measured in one population, what can we say about the effect in another?

The previous three chapters built the machinery to estimate causal effects cleanly within a single population: potential outcomes and the identification of the average treatment effect (Chapter 9), the adjustment methods that recover it from observational or reweighted data (Chapter 10), and the distinction between conditional and marginal estimands together with the collapsibility that separates them (Chapter 11). This chapter asks the next question. An effect is a number attached to a population. Move to a new population and the number can change, even when nothing about the biology, the treatment, or the conditional response has changed at all. When may we carry the number across unchanged, and when must we recompute it?

The answer is sharper, and more uncomfortable, than the folklore of indirect comparison suggests. The folklore says: if the same variables modify the treatment effect in both populations, and we have matched or adjusted for them, then the relative effect is “the same in any population” and may be transported at will. We will prove that this is true for some effect measures and false for others, and that the dividing line is exactly the property studied in Chapter 11. For a collapsible measure on its natural scale, the mean difference, the risk ratio, the rate ratio, a shared effect modifier structure does make the relative effect transportable. For a non-collapsible measure, the odds ratio and the hazard ratio, it does not: the marginal effect stays tied to the covariate distribution of the population in which it was averaged, so that the very same conditional model produces different marginal odds ratios in different populations. Because the odds ratio and the hazard ratio are the two effect measures HTA uses most, the result is a genuine paradox. The measures we most want to transport are precisely the ones that resist transport.

The chapter develops this in five movements. First we make the objects precise: a population is a probability distribution over covariates, an estimand is a functional of that distribution, and transportability is a statement that an estimand defined in a target population is recoverable from a source. We separate generalizability (does a sample represent its own source population?) from transportability (does a result move to a different population?). Second we define direct transportability, the estimand is numerically identical in source and target, and conditional transportability, the estimand is recoverable after adjusting for covariates, and we prove the transport g-formula that makes the latter operational. Third we dissect the pairwise population-adjusted indirect comparison (PAIC) and show that it secretly performs two transports, not one, and that standard guidance justifies only the first. Fourth we tie transportability to collapsibility and to a second, subtler condition, scale alignment, and prove the joint characterization of when a marginal relative effect is directly transportable. Fifth we work three numerical examples, continuous, binary, and time-to-event, whose every digit the reader can reproduce, and which exhibit the success of the collapsible aligned case and the two distinct ways transport fails.

The development follows the estimand-based framework of Chandler and Ishak (2026), which recasts population adjustment as a transportability problem, and draws the identification scaffolding from Hernán and Robins (2020), whose treatment of transport via a population indicator and the g-formula we adopt. Everything is proved. The only results imported without proof are the within-population identification theorems of Chapters 9 and 10 and the collapsibility facts of Chapter 11, all stated there in full.

Example 12.1 (A running example: one submission, three populations) Picture a health technology assessment (HTA) comparing two active drugs, \(B\) and \(C\), for a chronic inflammatory disease. Neither trial compared \(B\) against \(C\) head to head; each was run against a common comparator \(A\), so any statement about \(B\) versus \(C\) must be assembled indirectly.

• The index trial studied \(A\) versus \(B\) and reports individual patient data (IPD): we know each enrolled patient’s covariates and outcome. It recruited mostly younger, biologic-naive patients. Call its covariate distribution \(f_{AB}\).
• The comparator trial studied \(A\) versus \(C\) and reports only aggregate data (AgD): published summary means and event counts, no individual records. It recruited mostly older, biologic-experienced patients. Call its covariate distribution \(f_{AC}\).
• The reimbursement decision concerns neither trial’s enrollees but the decision population \(\mathcal P^{*}\): everyone who would receive the drug under the approved label, a broad mix of both kinds of patient.

To keep arithmetic in view, compress the single most important effect-modifying covariate into a binary indicator \(X\): set \(X=1\) for a biologic-experienced patient and \(X=0\) for a biologic-naive one. The three populations then differ only in how much weight they place on \(X=1\):

Population	Role in the comparison	Data available	\(P(X=1)\)
\(f_{AB}\) (index trial)	source: has outcomes on \(A\) and \(B\)	IPD	\(0.10\)
\(f_{AC}\) (comparator trial)	comparator: has outcomes on \(A\) and \(C\)	AgD	\(0.90\)
\(\mathcal P^{*}\) (decision population)	target: the effect we actually want	covariates only	\(0.50\)

Here is the whole chapter in one sentence. A population is not the roster of patients in a trial; it is the distribution of their covariates, summarized here by the single number \(P(X=1)\). A treatment-effect number is attached to one such distribution, and the chapter asks what happens to that number when we try to move it from one row of this table to another. For some effect measures the number survives the move unchanged; for others it does not. The two trial populations \(f_{AB}\) and \(f_{AC}\) return as \(\mathcal P_1\) and \(\mathcal P_2\) in the worked examples of Section 12.8, with exactly these covariate frequencies. \(\blacktriangleleft\)

12.2 Populations, estimands, and the structural model

Throughout, a population is a probability distribution. We make this the foundation.

Why a distribution, and not simply “the patients in trial \(X\)”? Because two trials with identical eligibility criteria can still enroll patients who differ systematically in age, severity, prior treatment, or biomarkers, and it is that covariate mix, not the list of names, that every formula below operates on. In Example 12.1 the three populations share the same disease and differ only in the single number \(P(X=1)\), and that difference is the entire input to the transport problem.

Before the formal objects start appearing in combination, here is a compact glossary of the symbols this chapter leans on. Each is defined in full at first use; this table is a quick reference, not a substitute.

Symbol	Read it as	Meaning
\(\mathcal P\), \(\mathcal P^{*}\)	“a population”, “the target population”	a distribution of covariates; \(\mathcal P^{*}\) is the one we want an answer for
\(f_{\mathcal P}(x)\)	“the covariate density”	how common covariate pattern \(x\) is in population \(\mathcal P\)
\(f_{AB}\), \(f_{AC}\)	“the index / comparator trial population”	covariate distributions of the two trials
\(\mu_t(x)\)	“the conditional mean”	average outcome under treatment \(t\) among patients with covariates \(x\)
\(m_t(\mathcal P)\)	“the marginal mean”	population-average outcome under \(t\) in \(\mathcal P\)
\(d_{ab(\mathcal P)}\)	“the marginal effect” (subscript is a population)	\(b\) versus \(a\) effect built from averaged means in \(\mathcal P\)
\(d_{ab}(x)\)	“the conditional effect” (argument is a covariate)	\(b\) versus \(a\) effect at one fixed covariate value \(x\)
\(g\), \(h\)	“model link, reporting transform”	scale the model is linear on; scale the effect is reported on
\(w(x)\)	“the transport weight”	\(f_{\mathcal P^{*}}(x)/f_{\mathcal P}(x)\), how much more common \(x\) is in the target
\(e_t(x)\)	“the (generalized) propensity”	\(P(T=t\mid X=x)\) in the source, the chance of getting treatment \(t\) at covariates \(x\)
\(\mathrm{logit}\), \(\mathrm{expit}\)	“logit, expit”	\(\mathrm{logit}(p)=\log\{p/(1-p)\}\) and its inverse \(\mathrm{expit}(z)=1/(1+e^{-z})\)

The two symbols easiest to confuse are \(d_{ab(\mathcal P)}\) and \(d_{ab}(x)\): a population in the subscript means the marginal effect (average first, then transform), while a covariate value in the argument means the conditional effect (transform at that value). That distinction drives the whole chapter.

A population \(\mathcal P\) is a probability distribution over the covariate space \(\mathcal X\), with density (or mass function) \(f_{\mathcal P}\) with respect to a fixed dominating measure. Two populations differ exactly when their covariate distributions differ. We attach to each individual the potential outcomes \(Y(t)\) of Definition 9.1 under treatments \(t\) drawn from a finite set, the observed treatment \(T\), the observed outcome \(Y\), and the covariate vector \(X\in\mathcal X\). We write \(\mathcal P^{*}\) for a target population and reserve unindexed probability statements for the population under discussion, as set in Notation.

The quantity that travels between populations is the conditional mean potential outcome \[ \mu_t(x)\;=\;\mathbb E\!\left[\,Y(t)\mid X=x\,\right], \tag{12.1}\] the average outcome under treatment \(t\) among individuals with covariates \(x\). Its building blocks, the conditional effect \(\tau(x)=\mu_1(x)-\mu_0(x)\) and the average effect \(\tau=\mathbb E[\tau(X)]\), are the estimands of Definition 9.4. The single structural hypothesis on which all of transportability rests is that \(\mu_t(\cdot)\) is a property of the biology, not of the population, and so is shared across populations.

Definition 12.1 (Structural invariance of the conditional model) Let \(S\) be a population-membership indicator, with \(S=s\) naming the population \(\mathcal P_s\). The conditional mean potential outcome model is structurally invariant across the populations under study if \[ \mathbb E\!\left[\,Y(t)\mid X=x,\,S=s\,\right]\;=\;\mu_t(x) \qquad\text{for every population } s \text{ and every } x\in\mathcal X, \tag{12.2}\] that is, the function \(\mu_t(\cdot)\) does not depend on \(s\). Equivalently, the potential outcome is mean independent of population membership given the covariates: \(\mathbb E[Y(t)\mid X,S]=\mathbb E[Y(t)\mid X]\). \(\blacktriangleleft\)

Reader translation. Once you know a patient’s covariates \(x\), learning which trial or population they came from tells you nothing more about their expected outcome under treatment \(t\). Biology responds to the covariates, not to the study logo. The assumption fails exactly when some covariate that changes the outcome is left out of \(x\) and is distributed differently across populations.

Invariance Equation 12.2 is the population-level analog of the conditional exchangeability of Definition 9.7. Conditional exchangeability says that, within a population, the potential outcome is mean independent of treatment given covariates, so that confounding is removed by conditioning on \(X\). Invariance says that, across populations, the potential outcome is mean independent of population membership given covariates, so that between-population differences in outcome are entirely explained by between-population differences in the covariate distribution. The whole of conditional transportability is the working out of this single sentence. It can fail, and we flag the failure mode now: if a variable that modifies the treatment effect is omitted from \(X\) and is distributed differently across populations, then \(\mu_t(\cdot)\) silently depends on \(s\) through that omitted variable, and Equation 12.2 is false. The covariate set \(X\) must be rich enough to capture every such variable. We assume invariance holds for the stated \(X\) unless we say otherwise.

Which covariates must be in \(X\)? Because invariance is a claim about one specific covariate set, it helps to sort candidate variables by the role they play.

• Effect modifiers change the size of the treatment effect. These must be in \(X\): an omitted effect modifier that is distributed differently across populations is precisely what breaks invariance and transport.
• Prognostic variables change the outcome but not the treatment effect. Including them is needed for the conditional mean \(\mu_t(\cdot)\) to be correct, but a prognostic variable that is not also an effect modifier does not by itself break transport of a relative effect on a collapsible scale.
• Trial-inclusion variables affect who enrolled. They matter for generalizability and for the overlap condition below, and matter for transport only insofar as they are also effect modifiers.
• Merely imbalanced, irrelevant variables differ across populations but neither modify the effect nor predict the outcome. Imbalance alone is not a reason to adjust for them.

Invariance versus SEMA (a preview). Readers who have run matching methods may be waiting for the shared effect modifier assumption (SEMA). It is a different assumption, and it arrives later, in Section 12.6; the distinction is worth fixing now. Structural invariance says the whole conditional mean function \(\mu_t(\cdot)\) is shared across populations. SEMA, once we have a model, will say something narrower: that the effect-modification part of that function is identical for the two active treatments \(B\) and \(C\). Invariance is assumed everywhere in this chapter; SEMA is an extra structural restriction layered on top, not a replacement for it. You do not need SEMA yet.

