26  Multilevel Unanchored Meta-Regression

26.1 Introduction

Every method in this book until now has needed a bridge. The anchored indirect comparison of Chapter 14 subtracts two relative effects that share a common comparator \(C\), and is exact only when the \(C\)-anchored contrasts are transitive (Theorem 14.1); Chapter 16 showed that an effect modifier distributed differently across the two trials breaks transitivity and that no manipulation of aggregate data can repair it (Theorem 16.1). Population adjustment in Part IV rebuilt one trial’s population in the image of the other, but matching-adjusted indirect comparison (MAIC, Chapter 18) and simulated treatment comparison (STC, Chapter 19) still reported their answer in the comparator population and, in their anchored form, still leaned on the shared comparator arm. Chapters 20 to 25 then built multilevel network meta-regression (ML-NMR), which fits one individual-level model across a connected network of individual patient data (IPD) and aggregate data (AgD) studies, integrates it over any covariate distribution, and so reports a contrast in any target population (Theorem 23.1). The connectivity was the last bridge standing: ML-NMR still required a path of shared treatments linking every study to every other.

This chapter removes that bridge. The setting is the unanchored disconnected network: individual patient data on one treatment, say the index treatment \(A\), aggregate data on a comparator \(B\), and no common comparator arm at all. The two studies are single-arm trials, or arms extracted from larger trials, that share no treatment. Such evidence is the daily reality of health-technology assessment (HTA) for single-arm oncology trials, rare-disease registries, and real-world cohorts, where a randomized head-to-head comparison does not exist and a connecting study cannot be found. Bucher cannot start, because there is nothing to anchor on. Standard network meta-analysis cannot start, because the network is disconnected. Unanchored MAIC and unanchored STC can produce a pairwise number, but only in the comparator population and only under the strong conditional constancy of absolute effects (Theorem 18.5, Theorem 19.6). What this chapter develops, following Chandler and Ishak (2026a), is the multilevel generalization of that pairwise idea: multilevel unanchored meta-regression (ML-UMR), which carries the integration machinery of ML-NMR, the Sobol’ quasi-Monte Carlo (QMC) rule and the Gaussian copula of Chapter 22, into the unanchored setting and so reports the unanchored contrast in any target population through one joint Bayesian likelihood.

The price of removing the anchor is a stronger identifying assumption, and naming it precisely is the central conceptual work of the chapter. An anchored comparison can tolerate effect modification, provided it is shared across the active treatments, because the shared modification cancels against the common comparator; this is the shared-effect-modifier assumption (SEMA, Definition 17.6) that powers ML-NMR. An unanchored comparison has nothing to cancel against. To let the IPD on treatment \(A\) inform the outcome under treatment \(B\) at an arbitrary covariate value, ML-UMR assumes that the entire covariate-outcome relationship, not merely its treatment-by-covariate part, is the same under both treatments. This is the shared prognostic factor assumption (SPFA): the prognostic slopes coincide, \(\boldsymbol\beta_A= \boldsymbol\beta_B\), which is equivalent to there being no effect modification at all. SPFA is the ML-UMR realization of conditional constancy of absolute effects (Definition 17.5); it is strictly stronger than SEMA (Proposition 26.1), and it is untestable from two single-arm studies. The chapter is honest about this throughout: ML-UMR extends the menu of target populations available for an unanchored comparison, at the cost of an assumption that the data cannot check.

The development proceeds in catalogue order. We fix the disconnected setting and its estimand (Definition 26.1), isolate the structural fact that a single-arm study cannot separate its baseline from its treatment effect, and state SPFA and its meaning (Definition 26.2). We then build the SPFA-invoking model and prove its identifiability from one IPD arm and one AgD arm (Theorem 26.1), relax SPFA when the comparator reports subgroup outcomes (Theorem 26.2), and specialize the aggregate integration to the unit hypercube (Theorem 26.3). The survival case occupies the center of the chapter: the proportional-hazards and accelerated-failure-time models, the integrated censored likelihood, and the target-population survival, hazard, and restricted mean survival time (Theorem 26.4, Theorem 26.7), with the reconstruction of pseudo-individual data from a published Kaplan-Meier curve made precise (Theorem 26.5). The joint individual-plus- aggregate likelihood (Theorem 26.6) ties the two data types into one posterior. We then locate ML-UMR relative to its pairwise ancestor: it reduces to unanchored STC in the comparator population and extends it elsewhere (Theorem 26.8), under an assumption strictly stronger than SEMA (Proposition 26.1). The chapter closes with the identifiability frontier and the inherent limits of unanchored comparison (Theorem 26.9), a fully worked binary example whose every number can be checked by hand, a compact survival illustration, notes, and exercises. This chapter corresponds to results M.1 to M.12 of the proof catalogue.

Throughout, \(g\) is the link function (Definition 7.2) on whose scale the linear predictor is built, \(A\) denotes the index treatment for which IPD are available, and \(B\) denotes the comparator for which only AgD are available. A population \(\mathcal{P}\) is a distribution over the covariate space \(\mathcal{X}\) with density \(f_{\mathcal{P}}\); the index trial’s population is \(\mathcal{P}_A\) with density \(f_A\) and the comparator’s is \(\mathcal{P}_B\) with density \(f_B\), and a general reporting target is \(\mathcal{P}^{*}\) with density \(f_{*}\) and mean \(\bar{\mathbf{x}}_{*}\). We write \(\boldsymbol\xi\) for the full parameter vector and use the relative-effect and aggregation notation of Notation and of Chapters 20 and 23.

26.2 A concrete scenario and a plain-language roadmap

Before any formalism, here is the clinical situation the chapter solves and the four-step logic that solves it. We return to this scenario at every transition, and the fully worked arithmetic appears in Section 26.15.

Example 26.1 (A single-arm oncology comparison (running example)) A sponsor seeks reimbursement for a new therapy \(A\) in advanced melanoma. The evidence is asymmetric:

• Index source (IPD on \(A\)). A single-arm trial of \(A\) with individual patient data: for each patient a covariate vector (for example age, prior lines of therapy, and a binary biomarker such as PD-L1 status) and an outcome (tumor response, yes or no; or a survival time).
• Comparator source (AgD on \(B\)). For standard chemotherapy \(B\) there are no patient rows, only a published report: the sample size, a table of baseline covariate means (and sometimes standard deviations and a correlation matrix), and an outcome summary, either a response rate or a Kaplan-Meier survival curve.
• No shared arm. The trial of \(A\) never gave \(B\), and the report on \(B\) never gave \(A\); there is no common comparator to anchor on. The network is disconnected.
• The HTA question. The reimbursement agency wants the effect of \(A\) versus \(B\) in a decision population \(\mathcal{P}^{*}\), whose covariate profile may match neither trial.

In the arithmetic of Section 26.15 we make this numerical: one binary biomarker, an index fit giving intercept \(\alpha_A=-0.5\) and prognostic slope \(\beta_{\mathrm{PF}}=1.0\) on the logit scale, and a comparator report of \(77\) responders out of \(200\) with biomarker prevalence \(0.5\). \(\blacktriangleleft\)

The data flow in words. ML-UMR answers the HTA question in four steps, each of which becomes a theorem below. Read this once now; the rest of the chapter simply makes each step precise.

1. Fit the index arm from its IPD. Regress the index outcome on the covariates to estimate one prognostic slope \(\boldsymbol\beta_{\mathrm{PF}}\) (how strongly the covariates move the outcome) and one index level \(\alpha_A\) (the fitted outcome of arm \(A\) at covariate zero). This is an ordinary generalized-linear-model fit using only the index data (Theorem 26.1).
2. Borrow the slope: the shared prognostic factor assumption. Assume arm \(B\) has the same covariate slope as arm \(A\) (Definition 26.2). This is the price of having no anchor, and, as the chapter stresses repeatedly, the two-study data cannot check it.
3. Locate the comparator level from its AgD. Use the single comparator outcome summary to pin the one remaining unknown, the comparator level \(\alpha_B\): choose \(\alpha_B\) so that arm \(B\)’s model-predicted average outcome, taken over the comparator’s covariate distribution, equals the reported comparator outcome (Theorem 26.1).
4. Integrate over any target and contrast. With both arm surfaces now known, average each arm’s outcome over any chosen target population \(\mathcal{P}^{*}\) and contrast on the link scale (Equation 26.1, Theorem 26.7).

The advertised advantage over the simpler pairwise methods is step 4: the freedom to choose \(\mathcal{P}^{*}\). The entire cost sits in step 2. Everything below unpacks these four steps and accounts carefully for what each assumes. Table 26.1 places ML-UMR beside the methods it generalizes; the one column that separates them is “reports contrast in,” and the price of the wider reach is the assumption column.

Table 26.1: Where ML-UMR sits among indirect-comparison methods. Anchored methods report in any population but need a connected network; unanchored MAIC and STC drop the network but report only in the comparator population; ML-UMR keeps the target-population freedom of ML-NMR without a network, paying with the stronger SPFA.

Method	Connected network?	Reports contrast in	Key assumption	What the data cannot check
Bucher / anchored NMA	required	any population, via the common comparator	transitivity; balanced effect modifiers	checkable when a closed loop exists
Unanchored MAIC	not required	comparator population only	constancy of absolute effects	the transported absolute level
Unanchored STC	not required	comparator population only	constancy of absolute effects	the transported absolute level
ML-NMR	required (connected)	any target population	SEMA: shared effect modification	little; effect modifiers are estimable from IPD
ML-UMR	not required	any target population	SPFA: no effect modification	the borrowed prognostic slope (Type 1)

A reader who wants only the practical map may read Example 26.1, Table 26.1, and the worked Section 26.15, then return to the theorems. The proofs are where the identifiability and equivalence claims summarized here are earned.

26.3 The unanchored disconnected network

We fix the data structures and the estimand, and expose the structural obstruction that distinguishes the unanchored case from everything before it.

Definition 26.1 (The unanchored disconnected two-study network) An unanchored disconnected network consists of two single-treatment data sources with no shared treatment:

1. an index source providing individual patient data on treatment \(A\): for each of \(n_A\) patients a covariate vector \(\mathbf{x}_i\in\mathcal{X}\subseteq\mathbb{R}^{d}\) and an outcome \(y_i\), drawn from a population \(\mathcal{P}_A\) with covariate density \(f_A\);

2. a comparator source providing aggregate data on treatment \(B\): a sample size \(n_B\), the marginal covariate means \(\bar{\mathbf{x}}_B=(\bar x_{B,1},\dots,\bar x_{B,d})\) (and, where reported, marginal standard deviations and a covariate correlation matrix \(\boldsymbol\Omega_B\)), and an outcome summary, for example an event count \(r_B\) for a binary outcome, a sample mean \(\bar y_B\) for a continuous outcome, or a reconstructed Kaplan-Meier curve for a time-to-event outcome, from a population \(\mathcal{P}_B\) with covariate density \(f_B\).

The treatments \(A\) and \(B\) share no common comparator arm; the network graph of Chapter 15 is disconnected. The estimand is the marginal relative effect of \(A\) versus \(B\) in a stated target population \(\mathcal{P}^{*}\), on the link scale \(g\), \[ \Delta_{AB}(\mathcal{P}^{*})\;:=\;g\!\big(\bar\theta_A(\mathcal{P}^{*})\big)-g\!\big(\bar\theta_B(\mathcal{P}^{*})\big), \qquad \bar\theta_k(\mathcal{P}^{*})=\int_{\mathcal{X}}g^{-1}\!\big(\eta_k(\mathbf{x})\big)\,f_{*}(\mathbf{x})\,d\mathbf{x}, \tag{26.1}\] where \(\eta_k(\mathbf{x})\) is the arm-\(k\) linear predictor and \(\bar\theta_k(\mathcal{P}^{*})\) is the arm-\(k\) marginal mean outcome in \(\mathcal{P}^{*}\). In the notation of Notation this is the target-population marginal contrast of Definition 23.1, written here with \(A\) as the subject so that a positive \(\Delta_{AB}\) favors the index treatment. \(\blacktriangleleft\)

Reader translation. Read Equation 26.1 from the inside out. For each arm, \(g^{-1}(\eta_k(\mathbf{x}))\) is that arm’s predicted outcome on the natural scale (a response probability for a binary outcome) at covariate value \(\mathbf{x}\); integrating against \(f_{*}\) averages this predicted outcome over the target population, giving the arm’s marginal mean \(\bar\theta_k(\mathcal{P}^{*})\); finally \(g\) (the logit, say) is applied to each average, and the two are subtracted. In one sentence: average each arm’s outcome over the target population first, then contrast on the link scale. The order is deliberate. Taking the link of the average, rather than the average of the conditional link-scale effect, is exactly what makes the marginal estimand differ from the conditional one; that gap is non-collapsibility (Definition 7.7), and the worked Section 26.15 displays it in two numbers you can check by hand. A novice reader may find it helpful to picture \(\mathcal{P}^{*}\) as a spreadsheet of covariate rows: fill in each arm’s predicted probability row by row, average the two columns, then compare the two averages on the logit scale.

The estimand Equation 26.1 is identical in form to the ML-NMR target estimand of Chapter 23 (Equation 23.5); what changes is the evidence available to identify the two arm linear predictors. The defining difficulty of the unanchored case is recorded next.

Lemma 26.1 (A single-arm study conflates baseline and treatment effect) Adopt the ML-NMR linear predictor of Definition 20.1, in which the arm-\(k\) linear predictor in study \(j\) is \(\eta_{jk}(\mathbf{x})=\mu_j+\mathbf{x}^\top\boldsymbol\beta_1+\gamma_k+\mathbf{x}^\top\boldsymbol\beta_{2,k}\), with \(\mu_j\) the study baseline and \(\gamma_k\) the treatment main effect. If a study contains a single arm \(k\), then the study baseline \(\mu_j\) and the treatment main effect \(\gamma_k\) enter the likelihood only through their sum \(\alpha:=\mu_j+\gamma_k\), and are not separately identified: any pair \((\mu_j,\gamma_k)\) and \((\mu_j+c,\gamma_k-c)\) with \(c\in\mathbb{R}\) induces the same distribution of the data.

Proof. With a single arm \(k\) present in the study, every observation in the study has linear predictor \(\eta(\mathbf{x})=\mu_j+\gamma_k+\mathbf{x}^\top(\boldsymbol\beta_1+\boldsymbol\beta_{2,k})\). The map \((\mu_j,\gamma_k)\mapsto\eta(\cdot)\) depends on \((\mu_j,\gamma_k)\) only through the sum \(\mu_j+\gamma_k\): replacing \((\mu_j,\gamma_k)\) by \((\mu_j+c,\gamma_k-c)\) leaves \(\mu_j+\gamma_k\), hence \(\eta(\mathbf{x})\) for every \(\mathbf{x}\), hence the conditional outcome law \(\pi_{\mathrm{Ind}}(\cdot\mid g^{-1}(\eta(\mathbf{x})))\), hence the likelihood, unchanged. The parameter \(c\) is therefore free, so \((\mu_j,\gamma_k)\) is not identified; only the linear combination \(\alpha=\mu_j+\gamma_k\) is. \(\square\)

Reader translation: what “identified” means, and why one arm cannot split the two pieces. To say a parameter is identified is to say the data pin it to a single value, even in the ideal limit of infinite sample size; an unidentified parameter is one the data leave genuinely undetermined, so that no estimator, however clever, can recover it. (This is the same notion introduced for regression in Part I; we lean on it heavily below, so keep the one-line meaning in mind.) The content of Lemma 26.1 is clinical, not just algebraic: from a single-arm trial you never observe the enrolled patients without the treatment, so you cannot tell how much of the observed outcome level is the treatment doing something (\(\gamma_k\)) and how much is merely the baseline prognosis of the patients who happened to enroll (\(\mu_j\)). Only their sum, the arm’s absolute outcome level \(\alpha=\mu_j+\gamma_k\), is visible; we call this sum the conflated intercept of the arm, and use the term freely below. An anchored two-arm study escapes the trap because both arms share the one study baseline \(\mu_j\), so differencing the arms cancels \(\mu_j\) and exposes \(\gamma_k\); the unanchored design has no second arm to difference against.

