19  Simulated Treatment Comparison

19.1 Introduction

The previous chapter repaired a failing indirect comparison by reweighting. Matching-adjusted indirect comparison (MAIC) leaves the outcomes untouched and bends the sample until its covariate moments match the target, so that a weighted average of the observed outcomes estimates what the trial would have found in the target population. This chapter develops the other half of the population-adjustment dichotomy. The simulated treatment comparison (STC) of Caro and Ishak (2010) leaves the sample untouched and bends a model: it fits an outcome regression to the individual patient data (IPD), then uses the fitted regression surface to predict what the outcomes would have been in the target population. Where MAIC is the indirect comparison analogue of inverse-probability weighting, STC is the analogue of outcome regression, or g-computation, of Chapter 10 (Theorem 10.4). The two methods stand in exactly the relation that weighting and standardization stand in throughout causal inference: dual routes to the same estimand, with complementary assumptions and complementary failure modes.

The setting is the anchored triangle of Chapter 14. An IPD trial, the index trial, compares an active treatment \(A\) against a common comparator \(C\) in a population \(\mathcal P_{AC}\), and supplies the full covariate, treatment, and outcome record of every patient. A second trial, available only in aggregate, the comparator trial, compares \(B\) against the same anchor \(C\) in a population \(\mathcal P_{BC}\), and supplies covariate summaries (means \(\bar{\mathbf x}_{BC}\), and where available a covariance \(\boldsymbol\Sigma_{BC}\)) together with a published relative effect \(\hat d_{BC}\). The clinical question is \(A\) versus \(B\). As in Chapter 14 the answer runs through the anchor: estimate the \(A\)-versus-\(C\) effect in the comparator population \(\mathcal P_{BC}\), then close the triangle by Bucher’s subtraction (Theorem 14.1). The entire job of STC is the first step, producing \(\hat d_{AC(\mathcal P_{BC})}\), the index trial’s relative effect transported to the comparator population. The bias that this corrects is the transitivity failure of Chapter 16: the two trials enroll different distributions of effect modifiers (Definition 11.3), so the unadjusted index effect \(\hat d_{AC(\mathcal P_{AC})}\) is the right contrast in the wrong population (Theorem 16.1).

STC predicts. It fits the conditional mean surface \(\mu_k(\mathbf x)=\mathbb E[Y(k)\mid X=\mathbf x]\) on the index IPD and integrates that surface over the comparator covariate distribution. This is g-computation with the integrating measure swapped from the source \(f_{AC}\) to the target \(f_{BC}\), and its validity rests on conditional transportability of the outcome surface (Theorem 12.2), not on overlap. A regression can be evaluated wherever its covariates are defined, so STC interpolates and even extrapolates where MAIC, confined to the convex hull of the IPD (the range of covariate combinations actually observed in the index trial; Theorem 18.8), cannot. The price is the mirror image of MAIC’s: STC is consistent only if the outcome model is correctly specified, while MAIC asks only that the matched moments be balanced and is indifferent to the functional form beyond them.

One technical phenomenon dominates the chapter, and it is the phenomenon of Chapter 11: non-collapsibility (Definition 7.7, Theorem 11.2). On a nonlinear link the relative effect for the average patient is not the average relative effect, and neither equals the marginal relative effect a trial reports. STC must therefore choose what it predicts. The historical and simplest choice, conditional STC, plugs the comparator’s mean covariate vector \(\bar{\mathbf x}_{BC}\) into the fitted model and reports the effect at that single synthetic patient. We prove that this is biased for the marginal estimand under any nonlinear link (Theorem 19.2), with a bias governed by the curvature of the inverse link and the covariate variance, and that it silently answers a different and usually unwanted question. The correct choice, marginal STC, integrates the fitted means over the comparator covariate distribution on the natural outcome scale and only then applies the link; we prove it consistent for the marginal effect under correct specification (Theorem 19.3). A closed-form counterexample (Theorem 19.4) exhibits the gap in a single line of arithmetic, and a fully worked logistic example (Example 19.3) carries an STC analysis from a fitted model to three distinct numbers: the biased plug-in, the correct target marginal, and the unadjusted source marginal.