Under invariance, a population contributes only its covariate distribution \(f_{\mathcal P}\), and the marginal mean potential outcome in \(\mathcal P\) is the average of \(\mu_t\) over that distribution: \[ m_t(\mathcal P)\;:=\;\mathbb E_{X\sim\mathcal P}\!\left[\mu_t(X)\right]\;=\;\int_{\mathcal X}\mu_t(x)\,f_{\mathcal P}(x)\,dx. \tag{12.3}\] For a binary outcome \(m_t(\mathcal P)\) is the population’s risk under treatment \(t\); for a survival summary it is the corresponding mean. Effects are contrasts of these marginal means on a chosen scale. Fix a strictly monotone, twice differentiable link \(g\) (the scale of the effect measure, Definition 7.2). The marginal relative effect of treatment \(b\) versus \(a\) in \(\mathcal P\) on scale \(g\) is, in the notation \(d_{ab(\mathcal P)}\) of Notation, \[ d_{ab(\mathcal P)}\;=\;g\!\left(m_b(\mathcal P)\right)-g\!\left(m_a(\mathcal P)\right), \tag{12.4}\] and the conditional relative effect at covariate value \(x\) is \[ d_{ab}(x)\;=\;g\!\left(\mu_b(x)\right)-g\!\left(\mu_a(x)\right). \tag{12.5}\] The marginal mean difference uses \(g=\mathrm{id}\), the marginal log odds ratio uses \(g=\mathrm{logit}\),¹ the marginal log risk ratio uses \(g=\log\). The marginal effect Equation 12.4 transforms the averaged means, while the population-average conditional effect \(\mathbb E_{X\sim\mathcal P}[d_{ab}(X)]\) averages the transformed means; these coincide for the identity scale and diverge for the logit, which is exactly the non-collapsibility of Definition 7.7 and Theorem 11.2. The marginal effect Equation 12.4 is the estimand of a pairwise population adjustment and the object whose transport we study.

Reader translation (marginal versus conditional). Two operations are in play, and they are not interchangeable. The marginal effect Equation 12.4 first averages the means over the population and then transforms with \(g\); the population-average conditional effect \(\mathbb E_{X\sim\mathcal P}[d_{ab}(X)]\) first transforms each stratum and then averages. For the identity scale the order does not matter, so the two agree. For a curved scale like the logit it does. A two-stratum toy makes the gap visible: let a population place equal weight on two covariate values whose control risks are \(0.2\) and \(0.5\).

Operation	Value
average the risks, then take \(\mathrm{logit}\)	\(\mathrm{logit}(0.35)=-0.619\)
take \(\mathrm{logit}\) of each risk, then average	\(\tfrac12(\mathrm{logit}\,0.2+\mathrm{logit}\,0.5)=\tfrac12(-1.386+0)=-0.693\)

The two numbers differ by \(0.074\), and that gap is precisely the non-collapsibility of Definition 7.7. Here it is small; it can be large, and the whole second half of the chapter is about when it forces a marginal effect to change from one population to the next.

Which effects are conditional, which are marginal, and why HTA wants marginal. A regression returns a conditional effect by default: a coefficient, or an odds ratio from a logistic fit, is the effect at a fixed covariate value, holding covariates equal across the comparison. A reimbursement decision needs a marginal effect: the average effect over the whole treated population, because a payer funds a population, not a single covariate stratum. For a collapsible measure the two coincide and the distinction is harmless. For the odds ratio and the hazard ratio they diverge, and a conditional odds ratio answers a different question (“for a patient like this one”) from the marginal odds ratio a population-level decision needs (“averaged over everyone we would treat”).

Remark (Generalizability versus transportability). With the membership indicator \(S\) in hand we can state the two faces of external validity precisely, which is the content of catalogue result N.1.²

Definition 12.2 (Generalizability) A study conducted on a sample is generalizable to its own source population \(\mathcal P_s\) if the estimand identified from the sample equals the estimand in \(\mathcal P_s\). Generalizability is a within-population property: it concerns sampling, that is, whether the enrolled sample is representative of the population it was drawn from. Formally, writing \(R=1\) for sample inclusion, the trial estimand generalizes to \(\mathcal P_s\) when \(\mathbb E[Y(t)\mid R=1,S=s]=\mathbb E[Y(t)\mid S=s]=m_t(\mathcal P_s)\), which holds when inclusion is mean independent of the potential outcome, \(R\perp Y(t)\mid S=s\), possibly after conditioning on measured covariates. \(\blacktriangleleft\)

Reader translation. Generalizability is a question about sampling within one population: did the enrolled patients represent the population they were drawn from? It is internal to a single study and says nothing about moving to a different population.

Definition 12.3 (Transportability) An estimand \(\psi(\mathcal P^{*})\) defined in a target population \(\mathcal P^{*}\) is transportable from a source population \(\mathcal P\) if \(\psi(\mathcal P^{*})\) is identified from the source outcome distribution together with the covariate distribution of the target. Transportability is a between-population property. Two modes are distinguished.

1. The estimand is directly transportable from \(\mathcal P\) to \(\mathcal P^{*}\) if it takes the same value in both, \(\psi(\mathcal P)=\psi(\mathcal P^{*})\), so the source value may be carried across with no adjustment.

2. The estimand is conditionally transportable from \(\mathcal P\) to \(\mathcal P^{*}\) via the covariate set \(X\) if it is recovered by re-standardizing the source conditional model to the target covariate distribution (made operational by the transport g-formula, Theorem 12.2), that is, if the structural invariance of Definition 12.1 holds for \(X\) together with the support condition below.

\(\blacktriangleleft\)

Reader translation. Transportability is about moving between populations, not sampling within one. Mode 1 (direct) is the lucky case: the number is literally identical in source and target, so carry it over untouched. Mode 2 (conditional) is the workhorse: keep the covariate-by-covariate response \(\mu_t(\cdot)\) learned in the source, but recompute the population average using the target’s covariate mix. In Example 12.1, mode 2 takes the response learned in the index trial \(f_{AB}\) and re-averages it over the decision population \(\mathcal P^{*}\).

Generalizability and transportability are independent properties. A trial may be perfectly generalizable to its own enrolled population (a representative sample) and yet its effect may fail to transport to a different population; conversely an effect may transport across populations that the trial never sampled. HTA lives in the gap between the two: trials are run to be internally valid and, ideally, generalizable to their protocol population, but decisions are made for a different population, so the operative question is transportability. The rest of the chapter is about the second of Definition 12.3’s two modes and the conditions under which it collapses into the first.

12.3 Direct transportability

Direct transportability is the free lunch of evidence synthesis: the source number applies to the target unchanged. It is the silent assumption of every unadjusted indirect comparison, of the Bucher subtraction (Theorem 14.1) and of standard network meta-analysis, all of which combine relative effects from different trials as though each were a population-free constant. The constancy of relative effects across trials that these methods require (Definition 14.2) is precisely direct transportability of the relative effect across the trial populations. When does it hold?

Theorem 12.1 (Sufficient conditions for direct transportability) Assume the structural invariance of Definition 12.1, and let \(\mathcal P_1,\mathcal P_2\) be two populations whose covariate supports both contain the support of the conditional effect’s dependence. The marginal relative effect \(d_{ab}\) of Equation 12.4 is directly transportable from \(\mathcal P_1\) to \(\mathcal P_2\), that is \(d_{ab(\mathcal P_1)}=d_{ab(\mathcal P_2)}\), under either of the following sufficient conditions.

1. Equal covariate distributions. \(f_{\mathcal P_1}=f_{\mathcal P_2}\) on \(\mathcal X\).

2. Constant conditional effect on a collapsible scale. The conditional relative effect is constant in the covariates, \(d_{ab}(x)=c\) for all \(x\), and the effect measure is collapsible on scale \(g\) in the sense of Definition 11.6, namely the marginal effect is a weighted average, with weights integrating to one, of the stratum-specific conditional effects.

Proof. For (a): if \(f_{\mathcal P_1}=f_{\mathcal P_2}\) then for each \(t\) the marginal means coincide, \(m_t(\mathcal P_1)=\int\mu_t(x)f_{\mathcal P_1}(x)\,dx=\int\mu_t(x)f_{\mathcal P_2}(x)\,dx=m_t(\mathcal P_2)\), because \(\mu_t\) is the same function in both populations by invariance Equation 12.2. Applying the link \(g\) and differencing, \(d_{ab(\mathcal P_1)}=g(m_b(\mathcal P_1))-g(m_a(\mathcal P_1)) =g(m_b(\mathcal P_2))-g(m_a(\mathcal P_2))=d_{ab(\mathcal P_2)}\).

For (b): collapsibility on scale \(g\) (Definition 11.6) provides, for each population \(\mathcal P\), nonnegative weights \(w_{\mathcal P}(x)\) with \(\int w_{\mathcal P}(x)\,dx=1\) such that the marginal effect is the weighted average of the conditional effects, \[ d_{ab(\mathcal P)}\;=\;\int_{\mathcal X} d_{ab}(x)\,w_{\mathcal P}(x)\,dx . \] This is established for the relevant scales in Lemma 12.1 below. Substituting the hypothesis \(d_{ab}(x)=c\), \[ d_{ab(\mathcal P)}\;=\;\int_{\mathcal X} c\,w_{\mathcal P}(x)\,dx\;=\;c\int_{\mathcal X} w_{\mathcal P}(x)\,dx\;=\;c, \] for every population \(\mathcal P\). In particular \(d_{ab(\mathcal P_1)}=c=d_{ab(\mathcal P_2)}\). \(\square\)

Why this matters. Condition (a) is the silent premise of every unadjusted indirect comparison and of the Bucher subtraction (Chapter 14): they treat a relative effect as a population-free constant, which is exactly (a). Condition (b) is the property that population adjustment (MAIC, STC, ML-NMR) works to engineer when (a) is unavailable. Naming the two tells you precisely what each family of methods is assuming.

Condition (a) is exact but rarely available: it is what holds when two trials happen to enroll identically distributed patients, the idealized situation in which unadjusted indirect comparison is unconditionally valid. Condition (b) is the substantive route and the one population adjustment aims for. It has two ingredients that the rest of the chapter pulls apart: a constant conditional effect, which a shared effect modifier structure will deliver for an anchored contrast,³ and collapsibility of the scale, which some effect measures have and others lack. The contrapositive is the engine of every failure we will meet: if the conditional effect is not constant in some covariate whose distribution differs across the populations, or if the measure is not collapsible, direct transportability can break.

Example 12.2 (Touching numbers early: an identity-scale effect transports) Take the two trial populations of Example 12.1, \(f_{AB}\) with \(P(X=1)=0.1\) and \(f_{AC}\) with \(P(X=1)=0.9\), a continuous outcome, and a treatment \(B\) that adds a constant \(3\) units to the mean at every covariate value, so the conditional effect is \(d_{BC}(x)=3\) at both \(x\) (a constant conditional effect on the identity scale). Condition (b) of Theorem 12.1 applies, because the identity scale is collapsible. Check it in one line: whatever the covariate frequencies, the marginal mean difference is \(\mathbb E_{\mathcal P}[\mu_B(X)-\mu_C(X)]=\mathbb E_{\mathcal P}[3]=3\), the same in \(f_{AB}\), in \(f_{AC}\), and in \(\mathcal P^{*}\). The full arithmetic with specific means is Section 12.8.1. Contrast it with the odds-ratio calculation of Section 12.8.2, where the identical setup on a logit scale produces different marginal effects in the two populations. \(\blacktriangleleft\)

We isolate the collapsibility ingredient, since it is the property that distinguishes the well-behaved measures.

Lemma 12.1 (Which scales are collapsible) Under structural invariance, write \(m_t=m_t(\mathcal P)\) for the marginal means in a population \(\mathcal P\).

1. The identity scale (\(g=\mathrm{id}\), the mean difference and risk difference) is collapsible with population weights: \(d_{ab(\mathcal P)}=m_b-m_a=\mathbb E_{X\sim\mathcal P}[\mu_b(X)-\mu_a(X)] =\int d_{ab}(x)f_{\mathcal P}(x)\,dx\).