This rank deficiency is the algebraic heart of “unanchored.” In a connected network the same treatment appears in at least two studies, and the across-study comparison of arms that share a treatment removes the baseline, leaving \(\gamma_k\) identified; this is exactly what the anchor provides. With no shared treatment there is no such comparison, and each single-arm source identifies only the absolute level of its own arm, the conflated intercept \(\alpha\). We therefore write the two arm linear predictors with arm-specific intercepts and arm-specific covariate slopes, \[ \eta_A(\mathbf{x})=\alpha_A+\mathbf{x}^\top\boldsymbol\beta_A, \qquad \eta_B(\mathbf{x})=\alpha_B+\mathbf{x}^\top\boldsymbol\beta_B, \tag{26.2}\] where \(\alpha_A=\mu_{\text{index}}+\gamma_A\) and \(\alpha_B=\mu_{\text{comp}}+\gamma_B\) each conflate a study baseline with a treatment main effect by Lemma 26.1, and \(\boldsymbol\beta_k=\boldsymbol\beta_1+\boldsymbol\beta_{2,k}\) is the total covariate slope on arm \(k\) in the decomposition of Definition 20.1. (Recall from Definition 20.1, restated here because it is the pivot of the whole chapter: \(\boldsymbol\beta_1\) is the prognostic slope, the covariate effect shared by every arm, the part of the outcome-covariate relationship that has nothing to do with treatment; and \(\boldsymbol\beta_{2,k}\) is the arm-\(k\) effect-modification slope, the treatment-by-covariate interaction that bends one arm’s covariate response relative to another. Their sum \(\boldsymbol\beta_k= \boldsymbol\beta_1+\boldsymbol\beta_{2,k}\) is the total slope that arm \(k\)’s own data actually trace out. Prognostic means “predicts outcome regardless of treatment”; effect-modifying means “changes the treatment contrast.” Keep both meanings in view; SPFA is a statement about the second one.) The whole identification problem is now visible: the IPD identify \((\alpha_A,\boldsymbol\beta_A)\), but the AgD supply only a covariate distribution and one or a few outcome summaries, far too little to identify the \((d+1)\)-dimensional \((\alpha_B,\boldsymbol\beta_B)\). Something must connect the two arms.

Remark. Why this is a transportability problem. Comparing \(A\) and \(B\) requires evaluating both arms in one population. The arm-\(A\) surface \(\mu_A(\mathbf{x})=g^{-1}(\eta_A(\mathbf{x}))\) is estimated where \(A\) was observed, namely \(\mathcal{P}_A\); to place it against \(B\) in \(\mathcal{P}_B\) or in any \(\mathcal{P}^{*}\) we must assume that the same conditional surface governs \(\mathcal{P}_B\) and \(\mathcal{P}^{*}\), an instance of conditional transportability (Theorem 12.2) and of the absolute-effect constancy of Definition 17.5. The general theory of Chapter 17 already told us the unanchored price: Theorem 17.2 requires constancy of absolute effects and balance of every imbalanced prognostic variable, not merely every effect modifier. SPFA is the parametric assumption that delivers exactly this. \(\blacktriangleleft\)

This setup corresponds to M.1 in the proof catalogue.

26.4 The shared prognostic factor assumption

We now state the assumption that connects the two arms and makes the unanchored comparison identifiable.

Definition 26.2 (Shared prognostic factor assumption (SPFA)) In the two-arm unanchored model Equation 26.2, the shared prognostic factor assumption is the hypothesis that the covariate-outcome slopes coincide across the two treatments, \[ \boldsymbol\beta_A=\boldsymbol\beta_B=:\boldsymbol\beta_{\mathrm{PF}}. \tag{26.3}\] The common value \(\boldsymbol\beta_{\mathrm{PF}}\) is the single shared prognostic slope; the subscript PF abbreviates prognostic factor. Equivalently, in the decomposition \(\boldsymbol\beta_k=\boldsymbol\beta_1+\boldsymbol\beta_{2,k}\) of Definition 20.1, SPFA states that the effect-modification vectors are equal, \(\boldsymbol\beta_{2,A}=\boldsymbol\beta_{2,B}\); and since in the two-treatment network one arm may be taken as the reference with \(\boldsymbol\beta_{2}=\mathbf{0}\), SPFA forces \(\boldsymbol\beta_{2,A}= \boldsymbol\beta_{2,B}=\mathbf{0}\), that is, no treatment-by-covariate interaction and no effect modification (Definition 11.3). Under SPFA the two arm linear predictors are \[ \eta_A(\mathbf{x})=\alpha_A+\mathbf{x}^\top\boldsymbol\beta_{\mathrm{PF}}, \qquad \eta_B(\mathbf{x})=\alpha_B+\mathbf{x}^\top\boldsymbol\beta_{\mathrm{PF}}, \tag{26.4}\] differing only in the conflated intercepts \(\alpha_A,\alpha_B\). \(\blacktriangleleft\)

Remark. SPFA in plain language (the one assumption that pays for everything). SPFA encodes a specific clinical belief: the way the covariates shape a patient’s outcome is the same whichever of the two treatments the patient receives. Put the other way, the treatment does not act more strongly in one covariate subgroup than another; there is no effect modification. A plausible case: two cytotoxic chemotherapies whose benefit is roughly uniform across age and performance status, so older and younger patients gain similarly. An implausible case: a targeted agent that works only in biomarker-positive patients, set against a chemotherapy that works regardless of biomarker; here the treatment contrast depends sharply on the biomarker, SPFA is false, and forcing it will bias any transported estimate. Two facts make SPFA the delicate point of the method. First, it is stronger than SEMA (Proposition 26.1): SEMA lets the two treatments modify the outcome as long as they modify it identically, because in an anchored design the shared modification cancels against the common comparator; SPFA has no comparator to cancel against and so must assume the modification is absent. Second, in the Type 1 setting (one comparator outcome summary) it is untestable: the data are equally consistent with SPFA and with an effect-modifying alternative (Theorem 26.9), so SPFA must be argued from external clinical knowledge, not checked against the two studies. Table 26.2 lays the two assumptions side by side. \(\blacktriangleleft\)

Table 26.2: SEMA versus SPFA. The anchored assumption tolerates effect modification because a common comparator cancels it; the unanchored assumption must forbid it outright. This is the single conceptual difference between Chapter 17’s SEMA and this chapter’s SPFA, made precise in Proposition 26.1.

Aspect	SEMA (anchored, ML-NMR)	SPFA (unanchored, ML-UMR)
Formal statement	\(\boldsymbol\beta_{2,A}=\boldsymbol\beta_{2,B}\)	\(\boldsymbol\beta_{2,A}=\boldsymbol\beta_{2,B}=\mathbf{0}\)
In words	both treatments modify the outcome identically	neither treatment modifies the outcome
Effect modification	allowed, provided shared	forbidden
Why the contrast is flat	shared modification cancels against the common comparator \(C\)	nothing to modify, so nothing to cancel
Constancy it requires	conditional constancy of relative effects	conditional constancy of absolute effects
Design it fits	connected, anchored network	disconnected, unanchored network
Testable from the data?	interaction estimable from IPD	not from two single-arm sources (Type 1)
Method it powers	ML-NMR, Bucher	ML-UMR, unanchored STC and MAIC

The single line that SPFA buys is the following: it makes the conditional contrast a constant, which is what lets the index IPD project the comparator arm everywhere.

Lemma 26.2 (SPFA is conditional constancy of absolute effects) Under SPFA Equation 26.3, the conditional contrast of \(A\) versus \(B\) on the link scale is free of the covariate, \[ \eta_A(\mathbf{x})-\eta_B(\mathbf{x})=\alpha_A-\alpha_B \qquad\text{for every }\mathbf{x}\in\mathcal{X}, \tag{26.5}\] and the arm-\(B\) conditional surface is determined by the arm-\(A\) surface and a single scalar level shift, \[ \eta_B(\mathbf{x})=\eta_A(\mathbf{x})-(\alpha_A-\alpha_B). \tag{26.6}\] Consequently, once \(\boldsymbol\beta_{\mathrm{PF}}\) is known from the index IPD, knowledge of one scalar, \(\alpha_B\), determines \(\eta_B(\mathbf{x})\), and hence \(g^{-1}(\eta_B(\mathbf{x}))\), at every covariate value, including values the comparator source never reported individually. This is the ML-UMR realization of conditional constancy of absolute effects (Definition 17.5).

Proof. Subtract the two predictors of Equation 26.4: \(\eta_A(\mathbf{x})-\eta_B(\mathbf{x})=(\alpha_A+\mathbf{x}^\top\boldsymbol\beta_{\mathrm{PF}})- (\alpha_B+\mathbf{x}^\top\boldsymbol\beta_{\mathrm{PF}})=\alpha_A-\alpha_B\), since the covariate terms are identical and cancel; this is Equation 26.5, free of \(\mathbf{x}\). Rearranging gives Equation 26.6. The arm-\(A\) surface \(\eta_A(\cdot)=\alpha_A+(\cdot)^\top\boldsymbol\beta_{\mathrm{PF}}\) is identified from the index IPD (a generalized linear model with full-rank design, Theorem 26.1 below), so \(\boldsymbol\beta_{\mathrm{PF}}\) is known; Equation 26.6 then expresses \(\eta_B\) in terms of the known \(\eta_A\) and the single unknown \(\alpha_A-\alpha_B\), equivalently the single unknown \(\alpha_B\). Evaluating \(g^{-1}\) at \(\eta_B(\mathbf{x})\) for any \(\mathbf{x}\) is therefore possible once \(\alpha_B\) is pinned, which Theorem 26.1 does from the comparator outcome summary. The constancy of the absolute conditional contrast across covariate values is precisely Definition 17.5 applied to the \(A\)-versus-\(B\) contrast. \(\square\)

Remark. Why SPFA and not merely SEMA. In the anchored triangle of Chapters 17 and 23, the active treatments \(A\) and \(B\) are each contrasted with a common comparator \(C\), and the shared-effect-modifier assumption (Definition 17.6) sets \(\boldsymbol\beta_{2,A}=\boldsymbol\beta_{2,B}\); the \(A\)-versus-\(B\) contrast then has effect-modification term \((\boldsymbol\beta_{2,A}-\boldsymbol\beta_{2,B})^\top\mathbf{x}=\mathbf{0}\), so the modification cancels against the anchor even when it is present and large. There is no anchor here to cancel against. The only way to make the contrast Equation 26.5 constant in \(\mathbf{x}\), and so to project the index surface onto the comparator arm at arbitrary covariate values, is to assume the modification is absent, \(\boldsymbol\beta_{2,A}=\boldsymbol\beta_{2,B}=\mathbf{0}\). This is why SPFA is strictly stronger than SEMA, a claim we make precise in Proposition 26.1. SPFA must be defended from external knowledge: that the natural history of the disease, the covariate-outcome relationship, is the same whichever treatment a patient receives. It is untestable from two single-arm sources, because there is no replicate of either treatment at a different covariate distribution against which to check the slope. \(\blacktriangleleft\)

These results correspond to M.2 in the proof catalogue.

26.5 The SPFA-invoking ML-UMR model

We assemble the model and prove that, under SPFA, it is identified from exactly the available evidence: a \((d+1)\)-dimensional index arm and a one-dimensional comparator outcome summary.

The identification argument in one breath. The theorem below is long, but its logic is the four-step roadmap of Section 26.2 made rigorous. In words: the index IPD estimate one slope \(\boldsymbol\beta_{\mathrm{PF}}\) and one intercept \(\alpha_A\) for arm \(A\) (an ordinary GLM fit, so both are identified whenever the covariates are not collinear); SPFA lets that same slope be reused for arm \(B\); the single comparator outcome summary then fixes the one remaining unknown, the comparator intercept \(\alpha_B\), because arm \(B\)’s average outcome rises monotonically with \(\alpha_B\) and so meets the observed value exactly once. That is the whole argument; the formal statement only names the regularity conditions that make each step go through. Two symbols recur that the chapter otherwise uses without introduction. \(\pi_{\mathrm{Ind}}\) is the individual-level outcome distribution, the GLM likelihood kernel for one patient (a Bernoulli response, say, or a censored survival contribution); \(\pi_{\mathrm{Agg}}\) is the aggregate-level distribution of a reported summary (a binomial event count, a normal sample mean). Both are inherited from Chapters 20 and 25. We write \(\theta_{\bullet B}\) for the comparator’s aggregate mean, the arm-\(B\) average outcome over the comparator covariate distribution, the bullet marking the covariate that has been averaged out.

Theorem 26.1 (The SPFA ML-UMR model and its identifiability) Adopt SPFA Equation 26.3, so the arm predictors are Equation 26.4 with parameter \(\boldsymbol\xi=(\alpha_A,\alpha_B,\boldsymbol\beta_{\mathrm{PF}},\boldsymbol\phi)\), where \(\boldsymbol\phi\) is any dispersion parameter. The model is:

Individual level (index \(A\), IPD). For \(i=1,\dots,n_A\) with covariates \(\mathbf{x}_i\), \[ Y_i^{(A)}\mid\mathbf{x}_i\ \sim\ \pi_{\mathrm{Ind}}\!\big(g^{-1}(\alpha_A+\mathbf{x}_i^\top\boldsymbol\beta_{\mathrm{PF}}),\,\boldsymbol\phi\big). \tag{26.7}\]

Aggregate level (comparator \(B\), AgD). The comparator outcome summary has mean equal to the aggregate response-scale average over the comparator covariate distribution, \[ \theta_{\bullet B}=\int_{\mathcal{X}}g^{-1}\!\big(\alpha_B+\mathbf{x}^\top\boldsymbol\beta_{\mathrm{PF}}\big)\,f_B(\mathbf{x})\,d\mathbf{x}, \qquad \text{summary}_B\ \sim\ \pi_{\mathrm{Agg}}(\theta_{\bullet B},n_B), \tag{26.8}\] the aggregate model of Definition 20.2 specialized to the comparator arm.

Suppose (i) the index design has full column rank, that is the matrix \([\,\mathbf{1}\ \ \mathbf{X}_A\,]\) with rows \((1,\mathbf{x}_i^\top)\) has rank \(d+1\), and the link \(g\) is strictly monotone and differentiable with \(g^{-1}\) strictly monotone; (ii) the comparator covariate density \(f_B\) is known (reported, or reconstructed by the copula method of Theorem 26.3); and (iii) the observed comparator outcome mean lies in the open range of \(g^{-1}\) (for the logit link this simply says the observed comparator proportion is strictly between \(0\) and \(1\); for the identity link it is no restriction at all). Then \(\boldsymbol\xi\) is identified: \((\alpha_A, \boldsymbol\beta_{\mathrm{PF}})\) are identified from the index likelihood Equation 26.7, and \(\alpha_B\) is the unique solution of \[ \Theta_B(\alpha_B):=\int_{\mathcal{X}}g^{-1}\!\big(\alpha_B+\mathbf{x}^\top\boldsymbol\beta_{\mathrm{PF}}\big)\,f_B(\mathbf{x})\,d\mathbf{x}\;=\;\theta_{\bullet B}^{\mathrm{obs}}, \tag{26.9}\] where \(\theta_{\bullet B}^{\mathrm{obs}}\) is the observed comparator mean. The target estimand Equation 26.1 is then identified in every population \(\mathcal{P}^{*}\) for which \(f_{*}\) is specified.

Proof. Identification of the index parameters. Restricted to the index source, Equation 26.7 is a generalized linear model with linear predictor \(\alpha_A+\mathbf{x}^\top\boldsymbol\beta_{\mathrm{PF}}\) and strictly monotone link \(g\). The mean function \(\mathbf{x}\mapsto g^{-1}(\alpha_A+\mathbf{x}^\top \boldsymbol\beta_{\mathrm{PF}})\) determines \((\alpha_A,\boldsymbol\beta_{\mathrm{PF}})\) uniquely: if two parameter pairs gave the same mean function then, applying the strictly monotone \(g\), the affine functions \(\alpha_A+\mathbf{x}^\top\boldsymbol\beta_{\mathrm{PF}}\) and \(\alpha_A'+\mathbf{x}^\top\boldsymbol\beta_{\mathrm{PF}}'\) would agree for all \(\mathbf{x}\) in the index support, and full column rank of \([\,\mathbf{1}\ \ \mathbf{X}_A\,]\) forces \(\alpha_A=\alpha_A'\) and \(\boldsymbol\beta_{\mathrm{PF}}=\boldsymbol\beta_{\mathrm{PF}}'\). This is the standard GLM identifiability already used for the IPD arm in Theorem 23.2. The dispersion \(\boldsymbol\phi\), when present, is identified from the index residual variation in the usual way.

Identification of the comparator intercept. With \(\boldsymbol\beta_{\mathrm{PF}}\) known from the previous step and \(f_B\) known by hypothesis (ii), the comparator aggregate mean Equation 26.8 is the function \(\alpha_B\mapsto\Theta_B(\alpha_B)\) of Equation 26.9. This is exactly the map \(\Theta\) of the aggregate-monotonicity lemma Lemma 23.2, with scalar shift \(c=\alpha_B\) and \(h(\mathbf{x})=\mathbf{x}^\top\boldsymbol\beta_{\mathrm{PF}}\). By that lemma \(\Theta_B\) is finite, continuous, and strictly increasing, hence injective, with range the open interval that is the range of \(g^{-1}\); by hypothesis (iii) the observed mean \(\theta_{\bullet B}^{\mathrm{obs}}\) lies in that range, so the equation \(\Theta_B(\alpha_B)=\theta_{\bullet B}^{\mathrm{obs}}\) has a unique solution. Thus \(\alpha_B\) is identified.