The plan is as follows. Section 19.3 sets up the outcome model and proves the maximum likelihood estimator (MLE) consistent and asymptotically normal (catalogue G.1), specializing the general theory of Chapter 5 (Theorem 5.6). Section 19.4 defines conditional STC and proves its plug-in bias (G.2). Section 19.5 defines marginal STC, proves its consistency, and identifies it as transported g-computation (G.3). Section 19.6 isolates non-collapsibility with a counterexample and its collapsible boundary (G.4). Section 19.7 is the worked example. Section 19.8 develops Bayesian g-computation (G.5). Section 19.9 states the anchored and unanchored STC consistency theorems under the population-adjustment conditions of Chapter 17 (G.6). Section 19.10 proves that under a linear model with first-moment matching MAIC and STC asymptotically coincide (G.7). This chapter corresponds to results G.1 to G.7 of the proof catalogue. We use the relative-effect notation of Notation and the link-position machinery of Chapter 14.

19.2 A concrete picture and the STC workflow

The introduction moved fast. Before any formula, fix a picture and keep it for the whole chapter.

A running clinical example. A new drug \(A\) is to be compared against a competitor drug \(B\) for an autoimmune disease. No trial has ever randomized \(A\) directly against \(B\). What we have is two separate randomized trials that share a common comparator, the standard of care \(C\):

  • the index trial \(AC\) randomizes \(A\) against \(C\) and reports individual patient data (IPD): for every patient we see the baseline covariates, the assigned arm, and the outcome;
  • the comparator trial \(BC\) randomizes \(B\) against \(C\) but is published only as aggregate data (AgD): covariate summaries (the fraction of patients with each feature) and a single headline effect \(\hat d_{BC}\).

Take a single binary covariate \(x\), a baseline risk marker, with \(x=1\) for a high-risk patient and \(x=0\) for a low-risk patient, and let the outcome be an unfavorable event (a disease flare within six months). The two trials enrolled different case mixes: the comparator trial \(BC\) recruited sicker patients, a fraction \(0.8\) high-risk, against \(0.5\) high-risk in the index trial \(AC\). Because risk also drives the event probability, this difference in case mix is precisely what makes a naive indirect comparison invalid; the two trials answer their questions in two different populations. The numbers in this paragraph are the numbers of the worked example Example 19.3, so everything below is a rehearsal for it. The coefficients are chosen for arithmetic transparency, not clinical realism.

The anchored triangle. The three treatments sit at the corners of a triangle. Two edges are supported by trials; the third, the one we want, is not.

     A - - - - - - - - - - - - - B      target: A vs B  (no direct trial)
      \                         /
 A vs C\   AC trial (IPD)      /  BC trial (aggregate)   B vs C
        \                     /
         \                   /
          \                 /
                  C
            common comparator (anchor)

STC does exactly one job: it produces the \(A\)-versus-\(C\) effect evaluated in the comparator population \(\mathcal P_{BC}\), written \(\hat d_{AC(\mathcal P_{BC})}\) (the left edge, transported to the population of the right edge). Bucher’s identity (Theorem 14.1) then closes the triangle, \(\hat d_{AB}=\hat d_{AC(\mathcal P_{BC})}-\hat d_{BC}\), because both contrasts now refer to the same population \(\mathcal P_{BC}\).

g-computation in two sentences. STC is the outcome-regression route, called g-computation or standardization in Chapter 10 (Theorem 10.4). g-computation fits a model for the outcome given treatment and covariates, then, to get the effect in a chosen population, predicts the outcome for every covariate profile in that population and averages the predictions. STC is this same recipe with one twist: the model is fitted in the index population \(\mathcal P_{AC}\), but the averaging is done over the target population \(\mathcal P_{BC}\).