2. The log scale (\(g=\log\), the log risk ratio and log rate ratio) is collapsible: with the reference-mean weights \(w_a(x)=\mu_a(x)f_{\mathcal P}(x)/m_a\), which are nonnegative and integrate to one, the marginal risk ratio is the weighted average of the conditional risk ratios, \(m_b/m_a=\int \big(\mu_b(x)/\mu_a(x)\big)\,w_a(x)\,dx\). Consequently a constant conditional risk ratio equals the marginal risk ratio in every population.

3. The logit scale (\(g=\mathrm{logit}\), the log odds ratio) and the log-hazard scale (\(g=\log\) applied to the hazard, the log hazard ratio) are not collapsible: there exist conditional models with a constant conditional effect whose marginal effect differs from that constant, as established in Theorem 11.2 and quantified in Theorem 12.6.

Proof. Part 1 is linearity of expectation: \(m_b-m_a=\int\mu_b f_{\mathcal P}-\int\mu_a f_{\mathcal P} =\int(\mu_b-\mu_a)f_{\mathcal P}=\int d_{ab}(x)f_{\mathcal P}(x)\,dx\), and \(d_{ab}(x)=\mu_b(x)-\mu_a(x)\) on the identity scale.

For part 2, the choice of weights is not arbitrary: \(w_a(x)\) puts mass on covariate stratum \(x\) in proportion to that stratum’s contribution \(\mu_a(x)f_{\mathcal P}(x)\) to the reference-arm risk \(m_a\), so strata where the reference arm has more events count for more, which is exactly the weighting that turns a ratio of averages into an average of ratios. The weights \(w_a(x)=\mu_a(x)f_{\mathcal P}(x)/m_a\) are nonnegative because \(\mu_a\ge 0\) and integrate to one because \(\int\mu_a(x)f_{\mathcal P}(x)\,dx=m_a\). Then \[ \int\frac{\mu_b(x)}{\mu_a(x)}\,w_a(x)\,dx =\frac1{m_a}\int\frac{\mu_b(x)}{\mu_a(x)}\,\mu_a(x)f_{\mathcal P}(x)\,dx =\frac1{m_a}\int\mu_b(x)f_{\mathcal P}(x)\,dx=\frac{m_b}{m_a}, \] so the marginal risk ratio \(m_b/m_a\) is the \(w_a\)-weighted average of the conditional risk ratios \(\mu_b(x)/\mu_a(x)\). If the conditional risk ratio is the constant \(R\), the weighted average of \(R\) is \(R\), hence \(m_b/m_a=R\) and \(\log(m_b/m_a)=\log R\) in every population; this is the collapsibility used in Theorem 12.1(b). Part 3 is the non-collapsibility of the odds ratio and hazard ratio, the subject of Theorem 11.2; the quantitative version, that the marginal log odds ratio is strictly attenuated and population dependent even when the conditional log odds ratio is constant, is proved in Theorem 12.6. \(\square\)

Why this matters. This lemma is the dividing line of the chapter. Mean differences, risk ratios, and rate ratios (identity and log scales) are collapsible, so a constant conditional effect passes to the margin unchanged and transports. Odds ratios and hazard ratios (logit and log-hazard scales) are not, and that single algebraic fact is why the measures HTA reports most are the ones that resist transport.

Remark (Direct collapsibility on the identity scale). Part 1 is the strongest form, sometimes called direct collapsibility: the marginal effect equals the population-average conditional effect with the population’s own weights, \(d_{ab(\mathcal P)} =\mathbb E_{X\sim\mathcal P}[d_{ab}(X)]\). Part 2 is weaker, the weights are tilted toward the reference risk, but it still delivers the one consequence direct transportability needs, that a constant conditional effect passes unchanged to the margin. It is exactly this weaker consequence, not the stronger identity, that the anchored contrast of Section 12.6 will exploit. The reason the weaker property suffices is simple: direct transportability only ever has to move a constant conditional effect to the margin, and a weighted average whose weights sum to one leaves a constant untouched whatever the weights are, so the reference-mean tilt cannot change the average of a constant. \(\blacktriangleleft\)

12.4 Conditional transportability and the transport g-formula

When direct transportability fails, the effect can still be recovered if we are willing to re-standardize the source model to the target covariate distribution. This is conditional transportability, and the identity that implements it is the transport g-formula, the population-level statement that re-weighting the source conditional model to the target distribution recovers the target marginal mean. It is the direct generalization of the g-computation identity of Theorem 10.4 from one population to two.

Theorem 12.2 (The transport g-formula) Let \(\mathcal P\) be a source population, available with outcome data, and \(\mathcal P^{*}\) a target population, available only as a covariate distribution. Assume

1. Consistency (Definition 9.3, Theorem 9.2): \(Y=Y(T)\);

2. Source exchangeability (Definition 9.7): within the source, \(T\perp Y(t)\mid X\), so that \(\mathbb E[Y\mid T=t,X=x,S=\text{src}]=\mu_t(x)\) identifies the conditional mean from source data;

3. Structural invariance (Definition 12.1): \(\mathbb E[Y(t)\mid X,S]=\mu_t(X)\);

4. Positivity and overlap (Definition 9.8): the target covariate support is contained in the source support, \(f_{\mathcal P^{*}}(x)>0\Rightarrow f_{\mathcal P}(x)>0\), and treatment \(t\) has positive probability throughout the source support.

Then the target marginal mean is identified by the transport g-formula \[ m_t(\mathcal P^{*})\;=\;\mathbb E_{X\sim\mathcal P^{*}}\!\left[\,\mathbb E\!\left[\,Y\mid T=t,\,X,\,S=\text{src}\,\right]\right] \;=\;\int_{\mathcal X}\mu_t(x)\,f_{\mathcal P^{*}}(x)\,dx, \tag{12.6}\] and equivalently by the re-weighting identity, with transportability weights \(w(x)=f_{\mathcal P^{*}}(x)/f_{\mathcal P}(x)\), \[ m_t(\mathcal P^{*})\;=\;\mathbb E_{X\sim\mathcal P}\!\left[\,w(X)\,\mu_t(X)\,\right] \;=\;\mathbb E_{X\sim\mathcal P}\!\left[\,w(X)\,\frac{\mathbf 1\{T=t\}\,Y}{e_t(X)}\,\right], \qquad e_t(x)=P(T=t\mid X=x,S=\text{src}). \tag{12.7}\]

Reader translation. The g-formula Equation 12.6 says: to get the target’s average outcome, take the response curve \(\mu_t(\cdot)\) fitted in the source and average it over the target’s covariates. The re-weighting form Equation 12.7 says the same thing a second way: rewrite that target average as a weighted average over the source, where each source patient is up- or down-weighted by \(w(x)=f_{\mathcal P^{*}}(x)/f_{\mathcal P}(x)\), the factor telling you how much more (or less) common their covariate pattern is in the target than in the source. That one sentence is the entire idea behind MAIC.

Reader translation of the overlap condition (iv). “The target support is contained in the source support” means every covariate pattern that occurs in the target must also occur in the source. In Example 12.1, if the decision population \(\mathcal P^{*}\) includes older patients with renal impairment whom the index trial \(f_{AB}\) excluded by protocol, the source model has never seen such patients and cannot predict their outcomes without extrapolation. Transport is identification only where the source has support; beyond it you are guessing.

Proof. We first identify the target marginal mean from target covariates and the shared conditional model. By the law of total expectation applied within the target population, \[ m_t(\mathcal P^{*})=\mathbb E[Y(t)\mid S=\text{tgt}] =\mathbb E_{X\sim\mathcal P^{*}}\!\left[\,\mathbb E[Y(t)\mid X,S=\text{tgt}]\,\right] =\mathbb E_{X\sim\mathcal P^{*}}\!\left[\mu_t(X)\right], \] the last step by structural invariance (iii). Within the source, consistency (i) and source exchangeability (ii) give \(\mathbb E[Y\mid T=t,X=x,S=\text{src}]=\mathbb E[Y(t)\mid T=t,X=x,S=\text{src}] =\mathbb E[Y(t)\mid X=x,S=\text{src}]=\mu_t(x)\), the second equality by (ii) and the third by invariance (iii). Substituting \(\mu_t(x)=\mathbb E[Y\mid T=t,X=x,S=\text{src}]\) into the previous display yields Equation 12.6. The integral form is the definition Equation 12.3 of \(m_t\) applied with the target density; positivity (iv) guarantees the inner conditional expectation is defined on the entire target support, so the integral is well defined.

For the re-weighting identity, change measure from the target density to the source density. In words, we are rewriting an average over the target population as a weighted average over the source, where the weight \(w(x)\) says how much more (or less) likely the covariate value \(x\) is in the target than in the source; this is the entire intuition behind matching-adjusted indirect comparison. Since \(f_{\mathcal P^{*}}(x)=w(x)f_{\mathcal P}(x)\) wherever \(f_{\mathcal P}(x)>0\), and this covers the target support by overlap (iv), \[ \int\mu_t(x)f_{\mathcal P^{*}}(x)\,dx=\int\mu_t(x)\,w(x)\,f_{\mathcal P}(x)\,dx =\mathbb E_{X\sim\mathcal P}\!\left[w(X)\mu_t(X)\right]. \] Finally replace \(\mu_t(X)\) by its inverse-probability representation inside the source. By Theorem 10.3, under (i), (ii), (iv), \(\mathbb E[\mathbf 1\{T=t\}Y/e_t(X)\mid X=x,S=\text{src}]=\mu_t(x)\), because \(\mathbb E[\mathbf 1\{T=t\}Y\mid X=x]=e_t(x)\,\mathbb E[Y\mid T=t,X=x]=e_t(x)\mu_t(x)\). Taking the source conditional expectation of the product \(w(X)\mathbf 1\{T=t\}Y/e_t(X)\) given \(X\) and then averaging over the source covariate distribution, \[ \mathbb E_{X\sim\mathcal P}\!\left[w(X)\frac{\mathbf 1\{T=t\}Y}{e_t(X)}\right] =\mathbb E_{X\sim\mathcal P}\!\left[w(X)\,\mu_t(X)\right]=m_t(\mathcal P^{*}), \] which is the right-hand identity of Equation 12.7. \(\square\)

Why this matters. This is the single identity that every population-adjustment method in the book computes; the methods differ only in whether they estimate the response curve, the weights, or both. Understand this one formula and MAIC, STC, and ML-NMR become three implementations of it.

The transport g-formula is implemented in three ways, each a method studied later in the book. The outcome-regression route fits the source conditional model \(\mu_t(\cdot)\) and averages its predictions over the target covariates, the population-level content of simulated treatment comparison (STC, Chapter 19) and of multilevel network meta-regression (ML-NMR, Chapters 20 to 23). The weighting route estimates the transportability weights \(w(\cdot)\) and forms the weighted source average, the population-level content of matching-adjusted indirect comparison (MAIC, Chapter 18). The doubly robust route combines the two through an augmented estimator that remains consistent if either the outcome model or the weight model is correct, the population-level content of the augmented inverse-probability estimator of Theorem 10.5. The choice among them is a question of efficiency and robustness to misspecification; all three target the same estimand Equation 12.6.

A number makes the weight concrete. Suppose \(X\) is binary with source frequency \(P(X=1)=0.3\) in \(\mathcal P\) and target frequency \(P(X=1)=0.6\) in \(\mathcal P^{*}\). Then the transport weights are \[ w(1)=\frac{f_{\mathcal P^{*}}(1)}{f_{\mathcal P}(1)}=\frac{0.6}{0.3}=2,\qquad w(0)=\frac{f_{\mathcal P^{*}}(0)}{f_{\mathcal P}(0)}=\frac{0.4}{0.7}=0.571, \] so a source patient with \(X=1\), twice as common in the target, counts double, while an \(X=0\) patient counts a little over half. As a check the weights average to one in the source: \(0.7(0.571)+0.3(2)=0.4+0.6=1\). Exercise Exercise 12.4 runs the full re-weighting identity on this setup.