Identification of the estimand. With \((\alpha_A,\alpha_B,\boldsymbol\beta_{\mathrm{PF}})\) identified, both arm predictors Equation 26.4 are determined, so the target arm means \(\bar\theta_k(\mathcal{P}^{*})= \int g^{-1}(\eta_k(\mathbf{x}))f_{*}(\mathbf{x})\,d\mathbf{x}\) are determined for any specified \(f_{*}\), and the contrast Equation 26.1 is their link-scale difference. The integrals are finite under the integrability hypothesis implicit in (iii) and the boundedness of \(g^{-1}\) for the standard links. \(\square\)

Why this matters. This is the payoff of the whole setup, and it is worth stating as a slogan: one IPD arm plus SPFA plus one aggregate number is exactly enough to place both treatments in any population. The index regression does the heavy lifting (a slope and a level), SPFA supplies the bridge (reuse the slope), and the comparator’s single reported outcome supplies the last degree of freedom (the comparator level). Nothing is over-determined and nothing is missing; the model is just identified. That tight accounting is also the warning label: with only one comparator number to spend, there is none left over to test SPFA.

Definition 26.3 (Type 1 and Type 2 ML-UMR models) Two data regimes recur throughout, and we name them once. The Type 1 model invokes SPFA (Definition 26.2) and uses a single overall comparator outcome summary; it identifies the comparator intercept \(\alpha_B\) only, reusing the index slope \(\boldsymbol\beta_{\mathrm{PF}}\), and is the model of Theorem 26.1. The Type 2 (relaxed) model drops SPFA, allows a distinct comparator slope \(\boldsymbol\beta_B\), and needs subgroup comparator outcomes to identify both \(\alpha_B\) and \(\boldsymbol\beta_B\) (Theorem 26.2). Type 1 is the default, because subgroup outcomes are rarely published; Type 2 is the principled remedy when effect modification cannot be ruled out and enough subgroups are reported. \(\blacktriangleleft\)

Table 26.3 records which piece of evidence supplies which parameter, and, crucially, whether it is observed, reconstructed, or merely assumed. The single row marked “assumed” is SPFA, and it is the row the data cannot audit in Type 1.

Table 26.3: What identifies what in ML-UMR. Read top to bottom, this is the four-step roadmap of Section 26.2 with each step’s data source and epistemic status made explicit. Only \(\alpha_A\), \(\boldsymbol\beta_{\mathrm{PF}}\), and \(\alpha_B\) are estimated from data; \(f_B\) is reconstructed; SPFA and \(f_{*}\) are supplied from outside the two studies.

Data source	What it identifies or supplies	Status
Index IPD (arm \(A\))	index level \(\alpha_A\) and prognostic slope \(\boldsymbol\beta_{\mathrm{PF}}\)	observed patient rows
SPFA (external belief)	that arm \(B\)’s slope equals \(\boldsymbol\beta_{\mathrm{PF}}\)	assumed; untestable in Type 1
Comparator outcome summary (arm \(B\))	comparator level \(\alpha_B\)	observed aggregate
Comparator covariate distribution \(f_B\)	the weighting of the arm-\(B\) aggregate integral	reported margins, reconstructed via copula
Comparator subgroup outcomes	comparator slope \(\boldsymbol\beta_B\) as well (Type 2)	rarely observed
Target distribution \(f_{*}\)	the population the contrast is reported in	chosen from the HTA question

Remark. The two families in closed and not-closed form. For a continuous outcome with the identity link and the linear model \(\mathbb{E}[Y\mid\mathbf{x}]=\alpha_k+\mathbf{x}^\top\boldsymbol\beta_{\mathrm{PF}}\), aggregation is exact (Theorem 20.3): \(\theta_{\bullet B}=\alpha_B+\bar{\mathbf{x}}_B^\top \boldsymbol\beta_{\mathrm{PF}}\), so Equation 26.9 solves in closed form, \[ \alpha_B=\bar y_B-\bar{\mathbf{x}}_B^\top\boldsymbol\beta_{\mathrm{PF}}, \] needing only the comparator covariate mean. For a binary outcome with the logit link, Bernoulli individual model and binomial aggregate model, \(\Theta_B(\alpha_B)=\int\operatorname{expit}(\alpha_B+ \mathbf{x}^\top\boldsymbol\beta_{\mathrm{PF}})f_B(\mathbf{x})\,d\mathbf{x}\) has no closed form (Theorem 20.5), and Equation 26.9 is solved numerically after the integral is evaluated by the QMC rule of Theorem 26.3; here the comparator covariate distribution, not just its mean, enters. This is the binary “Type 1” model of Chandler and Ishak (2026a); the reference software mlumr fits it as one joint Stan model rather than by the two-step solve, but the two-step solve is what the identifiability argument exhibits and what the worked example of Section 26.15 carries out by hand. \(\blacktriangleleft\)

Remark. Bayesian inference. In practice one does not separate the two steps. Place priors on \((\alpha_A,\alpha_B,\boldsymbol\beta_{\mathrm{PF}})\), for example weakly informative normals, and sample the posterior of the joint likelihood Theorem 26.6 by Hamiltonian Monte Carlo (Definition 8.9, Carpenter et al. (2017)); the QMC integral makes the comparator likelihood a smooth function of the parameters (Equation 22.20), so the gradient HMC needs is available. (Hamiltonian Monte Carlo is, for present purposes, a black-box sampler that draws from the posterior using its gradient; Definition 8.9 has the mechanics, and nothing below depends on them.) The reason to fit both blocks jointly, rather than run the two-step solve of the proof, is uncertainty propagation: the joint posterior carries the estimation error in \(\boldsymbol\beta_{\mathrm{PF}}\) from the index arm through into the uncertainty in \(\alpha_B\), and hence into every target-population contrast, whereas plugging a single point estimate of \(\boldsymbol\beta_{\mathrm{PF}}\) into a separate solve for \(\alpha_B\) would understate that uncertainty. The two-step identification argument is the guarantee that the posterior concentrates: the index data pin \((\alpha_A,\boldsymbol\beta_{\mathrm{PF}})\) and the comparator mean then pins \(\alpha_B\), so the likelihood has a unique maximizer in the limit. \(\blacktriangleleft\)

This theorem corresponds to M.3 in the proof catalogue.

26.6 Relaxing SPFA with comparator subgroups

SPFA fails whenever the covariate genuinely modifies the treatment effect, that is when \(\boldsymbol\beta_A\ne\boldsymbol\beta_B\). A single aggregate outcome mean cannot then identify the model, because it supplies one equation for the \((d+1)\) unknowns \((\alpha_B,\boldsymbol\beta_B)\). The remedy of Chandler and Ishak (2026a) is to demand more aggregate equations, by partitioning the comparator population into subgroups and using a reported outcome in each. Concretely, this means the comparator report must give, say, response rates separately for biomarker-positive and biomarker-negative patients, or Kaplan-Meier curves per risk stratum, each with its own covariate summary. The intuition is a counting argument any reader can hold onto: each subgroup outcome is one equation, the unknowns are the \(d+1\) numbers \((\alpha_B,\boldsymbol\beta_B)\), and you need at least as many genuinely different equations as unknowns. “Genuinely different” is the catch, and the theorem makes it precise. Such stratified reporting is the exception rather than the rule, which is why the Type 1 model of Theorem 26.1 remains the workhorse.

Theorem 26.2 (Identifiability of the relaxed model from comparator subgroups) Drop SPFA and allow a treatment-specific comparator slope: the arm predictors are \(\eta_A(\mathbf{x})=\alpha_A+\mathbf{x}^\top\boldsymbol\beta_A\) and \(\eta_B(\mathbf{x})=\alpha_B+\mathbf{x}^\top\boldsymbol\beta_B\) with possibly distinct \(\boldsymbol\beta_A,\boldsymbol\beta_B\in\mathbb{R}^{d}\). Suppose the comparator source reports outcomes for \(Z\) non-overlapping subgroups \(z=1,\dots,Z\), with known within-subgroup covariate densities \(f_B^{z}\) and observed aggregate means \(\theta_{\bullet B}^{z,\mathrm{obs}}\), modeled by \[ \theta_{\bullet B}^{z}(\alpha_B,\boldsymbol\beta_B)=\int_{\mathcal{X}}g^{-1}\!\big(\alpha_B+\mathbf{x}^\top\boldsymbol\beta_B\big)\,f_B^{z}(\mathbf{x})\,d\mathbf{x}, \qquad z=1,\dots,Z. \tag{26.10}\] Let \(\boldsymbol\psi=(\alpha_B,\boldsymbol\beta_B)\in\mathbb{R}^{d+1}\) and write \(w_z(\mathbf{x}):=(g^{-1})'(\alpha_B+\mathbf{x}^\top\boldsymbol\beta_B)>0\). If

1. (counting) \(Z\ge d+1\), and

2. (non-degeneracy) the \(Z\times(d+1)\) matrix \(M\) with rows \(\mathbf{m}_z^\top=\int w_z(\mathbf{x})\,(1,\mathbf{x}^\top)\,f_B^{z}(\mathbf{x})\,d\mathbf{x}\) has rank \(d+1\) at \(\boldsymbol\psi\),

then \(\boldsymbol\psi=(\alpha_B,\boldsymbol\beta_B)\) is locally identified by the \(Z\) subgroup equations Equation 26.10, given \(\alpha_A,\boldsymbol\beta_A\) from the index IPD. In particular two binary effect modifiers crossed into \(Z=4\) subgroups identify \(\boldsymbol\beta_B\in\mathbb{R}^{2}\). If instead \(Z<d+1\), the solution set of Equation 26.10 is a manifold of dimension at least \(d+1-Z\ge 1\), and \(\boldsymbol\psi\) is not identified.

Proof. Local identification when \(Z\ge d+1\) and \(M\) has full column rank. Define the map \(F:\mathbb{R}^{d+1}\to\mathbb{R}^{Z}\) by \(F(\boldsymbol\psi)=(\theta_{\bullet B}^{1}(\boldsymbol\psi),\dots, \theta_{\bullet B}^{Z}(\boldsymbol\psi))\). Each component is differentiable, and by differentiating under the integral sign (the integrand and its \(\boldsymbol\psi\)-gradient are bounded for the standard links, so dominated convergence applies), \[ \frac{\partial\theta_{\bullet B}^{z}}{\partial\alpha_B}=\int w_z(\mathbf{x})\,f_B^{z}(\mathbf{x})\,d\mathbf{x}, \qquad \frac{\partial\theta_{\bullet B}^{z}}{\partial\boldsymbol\beta_B}=\int w_z(\mathbf{x})\,\mathbf{x}\,f_B^{z}(\mathbf{x})\,d\mathbf{x}, \] so the Jacobian \(DF(\boldsymbol\psi)\) is exactly the matrix \(M\) with rows \(\mathbf{m}_z^\top=\int w_z (1,\mathbf{x}^\top)f_B^{z}\). By hypothesis (2) \(M\) has rank \(d+1\), its full column rank; since \(Z\ge d+1\) by hypothesis (1), \(DF(\boldsymbol\psi)\) has a left inverse, and the constant-rank form of the inverse function theorem gives a neighborhood of \(\boldsymbol\psi\) on which \(F\) is injective. Hence \(\boldsymbol\psi\) is the unique parameter in that neighborhood reproducing the observed subgroup means \((\theta_{\bullet B}^{z,\mathrm{obs}})_z\): it is locally identified. (The index parameters \(\alpha_A,\boldsymbol\beta_A\) are identified from the IPD as in Theorem 26.1, separately from the comparator block.)

Non-identification when \(Z<d+1\). The Jacobian \(DF(\boldsymbol\psi)=M\) is then a \(Z\times(d+1)\) matrix with \(Z<d+1\), so its rank is at most \(Z<d+1\) and its kernel has dimension at least \(d+1-Z\ge 1\). By the constant-rank theorem (or directly, by the implicit function theorem applied to the rank-\(Z\) system) the level set \(\{\boldsymbol\psi':F(\boldsymbol\psi')=F(\boldsymbol\psi)\}\) contains, near \(\boldsymbol\psi\), a smooth submanifold of dimension \(\ge d+1-Z\ge 1\); every point of it reproduces the same \(Z\) subgroup means. Distinct parameter values therefore give identical observable aggregate data, so \(\boldsymbol\psi\) is not identified. \(\square\)

Remark. What the subgroups must deliver. The non-degeneracy condition (2) is the requirement that the \(Z\) subgroups differ enough in their covariate distributions: the rows \(\mathbf{m}_z\) are the hazard-weighted subgroup moments of \((1,\mathbf{x})\), and they must span \(\mathbb{R}^{d+1}\). Two subgroups with identical covariate means contribute near-parallel rows and add little identifying power, exactly as two collinear regressors do in least squares. Crossing two binary effect modifiers gives four subgroups whose means sit at the four corners of a square in covariate space, which generically span \(\mathbb{R}^{3}\) and identify \((\alpha_B,\beta_{B,1},\beta_{B,2})\). This is the simulation design of Chandler and Ishak (2026a). \(\blacktriangleleft\)

Remark. Why Type 1 is the default. Subgroup-specific aggregate outcomes, an event rate or a Kaplan-Meier curve reported separately within each effect-modifier stratum, are rarely published. When they are not, only the single overall comparator mean is available, \(Z=1<d+1\), and by the theorem the relaxed model is not identified; one must invoke SPFA and fit the Type 1 model of Theorem 26.1. Even when a few subgroups are reported, the mlumr documentation warns that the comparator slope \(\boldsymbol\beta_B\) is “weakly identified by aggregate data,” meaning that \(M\), while of full rank, is ill-conditioned, so informative priors or many subgroups are needed for stable estimation. The relaxed model is the principled way to accommodate effect modification, but its data demands are stringent. \(\blacktriangleleft\)

Example 26.2 (Two subgroups identify a slope, but only if they differ) Take one binary covariate (\(d=1\)), the logit link, and the index parameters already fixed. The comparator now reports its response rate in two subgroups. In the favorable case the subgroups are the two covariate strata themselves: subgroup \(z=0\) is the biomarker-negative patients (covariate mean \(0\)) with observed response \(0.30\), and subgroup \(z=1\) is the biomarker-positive patients (covariate mean \(1\)) with observed response \(0.55\). Each subgroup integral Equation 26.10 is then a single point, so the two equations read \[ \operatorname{expit}(\alpha_B)=0.30,\qquad \operatorname{expit}(\alpha_B+\beta_B)=0.55, \] which solve exactly: \(\alpha_B=\operatorname{logit}(0.30)=-0.85\) and \(\alpha_B+\beta_B=\operatorname{logit}(0.55)=0.20\), so \(\beta_B=1.05\). Two aggregate numbers have identified both the comparator intercept and its slope, with no SPFA invoked. This is the Type 2 model, and the count \(Z=2=d+1\) meets the counting condition with room to spare on non-degeneracy, because the two subgroup means, \(0\) and \(1\), are as far apart as they can be.

Now spoil it. Suppose instead the two reported subgroups are “younger” and “older” patients, and the biomarker prevalence is almost the same in both, say covariate means \(0.48\) and \(0.52\). Each equation is now \(\theta_{\bullet B}^{z}\approx(1-\bar x_z)\operatorname{expit}(\alpha_B)+\bar x_z\operatorname{expit} (\alpha_B+\beta_B)\) with \(\bar x_z\) near \(0.5\) in both, so the two equations are nearly the same equation: together they pin the average level well, but say almost nothing about the slope \(\beta_B\), because shifting a little covariate mass between two nearly identical strata barely changes either subgroup’s outcome. The matrix \(M\) of Theorem 26.2 is then ill-conditioned; a small error in either reported rate swings the estimated \(\beta_B\) wildly. Counting (\(Z=d+1\)) is satisfied, yet non-degeneracy nearly fails, and only strong priors or a genuinely different third subgroup rescue the estimate. This is the arithmetic behind the mlumr warning that the relaxed comparator slope is “weakly identified by aggregate data.” \(\blacktriangleleft\)

This theorem corresponds to M.4 in the proof catalogue.

26.7 Aggregate marginalization by quasi-Monte Carlo

The aggregate integrals Equation 26.8 and Equation 26.10 are, except on the identity link, intractable in closed form. ML-UMR evaluates them with the exact pipeline ML-NMR uses, reconstructing the comparator covariate distribution from its reported margins and a correlation matrix and integrating on the unit hypercube. The result is a corollary of the hypercube-integration theorem of Chapter 22, recorded here in the unanchored notation.

Why an integral, and why the mean is not enough. Here is the practical reason for the machinery, in one paragraph. Arm \(B\)’s average outcome is \(\int g^{-1}(\alpha_B+\mathbf{x}^\top\boldsymbol\beta_{\mathrm{PF}}) f_B(\mathbf{x})\,d\mathbf{x}\). On the identity link \(g^{-1}\) is linear, so the integral equals \(g^{-1}\) evaluated at the mean covariate \(\bar{\mathbf{x}}_B\): the comparator’s reported means are all you need. On the logit link (and the hazard scale, and every nonlinear link) \(g^{-1}\) is curved, so the average of the outcome is not the outcome at the average covariate; the whole distribution of covariates matters, not just its mean. But the comparator paper reports margins, not patient rows. So the model fabricates a cloud of synthetic covariate points whose margins and correlations match what the paper reported, and averages the outcome over that cloud. The theorem below is the recipe for building the cloud, an inverse-CDF transform for each margin glued by a Gaussian copula for the correlations, together with a guarantee that a modest number of well-chosen points (a Sobol’ quasi-Monte Carlo net) makes the average accurate. A reader content with this picture may skim the theorem and rejoin at the discrete remark, where a single binary covariate collapses the integral to an exact two-point sum.