Remark. STC in five steps (the whole method, before the theory). Given the index IPD and the comparator summaries:

  1. Fit. Fit an outcome regression \(\mu_A(x)\), \(\mu_C(x)\) on the index IPD: a model for the event probability (or mean) given arm and covariates.
  2. Obtain target covariates. Get, or reconstruct, the covariate distribution \(f_{BC}\) of the comparator trial. With IPD this is the covariate table; with only summaries, simulate synthetic patients matching the published proportions, means, and (if reported) correlations.
  3. Predict. For each target covariate profile, use the fitted model to predict the event probability under \(A\) and under \(C\).
  4. Average, then link. Average those predicted probabilities over the target population on the natural outcome scale (probabilities, not log odds), and only then convert to the reported effect scale (for example the log odds ratio). This order is the crux of the chapter.
  5. Close the triangle. Combine the resulting \(\hat d_{AC(\mathcal P_{BC})}\) with the published \(\hat d_{BC}\) by Bucher’s subtraction (Theorem 14.1) to get \(\hat d_{AB}\).

The single most important warning of the chapter lives in step 4: there are two ways to summarize the fitted model in the target population, and for a nonlinear outcome model they give different answers.

Conditional STC (plug-in) Marginal STC (integrate/simulate)
Recipe plug the target mean covariate \(\bar{\mathbf x}_{BC}\) into the fitted model, read off the effect predict for every target covariate profile, average the predictions, then take the effect
Order of operations link, then average (contrast at the average patient) average, then link (average the outcomes, then contrast)
Data needed only the covariate means \(\bar{\mathbf x}_{BC}\) the full covariate distribution \(f_{BC}\)
Estimand effect for the single “average patient” (Equation 19.10) population-average (marginal) effect (Equation 19.9)
Correct for a population decision? only under a linear model; biased under any nonlinear link yes, under correct specification
Developed in Section 19.4 Section 19.5

The plug-in is historically the original STC of Caro and Ishak (2010), and it is tempting for a concrete reason: it needs only the published covariate means \(\bar{\mathbf x}_{BC}\), which every trial reports, not the full covariate distribution, which most trials do not. Its defect is that “the effect for the average patient” is not “the average effect” once the outcome model is nonlinear, a phenomenon called non-collapsibility (Chapter 11). Making that gap visible, and fixing it, is the spine of this chapter.

When to reach for STC. STC and MAIC (Chapter 18) solve the same transport problem by dual routes, and the choice between them is a choice of what to trust:

  • Reach for STC when you are willing to commit to an outcome model (a link and a functional form) and when the comparator population lies partly outside the covariate range of the index trial, so that weighting would break down. A regression can predict anywhere its covariates are defined.
  • Reach for MAIC when you do not want to commit to a functional form and the two populations overlap well, so that reweighting is stable. MAIC balances covariate moments and is indifferent to the shape of the outcome surface beyond them.

The trade-off, made precise in Section 19.10, is “trust a model” (STC) versus “trust an overlap” (MAIC). Keep it in mind as the assumptions accumulate.

A guide to the signs and scales. Two bookkeeping conventions recur and are worth tabulating once. First, the direction of a contrast. The GLM treatment coefficient orients the active arm against the reference arm; the relative-effect symbol of Notation orients the reference first. They differ only by a sign:

Quantity Definition Reads as
\(\Lambda(\mathbf x)\), \(\ell_{AC}\) \(\eta_A-\eta_C\) (active minus reference) GLM/model convention; positive means a higher linear predictor under \(A\)
\(d_{AC(\mathcal P)}\) \(\eta_C-\eta_A\) (reference first) Notation/Bucher convention; \(d_{AC}=-\Lambda\) at the marginal level

Nothing mathematical depends on the choice; we compute in the positive \(\ell_{AC}=\eta_A-\eta_C\) where it keeps arithmetic tidy and flip the sign only when handing the result to Bucher. Second, the scale. Every effect can be read on two scales, and the whole non-collapsibility story is that they do not commute with averaging:

Phrase (used interchangeably) Meaning
natural outcome scale = response scale the untransformed outcome: an event probability, a rate, a mean
link scale the transformed outcome \(g(\cdot)\): a log odds, a log rate, on which the model is linear

We say “natural outcome scale” for the untransformed scale throughout. “Averaging on the natural outcome scale, then linking” is marginal STC; “linking, then reading the contrast at the average covariate” is conditional STC. Finally, a compact glossary of the symbols this chapter leans on, all consistent with Notation:

Symbol Plain meaning
\(\mathcal P_{AC}\), \(\mathcal P_{BC}\) source (index) and target (comparator) populations
\(f_{BC}\) covariate distribution in the target population
\(\eta_k(\mathbf x)\) linear predictor (link-scale outcome) under arm \(k\)
\(\mu_k(\mathbf x)\) fitted event probability or mean under arm \(k\), on the natural outcome scale
\(\bar\mu_{k(\mathcal P)}\) that mean, averaged over population \(\mathcal P\) (a marginal mean)
\(d^{\mathrm{cond}}\) conditional-at-the-mean effect (the plug-in target)
\(d^{\mathrm{marg}}\) marginal (population-average) effect (the correct target)
\(\Lambda(\mathbf x)\) conditional relative effect on the link scale, \(\gamma+\mathbf x^\top\boldsymbol\beta_2\)

With the picture, the workflow, the two estimands, and the bookkeeping in hand, we now build the method rigorously, beginning with the regression that step 1 fits.

19.3 The outcome model and its estimation

This section carries out step 1 of the workflow: fitting the outcome regression. Why begin with a page of maximum likelihood theory in a chapter about STC? Because every downstream STC estimator, the plug-in of Section 19.4, the marginal estimator of Section 19.5, and their variances, is a function of the fitted coefficients \(\hat{\boldsymbol\xi}\). To push convergence and a limiting distribution through those functions we need two guarantees about \(\hat{\boldsymbol\xi}\) itself: that it converges to the truth (consistency) and that it has a known limiting spread (asymptotic normality). Theorem 19.1 supplies both; the reader who trusts that GLMs are well behaved may skim its proof and rejoin at Section 19.4.

Fix a strictly increasing, twice continuously differentiable link function \(g\) (Definition 7.2) with inverse \(g^{-1}\): the logit for binary outcomes, the log for rates and hazards, the identity for continuous outcomes. The index IPD consist of \(n\) independent records \((Y_i, T_i, \mathbf x_i)\), \(i=1,\dots,n\), drawn from \(\mathcal P_{AC}\), with arm indicator \(T_i\in\{A,C\}\) randomized and covariates \(\mathbf x_i\in\mathcal X\subseteq\mathbb R^p\).

Definition 19.1 (The STC outcome model) The outcome model is the generalized linear model that, with \(C\) as the reference arm, writes the conditional mean of the outcome on the link scale as a linear predictor \[ \eta_k(\mathbf x;\boldsymbol\xi) \;=\;\alpha \;+\; \mathbf x^\top\boldsymbol\beta_1 \;+\; \mathbf 1\{k=A\}\,\bigl(\gamma+\mathbf x^\top\boldsymbol\beta_2\bigr), \qquad k\in\{A,C\}, \tag{19.1}\] with intercept \(\alpha\), prognostic coefficients \(\boldsymbol\beta_1\) (Definition 11.5), treatment main effect \(\gamma\), and treatment-by-covariate interactions \(\boldsymbol\beta_2\) encoding effect modification (Definition 11.3). The full parameter is \(\boldsymbol\xi=(\alpha,\gamma,\boldsymbol\beta_1,\boldsymbol\beta_2)\in\mathbb R^{q}\), \(q=2p+2\). The fitted conditional mean under arm \(k\) is \[ \mu_k(\mathbf x;\boldsymbol\xi)\;=\;g^{-1}\!\bigl(\eta_k(\mathbf x;\boldsymbol\xi)\bigr), \qquad \mu_k(\mathbf x)\;:=\;\mu_k(\mathbf x;\boldsymbol\xi_0)\;=\;\mathbb E[Y(k)\mid X=\mathbf x], \tag{19.2}\] the last equality holding at the true parameter \(\boldsymbol\xi_0\) under correct specification. The conditional relative effect of \(A\) versus \(C\) at covariate value \(\mathbf x\) is the contrast of linear predictors, \[ \Lambda(\mathbf x;\boldsymbol\xi)\;=\;\eta_A(\mathbf x;\boldsymbol\xi)-\eta_C(\mathbf x;\boldsymbol\xi)\;=\;\gamma+\mathbf x^\top\boldsymbol\beta_2 . \tag{19.3}\] \(\blacktriangleleft\)