Remark (Normalized weights). The unnormalized re-weighting identity Equation 12.7 uses the density ratio \(w\), whose source mean is \(\mathbb E_{X\sim\mathcal P}[w(X)]=\int (f_{\mathcal P^{*}}/f_{\mathcal P})f_{\mathcal P}=1\). In a finite sample the weights are known only up to a constant, so practice uses the self-normalized, or Hájek,⁴ form \(\hat m_t=\big(\sum_i w_i\mathbf 1\{T_i=t\}Y_i/e_t(X_i)\big)/\big(\sum_i w_i\mathbf 1\{T_i=t\}/e_t(X_i)\big)\), which divides by the realized weight total and so requires only proportional weights. This is the estimator form recorded as catalogue result N.3 and the one MAIC uses. \(\blacktriangleleft\)

12.5 The two-step transport of a pairwise indirect comparison

We can now expose the structure that the introduction promised. A pairwise PAIC does not perform one transport; it performs two, and they require different justifications. The first is conditional transport and is what the method explicitly does. The second is direct transport and is usually left implicit. This is catalogue result N.4 and the organizing insight of Chandler and Ishak (2026).

The setting is an anchored indirect comparison of an active treatment \(B\) against an active treatment \(C\) through a common comparator \(A\). An index trial supplies individual patient data (IPD) on \(A\) and \(B\), drawn from a population with covariate density \(f_{AB}\). A comparator trial supplies aggregate data (AgD) on \(A\) and \(C\), drawn from a population with covariate density \(f_{AC}\). We want the relative effect of \(B\) versus \(C\) in a decision population \(\mathcal P^{*}\) that is, in general, neither of these. Write \(d_{1k}\) for the relative effect of treatment \(k\) versus the reference \(A\) on scale \(g\), so that the target estimand is \(d_{BC(\mathcal P^{*})}=d_{1C(\mathcal P^{*})}-d_{1B(\mathcal P^{*})}\) by the anchoring subtraction of Theorem 14.1.

Before the formal statement, here is the whole maneuver as a picture. A pairwise PAIC carries the comparison across two populations in sequence:

Step	What happens	Moves from	Moves to	Justified by	Estimand after the step
1 (explicit)	model or re-weight the index IPD to the comparator covariate mix	\(f_{AB}\) (index)	\(f_{AC}\) (comparator)	transport g-formula (Theorem 12.2)	\(d_{BC(AC)}\)
2 (implicit)	assume the comparator-population effect also holds in the decision population	\(f_{AC}\) (comparator)	\(\mathcal P^{*}\) (decision)	direct transportability (Theorem 12.1)	\(d_{BC(\mathcal P^{*})}\)

A MAIC or STC analysis performs and documents Step 1; Step 2 it performs silently, by reporting the Step 1 answer as though it were the decision-population answer. The theorem makes this precise.

Theorem 12.3 (The two-step decomposition of a pairwise PAIC) Under structural invariance (Definition 12.1), the assumptions of the transport g-formula (Theorem 12.2) applied to the index trial, and internal validity of the comparator trial, the pairwise PAIC produces the estimand \[ d_{BC(AC)}\;=\;d_{1C(AC)}-d_{1B(AC)}, \qquad d_{1B(AC)}=g\!\Big(\!\int\mu_B(x)f_{AC}(x)\,dx\Big)-g\!\Big(\!\int\mu_A(x)f_{AC}(x)\,dx\Big), \tag{12.8}\] the relative effect of \(B\) versus \(C\) in the comparator population \(f_{AC}\). Consequently, for a decision population \(\mathcal P^{*}\), the discrepancy between the target estimand and the PAIC estimand decomposes as \[ \underbrace{d_{BC(\mathcal P^{*})}-\hat d_{BC(AC)}}_{\text{total error}} \;=\; \underbrace{\big(d_{BC(\mathcal P^{*})}-d_{BC(AC)}\big)}_{\text{Step 2: direct-transport gap}} \;+\; \underbrace{\big(d_{BC(AC)}-\hat d_{BC(AC)}\big)}_{\text{Step 1: estimation/misspecification error}} . \tag{12.9}\] The Step 1 term vanishes in expectation under correct specification and overlap. The Step 2 term vanishes if and only if \(d_{BC}\) is directly transportable from \(f_{AC}\) to \(\mathcal P^{*}\) in the sense of Theorem 12.1.

Why this matters. A standard MAIC or STC report hands the decision maker \(d_{BC(AC)}\), the effect in the comparator trial’s population, whereas the reimbursement question is about \(d_{BC(\mathcal P^{*})}\), the effect in the decision population. Unless the two populations coincide there is a gap, and it is almost never stated in submissions. This is the chapter’s organizing insight.

Proof. Step 1 (conditional transport, index to comparator). The index IPD identify the conditional means \(\mu_A(\cdot)\) and \(\mu_B(\cdot)\) on the support of \(f_{AB}\), by source exchangeability and consistency (Theorem 12.2, hypotheses (i), (ii)). Treating \(f_{AC}\) as the target distribution and invoking the transport g-formula Equation 12.6 with the overlap hypothesis (iv) that the support of \(f_{AC}\) lies in the support of \(f_{AB}\), the marginal means of \(A\) and \(B\) standardized to the comparator population are identified as \(m_A(AC)=\int\mu_A(x)f_{AC}(x)\,dx\) and \(m_B(AC)=\int\mu_B(x)f_{AC}(x)\,dx\). Applying the link and differencing gives \(d_{1B(AC)}\) as displayed. The comparator trial is an internally valid randomized comparison of \(A\) and \(C\) in its own population \(f_{AC}\), so it identifies \(d_{1C(AC)}=g(m_C(AC))-g(m_A(AC))\) directly. Both relative effects are now defined in the same population \(f_{AC}\) and share the reference arm \(A\), so the anchoring subtraction of Theorem 14.1 applies within \(f_{AC}\) and yields Equation 12.8. The shared population is what licenses the subtraction: \(d_{BC}=d_{1C}-d_{1B}\) is a meaningful \(B\) versus \(C\) effect only when both contrasts are defined in the same population and share the reference arm \(A\); here both live in \(f_{AC}\), so their difference is a genuine effect in \(f_{AC}\) rather than a number attached to no population. Thus the population in which the pairwise PAIC estimand is defined is the comparator population, not the decision population.

Step 2 (implicit direct transport, comparator to decision population). The decision-relevant estimand is \(d_{BC(\mathcal P^{*})}\). The PAIC delivers \(d_{BC(AC)}\). Their difference is the Step 2 gap of Equation 12.9, and it is zero exactly when \(d_{BC(AC)}=d_{BC(\mathcal P^{*})}\), which is the definition of direct transportability of \(d_{BC}\) from \(f_{AC}\) to \(\mathcal P^{*}\) (Theorem 12.1). The total error decomposes by inserting and removing \(d_{BC(AC)}\): \(d_{BC(\mathcal P^{*})}-\hat d_{BC(AC)}=(d_{BC(\mathcal P^{*})}-d_{BC(AC)})+(d_{BC(AC)}-\hat d_{BC(AC)})\), which is Equation 12.9. The second bracket is the deviation of the estimator from its estimand, the ordinary Step 1 error from finite samples and from any misspecification of \(\mu_t(\cdot)\) or of the weights; under correct specification, overlap, and the consistency of the weighting or regression estimator it has mean zero. \(\square\)

The decomposition Equation 12.9 is the moral of the chapter in one line. The literature on MAIC and STC overwhelmingly studies the Step 1 term: model misspecification, poor overlap, effective sample size, variance inflation. The Step 2 term is usually invisible, because the estimand it concerns, \(d_{BC(\mathcal P^{*})}\) for a decision population \(\mathcal P^{*}\ne f_{AC}\), is not what the PAIC computes. Standard guidance closes the Step 2 gap by appeal to the shared effect modifier assumption (SEMA, Notation), the assumption that the same covariates modify the effects of \(B\) and \(C\) to the same degree, and asserts that under SEMA the active versus active contrast is “the same in any population.” The next three sections prove that this assertion is correct for collapsible measures and false for the odds ratio and the hazard ratio, so that the Step 2 gap is real precisely for the measures HTA uses most.

Why Step 1 vanishes but Step 2 need not. The Step 1 term \(d_{BC(AC)}-\hat d_{BC(AC)}\) is ordinary estimation error: if the outcome model or the weights are correctly specified, the comparator support overlaps the index support (condition (iv)), and the estimator is consistent (Chapters 18 to 20), then \(\hat d_{BC(AC)}\) converges to \(d_{BC(AC)}\) and the term has mean zero. It is the term the MAIC and STC literature studies in detail. The Step 2 term is not estimation error at all; it is a mismatch of estimands, the gap between the comparator population and the decision population, and no quantity of data from trials \(AB\) and \(AC\) can shrink it. Only knowledge of the decision population’s covariate distribution, together with a transport argument, can.

What this means for a reimbursement submission. Spell the scenario out with three distinct populations. Sponsor 1 holds IPD on its index trial \(AB\) and standardizes to the covariate distribution \(f_{AC}\) of the comparator trial it anchored on, so its reported \(d_{BC(AC)}\) is an effect in that comparator population. Sponsor 2, submitting the same drug elsewhere, anchored on a different comparator trial with a different covariate distribution and so reports an effect in a different population. A national HTA body, meanwhile, wants the effect in its own decision population \(\mathcal P^{*}\), defined by the label and reimbursement criteria, which matches neither comparator trial. For a collapsible measure the three may agree; for the odds ratio or hazard ratio they generally do not, and the disagreement is not anyone’s arithmetic mistake but a consequence of the estimand being tied to a population. A submission that wants to speak to \(\mathcal P^{*}\) must say what \(\mathcal P^{*}\) is in covariate terms and argue the Step 2 transport, not merely report the Step 1 number.

Remark (Why two sponsors can disagree). The two-step view explains a phenomenon often seen and often misattributed to error. Two sponsors comparing the same pair \(B\) versus \(C\) can produce different point estimates because each anchors on a different comparator trial, hence standardizes to a different comparator population \(f_{AC}\), hence reports \(d_{BC(AC)}\) for a different population. For a non-collapsible measure these are genuinely different estimands, not discordant estimates of one estimand. The disagreement is a feature of population dependence, not a symptom of analytic mistake. \(\blacktriangleleft\)

12.6 Collapsibility, scale alignment, and the anchored contrast

We now assemble the conditions under which the Step 2 gap closes. Two ingredients enter beyond SEMA. The first, collapsibility, we have already met. The second, scale alignment, is new and concerns the relationship between the scale on which the model is built and the scale on which the effect is reported. We treat alignment first, then prove the positive result, that a collapsible aligned measure transports under SEMA, and in Section 12.7 the negative result, that a non-collapsible measure does not.

Here is the destination before the derivations. Marginal transport of a relative effect under a shared effect modifier structure turns on three conditions, and each buys one link in a chain:

Condition	What it controls	What it buys
SEMA (shared effect modifiers)	the conditional active-versus-active contrast on the model scale	that contrast is the same constant at every covariate value
Scale alignment (\(h=g\))	model scale versus reporting scale	the constancy survives onto the reported scale
Collapsibility of the measure	how the marginal relates to the conditional	the constant conditional effect survives marginalization, so the marginal effect transports

A collapsible, aligned measure with SEMA (mean difference, risk ratio, rate ratio) gets all three links and transports both conditionally and marginally. The odds ratio and hazard ratio get the first two but fail the third, so they transport conditionally but not marginally. A misaligned pair breaks the second link, so nothing transports. The rest of the section proves each of these; Proposition 12.1 collects the verdicts in one table.