Theorem 26.3 (QMC marginalization of the comparator integral) Let the comparator covariate distribution \(f_B\) have continuous marginal distribution functions \(F_{B,1},\dots,F_{B,d}\) (Normal for a continuous covariate of reported mean and standard deviation, Bernoulli for a binary covariate of reported prevalence, and so on) and Gaussian-copula correlation matrix \(\boldsymbol\Omega_B=LL^\top\) (Definition 22.5). Let \(T:[0,1]^{d}\to\mathcal{X}\) be the transport map of Equation 22.17 built from these margins and \(L\). Then for any integrand \(\varphi:\mathcal{X}\to \mathbb{R}\), in particular the response-scale arm mean \(\varphi=g^{-1}\circ\eta_B\), \[ \int_{\mathcal{X}}\varphi(\mathbf{x})\,f_B(\mathbf{x})\,d\mathbf{x}=\int_{[0,1]^{d}}\varphi\big(T(\mathbf{u})\big)\,d\mathbf{u}, \tag{26.11}\] and the Sobol’ QMC estimate over a net \(P_N=\{\mathbf{u}^{(n)}\}_{n=1}^{N}\) (Definition 22.3), \[ \widehat\theta_{\bullet B}^{(N)}(\boldsymbol\xi)=\frac1N\sum_{n=1}^{N}g^{-1}\!\big(\alpha_B+T(\mathbf{u}^{(n)})^\top\boldsymbol\beta_{\mathrm{PF}}\big), \tag{26.12}\] satisfies, whenever the pullback \(\psi=g^{-1}\circ\eta_B\circ T\) has bounded Hardy-Krause variation (Definition 22.2; intuitively, \(\psi\) is not too wiggly on the cube, the multivariate analogue of a one-dimensional function of bounded total variation), \[ \big|\widehat\theta_{\bullet B}^{(N)}-\theta_{\bullet B}\big|\le V_{\mathrm{HK}}(\psi)\,D_N^{*}(P_N)=O\!\Big(V_{\mathrm{HK}}(\psi)\,\frac{(\log N)^{d}}{N}\Big). \tag{26.13}\] The same pipeline, with \(f_B\) replaced by the target density \(f_{*}\), evaluates the target arm means \(\bar\theta_k(\mathcal{P}^{*})\) of Equation 26.1.

Proof. The change-of-variables identity Equation 26.11 is part 1 of Theorem 22.6: the map \(T\) pushes the uniform law on \([0,1]^{d}\) forward to \(f_B\), by the inverse-CDF transform (Theorem 22.4) on each margin and Sklar’s theorem (Theorem 22.5) for the Gaussian copula, so the expectation of \(\varphi(\mathbf{X})\) under \(\mathbf{X}\sim f_B\) equals the cube integral of \(\varphi\circ T\). Substituting \(\varphi=g^{-1}\circ\eta_B\) and the Sobol’ net gives Equation 26.12. The error bound Equation 26.13 is the Koksma-Hlawka inequality (Theorem 22.3) applied to \(\psi\) on the cube with the point set \(P_N\), exactly as in part 2 of Theorem 22.6; for continuous margins each \(F_{B,i}^{-1}\) is continuous and, for a bounded smooth link such as the logit, \(\psi\) is smooth and bounded, hence of bounded Hardy-Krause variation, so the Sobol’ discrepancy rate \(D_N^{*}=O((\log N)^{d}/N)\) yields the stated rate. Replacing \(f_B\) by \(f_{*}\) changes only the margins and correlation fed into \(T\), leaving the argument verbatim. \(\square\)

Remark. Discrete margins and the exact finite sum. When a covariate is discrete, its inverse distribution function is a step function, and the pullback \(\psi\) is piecewise constant in that coordinate, so its Hardy-Krause variation can be large and the smooth-integrand rate Equation 26.13 need not hold. This is not a defect in low dimension: a single binary covariate makes the integral an exact two-point sum, \(\theta_{\bullet B}=(1-\bar x_B)\,g^{-1}(\alpha_B)+\bar x_B\,g^{-1}(\alpha_B+\beta_{\mathrm{PF}})\), with no QMC error at all, and the worked example of Section 26.15 uses this exact form. For several discrete covariates, or mixed discrete and continuous sets, one enumerates the discrete cells and applies QMC within each continuous slice, or uses a randomized (Owen-scrambled, Owen (1956)) Sobol’ net whose error behaves better on non-smooth integrands. The reference software defaults to \(N=64\) Sobol’ points and provides a doubling diagnostic, check_integration(), the empirical convergence check of Definition 24.4, to verify that the integration error is negligible against the Markov chain Monte Carlo error before the results are interpreted. \(\blacktriangleleft\)

Remark. Fixed nodes, smooth likelihood. As in ML-NMR, the integration nodes \(\mathbf{x}^{(n)}=T(\mathbf{u}^{(n)})\) depend only on the comparator’s reported margins and correlation, not on the parameters \(\boldsymbol\xi\); they are computed once. The aggregate mean Equation 26.12 is then a deterministic smooth function of \(\boldsymbol\xi\) with an exact gradient, which is what makes the joint posterior amenable to Hamiltonian Monte Carlo (Equation 22.20). Resampling the nodes at each step, as plain Monte Carlo would, would make the likelihood noisy and destroy the gradient; this is the same reason QMC, not Monte Carlo, is the integration rule throughout Part V. \(\blacktriangleleft\)

This corresponds to M.5 in the proof catalogue.

26.8 Time-to-event outcomes

Survival is the outcome that motivates ML-UMR in oncology HTA, where the comparator is a single-arm trial reporting a Kaplan-Meier curve. The general-likelihood ML-NMR framework of Chapter 25 already extended aggregation from means to arbitrary individual likelihoods (Theorem 25.7); ML-UMR uses that framework in the unanchored setting, with SPFA acting on the log-hazard scale. We state the model and its consequences, building directly on the survival results of Chapter 25.

The survival workflow, before the formalism. Picture the archetypal case from Example 26.1: a single-arm trial of a new agent in advanced melanoma provides IPD (each patient’s covariates and survival time, some censored), while the comparator is a published chemotherapy trial showing only a Kaplan-Meier curve and a numbers-at-risk table. The analysis is a five-stage pipeline, and it helps to see the whole pipeline before the theorems dissect it.

1. Reconstruct pseudo-individual event and censoring times from the comparator’s published Kaplan-Meier curve and at-risk table (Theorem 26.5). These pseudo-patients carry times but no covariates.
2. Attach a covariate cloud. Pair the reconstructed comparator times with the comparator’s reported covariate distribution \(f_B\), the same Gaussian-copula reconstruction used for binary outcomes (Theorem 26.3); the covariates are integrated out, not matched one-to-one to pseudo-patients.
3. Integrate the censored likelihood over that covariate cloud, arm by arm, to form each source’s contribution to one joint likelihood (Equation 26.15, Theorem 26.6).
4. Fit the posterior by HMC, borrowing the index prognostic slope for the comparator through SPFA on the log-hazard scale.
5. Read off target estimands. For any chosen population, push the posterior through the covariate integral to get marginal survival, marginal hazard, RMST, and median survival (Theorem 26.7).

A word on direction, because survival flips the sign convention of the binary example. There a higher response probability was better, so a positive \(\Delta_{AB}\) favored \(A\). With survival a lower hazard is better and a longer survival is better: a hazard ratio of \(A\) versus \(B\) below \(1\), or an RMST difference (\(A\) minus \(B\)) above \(0\), favors the index treatment. We keep each estimand’s numerator explicit and restate the direction at each use.

Theorem 26.4 (ML-UMR for time-to-event outcomes) Let the index and comparator hazards follow a parametric survival model with shared baseline shape and SPFA on the log-hazard scale.

Proportional hazards (PH). With a baseline hazard \(h_0(t)>0\) and SPFA prognostic slope \(\boldsymbol\beta_{\mathrm{PF}}\), \[ h_k(t\mid\mathbf{x})=h_0(t)\,\exp\!\big(\eta_k(\mathbf{x})\big), \qquad \eta_k(\mathbf{x})=\alpha_k+\mathbf{x}^\top\boldsymbol\beta_{\mathrm{PF}},\quad k\in\{A,B\}, \tag{26.14}\] which is the PH model Equation 25.5 with the single unanchored contrast \(\alpha_A-\alpha_B\) in place of the treatment effect and no effect-modification term.

Accelerated failure time (AFT). With an error distribution \(F_\epsilon\) and scale \(\sigma\), \(\log T=\eta_k(\mathbf{x})+\sigma\epsilon\), \(\epsilon\sim F_\epsilon\).

Then:

(a) Likelihood. The index contributes the right-censored individual likelihood Equation 25.12 at observed covariates; the comparator contributes, for each (possibly reconstructed) observation \((t_i,\delta_i)\) with unobserved covariates distributed as \(f_B\), the integrated censored likelihood \[ L_i^{(B)}(\boldsymbol\xi)=\int_{\mathcal{X}}h_B(t_i\mid\mathbf{x})^{\delta_i}\,S_B(t_i\mid\mathbf{x})\,f_B(\mathbf{x})\,d\mathbf{x}, \tag{26.15}\] well defined and finite by Theorem 25.5 and evaluated by the QMC rule Theorem 26.3.

(b) Conditional log hazard ratio. Under PH and SPFA the conditional log hazard ratio of \(A\) versus \(B\) is the time-invariant, covariate-free, population-free scalar \[ \log\frac{h_A(t\mid\mathbf{x})}{h_B(t\mid\mathbf{x})}=\eta_A(\mathbf{x})-\eta_B(\mathbf{x})=\alpha_A-\alpha_B \qquad\text{for all }t,\mathbf{x}, \tag{26.16}\] by Theorem 25.2 and Equation 26.5.

(c) Target-population marginal survival, hazard, RMST. For any target \(\mathcal{P}^{*}\), the marginal survival is the covariate average \(S_k(t\mid\mathcal{P}^{*})=\int S_k(t\mid\mathbf{x})f_{*}(\mathbf{x})\, d\mathbf{x}\); the marginal hazard is the survivor-weighted average Equation 25.7; and the restricted mean survival time is the collapsible average \(\operatorname{RMST}_k(\tau\mid\mathcal{P}^{*})= \int\operatorname{RMST}_k(\tau\mid\mathbf{x})f_{*}(\mathbf{x})\,d\mathbf{x}\) of Theorem 25.4.

(d) The marginal hazard ratio is time-varying. Even under the conditional PH and SPFA of Equation 26.14, the marginal hazard ratio \(h_A(t\mid\mathcal{P}^{*})/h_B(t\mid\mathcal{P}^{*})\) is in general not constant in \(t\), and is population-dependent, by Theorem 25.3.

(e) Scale misalignment: RMST is collapsible but not freely transportable. Although the RMST contrast is collapsible within any population, it is not transportable across populations from the log-hazard model. Precisely, under Equation 26.14 with \(\boldsymbol\beta_{\mathrm{PF}}\ne\mathbf{0}\), the conditional RMST contrast \(\mathbf{x}\mapsto\operatorname{RMST}_A(\tau\mid\mathbf{x})-\operatorname{RMST}_B(\tau\mid\mathbf{x})\) is a non-constant function of \(\mathbf{x}\), so its population average \[ \Delta\!\operatorname{RMST}_{AB}(\tau\mid\mathcal{P}^{*})=\int_{\mathcal{X}}\big[\operatorname{RMST}_A(\tau\mid\mathbf{x})-\operatorname{RMST}_B(\tau\mid\mathbf{x})\big]\,f_{*}(\mathbf{x})\,d\mathbf{x} \tag{26.17}\] depends on the whole target distribution \(f_{*}\) and differs across target populations even though the conditional log-hazard contrast Equation 26.16 is the single population-free scalar \(\alpha_A-\alpha_B\).

Proof. (a). The index term is Definition 25.4, derived from independent censoring in Proposition 25.1. For the comparator, the conditional kernel \(h_B(t\mid\mathbf{x})^{\delta} S_B(t\mid\mathbf{x})\) is a nonnegative function of \(\mathbf{x}\) that is a density (in \(t\)) when \(\delta=1\) and a probability (the survival) when \(\delta=0\); in either case it is integrable against \(f_B\) by Theorem 25.5, giving the well-defined Equation 26.15, and the QMC evaluation is Theorem 26.3 applied to this integrand. (This integrated censored likelihood is the \(\pi_{\mathrm{Ind}}=h^{\delta}S\) instance of the general-likelihood theorem Theorem 25.7.)

(b). By Theorem 25.2 the conditional log hazard ratio under PH is \(\eta_A(\mathbf{x})- \eta_B(\mathbf{x})\), free of \(t\) and of the baseline; under SPFA this equals \(\alpha_A-\alpha_B\) by Equation 26.5, free also of \(\mathbf{x}\).

(c). Marginal survival is by definition the covariate average of conditional survival, and RMST collapsibility is Theorem 25.4 (a Tonelli exchange of the time and covariate integrals); the survivor-weighted marginal hazard is Lemma 25.1. All three carry over verbatim with \(f_{\mathcal{P}}=f_{*}\).

(d). Apply Theorem 25.3: writing \(h_B(t\mid\mathbf{x})=h_0(t)e^{\eta_B(\mathbf{x})}\) as a frailty model (one in which each individual’s hazard is scaled by a fixed positive multiplier, the frailty) with frailty \(\rho(\mathbf{x})=e^{\mathbf{x}^\top\boldsymbol\beta_{\mathrm{PF}}}\) (non-degenerate when \(\boldsymbol\beta_{\mathrm{PF}}\ne\mathbf{0}\) and the covariate varies under \(f_{*}\)), the constant conditional ratio is \(\psi=e^{\alpha_A-\alpha_B}\), and the theorem gives a marginal hazard ratio that equals \(\psi\) only at \(t=0\) and is strictly attenuated toward the null for \(t>0\), hence time-varying; the attenuation depends on \(f_{*}\) through the survivor distribution Equation 25.8, hence is population-dependent.

(e). Fix two covariate values \(\mathbf{x},\mathbf{x}'\) with \(\mathbf{x}^\top\boldsymbol\beta_{\mathrm{PF}} \ne\mathbf{x}'^\top\boldsymbol\beta_{\mathrm{PF}}\) (possible since \(\boldsymbol\beta_{\mathrm{PF}}\ne \mathbf{0}\)). The conditional survival under arm \(k\) is, by Theorem 25.1, \(S_k(t\mid\mathbf{x})=\exp(-e^{\eta_k(\mathbf{x})}H_0(t))\) with \(H_0(t)=\int_0^t h_0\), so the conditional RMST is \(\operatorname{RMST}_k(\tau\mid\mathbf{x})=\int_0^{\tau}\exp(-e^{\alpha_k+\mathbf{x}^\top \boldsymbol\beta_{\mathrm{PF}}}H_0(s))\,ds\), a strictly decreasing function of the scalar \(e^{\mathbf{x}^\top\boldsymbol\beta_{\mathrm{PF}}}\) (larger frailty, smaller survival, smaller area). The contrast \[ \Delta(\mathbf{x}):=\operatorname{RMST}_A(\tau\mid\mathbf{x})-\operatorname{RMST}_B(\tau\mid\mathbf{x}) =\int_0^{\tau}\Big[e^{-e^{\alpha_A}r(\mathbf{x})H_0(s)}-e^{-e^{\alpha_B}r(\mathbf{x})H_0(s)}\Big]ds, \quad r(\mathbf{x}):=e^{\mathbf{x}^\top\boldsymbol\beta_{\mathrm{PF}}}, \] is a non-constant function of \(r(\mathbf{x})\) whenever \(\alpha_A\ne\alpha_B\): as \(r(\mathbf{x})\downarrow0\) both exponentials tend to \(1\) and \(\Delta\to0\), while at a finite \(r(\mathbf{x})\) the two areas differ, so \(\Delta(\mathbf{x})\) is not constant in \(\mathbf{x}\). Therefore its average Equation 26.17 against \(f_{*}\) is a genuine functional of the distribution of \(r(\mathbf{X})\) under \(f_{*}\), not a function of a single moment, and two targets with different covariate distributions give different \(\Delta\!\operatorname{RMST}_{AB}(\tau\mid\mathcal{P}^{*})\), even though the conditional log-hazard contrast Equation 26.16 is the same scalar \(\alpha_A-\alpha_B\) in every population. \(\square\)