Reader’s translation. The model says that each patient’s outcome, on the link scale, is a baseline level (\(\alpha\)), plus a prognostic tilt for how sick they are (\(\mathbf x^\top\boldsymbol\beta_1\)), plus, only if they received the active drug, a treatment bump (\(\gamma\)) that may itself depend on their covariates (\(\mathbf x^\top\boldsymbol\beta_2\), effect modification). Set \(\boldsymbol\beta_2=\mathbf 0\) and every patient gets the same treatment bump; let it be nonzero and the drug helps some patients more than others. The conditional relative effect Equation 19.3 is just that treatment bump on the link scale.

Writing \(z_i=\bigl(1,\ \mathbf 1\{T_i=A\},\ \mathbf x_i^\top,\ \mathbf 1\{T_i=A\}\,\mathbf x_i^\top\bigr)^\top\in\mathbb R^{q}\) for the design row of patient \(i\), the linear predictor is \(\eta_i=z_i^\top\boldsymbol\xi\), so Equation 19.1 is an ordinary GLM with model matrix \(Z=[z_1,\dots,z_n]^\top\). The conditional relative effect Equation 19.3 is constant in \(\mathbf x\) exactly when \(\boldsymbol\beta_2=\mathbf 0\), that is, when no measured covariate modifies the treatment effect on the link scale; this is the link-scale form of the effect-modifier definition of Chapter 11.

Remark. Orientation of the contrast. Equation Equation 19.3 is the effect of the active arm relative to the reference arm, \(\eta_A-\eta_C\), which is the sign convention of the GLM treatment coefficient and of the STC literature. The relative-effect symbol of Notation and Chapter 14 places the reference first, \(d_{AC(\mathcal P)}=\eta_{C(\mathcal P)}-\eta_{A(\mathcal P)}\), so the marginal version of Equation 19.3 enters Bucher’s identity as \(d_{AC}=-\Lambda\) at the marginal level. The overall sign is bookkeeping, exactly as in the index-order remark of Chapter 14, and it changes none of the bias, consistency, or non-collapsibility statements below, each of which is a statement about a single arm or about the difference of two arms on a fixed scale. To keep the worked arithmetic in positive numbers we report the active-versus-reference log odds-ratio \(\ell_{AC}:=\eta_{A}-\eta_{C}\) where convenient and note \(d_{AC}=-\ell_{AC}\). \(\blacktriangleleft\)