Throughout we use the standard outcome model for an anchored network, the individual-level network meta-regression with reference treatment \(A\). On the linear-predictor scale, the model writes the conditional mean through a link \(g\) as \(g(\mu_k(x))=\eta_k(x)\) with \[ \eta_A(x)=\alpha+x^\top\boldsymbol\beta_1,\qquad \eta_k(x)=\alpha+x^\top\boldsymbol\beta_1+\gamma_k+x^\top\boldsymbol\beta_{2,k}\quad(k=B,C), \tag{12.10}\] where \(\alpha\) is the baseline, \(\boldsymbol\beta_1\) the prognostic (main) covariate effects, \(\gamma_k\) the treatment main effect of \(k\) versus \(A\), and \(\boldsymbol\beta_{2,k}\) the treatment-by-covariate interactions, the effect modification of \(k\) versus \(A\) (Definition 11.3, Definition 11.5, Notation).

Reader translation. The linear predictor \(\eta_k(x)\) is the covariate combination that the link maps the mean to; the \(g\)-scale (or model scale) is the scale on which \(g(\mu)\) is linear in the covariates, here the scale on which model Equation 12.10 is additive. Reading the model left to right for treatment \(k\): start at the baseline \(\alpha\); add the prognostic pull of the covariates \(x^\top\boldsymbol\beta_1\), which shifts everyone, treated or not; add the treatment’s own effect \(\gamma_k\); and finally add the interaction \(x^\top\boldsymbol\beta_{2,k}\), the part of the treatment effect that depends on who the patient is.

The shared effect modifier assumption (SEMA) is the equality of the interaction vectors for the two active treatments, \[ \text{SEMA:}\qquad \boldsymbol\beta_{2,B}=\boldsymbol\beta_{2,C}. \tag{12.11}\]

12.6.1 Scale alignment

The model Equation 12.10 is built on the \(g\)-scale: the linear predictor, and with it SEMA, lives on the scale where \(g(\mu)\) is linear in covariates. The effect we want to report, however, need not live on the same scale. We might fit a proportional-hazards model on the log-hazard scale but report a difference in restricted mean survival time⁵ on the time scale; we might fit a logistic model on the logit scale but report a risk difference on the probability scale. Write \(h\) for the effect-measure transform, so the reported conditional effect of \(b\) versus \(a\) is \(h(\mu_b(x))-h(\mu_a(x))\). Alignment is the agreement of the two scales.

Model scale versus reporting scale (in plain terms). Ask one question: is the effect you plan to report read off the same scale the model was built on? Two aligned pairs, where the answer is yes: a logistic model reported as a log odds ratio, and a proportional-hazards model reported as a log hazard ratio. Two misaligned pairs, where the answer is no: a logistic model reported as a risk difference, and a proportional-hazards model reported as an RMST difference. The definition below names the composition \(h\circ g^{-1}\) that measures the mismatch.

Definition 12.4 (Scale alignment) A model with link \(g\) and an effect measure with transform \(h\) are scale aligned if the composition \(h\circ g^{-1}\) is the identity on the range of the linear predictor, equivalently if \(h=g\) as functions on the mean space. When \(h\circ g^{-1}\) is not the identity the model and the measure are scale misaligned. \(\blacktriangleleft\)

Reader translation. Alignment asks one practical question: did you build the model on the same scale you plan to report the effect on? A logit model reported as an odds ratio is aligned (\(h=g\)). A logit model reported as a risk difference is misaligned (\(h\circ g^{-1}=\mathrm{expit}\), not the identity). When the two scales disagree, the covariate distribution sneaks back into the reported contrast even if the model had a perfectly shared effect modifier structure.

The aligned pairs are the ones where the effect is read directly off the linear predictor: identity link with mean difference, logit link with log odds ratio, log link with log risk ratio, log-hazard link with log hazard ratio. The misaligned pairs are the ones where a transform intervenes: logit link with risk difference, where \(h\circ g^{-1}=\mathrm{expit}\), and log-hazard link with restricted mean survival time, where \(h\circ g^{-1}\) exponentiates the linear predictor and then integrates a survival curve. The following theorem shows that alignment is exactly what makes the conditional effect on the reported scale inherit the constancy that SEMA confers on the linear predictor.

Theorem 12.4 (Scale alignment controls covariate dependence of the reported contrast) Consider the anchored model Equation 12.10 under SEMA Equation 12.11, and an effect measure with transform \(h\). Let \(\psi=h\circ g^{-1}\).

1. Aligned case. If \(h=g\) (so \(\psi=\mathrm{id}\)), the conditional active versus active contrast on the reported scale is constant in the covariates, \[ h(\mu_B(x))-h(\mu_C(x))=\eta_B(x)-\eta_C(x)=\gamma_B-\gamma_C\qquad\text{for all }x. \tag{12.12}\]

2. Misaligned case. If \(\psi=h\circ g^{-1}\) is not affine, then there exist conditional models of the form Equation 12.10 satisfying SEMA for which the conditional reported contrast \(h(\mu_B(x))-h(\mu_C(x))\) is a non-constant function of \(x\). Hence under misalignment a shared effect modifier structure on the model scale does not make the reported effect constant in the covariates, and the marginal reported effect depends on the population.

Why this matters. If you fit on one scale and report on another, a shared effect modifier structure stops protecting you. Alignment (part i) keeps the reported contrast constant across covariates; a mismatch (part ii) manufactures covariate dependence out of nothing, so the reported effect starts to depend on the population even with no interaction in the model at all.

Proof. (i). Under SEMA, subtracting the two linear predictors in Equation 12.10 cancels the baseline, the prognostic term, and, because \(\boldsymbol\beta_{2,B}=\boldsymbol\beta_{2,C}\), the interaction term: \[ \eta_B(x)-\eta_C(x)=(\gamma_B-\gamma_C)+x^\top(\boldsymbol\beta_{2,B}-\boldsymbol\beta_{2,C})=\gamma_B-\gamma_C, \] a constant in \(x\). With \(h=g\), \(h(\mu_k(x))=g(\mu_k(x))=\eta_k(x)\), so the reported contrast equals the linear-predictor contrast, giving Equation 12.12.

(ii). Write the reported conditional means through \(\psi\): since \(\mu_k(x)=g^{-1}(\eta_k(x))\), \(h(\mu_k(x))=h(g^{-1}(\eta_k(x)))=\psi(\eta_k(x))\). By part (i)’s cancellation under SEMA, \(\eta_B(x)=\eta_C(x)+c\) with \(c=\gamma_B-\gamma_C\). Hence the reported conditional contrast is \[ h(\mu_B(x))-h(\mu_C(x))=\psi(\eta_C(x)+c)-\psi(\eta_C(x))=:\Delta\big(\eta_C(x)\big), \qquad \Delta(\eta):=\psi(\eta+c)-\psi(\eta). \] Before the general argument, see it on a concrete \(\psi\). Take \(\psi=\mathrm{expit}\), the logit-model, risk-difference case. Then \(\psi'(\eta)=\mathrm{expit}(\eta)\{1-\mathrm{expit}(\eta)\}\) is largest at \(\eta=0\) and shrinks toward \(0\) as \(|\eta|\) grows, so \(\Delta(\eta)=\mathrm{expit}(\eta+c)-\mathrm{expit}(\eta)\) is large for patients whose linear predictor sits near \(0\) and small for those far out in the tails: the reported risk-difference contrast is bigger for some covariate patterns than others, which is covariate dependence. The general argument formalizes this. Recall that a function is affine exactly when its second derivative is identically zero, so “\(\psi\) is not affine” means \(\psi''\ne 0\) somewhere. Because \(\psi\) is not affine, there is a point \(\eta^{*}\) with \(\psi''(\eta^{*})\ne 0\); by continuity \(\psi''\ne 0\) on a neighborhood, so \(\psi'\) is strictly monotone there, and for a suitable nonzero \(c\) the difference \(\Delta'(\eta)=\psi'(\eta+c)-\psi'(\eta)\ne 0\) for \(\eta\) near \(\eta^{*}\), making \(\Delta\) strictly monotone, hence non-constant, on that neighborhood. Choose the prognostic effects \(\boldsymbol\beta_1\ne\mathbf 0\) and a covariate \(x\) that ranges so that \(\eta_C(x)=\alpha+x^\top\boldsymbol\beta_1+\gamma_C\) takes at least two values inside that neighborhood. Then \(\Delta(\eta_C(x))\) takes at least two distinct values, so the reported conditional contrast is non-constant in \(x\). A non-constant conditional contrast is genuine effect modification on the reported scale, and by the contrapositive of Theorem 12.1(b) the marginal reported effect is not invariant across populations: two populations that place different weight on the covariate values where \(\Delta\) differs yield different population-average reported contrasts, hence different marginal effects. \(\square\)

Part (ii) deserves emphasis because it is counterintuitive: scale misalignment manufactures effect modification. Even a model with a perfectly shared effect modifier structure on its own scale, even one with no interactions at all, will exhibit a covariate-dependent contrast once that contrast is read on a misaligned scale. The covariate enters the reported contrast not through an interaction term but through the prognostic gradient \(x^\top\boldsymbol\beta_1\), which the nonlinear \(\psi\) refuses to cancel. The restricted mean survival time example of Section 12.8 makes this concrete.

12.6.2 The collapsible aligned case transports

We can now state and prove the positive result: under SEMA and scale alignment, a collapsible measure’s marginal active versus active effect is directly transportable. This is the rigorous version of the folklore, and it is correct, with the two qualifications the folklore omits.

Theorem 12.5 (Direct transportability of a collapsible aligned marginal effect) Consider the anchored model Equation 12.10 under structural invariance. Suppose

1. SEMA holds, \(\boldsymbol\beta_{2,B}=\boldsymbol\beta_{2,C}\);

2. scale alignment holds, \(h=g\) (Definition 12.4);

3. the effect measure is collapsible on scale \(g\) (Definition 11.6, Lemma 12.1), for instance the mean difference on the identity scale or the log risk ratio on the log scale.

Then the conditional active versus active contrast is constant, \(d_{BC}(x)=\gamma_B-\gamma_C\) for all \(x\), and the marginal active versus active effect is directly transportable across all populations: \[ d_{BC(\mathcal P)}=\gamma_B-\gamma_C\qquad\text{for every population }\mathcal P . \tag{12.13}\] In particular the Step 2 gap of Theorem 12.3 closes, and the pairwise PAIC estimand \(d_{BC(AC)}\) equals the decision-relevant estimand \(d_{BC(\mathcal P^{*})}\).

Proof. By scale alignment (ii) the reported and model scales coincide, so the conditional contrast on the reported scale is the linear-predictor contrast, which by SEMA (i) and the cancellation of Theorem 12.4(i) equals the constant \(\gamma_B-\gamma_C\): \[ d_{BC}(x)=g(\mu_B(x))-g(\mu_C(x))=\eta_B(x)-\eta_C(x)=\gamma_B-\gamma_C\quad\text{for all }x. \] Now apply Theorem 12.1(b): the conditional effect is the constant \(c=\gamma_B-\gamma_C\) and the measure is collapsible (iii), so the marginal effect equals \(c\) in every population, \(d_{BC(\mathcal P)}=\gamma_B-\gamma_C\), which is Equation 12.13. Since this holds for every population, in particular \(d_{BC(AC)}=\gamma_B-\gamma_C=d_{BC(\mathcal P^{*})}\), so the Step 2 gap \(d_{BC(\mathcal P^{*})}-d_{BC(AC)}\) of Equation 12.9 is zero. \(\square\)

Why this matters. This is the rigorous form of the folklore “a shared effect modifier makes relative effects transportable.” The folklore is true, but only with the two pieces of fine print it omits: the measure must be collapsible and reported on its own scale. For the mean difference, risk ratio, and rate ratio this holds and the Step 2 gap closes; for the odds ratio and hazard ratio it does not.