Remark. Part (e) in two numbers. The scale-misalignment proof is the densest block in the chapter, so here is a concrete check with the exponential model of the survival coda (Section 26.15): \(h_k(t\mid x)= \lambda_k e^{\beta x}\), \(\beta=\log 2\), \(\lambda_A=0.10\), \(\lambda_B=0.15\), horizon \(\tau=10\). The conditional RMST contrast \(\Delta(x)=\operatorname{RMST}_A(\tau\mid x)-\operatorname{RMST}_B(\tau\mid x)\) with \(\operatorname{RMST}_k(\tau\mid x)=(1-e^{-\lambda_k e^{\beta x}\tau})/(\lambda_k e^{\beta x})\) evaluates to \(\Delta(0)=6.321-5.179=1.142\) at \(x=0\) and \(\Delta(1)=4.323-3.167=1.156\) at \(x=1\): not the same number. Because \(\Delta(x)\) varies with \(x\), its population average shifts with the covariate mix, giving a target-population RMST contrast of \(0.5(1.142)+0.5(1.156)=1.149\) under \(P(X=1)=0.5\) but \(0.7(1.142)+0.3(1.156)=1.146\) under \(P(X=1)=0.3\). The gap is small in this benign example, but it is genuine, and it grows with the strength of the covariate effect \(\beta\) and the horizon \(\tau\). The conditional log hazard ratio, by contrast, is the single number \(\log(\lambda_B/\lambda_A)\) in every population: that is the transportable object, and the RMST contrast is not. \(\blacktriangleleft\)

Remark. Which estimand to report. Part (b) gives the clean transportable object: the conditional log hazard ratio, a single number under SPFA, valid in every population. Part (d) warns that the marginal hazard ratio, the quantity a single Cox fit reports, is time-varying and population-dependent, hence not a transportable parameter; the mlumr default reports the scalar marginal hazard ratio as its value at the start of follow-up, \(t=0\), and exposes the full curve through predict(type = "loghr"). Part (e) is the subtle one, and is the unanchored instance of the scale-misalignment caveat flagged after Theorem 25.4 in Chapter 25: the RMST contrast is collapsible and clinically interpretable, but because the model is fitted on the log-hazard scale while RMST lives on the time scale, transporting the RMST contrast to a population with a different covariate distribution requires that target distribution and is not licensed by SPFA on the log-hazard scale alone. The practical recommendation, following Chandler and Ishak (2026c), is to report the conditional log hazard ratio as the primary transportable summary and the target-population RMST as a collapsible read-out in each named population. \(\blacktriangleleft\)

Remark. Baselines and distributions. The reference implementation supports exponential, Weibull, and Gompertz PH models and their AFT counterparts (exponential-AFT, Weibull-AFT, log-normal, log-logistic, gamma, generalized gamma), plus flexible M-spline and piecewise-exponential baselines with a random-walk smoothing prior, following the multinma survival models of Phillippo et al. (2025). The choice of baseline shape is orthogonal to the SPFA structure: SPFA constrains the covariate slope, the baseline \(h_0(t)\) is shared across arms by the PH assumption, and the unanchored contrast is carried entirely by \(\alpha_A-\alpha_B\). \(\blacktriangleleft\)

This theorem corresponds to M.6 in the proof catalogue.

26.9 Reconstructing pseudo-individual data from a Kaplan-Meier curve

The comparator likelihood Equation 26.15 is a product over individual observations \((t_i,\delta_i)\), but a published survival study reports a Kaplan-Meier curve and a numbers-at-risk table, not individual times. We reconstruct pseudo-individual data that reproduce the published curve, and prove that the reconstruction is exact in the idealized noiseless case.

Recall the Kaplan-Meier (product-limit) estimator. With distinct observed event times \(t_1<t_2<\cdots<t_m\), \(n_j\) patients at risk just before \(t_j\), and \(d_j\) events at \(t_j\), the estimator is \[ \widehat S(t)=\prod_{j:\,t_j\le t}\Big(1-\frac{d_j}{n_j}\Big), \qquad \widehat S(t)=1\text{ for }t<t_1 . \tag{26.18}\] Between consecutive event times the number at risk decreases by the events at \(t_j\) and by the patients censored in \([t_j,t_{j+1})\), say \(c_j\) of them: \(n_{j+1}=n_j-d_j-c_j\).

Theorem 26.5 (Exact reconstruction of pseudo-IPD from a Kaplan-Meier curve) Suppose the published report provides, without digitization error, the step survival curve Equation 26.18 (equivalently the event times \(t_1<\cdots<t_m\) and the survival values \(\widehat S(t_j)\)) together with the numbers at risk \(n_1,\dots,n_m\) at the event times and the initial sample size \(n_B=n_1+(\text{any pre-}t_1\text{ censoring})\). Then:

1. The event counts are uniquely determined by the curve and the risk set, \[ d_j=n_j\Big(1-\frac{\widehat S(t_j)}{\widehat S(t_{j-1})}\Big),\qquad j=1,\dots,m,\ \ \widehat S(t_0):=1, \tag{26.19}\] and the censoring counts are determined by the risk-set recursion \(c_j=n_j-d_j-n_{j+1}\) (with \(n_{m+1}\) the final reported at-risk number).

2. Any pseudo-sample \(\{(t_i,\delta_i)\}_{i=1}^{n_B}\) that realizes these per-interval event counts \(d_j\) and censoring counts \(c_j\) (placing \(d_j\) events exactly at \(t_j\) and \(c_j\) censorings anywhere in \([t_j,t_{j+1})\)) has Kaplan-Meier estimator identical to the published Equation 26.18 at every \(t\).

3. The reconstruction preserves the marginal survival distribution exactly but discards individual-level covariate information: the pseudo-sample carries no covariate values, so it informs only the marginal \((t,\delta)\) law and not any covariate-stratified survival.

Proof. (1). From the product form Equation 26.18, consecutive values satisfy \(\widehat S(t_j)/\widehat S(t_{j-1})=1-d_j/n_j\), because the product for \(\widehat S(t_j)\) is the product for \(\widehat S(t_{j-1})\) times the single extra factor \((1-d_j/n_j)\). Solving for \(d_j\) gives Equation 26.19; the value is uniquely determined because \(n_j>0\) and the ratio \(\widehat S(t_j)/\widehat S(t_{j-1})\in(0,1]\) are both given. With \(d_j\) known and the at-risk numbers \(n_j,n_{j+1}\) given, the risk-set conservation \(n_{j+1}=n_j-d_j-c_j\) solves uniquely for \(c_j=n_j-d_j- n_{j+1}\). (These are nonnegative integers up to the rounding inherent in reading \(\widehat S\); the Guyot algorithm rounds and redistributes to keep the counts integer and the at-risk numbers matched.)

(2). Construct a pseudo-sample with exactly \(d_j\) events at time \(t_j\) and \(c_j\) censorings at arbitrary times in \([t_j,t_{j+1})\). Its Kaplan-Meier estimator is computed from Equation 26.18 using its own event times and risk sets. At time \(t_1\) the pseudo-sample has \(n_B\) at risk and \(d_1\) events, matching \(n_1,d_1\). Inductively, if the pseudo-sample has \(n_j\) at risk just before \(t_j\) (matching the published \(n_j\)), then it has \(d_j\) events at \(t_j\) and \(c_j\) censorings in \([t_j,t_{j+1})\) by construction, so just before \(t_{j+1}\) it has \(n_j-d_j-c_j=n_{j+1}\) at risk, matching the published \(n_{j+1}\); the censorings in the open interval contribute no event factor and only reduce the risk set. Hence at every event time the pseudo-sample has the same \((n_j,d_j)\) as the published data, so its product-limit estimator equals Equation 26.18 factor by factor, at every \(t\). (The placement of censorings within an interval is irrelevant to the estimator, which only counts how many remain at risk at the next event time, so the reconstruction is unique as a curve even though the exact censoring times are not.)

(3). The pseudo-sample is a list of times and event indicators with no covariates attached, so any statistic that depends only on the marginal \((t,\delta)\) distribution (the survival curve, the marginal hazard, the RMST) is reproduced, while any covariate-conditional statistic is not recoverable from it. This is the standard limitation: the reconstruction is faithful at the margin and silent about within-strata structure. \(\square\)

Example 26.3 (A three-event Kaplan-Meier inversion by hand) Suppose the comparator report gives \(n_B=20\) patients, event times \(t_1<t_2<t_3\), at-risk numbers \(n_1=20\), \(n_2=14\), \(n_3=10\) (and \(n_4=8\) after \(t_3\)), and the published step-curve values \(\widehat S(t_1)=0.800\), \(\widehat S(t_2)=0.629\), \(\widehat S(t_3)=0.503\). Apply Equation 26.19 with \(\widehat S(t_0):=1\): \[ d_1=20\Big(1-\tfrac{0.800}{1}\Big)=4,\quad d_2=14\Big(1-\tfrac{0.629}{0.800}\Big)=3,\quad d_3=10\Big(1-\tfrac{0.503}{0.629}\Big)=2, \] and the risk-set recursion \(c_j=n_j-d_j-n_{j+1}\) gives the censoring counts \(c_1=20-4-14=2\), \(c_2=14-3-10=1\), \(c_3=10-2-8=0\). So the published curve and at-risk table pin down exactly \(4\) events then \(2\) censorings, \(3\) events then \(1\) censoring, then \(2\) events with \(8\) still at risk. Any pseudo-sample placing events and censorings this way reproduces the curve factor by factor (part 2 of the theorem). But note what the pseudo-sample does not have: a covariate for any of these \(20\) patients. The reconstruction recovers the marginal survival times and nothing about who was biomarker-positive or how old anyone was. That is why the survival likelihood Equation 26.15 still integrates over the reported covariate distribution \(f_B\): the pseudo-times supply the \((t,\delta)\) margin, \(f_B\) supplies the covariate cloud, and the two are combined but never matched patient by patient. \(\blacktriangleleft\)

Remark. From idealized inversion to the Guyot algorithm. Theorem Theorem 26.5 is the exact case in which the at-risk number is known at every event time. In practice the numbers at risk are reported only at a coarse grid of follow-up times, and the survival values are read off a figure with digitization error. The algorithm of Guyot and colleagues (Guyot et al. 2012) solves the same inversion under these realistic constraints: it allocates censorings between reported at-risk times so that the implied risk sets match the table, then applies Equation 26.19 event by event, iterating to reconcile the curve drops with the at-risk evolution. The output is pseudo-IPD whose Kaplan-Meier curve matches the published one to within digitization noise; Chandler and Ishak (2026c) and the mlumr function set_agd_surv() implement it for the comparator arm. Sensitivity analyses over several independent digitizations quantify the reconstruction uncertainty, which is the price the unanchored survival comparison pays for having only a published figure. \(\blacktriangleleft\)

This theorem corresponds to M.7 in the proof catalogue, and supplies the reconstruction whose existence Chapter 25 invoked in the remark following Theorem 25.6.

26.10 The joint individual-plus-aggregate likelihood

The two data sources are combined into one likelihood, and one posterior. The factorization is the unanchored instance of the aggregate arm likelihood of Chapter 25.

Theorem 26.6 (The joint ML-UMR likelihood) Assume the index observations are conditionally independent given their covariates and \(\boldsymbol\xi\), and the comparator observations are conditionally independent given their (unobserved) covariates and \(\boldsymbol\xi\), with comparator covariates drawn independently from \(f_B\). Then the joint likelihood of the index IPD \(\mathbf{Y}^{(A)}\) and the comparator data \(\mathbf{Y}^{(B)}\) factorizes as \[ L_{\mathrm{joint}}(\boldsymbol\xi)=\underbrace{\prod_{i=1}^{n_A}\pi_{\mathrm{Ind}}\!\big(y_i^{(A)}\mid g^{-1}(\eta_A(\mathbf{x}_i)),\boldsymbol\phi\big)}_{\text{index IPD, covariates observed}}\;\cdot\;\underbrace{\prod_{i=1}^{n_B}\int_{\mathcal{X}}\pi_{\mathrm{Ind}}\!\big(y_i^{(B)}\mid g^{-1}(\eta_B(\mathbf{x})),\boldsymbol\phi\big)\,f_B(\mathbf{x})\,d\mathbf{x}}_{\text{comparator AgD, covariates integrated}}, \tag{26.20}\] where for survival the kernel is the censored likelihood \(h^{\delta}S\) and the comparator factors use the reconstructed pseudo-IPD of Theorem 26.5. With a prior \(\pi(\boldsymbol\xi)\) the posterior is \(\pi(\boldsymbol\xi\mid\mathcal{D})\propto L_{\mathrm{joint}}(\boldsymbol\xi)\, \pi(\boldsymbol\xi)\), and the two sources are weighted automatically by their information content.

Proof. Index factor. In the index source the covariates \(\mathbf{x}_i\) are observed, so each individual contributes its conditional likelihood \(\pi_{\mathrm{Ind}}(y_i^{(A)}\mid g^{-1}(\eta_A(\mathbf{x}_i)), \boldsymbol\phi)\) evaluated at the known covariate; conditional independence makes the index likelihood the product of these, the first factor of Equation 26.20.

Comparator factor. In the comparator source the covariates are unobserved. By Theorem 25.5, the marginal likelihood of one comparator observation \(y_i^{(B)}\), integrating out its covariate distributed as \(f_B\), is \(\int_{\mathcal{X}}\pi_{\mathrm{Ind}}(y_i^{(B)}\mid g^{-1}(\eta_B(\mathbf{x})),\boldsymbol\phi)f_B(\mathbf{x})\,d\mathbf{x}\); by Theorem 25.6, conditional independence of the comparator outcomes given covariates together with independent covariate draws makes the comparator likelihood the product of these marginal contributions, the second factor of Equation 26.20. (For a sufficient summary, a binomial event count or a normal sample mean, the product collapses to the corresponding aggregate one-line likelihood with mean \(\theta_{\bullet B}\), as in Definition 20.2; for survival no single summary is sufficient and the product runs over the reconstructed pseudo-IPD.)

Joint factorization and posterior. The index and comparator sources are independent given \(\boldsymbol\xi\) (different patients in different studies), so their joint likelihood is the product of the two factors, which is Equation 26.20. Multiplying by the prior and normalizing gives the posterior. \(\square\)

Why this matters. This single likelihood is what makes ML-UMR a model rather than a two-step recipe: the index IPD and the comparator AgD enter one posterior, so uncertainty flows automatically from each source into the final contrast, and the same fitted object yields the estimand in every target population. It is the unanchored counterpart of the ML-NMR likelihood of Chapter 20, with the comparator arm’s covariates integrated out instead of observed. The next remark is a caution against over-reading this: more comparator data sharpen \(\alpha_B\) but never test the borrowed slope.

Remark. Automatic information weighting, and what it is not. The posterior weights each source by its likelihood curvature: a large index sample sharpens \((\alpha_A,\boldsymbol\beta_{\mathrm{PF}})\), and a large comparator sample (more events, more at-risk follow-up) sharpens \(\alpha_B\) through the aggregate factor. This is unlike MAIC and STC, which condition on the index IPD and use the comparator data only for reweighting or for the comparator-arm value; ML-UMR puts both data types into one likelihood and lets the posterior balance them. What the weighting is not is a substitute for the identifying assumption: the comparator factor informs only the one scalar \(\alpha_B\) (Type 1) or the \((d+1)\)-vector \((\alpha_B, \boldsymbol\beta_B)\) when subgroups are reported (Type 2, Theorem 26.2); the slope \(\boldsymbol\beta_{\mathrm{PF}}\) that projects the index surface onto the comparator arm comes from the IPD under SPFA, and no amount of comparator data tests that projection in the Type 1 model. The reference model also has no random-effects term for between-study heterogeneity: the comparator rows are treated as one population, a limitation recorded in the mlumr documentation. \(\blacktriangleleft\)

This theorem corresponds to M.8 in the proof catalogue.

26.11 Marginal survival, hazard, and RMST in target populations

From the joint posterior we read off any target-population estimand by pushing the posterior through the integration pipeline. We state the construction and its convergence.

Theorem 26.7 (Target-population survival estimands from the posterior) Let \(\pi(\boldsymbol\xi\mid\mathcal{D})\) be the ML-UMR posterior Theorem 26.6, and fix a target population \(\mathcal{P}^{*}\) with covariate density \(f_{*}\) and a horizon \(\tau\). For each arm \(k\in\{A,B\}\) define the functionals \[ S_k(t\mid\mathcal{P}^{*};\boldsymbol\xi)=\!\int S_k(t\mid\mathbf{x};\boldsymbol\xi)f_{*}(\mathbf{x})d\mathbf{x}, \quad \operatorname{RMST}_k(\tau\mid\mathcal{P}^{*};\boldsymbol\xi)=\!\int_0^{\tau}\! S_k(t\mid\mathcal{P}^{*};\boldsymbol\xi)\,dt, \tag{26.21}\] and the marginal hazard \(h_k(t\mid\mathcal{P}^{*};\boldsymbol\xi)\) by the survivor-weighted formula Equation 25.7. Then:

1. Each functional is a continuous (indeed smooth) function of \(\boldsymbol\xi\), so its posterior is the well-defined pushforward of \(\pi(\boldsymbol\xi\mid\mathcal{D})\), summarized by posterior draws.

2. The estimands of the table below are obtained from these functionals; the survival, RMST, and median are collapsible read-outs (averages of conditional quantities), while the marginal hazard ratio and its log are non-collapsible and time-varying by Theorem 26.4(d).