19.3.1 Estimation and its large-sample behavior

The model is fitted by maximum likelihood. For an outcome whose conditional law is a one-parameter exponential-family member with mean \(\mu_i=g^{-1}(z_i^\top\boldsymbol\xi)\), the log-likelihood is \[ \ell_n(\boldsymbol\xi)\;=\;\sum_{i=1}^n \ell_i(\boldsymbol\xi), \qquad \ell_i(\boldsymbol\xi)=\frac{Y_i\vartheta_i-b(\vartheta_i)}{a(\phi)}+c(Y_i,\phi), \tag{19.4}\] where \(\vartheta_i\) is the natural parameter, related to the mean by \(\mu_i=b'(\vartheta_i)\) and to the linear predictor by the link \(g(\mu_i)=z_i^\top\boldsymbol\xi\), and \(a(\phi)>0\) is the dispersion. The score and observed information are \[ \mathbf s_n(\boldsymbol\xi)=\nabla\ell_n(\boldsymbol\xi)=\sum_i \frac{Y_i-\mu_i}{a(\phi)\,V(\mu_i)\,g'(\mu_i)}\,z_i, \qquad \mathcal I_n(\boldsymbol\xi)=\sum_i W_i\,z_iz_i^\top, \tag{19.5}\] with variance function \(V(\mu)=b''(\vartheta)\) and GLM working weight \(W_i=\bigl[a(\phi)\,V(\mu_i)\,g'(\mu_i)^2\bigr]^{-1}>0\) (the standard iteratively reweighted least squares weight; see McCullagh and Nelder (1989), Chapter 2). We record the asymptotics as catalogue result G.1.

Theorem 19.1 (Consistency and asymptotic normality of the outcome MLE) Suppose the outcome model Definition 19.1 is correctly specified, with true parameter \(\boldsymbol\xi_0\) in the interior of a compact parameter set \(\Xi\subseteq\mathbb R^{q}\), and that

  1. the records \((Y_i,T_i,\mathbf x_i)\) are independent and identically distributed from \(\mathcal P_{AC}\), with \(\mathbb E[Y(k)\mid X=\mathbf x]=g^{-1}(\eta_k(\mathbf x;\boldsymbol\xi_0))\);
  2. the covariates are bounded and the limiting design second moment \(\mathbf M:=\mathbb E[\,W\,zz^\top\,]\) is finite and positive definite, which requires both arms to be present with positive probability and the covariates to vary within the active arm so that the columns of \(z\) corresponding to \(\alpha,\gamma,\boldsymbol\beta_1,\boldsymbol\beta_2\) are not collinear;
  3. the link \(g\) is twice continuously differentiable with \(g'>0\), and the dispersion \(a(\phi)\) is fixed or estimated by a \(\sqrt n\)-consistent estimator.

Then the maximum likelihood estimator \(\hat{\boldsymbol\xi}\) exists with probability tending to one, is consistent, \(\hat{\boldsymbol\xi}\xrightarrow{p}\boldsymbol\xi_0\), and is asymptotically normal, \[ \sqrt n\,(\hat{\boldsymbol\xi}-\boldsymbol\xi_0)\ \xrightarrow{d}\ \mathcal N\!\bigl(\mathbf 0,\ \mathcal I(\boldsymbol\xi_0)^{-1}\bigr), \qquad \mathcal I(\boldsymbol\xi_0)=\mathbf M, \tag{19.6}\] where \(\mathcal I(\boldsymbol\xi_0)\) is the per-observation Fisher information. (Corresponds to G.1 in the proof catalogue.)

Proof. The result is the master maximum likelihood theorem of Chapter 5 (Theorem 5.6) applied to the estimating function Equation 19.5; the work is to verify its hypotheses for the GLM Equation 19.4. We take \(a(\phi)\) known to lighten notation; a \(\sqrt n\)-consistent \(\hat\phi\) enters the mean parameters only through a block-diagonal information and does not change the limit for \(\hat{\boldsymbol\xi}\).

Identifiability. The map \(\boldsymbol\xi\mapsto(\eta_1,\dots,\eta_n)=(z_1^\top\boldsymbol\xi,\dots)\) is injective on \(\mathbb R^q\) iff the design rows span \(\mathbb R^q\), equivalently iff \(\mathbb E[zz^\top]\succ0\). Hypothesis 2 grants this: an intercept needs a constant column; \(\gamma\) needs \(\Pr(T=A)\in(0,1)\) so the column \(\mathbf 1\{T=A\}\) is nonconstant; \(\boldsymbol\beta_1\) needs covariate variation; \(\boldsymbol\beta_2\) needs covariate variation within the active arm. Since \(g\) is strictly increasing, distinct linear predictors give distinct mean vectors, so distinct \(\boldsymbol\xi\) give distinct distributions and the parameter is identified.