This is the deepest structural fact of anchored indirect comparison, and it is worth naming the mechanism. Individually, the contrast of \(B\) versus the common comparator \(A\) need not have a constant conditional value: the interaction \(\boldsymbol\beta_{2,B}\) makes \(d_{1B}(x)\) vary with \(x\), and on the log scale even a constant conditional \(B\) versus \(A\) risk ratio is not directly collapsible with population weights. But the difference of the two active versus comparator contrasts cancels the shared interaction, and on a collapsible aligned scale the cancellation survives marginalization. The anchored contrast inherits a transportability that neither of its halves possesses. We record the conditional analog, which needs one fewer hypothesis.

Corollary 12.1 (Direct transportability of the conditional contrast) Under hypotheses (i) SEMA and (ii) scale alignment of Theorem 12.5, but without requiring collapsibility, the population-average conditional active versus active effect is directly transportable: \(\mathbb E_{X\sim\mathcal P}[d_{BC}(X)]=\gamma_B-\gamma_C\) for every population \(\mathcal P\).

Proof. By Theorem 12.4(i) the conditional contrast \(d_{BC}(x)=\gamma_B-\gamma_C\) is constant in \(x\) under SEMA and alignment. Its population average is the average of a constant, \(\mathbb E_{X\sim\mathcal P}[\gamma_B-\gamma_C]=\gamma_B-\gamma_C\), the same in every population. No collapsibility property of the scale is used, because the population-average conditional effect averages the contrast directly rather than transforming averaged means. \(\square\)

Why this matters. Conditional estimands, which ML-NMR and STC can target, transport under just SEMA and alignment; marginal estimands, which MAIC must target, need collapsibility on top. That extra requirement is the transportability advantage of a conditional target. A concrete preview: in the binary example of Section 12.8.2 the conditional log odds ratio is the constant \(c=2\) at both covariate values, so by this corollary it transports; yet the marginal log odds ratio there comes out \(1.175\) in one population and \(0.622\) in the other, so it does not. The corollary explains the first fact; Theorem 12.6 explains the second.

The corollary is the reason that estimands targeting the conditional effect, as ML-NMR and STC can, enjoy a transportability advantage over those targeting the marginal effect, as MAIC must. The conditional effect needs two conditions, SEMA and alignment; the marginal effect, for a non-collapsible measure, needs a third that it cannot have. We turn to that obstruction now.

12.7 Why a shared effect modifier assumption is not enough

For a non-collapsible measure the cancellation of Theorem 12.5 happens on the linear-predictor scale but does not survive the passage to the margin, because marginalization and the link do not commute. The result is that the marginal odds ratio, even with SEMA and alignment in force and the conditional odds ratio held perfectly constant, drifts with the covariate distribution. This is catalogue result N.11, the insufficiency of SEMA, and it rests on the strict convexity that defines non-collapsibility.

Theorem 12.6 (SEMA and alignment are sufficient if and only if the measure is collapsible) Consider the anchored model Equation 12.10 under SEMA Equation 12.11 and scale alignment \(h=g\), so that the conditional active versus active contrast is the constant \(c=\gamma_B-\gamma_C\) (Theorem 12.4(i)). The marginal active versus active effect \(d_{BC(\mathcal P)}\) is directly transportable across all populations if and only if the effect measure is collapsible on scale \(g\).

In particular, for the odds ratio (logit link) and the hazard ratio (log-hazard link), which are non-collapsible (Theorem 11.2), the marginal effect is not directly transportable under SEMA and alignment: there exist populations \(\mathcal P_1\ne\mathcal P_2\) with \(d_{BC(\mathcal P_1)}\ne d_{BC(\mathcal P_2)}\). Concretely, when the conditional log odds ratio is the nonzero constant \(c\) and the reference risk \(\mu_C(X)\) is non-degenerate, the marginal log odds ratio is strictly attenuated, \[ 0<\big|\,\mathrm{logit}\,m_B(\mathcal P)-\mathrm{logit}\,m_C(\mathcal P)\,\big|<|c|, \tag{12.14}\] and its value depends on the distribution of \(\mu_C(X)\).

Reader translation. The odds ratio is a curved (nonlinear) summary of risks. Averaging risks across a population and then forming the odds ratio is not the same as forming the odds ratio in each stratum and then averaging; the curvature opens a gap, and the size of the gap depends on how spread out the risks are, which is a property of the population. So even with the conditional odds ratio pinned to a constant, the marginal odds ratio drifts as the covariate distribution changes. The proof turns this into an exact strict inequality using Jensen’s inequality.

Proof. Sufficiency (collapsible \(\Rightarrow\) transportable). This is Theorem 12.5: with the conditional contrast constant at \(c\) and the measure collapsible, \(d_{BC(\mathcal P)}=c\) for every \(\mathcal P\).

Necessity (non-collapsible \(\Rightarrow\) not transportable), shown for the odds ratio. Work on the logit scale, \(g=h=\mathrm{logit}\). SEMA and alignment give a constant conditional log odds ratio \(\mathrm{logit}\,\mu_B(x)-\mathrm{logit}\,\mu_C(x)=c\) for all \(x\), equivalently \[ \mu_B(x)=\varphi\big(\mu_C(x)\big),\qquad \varphi(p):=\mathrm{expit}\big(\mathrm{logit}\,p+c\big)=\frac{e^{c}p}{1+(e^{c}-1)p}, \] the map that adds \(c\) on the logit scale. We show \(\varphi\) is strictly concave on \((0,1)\) when \(c>0\). Writing \(k=e^{c}>1\), \(\varphi(p)=kp/(1+(k-1)p)\), so \[ \varphi'(p)=\frac{k}{\big(1+(k-1)p\big)^{2}}>0,\qquad \varphi''(p)=\frac{-2k(k-1)}{\big(1+(k-1)p\big)^{3}}<0, \] since \(k>1\) makes the numerator of \(\varphi''\) negative and the denominator positive on \((0,1)\). Thus \(\varphi\) is strictly increasing and strictly concave. Now compute the marginal means. With \(m_C=\mathbb E_{X\sim\mathcal P}[\mu_C(X)]\) and \(m_B=\mathbb E_{X\sim\mathcal P}[\mu_B(X)] =\mathbb E_{X\sim\mathcal P}[\varphi(\mu_C(X))]\), strict Jensen’s inequality (Theorem 4.6) applied to the strictly concave \(\varphi\) and the non-degenerate random variable \(\mu_C(X)\) gives a strict inequality \[ m_B=\mathbb E\big[\varphi(\mu_C(X))\big]<\varphi\big(\mathbb E[\mu_C(X)]\big)=\varphi(m_C)=\mathrm{expit}\big(\mathrm{logit}\,m_C+c\big). \] Applying the strictly increasing logit to both sides, \(\mathrm{logit}\,m_B<\mathrm{logit}\,m_C+c\), that is, the marginal log odds ratio satisfies \(\mathrm{logit}\,m_B-\mathrm{logit}\,m_C<c\). For the lower bound, \(c>0\) and \(\varphi(p)>p\) on \((0,1)\) give \(m_B=\mathbb E[\varphi(\mu_C(X))]>\mathbb E[\mu_C(X)]=m_C\), hence \(\mathrm{logit}\,m_B>\mathrm{logit}\,m_C\) and the marginal log odds ratio is positive. Together these are the strict attenuation Equation 12.14 (the case \(c<0\) is symmetric, with \(\varphi\) strictly convex).

Finally, the marginal log odds ratio depends on the covariate distribution. Take \(\mathcal P_1\) degenerate at a single covariate value \(x_0\): then \(\mu_C(X)=\mu_C(x_0)\) is constant, Jensen holds with equality, and \(\mathrm{logit}\,m_B-\mathrm{logit}\,m_C=c\). Take \(\mathcal P_2\) any non-degenerate distribution of \(\mu_C(X)\): then the strict inequality gives \(\mathrm{logit}\,m_B-\mathrm{logit}\,m_C<c\). Hence \(d_{BC(\mathcal P_1)}=c\ne d_{BC(\mathcal P_2)}\), so \(d_{BC}\) is not directly transportable, proving necessity for the odds ratio. The hazard ratio is non-collapsible by the same mechanism, with the mixture-of-survival argument of Theorem 11.2 playing the role of strict Jensen; we do not repeat it. \(\square\)

Why this matters. This is why two HTA submissions built on the very same trials can report different odds ratios and both be correct: SEMA makes the conditional odds ratio portable, but HTA usually reports the marginal odds ratio, and that one is not portable. The attenuation formula Equation 12.14 is not a rounding artifact; it is structural, and it says the reported number belongs to the population it was averaged in.

Why prove necessity, not just show an example? A single counterexample would establish only that the marginal odds ratio can fail to transport. Necessity says something stronger: under SEMA and alignment, for a non-collapsible measure there is no configuration, no clever choice of constants, that rescues marginal transport, because the obstruction is the curvature of \(\varphi\) itself, which SEMA cannot touch. That is the difference between “sometimes fails” and “must fail,” and it is why the result is stated as an if and only if.

The proof locates the obstruction precisely. The strict concavity of the logit-shift map \(\varphi\) is the analytic essence of odds-ratio non-collapsibility, and it is what forces a strict Jensen gap between the marginal mean \(m_B\) and the value \(\varphi(m_C)\) that a collapsible measure would have produced. The size of that gap is governed by the spread of \(\mu_C(X)\), which is a property of the covariate distribution, so the marginal log odds ratio is a function of the population. There is no choice of \(c\), no shared effect modifier structure, that removes the curvature of \(\varphi\). SEMA controls the conditional effect; it has no purchase on the marginalization.

Collecting Theorem 12.5, Corollary 12.1, and Theorem 12.6 gives the clean characterization, catalogue results N.8 to N.10.

Proposition 12.1 (Transportability of common effect measures under SEMA) Assume SEMA and structural invariance. For each effect measure below, with its natural model link, the conditional active versus active effect is directly transportable when SEMA and scale alignment hold, and the marginal active versus active effect is directly transportable when in addition the measure is collapsible. The resulting status is:

Effect measure	Model link \(g\)	Reported scale \(h\)	Aligned?	Collapsible?	Conditional transports?	Marginal transports?
Mean difference	identity	identity	yes	yes	yes	yes
Risk difference	identity	identity	yes	yes	yes	yes
Log risk ratio	log	log	yes	yes	yes	yes
Log rate ratio	log	log	yes	yes	yes	yes
RMST difference (RMST-scale SEMA)	identity (RMST)	identity	yes	yes	yes	yes
Log odds ratio	logit	logit	yes	no	yes	no
Log hazard ratio	log-hazard	log-hazard	yes	no	yes	no
Risk difference (from logit model)	logit	identity	no	yes	no	no
RMST difference (log-hazard SEMA)	log-hazard	identity	no	yes	no	no

Proof. Each row is an instance of the three theorems. A “yes” in the conditional column is Corollary 12.1, valid whenever the measure is scale aligned (so that SEMA makes the conditional contrast constant). A “yes” in the marginal column additionally requires collapsibility, by Theorem 12.5; a “no” in the marginal column with alignment present is the non-collapsibility obstruction of Theorem 12.6 (odds ratio, hazard ratio rows). The two misaligned rows have a “no” already in the conditional column by Theorem 12.4(ii): the reported contrast is covariate dependent even under SEMA, so neither the conditional nor the marginal effect transports, regardless of collapsibility. \(\square\)

Why this matters. This one table is the chapter’s practical output: for any effect measure it tells you whether you may carry the number to a new population unchanged. The two rows HTA cares about most, the log odds ratio and the log hazard ratio, sit in the “no” column for marginal transport. Read the table before trusting any indirect-comparison number that has crossed populations.

The table exhibits the paradox in its starkest form. The mean difference, risk ratio, and rate ratio, all collapsible and naturally aligned, transport both conditionally and marginally under SEMA. The log odds ratio and log hazard ratio, the two effect measures that dominate HTA, transport conditionally under SEMA but not marginally, because non-collapsibility breaks the third condition. And a risk difference computed from a logistic model, or a restricted mean survival time computed from a proportional-hazards model, fails to transport at all, because scale misalignment breaks the first. The measures we report most are the measures that travel least.