Estimand	Formula
Marginal survival	\(S_k(t\mid\mathcal{P}^{*})=\int S_k(t\mid\mathbf{x})f_{*}(\mathbf{x})\,d\mathbf{x}\)
Marginal hazard	\(h_k(t\mid\mathcal{P}^{*})=\dfrac{\int h_k(t\mid\mathbf{x})S_k(t\mid\mathbf{x})f_{*}}{\int S_k(t\mid\mathbf{x})f_{*}}\)
RMST	\(\operatorname{RMST}_k(\tau\mid\mathcal{P}^{*})=\int_0^{\tau}S_k(t\mid\mathcal{P}^{*})\,dt\)
RMST contrast	\(\Delta\!\operatorname{RMST}_{AB}(\tau\mid\mathcal{P}^{*})=\operatorname{RMST}_A-\operatorname{RMST}_B\)
Marginal log HR at \(t\)	\(\log\!\big[h_A(t\mid\mathcal{P}^{*})/h_B(t\mid\mathcal{P}^{*})\big]\)
Median survival	the \(t\) solving \(S_k(t\mid\mathcal{P}^{*})=\tfrac12\)

3. Computed by QMC over \(N\) Sobol’ nodes for the inner covariate integral and \(S\) posterior draws for the outer parameter integral, the posterior mean of any of these estimands converges, as \(N\to\infty\) and \(S\to\infty\), to the true posterior mean, with QMC error \(O((\log N)^{d}/N)\) (Theorem 26.3) and Monte Carlo error \(O(S^{-1/2})\).

Proof. (1). For fixed \(\mathbf{x}\) and \(t\), \(S_k(t\mid\mathbf{x};\boldsymbol\xi)=\exp(-e^{\eta_k(\mathbf{x}; \boldsymbol\xi)}H_0(t;\boldsymbol\xi))\) is a smooth function of \(\boldsymbol\xi\) (composition of the smooth exponential with the affine \(\eta_k\) and the parametric cumulative baseline \(H_0\)), bounded in \([0,1]\). Integrating against the probability measure \(f_{*}\) preserves smoothness and boundedness by differentiation under the integral sign (dominated convergence, the derivative bounded by an integrable envelope), so \(S_k(t\mid\mathcal{P}^{*};\boldsymbol\xi)\) is smooth in \(\boldsymbol\xi\); integrating over \(t\in[0,\tau]\) gives the smooth RMST functional by Tonelli (Theorem 4.8, Theorem 25.4). A smooth function of \(\boldsymbol\xi\) has a well-defined pushforward law under the posterior, estimated by applying the functional to each posterior draw.

(2). The marginal survival, RMST, RMST contrast, and median are averages or monotone read-outs of conditional survival, hence collapsible by Theorem 25.4; the marginal hazard is the survivor-weighted average Lemma 25.1, and its ratio across arms is non-collapsible and time-varying by Theorem 26.4(d). The table simply records these formulas.

(3). For each posterior draw \(\boldsymbol\xi^{(s)}\), the inner integral \(S_k(t\mid\mathcal{P}^{*}; \boldsymbol\xi^{(s)})\) is approximated by the QMC average \(\frac1N\sum_n S_k(t\mid\mathbf{x}^{(n)}; \boldsymbol\xi^{(s)})\) over Sobol’ nodes \(\mathbf{x}^{(n)}=T(\mathbf{u}^{(n)})\) for \(f_{*}\), with error \(O((\log N)^{d}/N)\) by Theorem 26.3 (the survival integrand is smooth and bounded). The posterior mean is then the average over draws, \(\frac1S\sum_s(\cdot)\), with Monte Carlo error \(O(S^{-1/2})\) by the central limit theorem for the (stationary) posterior sample, exactly as in Bayesian g-computation (Definition 19.5). The two errors add, and both vanish as \(N,S\to\infty\). \(\square\)

Which of the six estimands should a health researcher actually report? Table 26.4 pairs each with the clinical question it answers and flags whether it travels across populations. The rule of thumb, defended in the remark below, is to report the conditional log hazard ratio as the transportable summary and the RMST (or median) as the clinically interpretable read-out in each named target population, and to treat the marginal hazard ratio with care, because it is neither constant in time nor portable across populations.

Table 26.4: Which survival estimand to report. Everything except the conditional log hazard ratio is population-specific and must be recomputed for each target \(\mathcal{P}^{*}\); the RMST and median are at least collapsible (clean averages of conditional quantities), while the marginal hazard ratio is neither collapsible nor time-constant.

Estimand	Clinical question it answers	Transportable?
Marginal survival \(S_k(t\mid\mathcal{P}^{*})\)	what fraction of this population is alive at \(t\) under arm \(k\)	population-specific
Median survival	the time by which half this population has had the event	population-specific
RMST \(\operatorname{RMST}_k(\tau\mid\mathcal{P}^{*})\)	average event-free time up to horizon \(\tau\), per arm	collapsible, but population-specific
RMST contrast	average event-free time gained on \(A\) versus \(B\) up to \(\tau\)	collapsible; differs across populations (Theorem 26.4)
Marginal hazard ratio at \(t\)	instantaneous risk ratio in this population at \(t\)	not transportable; time-varying
Conditional log HR \(\alpha_A-\alpha_B\)	the per-patient log risk ratio, identical for all under SPFA	the single transportable scalar

Remark. The whole point of the marginalization. Parts (1) and (3) are what let ML-UMR report a survival contrast in a population represented by neither trial: choose any \(f_{*}\), push the posterior through Equation 26.21, and summarize. The honest cost is part (2) together with Theorem 26.4(e): the marginal hazard ratio is not a single transportable number, and the RMST contrast, though collapsible, must be recomputed for each target distribution. The transportable scalar is the conditional log hazard ratio \(\alpha_A-\alpha_B\) of Equation 26.16, valid in every population under SPFA. \(\blacktriangleleft\)

This theorem corresponds to M.9 in the proof catalogue.

26.12 ML-UMR and unanchored STC

ML-UMR has a pairwise ancestor. We show that, in the comparator population, the SPFA model reduces exactly to unanchored STC, and that its extension to other targets is what the multilevel formulation buys, at the cost of the stronger assumption.

Put plainly, and this is the motivation worth holding before the formalism: we want to know exactly what ML-UMR buys over the simpler method a health researcher may already know. So we prove the two agree where STC is valid, the comparator population, and pinpoint where they part company, every other target. The agreement is reassuring (ML-UMR is not doing anything exotic at the comparator); the extension to other populations is the entire value added; and SPFA is the price of that extension. A reader who trusts this paragraph can read the theorem as its proof.

Theorem 26.8 (Equivalence with unanchored STC at the comparator, extension elsewhere) Adopt SPFA and suppose the comparator AgD provide a sufficient outcome summary (an event count, a sample mean) so that the comparator likelihood factor of Theorem 26.6 is the aggregate model with mean \(\theta_{\bullet B}\), and let \(\hat{\boldsymbol\xi}\) be the (limiting) maximizer of the joint likelihood. Then:

1. (Calibration identity.) At \(\hat{\boldsymbol\xi}\) the fitted comparator aggregate mean equals the observed comparator outcome, \(\theta_{\bullet B}(\hat{\boldsymbol\xi})=\theta_{\bullet B}^{\mathrm{obs}}\); equivalently \(\bar\theta_B(\mathcal{P}_B)=\theta_{\bullet B}^{\mathrm{obs}}\).

2. (Equivalence at the comparator population.) With target \(\mathcal{P}^{*}=\mathcal{P}_B\), the ML-UMR estimand Equation 26.1 equals the unanchored marginal STC estimator of Definition 19.4, namely \[ \Delta_{AB}(\mathcal{P}_B)=g\!\big(\bar\theta_A(\mathcal{P}_B)\big)-g\!\big(\theta_{\bullet B}^{\mathrm{obs}}\big), \qquad \bar\theta_A(\mathcal{P}_B)=\int_{\mathcal{X}}g^{-1}\!\big(\eta_A(\mathbf{x})\big)f_B(\mathbf{x})\,d\mathbf{x}, \tag{26.22}\] the index surface integrated over the comparator covariate distribution, contrasted on the link scale with the observed comparator outcome. Under the conditions of Theorem 19.3 and the unanchored condition of Theorem 19.6(2), both are consistent for \(\Delta_{AB}(\mathcal{P}_B)\).

3. (Strict extension elsewhere.) For a target \(\mathcal{P}^{*}\ne\mathcal{P}_B\), ML-UMR produces \(\Delta_{AB}(\mathcal{P}^{*})\) by integrating both arms against \(f_{*}\) using the fitted \(\alpha_B\) and \(\boldsymbol\beta_{\mathrm{PF}}\), whereas unanchored STC, lacking a model for arm \(B\) beyond the observed comparator outcome, cannot move arm \(B\) off \(\mathcal{P}_B\). The extension uses SPFA, which is strictly stronger than the assumption STC needs for its native comparator-population estimand (Proposition 26.1).

Proof. (1). Write the comparator factor as the aggregate model \(\pi_{\mathrm{Agg}}(\text{summary}_B\mid \theta_{\bullet B}(\boldsymbol\xi),n_B)\), a one-parameter exponential-family likelihood in the mean \(\theta_{\bullet B}\) (binomial in the event proportion, normal in the sample mean). Its score in \(\theta_{\bullet B}\) vanishes exactly when the fitted mean equals the observed sufficient statistic, the standard mean-matching property of the exponential family; since \(\boldsymbol\beta_{\mathrm{PF}}\) is fixed by the index factor and \(\theta_{\bullet B}\) is a strictly monotone function of the free parameter \(\alpha_B\) (Lemma 23.2), the joint maximizer sets \(\theta_{\bullet B}(\hat{\boldsymbol\xi}) =\theta_{\bullet B}^{\mathrm{obs}}\). By Equation 26.8, \(\theta_{\bullet B}=\bar\theta_B(\mathcal{P}_B)\), giving the identity.

(2). At target \(\mathcal{P}_B\) the ML-UMR arm means are \(\bar\theta_A(\mathcal{P}_B)=\int g^{-1}(\eta_A) f_B\) and \(\bar\theta_B(\mathcal{P}_B)=\theta_{\bullet B}^{\mathrm{obs}}\) by part (1), so the estimand Equation 26.1 is \(g(\bar\theta_A(\mathcal{P}_B))-g(\theta_{\bullet B}^{\mathrm{obs}})\), which is Equation 26.22. The unanchored marginal STC estimator of Definition 19.4 fits the outcome surface on the index IPD, predicts arm \(A\) in the comparator population as \(\hat\mu_A^{\mathrm{STC}}=\int g^{-1} (\eta_A)f_B=\bar\theta_A(\mathcal{P}_B)\), and contrasts it on the link scale with the comparator’s own observed outcome \(g(\theta_{\bullet B}^{\mathrm{obs}})\); this is exactly Equation 26.22. Hence the two coincide. Consistency of the arm-\(A\) prediction is Theorem 19.3 under correct specification and transportability of the index surface to \(\mathcal{P}_B\); the unanchored validity of using the comparator’s observed outcome as the arm-\(B\) value is Theorem 19.6(2) under constancy of absolute effects, which SPFA supplies. The Bayesian version of Equation 26.22 is the pushforward of the posterior through the same functional, the Bayesian g-computation of Definition 19.5.

(3). For \(\mathcal{P}^{*}\ne\mathcal{P}_B\), ML-UMR evaluates \(\bar\theta_B(\mathcal{P}^{*})=\int g^{-1} (\alpha_B+\mathbf{x}^\top\boldsymbol\beta_{\mathrm{PF}})f_{*}(\mathbf{x})\,d\mathbf{x}\) with the fitted \(\alpha_B\) and the SPFA slope; this requires the arm-\(B\) conditional surface at covariate values typical of \(\mathcal{P}^{*}\), which Lemma 26.2 supplies only because SPFA fixes the comparator slope equal to the index slope. Unanchored STC, having fit no outcome model for treatment \(B\) (no comparator IPD), has no value for arm \(B\) away from the observed comparator outcome, so it cannot evaluate \(\bar\theta_B(\mathcal{P}^{*})\); its estimand is defined only at \(\mathcal{P}_B\). Thus ML-UMR strictly extends the set of admissible targets, and the extension is powered by SPFA, which Proposition 26.1 shows is strictly stronger than the shared-effect-modifier assumption. \(\square\)

Remark. The honest summary. Chandler and Ishak (2026a) state in their conclusions that ML-UMR “simplifies to unanchored STC in the comparator population”; part (2) is that statement, proved. The value added by ML-UMR is not at the comparator, where it agrees with the older and assumption-lighter STC, but at other target populations, where STC is silent. That added scope is bought entirely with SPFA. Whether the extra targets are worth the stronger, untestable assumption is an application-specific judgment, and the safe default, when SPFA is doubtful, is to report only the comparator-population contrast, where ML-UMR and unanchored STC coincide. \(\blacktriangleleft\)

This theorem corresponds to M.10 in the proof catalogue.

26.13 SPFA versus SEMA

We now prove precisely the sense in which the unanchored assumption is stronger than the anchored one.

Proposition 26.1 (SPFA is strictly stronger than SEMA) Consider the linear-predictor model of Definition 20.1 with treatments carrying interaction vectors \(\boldsymbol\beta_{2,k}\), and the two assumptions, stated for the \(A\)-versus-\(B\) contrast: \[ \text{SEMA: }\ \boldsymbol\beta_{2,A}=\boldsymbol\beta_{2,B}, \qquad \text{SPFA: }\ \boldsymbol\beta_{2,A}=\boldsymbol\beta_{2,B}=\mathbf{0}. \] Then:

1. (SPFA \(\Rightarrow\) SEMA.) SPFA implies SEMA.

2. (Anchored cancellation under SEMA.) In the anchored triangle with common comparator \(C\) (\(\boldsymbol\beta_{2,C}=\mathbf{0}\)), SEMA makes the conditional \(A\)-versus-\(B\) contrast free of the covariate, \(\eta_A(\mathbf{x})-\eta_B(\mathbf{x})=\gamma_A-\gamma_B\), even when the shared interaction \(\boldsymbol\beta_{2,A}=\boldsymbol\beta_{2,B}\) is nonzero, because the modification cancels against the anchor.

3. (Unanchored necessity of SPFA.) In the unanchored two-treatment network, the conditional \(A\)-versus-\(B\) contrast \(\eta_A(\mathbf{x})-\eta_B(\mathbf{x})=(\gamma_A-\gamma_B)+\mathbf{x}^\top (\boldsymbol\beta_{2,A}-\boldsymbol\beta_{2,B})\) is free of \(\mathbf{x}\) if and only if \(\boldsymbol\beta_{2,A}=\boldsymbol\beta_{2,B}\); and the arm-\(B\) absolute surface is determined by the index IPD (which identifies only \(\boldsymbol\beta_A=\boldsymbol\beta_1+\boldsymbol\beta_{2,A}\), with one arm taken as reference so \(\boldsymbol\beta_{2,A}=\mathbf{0}\)) if and only if \(\boldsymbol\beta_{2,B}=\mathbf{0}\), that is SPFA.

4. (Strictness.) The implication in (1) is strict: there exist data-generating processes satisfying SEMA but not SPFA, namely any with a shared nonzero interaction \(\boldsymbol\beta_{2,A}=\boldsymbol\beta_{2,B} \ne\mathbf{0}\).

Proof. (1). If \(\boldsymbol\beta_{2,A}=\boldsymbol\beta_{2,B}=\mathbf{0}\) then in particular \(\boldsymbol\beta_{2,A}=\boldsymbol\beta_{2,B}\), so SPFA implies SEMA.

(2). In the anchored model the active-versus-comparator predictors are \(\eta_A(\mathbf{x})=\mu+ \mathbf{x}^\top\boldsymbol\beta_1+\gamma_A+\mathbf{x}^\top\boldsymbol\beta_{2,A}\) and \(\eta_B(\mathbf{x})= \mu+\mathbf{x}^\top\boldsymbol\beta_1+\gamma_B+\mathbf{x}^\top\boldsymbol\beta_{2,B}\) (sharing the study baseline and prognostic term as contrasts against \(C\)). Subtracting, \(\eta_A(\mathbf{x})-\eta_B(\mathbf{x})=(\gamma_A-\gamma_B)+\mathbf{x}^\top(\boldsymbol\beta_{2,A}- \boldsymbol\beta_{2,B})\); under SEMA the bracket \(\boldsymbol\beta_{2,A}-\boldsymbol\beta_{2,B}=\mathbf{0}\) vanishes, leaving \(\gamma_A-\gamma_B\), free of \(\mathbf{x}\), regardless of the common value \(\boldsymbol\beta_{2,A}=\boldsymbol\beta_{2,B}\), which may be any nonzero vector. The anchor (the common \(C\), against which both contrasts are formed) is what allows the nonzero modification to cancel.