12.8 Worked examples

We exhibit all three phenomena with a single covariate structure, a binary effect modifier \(X\in\{0,1\}\) distributed differently in two populations, \(\mathcal P_1\) with \(P(X=1)=0.1\) and \(\mathcal P_2\) with \(P(X=1)=0.9\). In every example SEMA holds and the scales are arranged so that the conditional active versus active effect, \(B\) versus \(C\), is constant across \(x\). The question is whether the marginal effect is the same in \(\mathcal P_1\) and \(\mathcal P_2\). The continuous case says yes; the binary and survival cases say no, for the two distinct reasons of Theorem 12.6 and Theorem 12.4(ii).

12.8.1 Continuous outcome: the mean difference transports

Let the outcome be continuous with an identity-link model and a binary effect modifier. Take conditional means under the comparator \(C\) equal to \(\mu_C(0)=2\), \(\mu_C(1)=7\), and a constant active versus active benefit of \(\gamma_B-\gamma_C=3\) on the identity scale, so \(\mu_B(x)=\mu_C(x)+3\), giving \(\mu_B(0)=5\), \(\mu_B(1)=10\). SEMA holds because the active versus active contrast carries no interaction; the conditional mean difference is \(d_{BC}(x)=\mu_B(x)-\mu_C(x)=3\) for both \(x\). The identity scale is aligned and collapsible, so Theorem 12.5 predicts the marginal mean difference equals \(3\) in every population. Verify by hand. In \(\mathcal P_1\), \[ m_C(\mathcal P_1)=0.9(2)+0.1(7)=2.5,\quad m_B(\mathcal P_1)=0.9(5)+0.1(10)=5.5,\quad d_{BC(\mathcal P_1)}=5.5-2.5=3. \] In \(\mathcal P_2\), \[ m_C(\mathcal P_2)=0.1(2)+0.9(7)=6.5,\quad m_B(\mathcal P_2)=0.1(5)+0.9(10)=9.5,\quad d_{BC(\mathcal P_2)}=9.5-6.5=3. \] The marginal mean difference is \(3\) in both populations, as it must be, since \(d_{BC(\mathcal P)}=\mathbb E_{\mathcal P}[\mu_B(X)-\mu_C(X)]=\mathbb E_{\mathcal P}[3]=3\) for any covariate distribution. The collapsible aligned measure transports.

What a researcher would report. The mean difference is \(3\) units, and it is the same \(3\) in the index population \(f_{AB}\), the comparator population \(f_{AC}\), and the decision population \(\mathcal P^{*}\). For this collapsible, aligned measure the analyst can report a single number and attach it to any of the three populations without qualification; nothing goes wrong when it is carried to the decision population. This is the best case, and it is why mean differences and risk ratios are the easy measures for indirect comparison.

12.8.2 Binary outcome: the marginal odds ratio does not transport

Keep the same binary \(X\) and the same two populations, but now a binary outcome with a logit-link model. On the logit scale set the comparator linear predictors \(\eta_C(0)=-3\), \(\eta_C(1)=3\) (a strong prognostic gradient: the covariate raises the log-odds by \(6\)), and a constant active versus active log odds ratio \(c=\gamma_B-\gamma_C=2\), so \(\eta_B(x)=\eta_C(x)+2\), giving \(\eta_B(0)=-1\), \(\eta_B(1)=5\). SEMA holds and the logit scale is aligned, so the conditional log odds ratio is the constant \(c=2\), an odds ratio of \(e^{2}=7.389\), at both covariate values. The risks \(\mu_k(x)=\mathrm{expit}(\eta_k(x))\) are \[ \mu_C(0)=0.047426,\quad \mu_C(1)=0.952574,\quad \mu_B(0)=0.268941,\quad \mu_B(1)=0.993307, \] each \(\mathrm{expit}(z)=1/(1+e^{-z})\) computed to six places. Now marginalize. In \(\mathcal P_1\), \(P(X=1)=0.1\): \[ m_C(\mathcal P_1)=0.9(0.047426)+0.1(0.952574)=0.137941,\qquad m_B(\mathcal P_1)=0.9(0.268941)+0.1(0.993307)=0.341378. \] The marginal log odds ratio is \[ d_{BC(\mathcal P_1)}=\mathrm{logit}(0.341378)-\mathrm{logit}(0.137941)=(-0.65716)-(-1.83250)=1.17534, \] a marginal odds ratio of \(e^{1.17534}=3.239\). In \(\mathcal P_2\), \(P(X=1)=0.9\): \[ m_C(\mathcal P_2)=0.1(0.047426)+0.9(0.952574)=0.862060,\qquad m_B(\mathcal P_2)=0.1(0.268941)+0.9(0.993307)=0.920871, \] \[ d_{BC(\mathcal P_2)}=\mathrm{logit}(0.920871)-\mathrm{logit}(0.862060)=2.45423-1.83250=0.62173, \] a marginal odds ratio of \(e^{0.62173}=1.862\). The conditional odds ratio is \(7.389\) in both populations, yet the marginal odds ratio is \(3.24\) in \(\mathcal P_1\) and \(1.86\) in \(\mathcal P_2\). SEMA holds, the scale is aligned, the conditional effect is pinned to a constant, and still the marginal effect nearly halves when the covariate distribution shifts. Both marginal values lie strictly between \(1\) (log odds ratio \(0\)) and \(7.389\) (log odds ratio \(2\)), the strict attenuation Equation 12.14 of Theorem 12.6, and they differ from each other because the spread of \(\mu_C(X)\) differs between the populations. This single calculation is the chapter’s central message in numbers: a marginal odds ratio is not a property the treatments carry from population to population.

What a researcher would report, and what would go wrong. A MAIC or STC standardizing the index trial to the comparator population \(f_{AC}\) (here \(P(X=1)=0.9\)) would report a marginal odds ratio of about \(1.86\). That number is correct for \(f_{AC}\). Carried unchanged to a younger decision population like \(f_{AB}\) (here \(P(X=1)=0.1\)) it is simply wrong: the right marginal odds ratio there is \(3.24\). Nothing was mis-estimated; the number was moved to a population it does not describe. A report should therefore state which population its odds ratio belongs to, and either argue that the decision population matches it or redo the standardization for the decision population.

Remark (What a degenerate population shows). The proof of Theorem 12.6 predicts that a population concentrated at one covariate value recovers the conditional value. Check it: at the point mass \(X\equiv 0\), the marginal log odds ratio is \(\mathrm{logit}(\mu_B(0))-\mathrm{logit}(\mu_C(0))=\eta_B(0)-\eta_C(0)=(-1)-(-3)=2=c\), and at \(X\equiv 1\) it is \(\eta_B(1)-\eta_C(1)=5-3=2=c\). Only when the covariate genuinely varies, as in \(\mathcal P_1\) and \(\mathcal P_2\), does the Jensen gap open and the marginal effect attenuate below \(c\) and become population dependent. \(\blacktriangleleft\)

12.8.3 Time-to-event outcome: scale misalignment breaks transport

Finally a survival outcome modeled by proportional hazards, where the effect is reported as a difference in restricted mean survival time (RMST). This is the misaligned case of Theorem 12.4(ii): SEMA is imposed on the log-hazard scale, but RMST lives on the time scale, and \(h\circ g^{-1}\) is not the identity. Take an exponential baseline so the integrals are elementary. Let the comparator hazards at the two covariate values be \(\lambda_C(0)=0.5\) and \(\lambda_C(1)=2.0\), and let \(B\) halve the hazard, a constant hazard ratio of \(0.5\), so \(\lambda_B(x)=0.5\,\lambda_C(x)\), giving \(\lambda_B(0)=0.25\), \(\lambda_B(1)=1.0\). On the log-hazard scale the active versus active contrast is \(\log 0.5=-0.69315\) at both covariate values: SEMA holds and the conditional hazard ratio is the constant \(0.5\). Restrict at time \(\tau=2\). For an exponential hazard \(\lambda\) the survival is \(S(t)=e^{-\lambda t}\) and the RMST is \[ \mathrm{RMST}(\lambda)=\int_0^{\tau}e^{-\lambda t}\,dt=\frac{1-e^{-\lambda\tau}}{\lambda}. \] Watch for the following, because it is the whole point and it happens before any marginalization: the conditional hazard ratio is a constant \(0.5\) at both covariate values, yet we are about to find that the conditional RMST difference is already unequal at \(x=0\) and \(x=1\). The misaligned scale has made the reported conditional contrast covariate dependent on its own, with no help from the covariate distribution. In the binary example the conditional contrast stayed constant and only marginalization broke transport; here the breakage is visible one step earlier. Compute the four conditional RMSTs: \[ \begin{aligned} \mathrm{RMST}_C(0)&=\tfrac{1-e^{-1.0}}{0.5}=1.264241, & \mathrm{RMST}_B(0)&=\tfrac{1-e^{-0.5}}{0.25}=1.573877,\\ \mathrm{RMST}_C(1)&=\tfrac{1-e^{-4.0}}{2.0}=0.490842, & \mathrm{RMST}_B(1)&=\tfrac{1-e^{-2.0}}{1.0}=0.864665. \end{aligned} \] The conditional RMST difference is therefore \[ d_{BC}(0)=1.573877-1.264241=0.309636,\qquad d_{BC}(1)=0.864665-0.490842=0.373823, \] not constant in \(x\), even though the conditional hazard ratio is the constant \(0.5\). This is exactly Theorem 12.4(ii): the nonlinear map from log-hazard to RMST refuses to cancel the covariate, which enters through the prognostic gradient \(\lambda_C(0)\ne\lambda_C(1)\). Because the conditional RMST difference varies in \(x\), the marginal RMST difference depends on the population. The RMST difference is collapsible, so the marginal is the population-average of the conditional (each marginal RMST is the average of the conditional survival curves integrated, by Fubini), and we average: \[ d_{BC(\mathcal P_1)}=0.9(0.309636)+0.1(0.373823)=0.316055,\qquad d_{BC(\mathcal P_2)}=0.1(0.309636)+0.9(0.373823)=0.367404. \] The marginal RMST difference is \(0.316\) in \(\mathcal P_1\) and \(0.367\) in \(\mathcal P_2\): a collapsible measure that still fails to transport, because the misaligned scale has manufactured effect modification on the reported scale. The contrast with the binary example is instructive. There, the conditional effect was constant and the failure came from non-collapsibility of the marginalization (Theorem 12.6); here, the measure is collapsible and the marginalization is innocent, but the conditional effect is itself covariate dependent because of scale misalignment (Theorem 12.4). Two different theorems, two different mechanisms, the same operational conclusion: the effect is comparator-population specific and the Step 2 transport of Theorem 12.3 is not free.

What a researcher would report. An RMST difference standardized to \(f_{AC}\) is about \(0.367\) units of extra event-free time; standardized to \(f_{AB}\) it is about \(0.316\). Even though RMST is collapsible, the number is population-specific, because the proportional-hazards model was fitted on the log-hazard scale while the effect is reported on the time scale. Reporting an RMST difference from a hazard model therefore carries the same “which population?” caveat as an odds ratio, for a different underlying reason.

12.9 Notes and references

The estimand-based theory of transportability for population-adjusted indirect comparisons developed here is that of Chandler and Ishak (2026), which reframes MAIC, STC, and their kin as solving a transportability problem and isolates the three conditions, SEMA, scale alignment, and collapsibility, that govern direct transportability of a marginal relative effect. The chapter expands every result of that paper’s estimand-based core into a full proof: Theorem 12.2 (the transport g-formula), Theorem 12.3 (the two-step decomposition and bias split), Theorem 12.5 and Corollary 12.1 (the positive transport results), Theorem 12.4 (the role of scale alignment), and Theorem 12.6 (the insufficiency of SEMA for non-collapsible marginal effects). These correspond, in order, to catalogue results N.3, N.4, N.6 with N.8, N.9, N.7, and N.8 with N.11 in proofs/N_transportability.md; Definition 12.2 and Definition 12.3 formalize N.1 and N.2, Lemma 12.1 and Proposition 12.1 cover N.5 and N.10, and the HTA discussion threaded through Section 12.5 and Section 12.7 is N.12.