(3). The contrast expression is the same algebra as in (2). It is constant in \(\mathbf{x}\) iff the linear term \(\mathbf{x}^\top(\boldsymbol\beta_{2,A}-\boldsymbol\beta_{2,B})\) is identically zero, iff \(\boldsymbol\beta_{2,A}=\boldsymbol\beta_{2,B}\) (taking \(\mathbf{x}\) to range over a set spanning \(\mathbb{R}^{d}\)). Now the unanchored index IPD is single-arm and identifies only the total slope \(\boldsymbol\beta_A=\boldsymbol\beta_1+\boldsymbol\beta_{2,A}\) (by Lemma 26.1 there is no within-network contrast to separate \(\boldsymbol\beta_1\) from \(\boldsymbol\beta_{2,A}\)); fixing the index arm as reference sets \(\boldsymbol\beta_{2,A}=\mathbf{0}\), so \(\boldsymbol\beta_A=\boldsymbol\beta_1\). The comparator slope is \(\boldsymbol\beta_B=\boldsymbol\beta_1+\boldsymbol\beta_{2,B}\). The arm-\(B\) surface is determined by the index-identified \(\boldsymbol\beta_1\) alone (so that knowing one scalar \(\alpha_B\) pins \(\eta_B\) everywhere, Lemma 26.2) iff \(\boldsymbol\beta_B=\boldsymbol\beta_1\), iff \(\boldsymbol\beta_{2,B}=\mathbf{0}\), which is SPFA. Without it, \(\boldsymbol\beta_{2,B}\) is unidentified from the single comparator mean (Theorem 26.9(3) below), and the projection fails.

(4). Take any model with \(\boldsymbol\beta_{2,A}=\boldsymbol\beta_{2,B}=\mathbf{c}\ne\mathbf{0}\). It satisfies SEMA (\(\boldsymbol\beta_{2,A}=\boldsymbol\beta_{2,B}\)) but violates SPFA (\(\boldsymbol\beta_{2,A} \ne\mathbf{0}\)). Hence the set of SPFA-compliant processes is a proper subset of the SEMA-compliant ones: SPFA is strictly stronger. \(\square\)

Remark. One picture. SEMA says the two active treatments modify the outcome identically, so their contrast is flat; SPFA says they do not modify it at all. The anchored design exploits a common comparator to cancel shared modification, so it can live with SEMA; the unanchored design has nothing to cancel against and must assume there is nothing to cancel, which is SPFA. The same gap is the difference between conditional constancy of relative effects (Definition 17.4), which anchored adjustment needs, and conditional constancy of absolute effects (Definition 17.5), which unanchored adjustment needs; SPFA is the parametric face of the latter, as established in Lemma 26.2. The Chandler-Ishak posters describe SPFA as “a limitation inherent to unanchored comparisons,” and this proposition is the precise statement of that remark. \(\blacktriangleleft\)

This proposition corresponds to M.11 in the proof catalogue.

26.14 Identifiability and the limits of unanchored comparison

We collect the identifiability frontier into one theorem and then survey the limits that no estimator can overcome.

Theorem 26.9 (Identifiability frontier of ML-UMR) In the unanchored two-treatment network, with the index parameters \((\alpha_A,\boldsymbol\beta_A)\) identified from the IPD, the comparator block has the following identifiability:

1. SPFA, overall AgD (one aggregate mean). \((\alpha_B,\boldsymbol\beta_{\mathrm{PF}})\) identified (\(\boldsymbol\beta_{\mathrm{PF}}\) from IPD, \(\alpha_B\) from the single mean): the model is just identified (Theorem 26.1).

2. SPFA, subgroup AgD (\(Z\ge2\) means). Over-identified: \(\alpha_B\) solves any one subgroup equation, and the remaining \(Z-1\) equations are testable constraints (a goodness-of-fit check on SPFA).

3. No SPFA, overall AgD. \((\alpha_B,\boldsymbol\beta_B)\in\mathbb{R}^{d+1}\) is not identified from a single aggregate mean: the solution set of \(\Theta_B(\alpha_B,\boldsymbol\beta_B)=\theta_{\bullet B}^{\mathrm{obs}}\) is a \(d\)-dimensional manifold.

4. No SPFA, subgroup AgD with \(Z\ge d+1\). \((\alpha_B,\boldsymbol\beta_B)\) identified under the non-degeneracy of Theorem 26.2.

Proof. (1) is Theorem 26.1: \(\boldsymbol\beta_{\mathrm{PF}}\) from the full-rank IPD design and \(\alpha_B\) as the unique root of the strictly monotone \(\Theta_B\) (Lemma 23.2), one unknown matched by one equation.

(2). Under SPFA, \(\boldsymbol\beta_{\mathrm{PF}}\) is known from the IPD and only \(\alpha_B\) is free, so each subgroup equation \(\Theta_B^{z}(\alpha_B)=\theta_{\bullet B}^{z,\mathrm{obs}}\) determines \(\alpha_B\) by monotonicity; with \(Z\ge2\) subgroups the system is over-determined in the single unknown \(\alpha_B\), and the \(Z-1\) residual equations are constraints that hold only if SPFA is correct, hence a test of SPFA. In practice the test is operationalized as a posterior predictive check: fit the SPFA model, then compare each observed subgroup outcome against its posterior predictive distribution under the fitted single \(\alpha_B\); a systematic discrepancy (some subgroups over-predicted, others under-predicted, in a pattern aligned with a covariate) is evidence against SPFA. The parametric counterpart is a likelihood-ratio or score test of the Type 1 model nested in the relaxed Type 2 model of Theorem 26.2.

(3). With \(\boldsymbol\beta_B\) free, the map \((\alpha_B,\boldsymbol\beta_B)\mapsto\Theta_B(\alpha_B, \boldsymbol\beta_B)\) goes from \(\mathbb{R}^{d+1}\) to \(\mathbb{R}\); its gradient \(\int(g^{-1})'(\alpha_B+ \mathbf{x}^\top\boldsymbol\beta_B)(1,\mathbf{x}^\top)f_B\) is a nonzero row vector (the integrand is positive), so by the implicit function theorem the level set \(\{\Theta_B=\theta_{\bullet B}^{\mathrm{obs}}\}\) is a smooth manifold of dimension \((d+1)-1=d\ge1\) near any solution. Every point of it reproduces the same observed mean, so distinct parameters give identical data: not identified. (This is the \(Z=1<d+1\) case of Theorem 26.2.)

(4) is Theorem 26.2 with \(Z\ge d+1\) and the full-rank weighted-moment matrix \(M\). \(\square\)

Three structural limits are shared by every unanchored method, ML-UMR included, and follow from the theory already developed.

Remark. Inherent limits. First, the identifying assumption is untestable in the overall-AgD case: by Theorem 26.9(3) a single comparator mean cannot distinguish SPFA from any effect-modifying alternative, so SPFA must be defended externally; it becomes testable only with subgroup AgD (Theorem 26.9(2)). Second, no unanchored method can remove confounding by unmeasured prognostic factors: the projection Equation 26.6 transports the index surface in the measured covariates only, and an imbalanced prognostic variable omitted from \(\mathbf{x}\) biases the absolute arm positions exactly as in the necessity direction of Theorem 17.2. Third, the comparator covariate distribution \(f_B\) is reconstructed from reported margins and an assumed correlation (Theorem 26.3); misspecifying it biases the aggregate integral and hence \(\alpha_B\). For survival, the Kaplan-Meier reconstruction (Theorem 26.5) adds digitization noise on top. \(\blacktriangleleft\)

Remark. Empirical behavior under SPFA violation (simulation evidence, not theorems). The Chandler-Ishak studies report simulation results under specific data-generating mechanisms; we record them as evidence, not as proved statements, and the worked example of Section 26.15 derives the qualitative pattern by hand. For binary outcomes (Chandler and Ishak 2026a), with about 1000 patients per arm and two correlated binary prognostic factors, the SPFA model estimating the comparator-population log odds ratio shows absolute bias below about \(0.009\) and 95% interval coverage near nominal across no-, weak-, and strong-modification scenarios; the index-population (transported) estimate degrades as modification strengthens, with bias around \(-0.06\) (about \(-11\%\)) under weak modification and around \(-0.29\) (about \(-67\%\)) under strong modification, and coverage falling accordingly. The relaxed model (Theorem 26.2) restores small bias and near-nominal coverage when subgroup AgD are available. For survival outcomes (Chandler and Ishak 2026c), RMST estimands are near-unbiased with near-nominal coverage when SPFA holds, degrade modestly under weak modification, and show substantial bias (often exceeding 25% to 30%) and undercoverage in the index population under strong modification. The robust pattern, derived analytically in Section 26.15, is that the comparator- population estimand is anchored by the observed comparator outcome and so resists SPFA violation, while non-comparator targets are fragile because the wrong slope is extrapolated to a different covariate distribution. Different mechanisms can give different profiles; these numbers are illustrative, not universal. \(\blacktriangleleft\)

Remark. When to use ML-UMR. First, a triage against ML-NMR. If even one study connects the index and comparator treatments through a common comparator, use the anchored machinery: Bucher (Chapter 14), network meta-analysis (Chapter 15), or connected ML-NMR (Chapters 20 to 25), all of which need only SEMA and let the IPD check for effect modification rather than assuming it away. Reach for ML-UMR only when the network is genuinely disconnected, so that no anchor exists and SPFA becomes unavoidable. The one-line rule: prefer ML-NMR whenever a connecting study exists; use ML-UMR when it does not (see Table 26.1). That settled, the method is appropriate when individual data exist for the index treatment, only aggregate data exist for the comparator, no connected (anchored) network is available, SPFA is defensible from clinical knowledge, and the comparator’s covariate summaries are reported. Where SPFA is doubtful, the defensible report is the comparator-population contrast (where ML-UMR coincides with unanchored STC, Theorem 26.8), interpreted as population-specific rather than transportable, in line with the NICE guidance of Phillippo et al. (2016). The mlumr cautions, no between-study heterogeneity term, weak identification of the relaxed comparator slope, non-collapsibility of the hazard ratio, and the need to verify integration convergence, are the practical face of these limits. \(\blacktriangleleft\)

This theorem corresponds to M.12 in the proof catalogue.

26.15 A worked unanchored example

We carry a complete binary ML-UMR analysis through by hand, choosing one binary covariate so that every integral is an exact two-point sum (Theorem 26.3, discrete remark) and every number can be checked with a calculator. This is the running example of Example 26.1 made numerical. The covariate \(X\in\{0,1\}\) is a binary biomarker (say PD-L1 positivity, with \(X=1\) for biomarker-positive); the outcome is a binary response (a good outcome, so a higher probability is better, and biomarker-positive patients respond better under the fitted index model); the link is the logit, \(g= \operatorname{logit}\), \(g^{-1}=\operatorname{expit}\).

Inputs. Fitting the index IPD on treatment \(A\) returns the prognostic model \[ \eta_A(x)=\alpha_A+\beta_{\mathrm{PF}}\,x,\qquad \hat\alpha_A=-0.5,\ \ \hat\beta_{\mathrm{PF}}=1.0, \] so the index conditional response probabilities are \(p_A(0)=\operatorname{expit}(-0.5)=0.3775\) and \(p_A(1)=\operatorname{expit}(0.5)=0.6225\). The comparator AgD on treatment \(B\) report \(n_B=200\) patients, \(r_B=77\) responders (crude proportion \(\theta_{\bullet B}^{\mathrm{obs}}=0.385\)), and biomarker prevalence \(\bar x_B=P(X=1\mid\mathcal{P}_B)=0.50\). The index population has biomarker prevalence \(\bar x_A=P(X=1\mid\mathcal{P}_A)=0.30\). We use the constants \(\operatorname{expit}(-1)=0.2689\), \(\operatorname{expit}(-0.5)=0.3775\), \(\operatorname{expit}(0)=0.5\), \(\operatorname{expit}(0.5)=0.6225\).

The whole analysis lives in the two rows \(x=0\) and \(x=1\); Table 26.5 collects every conditional probability the steps below use, including the SPFA-violation column of Step 6. Reading it now makes each step a lookup rather than a fresh computation.

Table 26.5: Every number in the worked example, by covariate stratum. The fitted and true arm-\(B\) columns both average to the observed comparator response \(0.385\) under the comparator weights \((0.5,0.5)\), which is why the comparator-population contrast is unbiased; they diverge pointwise, so under the index weights \((0.7,0.3)\) they average to different values (\(0.3383\) versus \(0.2989\)), which is the index-population bias of Step 6.

Covariate \(x\)	weight in \(\mathcal{P}_B\)	weight in \(\mathcal{P}_A\)	\(p_A(x)\)	fitted \(p_B(x)\) under SPFA	true \(p_B(x)\), slope \(2\)
\(x=0\) (biomarker-negative)	\(0.5\)	\(0.7\)	\(0.3775\)	\(0.2689\)	\(0.1694\)
\(x=1\) (biomarker-positive)	\(0.5\)	\(0.3\)	\(0.6225\)	\(0.5000\)	\(0.6011\)

Step 1: identify \(\alpha_B\) (Theorem 26.1, Equation 26.9). Under SPFA the comparator shares the slope \(\beta_{\mathrm{PF}}=1.0\). With the binary covariate the aggregate mean is the exact two-point sum \[ \Theta_B(\alpha_B)=(1-\bar x_B)\,\operatorname{expit}(\alpha_B)+\bar x_B\,\operatorname{expit}(\alpha_B+1) =0.5\,\operatorname{expit}(\alpha_B)+0.5\,\operatorname{expit}(\alpha_B+1). \] This is strictly increasing in \(\alpha_B\) (Lemma 23.2), so the equation \(\Theta_B(\alpha_B)=0.385\) has a unique root. Bracketing: \(\Theta_B(-1.0)=0.5(0.2689)+0.5(0.5)=0.3845\) and \(\Theta_B(-0.9)=0.5(0.2891)+0.5(0.5250)=0.4070\), so the root lies just above \(-1.0\); to the precision of the reported proportion, \(\hat\alpha_B=-1.00\). (The fitted aggregate mean \(\Theta_B(-1.00)=0.3845\) matches the observed \(0.385\) to within rounding, the exact-calibration identity of Theorem 26.8(1).)

Step 2: the conditional contrast. Under SPFA the conditional log odds ratio of \(A\) versus \(B\) is the single number, free of \(x\) and of population, \[ \eta_A(x)-\eta_B(x)=\hat\alpha_A-\hat\alpha_B=-0.5-(-1.0)=0.5 \qquad(\text{@eq-spfa-constant-contrast}). \]

Step 3: the estimand in the comparator population \(\mathcal{P}_B\) (\(\bar x_B=0.5\)). The arm means are exact two-point sums: \[ \bar\theta_A(\mathcal{P}_B)=0.5\,\operatorname{expit}(-0.5)+0.5\,\operatorname{expit}(0.5)=0.5(0.3775)+0.5(0.6225)=0.5000, \] \[ \bar\theta_B(\mathcal{P}_B)=0.5\,\operatorname{expit}(-1.0)+0.5\,\operatorname{expit}(0)=0.5(0.2689)+0.5(0.5)=0.3845, \] the latter equal to the observed comparator proportion, as it must be. The marginal log odds ratio is \[ \Delta_{AB}(\mathcal{P}_B)=\operatorname{logit}(0.5000)-\operatorname{logit}(0.3845)=0-(-0.4706)=0.4706 . \]

Step 4: the estimand in the index population \(\mathcal{P}_A\) (\(\bar x_A=0.3\)). Now integrate against the index covariate distribution: \[ \bar\theta_A(\mathcal{P}_A)=0.7(0.3775)+0.3(0.6225)=0.4510, \qquad \bar\theta_B(\mathcal{P}_A)=0.7(0.2689)+0.3(0.5)=0.3383, \] \[ \Delta_{AB}(\mathcal{P}_A)=\operatorname{logit}(0.4510)-\operatorname{logit}(0.3383)=(-0.1966)-(-0.6711)=0.4745 . \]

Step 5: read the non-collapsibility. The conditional log odds ratio is \(0.5\) in every population (Step 2), but the marginal log odds ratio is \(0.4706\) in \(\mathcal{P}_B\) and \(0.4745\) in \(\mathcal{P}_A\), both attenuated toward the null \(0\) and differing slightly between populations. This is exactly the collapsibility dichotomy of Theorem 23.1(c): on the non-collapsible logit scale the marginal estimand depends on the whole target distribution, while the conditional contrast does not. The attenuation and its population dependence are the Definition 7.7 phenomenon, here visible in two hand-computed numbers.

Step 6: robustness at the comparator, fragility at the index, under SPFA violation. Suppose the true comparator model has a steeper slope, \(\beta_B^{\mathrm{true}}=2.0\ne\beta_{\mathrm{PF}}=1.0\) (strong effect modification), with a true intercept calibrated so that the comparator-population proportion is still \(0.385\) at \(\bar x_B=0.5\): solving \(0.5\,\operatorname{expit}(\alpha)+0.5\,\operatorname{expit}(\alpha+2)= 0.385\) gives \(\alpha_B^{\mathrm{true}}=-1.59\). Then:

• Comparator population. The true arm-\(B\) mean in \(\mathcal{P}_B\) is \(0.385\) by construction, identical to the SPFA fit of Step 3; the true arm-\(A\) mean is \(0.5\) (arm \(A\) is correctly specified). So the true \(\Delta_{AB}(\mathcal{P}_B)=0.4706\) equals the SPFA estimate: no bias at the comparator, because the observed comparator outcome anchors the arm-\(B\) mean there regardless of the slope.