The identification scaffolding is from Hernán and Robins (2020), whose chapters on transportability use a population indicator \(S\) and a transport g-formula identical in structure to Equation 12.6; the within-population identification it rests on, Theorem 9.2, Definition 9.7, Definition 9.8, Theorem 10.4, and Theorem 10.3, is proved in Chapters 9 and 10, and the doubly robust augmentation alluded to after Theorem 12.2 is Theorem 10.5. The collapsibility facts, Definition 11.6, Definition 7.7, Example 7.5, and the non-collapsibility of the odds ratio and hazard ratio Theorem 11.2, are from Chapters 7 and 11; the strict-concavity argument used in Theorem 12.6 is the quantitative form of that non-collapsibility and rests on Jensen’s inequality Theorem 4.6. The constancy of relative effects that unanchored and standard indirect comparisons assume, Definition 14.2, is direct transportability of the relative effect in the sense of Theorem 12.1, and its failure under effect modification is the subject of Theorem 16.1 and the ecological bias of Definition 16.2 in Chapter 16. The shared effect modifier assumption itself is developed as part of the population adjustment theory of Chapter 17; here it is used as a stated structural condition, Equation 12.11, and the methods that operationalize the transport g-formula, MAIC, STC, and ML-NMR, are the subjects of Chapters 18 to 23. The current NICE Decision Support Unit and EU joint guidance that treat SEMA as sufficient for transport are documented in Phillippo et al. (2016); Theorem 12.6 is the precise sense in which that treatment is incomplete for non-collapsible marginal effects, and Phillippo et al. (2020) is the modeling framework that makes the target population, and hence the Step 2 transport, explicit.

Machine-checked status. In one sentence for readers outside formal methods: none of this chapter’s own results have yet been checked by the companion computer-proof system, and this paragraph only records which neighboring facts have been. None of this chapter’s owned results has a counterpart in the ITC_Coq development. The Coq tree formalizes the algebraic core of indirect comparison, Bucher’s identity and unbiasedness in theories/Bucher.v, propensity-score balancing, inverse-probability and doubly robust estimation, and the anchored cancellation of prognostic variables in theories/EffectModifiers.v, but the transportability propositions here, which combine SEMA, collapsibility, and scale alignment across populations, are pen-and-paper by design, as the companion manuscript notes. The closest machine-checked relative is the collapsibility of the linear-model effect, Lemma STC_collapse_linear in theories/STC.v, which proves the identity-scale case of Lemma 12.1(1) from linearity of expectation; it is owned and tagged by Chapter 19. No tag is added to any result in this chapter.

12.10 Exercises

A starred exercise, marked \(\star\), has a full worked solution in Appendix F. The unstarred exercises are arranged from applied identification and interpretation, through computation, to proof, so a reader can stop at the level that suits them.

Exercise 12.1 (Warm-up, interpretation) Name the three populations. A sponsor submits an indirect comparison of drug \(B\) against drug \(C\), both anchored on a common comparator \(A\). The sponsor owns individual patient data from a trial of \(A\) versus \(B\) that enrolled mostly younger, biologic-naive patients, and uses published aggregate results from a trial of \(A\) versus \(C\) that enrolled mostly older, biologic-experienced patients. The national HTA body must decide about all label-eligible patients, a mix of both. (a) Identify, in the language of this chapter, the source population, the comparator population, and the decision population, and say which supplies IPD and which supplies AgD. (b) Using Theorem 12.3, state which population the standard pairwise PAIC estimand \(d_{BC(AC)}\) is attached to. (c) In one sentence, say what extra information the sponsor would need before claiming the result applies to the decision population.

Exercise 12.2 (Interpretation) Reading a MAIC report. A submitted MAIC reports a marginal odds ratio for \(B\) versus \(C\) “standardized to the comparator trial population,” and a reviewer asks whether it applies to the national decision population. (a) Using the two-step decomposition Equation 12.9, explain which of the two terms the MAIC has controlled and which it has left untouched. (b) Explain, citing Theorem 12.6, why “the shared effect modifier assumption holds, so the odds ratio is the same in any population” is not a valid answer for a marginal odds ratio. (c) State one condition on the two populations under which the reported number would in fact apply unchanged to the decision population.

Exercise 12.3 (Computation) Direct transportability of a constant conditional effect. Suppose the conditional mean difference is the constant \(\tau(x)=4\) for all \(x\), and a covariate \(X\) has mean \(\bar x_1=0\) in population \(\mathcal P_1\) and \(\bar x_2=10\) in population \(\mathcal P_2\). Using Theorem 12.1, show that the marginal mean difference is \(4\) in both populations, and explain why the difference in covariate means is irrelevant. Then show by counterexample that the same conclusion fails for the marginal odds ratio when the model is logistic with the same constant conditional log odds ratio.

Exercise 12.4 (Computation) Transport g-formula and weights. Let \(X\in\{0,1\}\) with source probabilities \(P_{\mathcal P}(X=1)=0.3\) and target probabilities \(P_{\mathcal P^{*}}(X=1)=0.6\), and let the source conditional means under treatment \(t\) be \(\mu_t(0)\) and \(\mu_t(1)\). (a) Write the transportability weights \(w(0),w(1)\) as density ratios and verify \(\mathbb E_{X\sim\mathcal P}[w(X)]=1\). (b) Show by direct computation that the re-weighting identity Equation 12.7 and the g-formula Equation 12.6 give the same target marginal mean \(m_t(\mathcal P^{*})\). (c) Explain what the overlap condition (iv) of Theorem 12.2 requires here and what goes wrong if \(P_{\mathcal P}(X=1)=0\).

Exercise 12.5 (Interpretation) The two sponsors. Two sponsors each compare \(B\) versus \(C\) anchored on \(A\), using comparator trials with covariate distributions \(f_{AC}^{(1)}\) and \(f_{AC}^{(2)}\) that differ. The conditional log odds ratio of \(B\) versus \(C\) is the constant \(c=0.7\) (SEMA and alignment hold). Using Theorem 12.3 and Theorem 12.6, explain why the two sponsors’ marginal log odds ratios \(d_{BC(AC^{(1)})}\) and \(d_{BC(AC^{(2)})}\) can differ, identify which of the two bias terms in Equation 12.9 is responsible, and state what additional information would be needed to transport either estimate to a third, decision-relevant population.

Exercise 12.6 (Computation) Risk ratio transports where the odds ratio does not. Consider a log-link binomial model with a binary \(X\), constant conditional log risk ratio \(\log R\) for \(B\) versus \(C\), and reference risks \(\mu_C(0)=0.05\), \(\mu_C(1)=0.20\). (a) Show that the marginal risk ratio equals \(R\) in every population, using Lemma 12.1(2). (b) Take \(R=1.5\) and the two populations \(P(X=1)=0.1\) and \(P(X=1)=0.9\); compute the marginal risks under \(B\) and \(C\) and confirm the marginal risk ratio is \(1.5\) in both. (c) Contrast this with Section 12.8.2 and explain in one sentence why the odds ratio behaves differently.

Exercise 12.7 (\(\star\), proof) Strict attenuation of the marginal odds ratio. Let the conditional log odds ratio be the constant \(c>0\), so \(\mu_B(x)=\varphi(\mu_C(x))\) with \(\varphi(p)=\mathrm{expit}(\mathrm{logit}\,p+c)\). (a) Prove that \(\varphi\) is strictly increasing and strictly concave on \((0,1)\) by computing \(\varphi'\) and \(\varphi''\). (b) Using Jensen’s inequality Theorem 4.6, prove the strict attenuation \(0<\mathrm{logit}\,m_B-\mathrm{logit}\,m_C<c\) whenever \(\mu_C(X)\) is non-degenerate. (c) Show that the attenuation gap \(c-(\mathrm{logit}\,m_B-\mathrm{logit}\,m_C)\) increases as the spread of \(\mu_C(X)\) increases, in the sense that it is zero for a point mass and positive otherwise, and use this to conclude that the marginal log odds ratio is population dependent. (d) Verify (b) numerically on the \(\mathcal P_1,\mathcal P_2\) of Section 12.8.2.

Exercise 12.8 (\(\star\), computation and proof) Scale misalignment manufactures effect modification. Consider a logistic model (\(g=\mathrm{logit}\)) with no interactions at all, \(\eta_k(x)=\alpha+x^\top\boldsymbol\beta_1+\gamma_k\), so that SEMA holds trivially, and report the effect as a risk difference (\(h=\mathrm{id}\)). (a) Show \(h\circ g^{-1}=\mathrm{expit}\) and that the conditional risk difference \(\mathrm{expit}(\eta_B(x))-\mathrm{expit}(\eta_C(x))\) is non-constant in \(x\) whenever \(\boldsymbol\beta_1\ne\mathbf 0\), instantiating Theorem 12.4(ii). (b) With scalar \(x\in\{0,1\}\), \(\alpha=-2\), \(\beta_1=3\), \(\gamma_C=0\), \(\gamma_B=1.5\), compute the conditional risk differences at \(x=0\) and \(x=1\) and the marginal risk differences in \(P(X=1)=0.2\) and \(P(X=1)=0.8\). (c) Explain why this failure occurs even though the risk difference is a collapsible measure.

Exercise 12.9 (Proof) RMST transport under an RMST-scale assumption. The last row of Proposition 12.1 claims that an RMST difference transports if SEMA is imposed on the RMST (time) scale rather than the log-hazard scale. (a) State precisely what “RMST-scale SEMA” means for the conditional RMST difference \(d_{BC}(x)\). (b) Prove that under it the marginal RMST difference is directly transportable, citing Theorem 12.5 and the collapsibility of the RMST difference. (c) Explain why this structural assumption is restrictive in practice: what would have to be true of the baseline hazard and the covariate effects for a proportional-hazards model to also satisfy RMST-scale SEMA?

Exercise 12.10 (Interpretation) The conditional advantage. Using Corollary 12.1 and Theorem 12.6, explain why an analysis targeting the population-average conditional log odds ratio needs only SEMA and scale alignment to transport, while one targeting the marginal log odds ratio additionally needs collapsibility, which the odds ratio lacks. Then state the scientific cost of preferring the conditional estimand: what question does it answer, and why is it not interchangeable with the marginal estimand a decision maker may actually want?

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.

Hernán, Miguel A., and James M. Robins. 2020. Causal Inference: What If. Chapman & Hall/CRC.

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 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.

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1. Recall the two inverse maps of logistic regression: \(\mathrm{logit}(p)=\log\{p/(1-p)\}\) sends a probability in \((0,1)\) to a real number, and its inverse \(\mathrm{expit}(z)=1/(1+e^{-z})\) sends a real number back to a probability. Both are used throughout this chapter.

2. Labels such as N.1, N.3, and N.4 point to numbered results in the companion proof catalogue for this Part; they let the formal development be cross-checked against the chapter. A first-time reader can safely ignore the codes.

3. An anchored indirect comparison of \(B\) versus \(C\) runs through a comparator \(A\) present in both trials, so study-level factors that shift both arms cancel; an unanchored comparison pits single arms against each other with no shared comparator and must assume far more. Anchoring is developed in Chapter 14.

4. Named after the statistician Jaroslav Hájek; “self-normalized” means dividing by the sum of the weights, so only the relative weights matter and any unknown overall scale cancels.

5. Restricted mean survival time up to a horizon \(\tau\) is the expected event-free time in \([0,\tau]\), equivalently the area under the survival curve out to \(\tau\). An RMST difference is measured in units of time and is often easier for patients and payers to interpret than a hazard ratio.