• Index population. The true arm-\(B\) mean in \(\mathcal{P}_A\) (\(\bar x_A=0.3\)) is \(0.7\,\operatorname{expit}(-1.59)+0.3\,\operatorname{expit}(0.41)=0.7(0.1694)+0.3(0.6011)=0.2989\), whereas the SPFA model (slope \(1.0\), \(\hat\alpha_B=-1.0\)) gives \(\bar\theta_B(\mathcal{P}_A)=0.3383\) from Step 4. The true index-population log odds ratio is \(\operatorname{logit}(0.4510)-\operatorname{logit}(0.2989)= (-0.1966)-(-0.8525)=0.6559\), while the SPFA estimate is \(0.4745\). The bias is \(0.4745-0.6559=-0.1814\), about \(-28\%\): substantial bias at the index population, because the wrong slope is extrapolated from the comparator’s covariate distribution (mean \(0.5\)) to the index’s (mean \(0.3\)).

This is the central empirical pattern of Theorem 26.9 derived by hand: SPFA-invoking ML-UMR is robust where it is anchored by data (the comparator population) and fragile where it must extrapolate (any other target), and the fragility grows with the strength of the violated effect modification and with the distance between the covariate distributions. When SPFA is doubtful, report only \(\Delta_{AB}(\mathcal{P}_B)\).

Remark. A survival coda. The same phenomena appear on the time scale. Take an exponential PH model with one binary covariate, \(h_k(t\mid x)=\lambda_k\,e^{\beta x}\), \(\beta=\log2\) (here \(X=1\) marks a poor-prognosis stratum, for example elevated LDH, that doubles the hazard; lower hazard is better, so on this time-to-event scale \(X=1\) is unfavorable, unlike the favorable response biomarker of the binary example), index rate \(\lambda_A=0.10\) and comparator rate \(\lambda_B=0.15\), so the conditional hazard ratio of \(B\) versus \(A\) is the constant \(\lambda_B/\lambda_A=1.5\) at every \(t\) and \(x\) (SPFA on the log-hazard scale, Equation 26.16). In a population with \(P(X=1)=0.5\), the marginal survival is \(\bar S_k(t)=0.5\,e^{-\lambda_k t}+0.5\,e^{-2\lambda_k t}\), and the marginal hazard ratio \(\bar h_B(t)/\bar h_A(t)\) equals \(1.500\) at \(t=0\), dips to \(1.438\) at \(t=5\) and \(1.398\) at \(t=10\), then drifts back to \(1.491\) by \(t=50\): time-varying, attenuated toward \(1\), exactly Theorem 26.4(d) and Theorem 25.3. The restricted mean survival times to \(\tau=10\) are \(\operatorname{RMST}_A=5.32\) and \(\operatorname{RMST}_B=4.17\) (each the exact integral \(0.5\frac{1-e^{-\lambda\tau}}{\lambda}+0.5\frac{1-e^{-2\lambda\tau}}{2\lambda}\)), a collapsible difference of \(1.15\) time units favoring the index treatment, the collapsible read-out of Theorem 26.7. Recomputing the RMST difference in a population with \(P(X=1)=0.3\) gives a different value, the scale- misalignment of Theorem 26.4(e): collapsible within each population, but not the same number across them. \(\blacktriangleleft\)

26.16 Notes and references

Multilevel unanchored meta-regression is the contribution of Chandler and Ishak (2026a), who introduced the binary and continuous SPFA and relaxed-SPFA models and the simulation evidence summarized in Section 26.14, and of Chandler and Ishak (2026c), who extended the method to time-to-event outcomes with parametric and flexible baselines and the Kaplan-Meier reconstruction of the comparator arm. The method is an extension beyond the 2019 thesis of Phillippo (2019): it inherits the multilevel network meta-regression framework of Phillippo et al. (2020) and its general-likelihood extension of Phillippo et al. (2025) (Chapters 20 to 25), specifically the Sobol’ quasi-Monte Carlo integration (Definition 22.3, Sobol’ (1967); Owen (1956)) with a Gaussian copula (Definition 22.5, Theorem 22.5, Sklar (1959)) developed in Chapter 22, and reuses that machinery in the disconnected setting. The reference implementation is the mlumr R package of Sofi-Mahmudi and Chandler, which fits the SPFA and relaxed models by Hamiltonian Monte Carlo in Stan (Carpenter et al. (2017)), alongside frequentist STC and a naive benchmark, and whose documentation supplies the practical cautions recorded in Section 26.14; it is an adaptation of the GPL-3 multinma package of Phillippo. The transportability framing of population-adjusted indirect comparison that situates the unanchored estimand, conditional constancy of absolute effects (Definition 17.5) versus relative effects (Definition 17.4), is developed by Chandler and Ishak (2026b) and in Chapter 17; the NICE Technical Support Document guidance is Phillippo et al. (2016).

The structural results of this chapter are new statements of the catalogue items M.1 to M.12, proved in the book’s notation. The single-arm conflation (Lemma 26.1) and the SPFA-as- absolute-constancy lemma (Lemma 26.2) make explicit why the unanchored design needs the stronger assumption; the identifiability theorems (Theorem 26.1, Theorem 26.2, Theorem 26.9) rest on the aggregate-monotonicity lemma Lemma 23.2 of Chapter 23 and an inverse-function-theorem argument for the relaxed model; the equivalence with unanchored STC (Theorem 26.8) specializes the marginal STC consistency of Chapter 19 (Theorem 19.3, Theorem 19.6) and the generality theorem of Chapter 23 (Theorem 23.4), and the strict-strengthening proposition (Proposition 26.1) makes precise the relationship to the shared-effect-modifier assumption (Definition 17.6). The survival development (Theorem 26.4, Theorem 26.7) builds entirely on the general-likelihood survival results of Chapter 25, the survival-hazard identity (Theorem 25.1), the conditional and marginal hazard ratios (Theorem 25.2, Theorem 25.3), the censored likelihood (Definition 25.4, Proposition 25.1), the integrated likelihoods (Theorem 25.5, Theorem 25.6), and the collapsible restricted mean survival time (Theorem 25.4); the scale-misalignment result Theorem 26.4(e) is the unanchored counterpart of the caveat flagged after Theorem 25.4. The Kaplan-Meier reconstruction (Theorem 26.5) is the exact-inversion form of the algorithm of Guyot and colleagues (Guyot et al. 2012), whose realistic version handles coarse at-risk tables and digitization error.

There is no machine-checked counterpart for this chapter. The companion Coq development (~/Documents/GitHub/ITC_Coq/theories/) does not formalize ML-UMR: there is no ML_UMR.v, and ML-NMR itself is unformalized for the reasons given in Chapter 25’s notes (the integration and survival content exceed the development’s analysis machinery). The algebraic core closest to formalization is the contrast-cancellation phenomenon of Proposition 26.1, which parallels the anchored cancellation in theories/EffectModifiers.v, and the STC reduction of Theorem 26.8, whose comparator- population identity is the unanchored reading of theories/STC.v’s STC_consistent and methods_agree_when_correct; but the unanchored setting requires absolute-effect transport rather than contrast transport, a different algebraic structure that is not currently mechanized. The proofs above are therefore pen-and-paper and self-contained. This chapter corresponds to results M.1 (the unanchored setting), M.2 (SPFA), M.3 (the SPFA model and identifiability), M.4 (the relaxed model), M.5 (QMC marginalization), M.6 (time-to-event), M.7 (pseudo-IPD reconstruction), M.8 (the joint likelihood), M.9 (target-population survival), M.10 (equivalence with unanchored STC), M.11 (SPFA versus SEMA), and M.12 (the identifiability frontier and limits) of the proof catalogue in proofs/M_ml_umr.md.

26.17 Exercises

Throughout, an exercise marked \(\star\) has a full worked solution in Appendix F. The first three exercises are conceptual warm-ups that use the tables of this chapter and need no derivation; the remainder build the computations and proofs.

Exercise 26.1 (Warm-up: read the data picture.) A mock HTA dossier reports an index single-arm trial of drug \(A\) with patient-level age, biomarker status, and response; and for the comparator \(B\), only the overall response rate, the sample size, and the mean age and biomarker prevalence (no subgroup outcomes, no covariate correlation). Using Table 26.3, state for each of \(\alpha_A\), \(\boldsymbol\beta_{\mathrm{PF}}\), \(\alpha_B\), \(\boldsymbol\beta_B\), \(f_B\), and \(f_{*}\) whether this evidence identifies it, reconstructs it, or leaves it to assumption, and say whether a Type 1 or a Type 2 model is possible (Definition 26.3). What single additional item would the report need to make the relaxed model identifiable?

Exercise 26.2 (Interpretation: why SPFA cannot be tested.) In one short paragraph, explain to a clinical colleague why the Type 1 ML-UMR model cannot check SPFA from the two studies (cite Theorem 26.9), and why an anchored comparison can tolerate effect modification that an unanchored one cannot (cite Table 26.2). Then give one clinical scenario where SPFA is plausible and one where it is clearly false, and say which target-population estimate would be most, and which least, trustworthy in the false case (cite Step 6 of Section 26.15).

Exercise 26.3 (Interpretation: choose the method.) For each evidence set, decide whether to use anchored NMA or ML-NMR, Type 1 ML-UMR, Type 2 relaxed ML-UMR, or only unanchored STC, and justify in one sentence using Table 26.1: (a) two single-arm trials sharing no comparator, the comparator reporting only an overall response rate; (b) the same, but the comparator reports response rates in four biomarker-by-age cells; (c) an index and a comparator trial that each contain a common placebo arm; (d) an unanchored pair in which SPFA is clinically indefensible and only the comparator population matters for the decision.

Exercise 26.4 The conflation, concretely. In a single-arm index study with the logit link and no covariates, the model is \(\operatorname{logit}P(Y=1)=\mu+\gamma_A\). Show that the likelihood depends on \((\mu,\gamma_A)\) only through \(\alpha=\mu+\gamma_A\), that the maximum likelihood estimate of \(\alpha\) is \(\operatorname{logit}(\bar y)\) with \(\bar y\) the observed response proportion, and that \(\mu\) and \(\gamma_A\) are individually unidentified (Lemma 26.1). Explain in one sentence why an anchored two-arm study does not have this problem.

Exercise 26.5 (\(\star\)) Identifying the comparator intercept. With the logit link, SPFA slope \(\beta_{\mathrm{PF}}=0.8\), a single binary covariate with comparator prevalence \(\bar x_B=0.4\), and an observed comparator response proportion \(\theta_{\bullet B}^{\mathrm{obs}}=0.30\), write the two-point aggregation equation \(\Theta_B(\alpha_B)=0.30\) explicitly and solve it for \(\alpha_B\) to two decimal places by bracketing and linear interpolation (Theorem 26.1, Lemma 23.2). Then compute the comparator- population marginal response under arm \(B\) and verify it equals \(0.30\). Using an index fit \(\alpha_A=-0.2\), compute the conditional and the comparator-population marginal log odds ratios of \(A\) versus \(B\), and comment on the gap.

Exercise 26.6 Continuous outcomes need only the mean. For a continuous outcome with the identity link and the linear model \(\mathbb{E}[Y\mid x]=\alpha_k+\mathbf{x}^\top\boldsymbol\beta_{\mathrm{PF}}\), prove that \(\alpha_B=\bar y_B-\bar{\mathbf{x}}_B^\top\boldsymbol\beta_{\mathrm{PF}}\) exactly (using Theorem 20.3), so that ML-UMR for a continuous outcome under SPFA needs only the comparator covariate mean, not its full distribution. Deduce that for the identity link the target estimand \(\Delta_{AB}(\mathcal{P}^{*})=\alpha_A-\alpha_B\) is the same in every target population, and explain why this collapsibility fails for the logit link.

Exercise 26.7 Counting subgroups. In the relaxed model with \(d=2\) continuous covariates, you have \(Z=3\) comparator subgroups. State whether \((\alpha_B,\boldsymbol\beta_B)\in\mathbb{R}^{3}\) is identified, citing the counting condition of Theorem 26.2, and describe a configuration of subgroup covariate means for which the non-degeneracy condition (full rank of the weighted-moment matrix \(M\)) holds and one for which it fails. What changes if a fourth subgroup is added?

Exercise 26.8 ML-UMR equals unanchored STC at the comparator. Using the calibration identity Theorem 26.8(1) and the marginal STC definition Definition 19.4, show by direct substitution that at \(\mathcal{P}^{*}=\mathcal{P}_B\) the SPFA ML-UMR estimand and the unanchored marginal STC estimator are the same expression Equation 26.22. Then exhibit, in one or two sentences, why STC cannot produce the estimand at a target \(\mathcal{P}^{*}\ne\mathcal{P}_B\) while ML-UMR can, and name the assumption that powers the difference.

Exercise 26.9 Time-varying marginal hazard ratio. In the exponential PH survival coda of Section 26.15 (\(h_k(t\mid x)=\lambda_k e^{\beta x}\), \(\beta=\log2\), \(\lambda_A=0.10\), \(\lambda_B=0.15\), \(P(X=1)=0.5\)), derive the closed form \(\bar h_k(t)=\lambda_k\frac{(1-\pi)+2\pi e^{-\lambda_k t}}{(1-\pi)+\pi e^{-\lambda_k t}}\) with \(\pi=0.5\) (using Lemma 25.1), and confirm by direct computation the marginal hazard ratios \(1.500\) at \(t=0\), \(1.398\) at \(t=10\), and the limit \(\lambda_B/\lambda_A\) as \(t\to\infty\). Explain in terms of survivor selection (Theorem 25.3) why the ratio dips below the conditional value for intermediate \(t\) and returns toward it as \(t\to\infty\).

Exercise 26.10 (\(\star\)) Collapsible but not transportable, unanchored. For the exponential PH model of the survival coda, write \(\operatorname{RMST}_k(\tau\mid x)=\frac{1-\exp(-\lambda_k e^{\beta x}\tau)}{\lambda_k e^{\beta x}}\) and hence the population RMST contrast \(\Delta\!\operatorname{RMST}_{AB}(\tau\mid\mathcal{P}^{*})\) as a two-point average over \(P(X=1)=\pi^{*}\). Prove that this contrast is collapsible within each population (Theorem 25.4) but differs between two populations with \(\pi^{*}=0.5\) and \(\pi^{*}=0.3\), and give the numerical difference at \(\tau=10\). Relate your finding to Theorem 26.4(e) and contrast it with the conditional log hazard ratio, which is the single number \(\log(\lambda_B/\lambda_A)\) in every population.

Exercise 26.11 Where the bias goes. Reproduce the SPFA-violation calculation of Step 6 of Section 26.15 with a weaker true comparator slope \(\beta_B^{\mathrm{true}}=1.5\) (instead of \(2.0\)): find the calibrated true intercept \(\alpha_B^{\mathrm{true}}\) from \(\Theta_B^{\mathrm{true}}(\alpha)=0.385\) at \(\bar x_B=0.5\), then compute the true and SPFA-estimated index-population (\(\bar x_A=0.3\)) marginal log odds ratios and the bias. Confirm that the comparator-population bias is zero and that the index-population bias is smaller in magnitude than the \(-0.18\) of the strong-violation case, illustrating Theorem 26.9 and the empirical pattern that bias grows with the strength of the violated modification.

Solutions to the starred exercises Exercise 26.5 and Exercise 26.10 appear in Appendix F.

Carpenter, Bob, Andrew Gelman, Matthew D. Hoffman, et al. 2017. “Stan: A Probabilistic Programming Language.” Journal of Statistical Software 76 (1).

Chandler, Conor, and K. Jack Ishak. 2026a. “Anchors Away: Navigating Unanchored Indirect Comparisons with Multilevel Unanchored Meta-Regression.” arXiv Preprint arXiv:2606.20341.

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

Chandler, Conor, and K. Jack Ishak. 2026c. “Surviving Unanchored Indirect Comparisons: Multilevel Unanchored Meta-Regression for Survival Analyses.” Presented at ISPOR 2026.

Guyot, Patricia, A. E. Ades, Mario J. N. M. Ouwens, and Nicky J. Welton. 2012. “Enhanced Secondary Analysis of Survival Data: Reconstructing the Data from Published Kaplan-Meier Survival Curves.” BMC Medical Research Methodology 12: 9.

Owen, Donald B. 1956. “Tables for Computing Bivariate Normal Probabilities.” The Annals of Mathematical Statistics 27 (4): 1075–90.

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

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

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

Phillippo, David M., Sofia Dias, A. E. Ades, and Nicky J. Welton. 2025. “Multilevel Network Meta-Regression for General Likelihoods.” Journal of the Royal Statistical Society: Series A, ahead of print. https://doi.org/10.1093/jrsssa/qnaf169.

Sklar, Abe. 1959. “Fonctions de répartition à \(n\) Dimensions Et Leurs Marges.” Publications de l’Institut de Statistique de l’Université de Paris 8: 229–31.

Sobol’, Ilya M. 1967. “On the Distribution of Points in a Cube and the Approximate Evaluation of Integrals.” USSR Computational Mathematics and Mathematical Physics 7 (4): 86–112.