The score has mean zero at the truth. Under correct specification, \(\mathbb E[Y_i\mid z_i]=\mu_i(\boldsymbol\xi_0)\), so each summand of Equation 19.5 has conditional mean \[ \mathbb E\!\left[\frac{Y_i-\mu_i(\boldsymbol\xi_0)}{a(\phi)V(\mu_i)g'(\mu_i)}\,z_i\ \middle|\ z_i\right]=\mathbf 0, \] and hence unconditional mean zero by the tower property (the law of iterated expectations, \(\mathbb E[\mathbb E[Y\mid X]]=\mathbb E[Y]\)). Thus \(\boldsymbol\xi_0\) solves the population score equation \(\mathbb E[\mathbf s_i(\boldsymbol\xi_0)]=\mathbf 0\), the defining property of the estimand for an M-estimator (an estimator defined as the zero of an estimating function, here the score \(\mathbf s_n\)).

Information and the second-moment matrix. Differentiating Equation 19.5 and taking expectations, the cross-product terms in \(Y_i-\mu_i\) vanish in expectation at \(\boldsymbol\xi_0\), leaving the Fisher information per observation \[ \mathcal I(\boldsymbol\xi_0)=\mathbb E\!\left[-\nabla^2\ell_i(\boldsymbol\xi_0)\right]=\mathbb E\!\left[\,W_i\,z_iz_i^\top\,\right]=\mathbf M, \] which is finite (bounded covariates, continuous weights) and positive definite by hypothesis 2 together with \(W_i>0\). This is the information equality for a correctly specified exponential family.

Existence, uniqueness, and consistency. For the canonical link the log-likelihood Equation 19.4 is concave in \(\boldsymbol\xi\) because \(b\) is convex and \(\eta_i\) is linear in \(\boldsymbol\xi\), so any stationary point is the unique global maximizer; for a noncanonical but smooth strictly increasing link the log-likelihood is locally concave near \(\boldsymbol\xi_0\) and the estimating function \(\mathbf s_n\) is, by the uniform law of large numbers on the compact \(\Xi\), uniformly close to its expectation, whose unique zero is \(\boldsymbol\xi_0\). By the standard M-estimation consistency argument (the zero of an estimating function that converges uniformly to a function with a well-separated unique zero is consistent; Vaart (1998), Theorem 5.9), \(\hat{\boldsymbol\xi}\xrightarrow{p}\boldsymbol\xi_0\).

Asymptotic normality. A first-order Taylor expansion of \(\mathbf 0=\mathbf s_n(\hat{\boldsymbol\xi})\) about \(\boldsymbol\xi_0\) gives \[ \mathbf 0=\frac1{\sqrt n}\mathbf s_n(\boldsymbol\xi_0)+\Bigl(\frac1n\nabla\mathbf s_n(\tilde{\boldsymbol\xi})\Bigr)\sqrt n(\hat{\boldsymbol\xi}-\boldsymbol\xi_0), \] for \(\tilde{\boldsymbol\xi}\) on the segment between \(\hat{\boldsymbol\xi}\) and \(\boldsymbol\xi_0\). The score summands are independent, mean zero, with per-observation covariance \(\mathcal I(\boldsymbol\xi_0)=\mathbf M\) (the information equality), so by the multivariate central limit theorem \(n^{-1/2}\mathbf s_n(\boldsymbol\xi_0)\xrightarrow{d}\mathcal N(\mathbf 0,\mathbf M)\). By consistency and the uniform law of large numbers \(-n^{-1}\nabla\mathbf s_n(\tilde{\boldsymbol\xi})\xrightarrow{p}\mathbf M\) (the observed information per observation converging to the Fisher information \(\mathbf M\)), which is invertible. Solving for \(\sqrt n(\hat{\boldsymbol\xi}-\boldsymbol\xi_0)\) and applying Slutsky’s theorem, \[ \sqrt n(\hat{\boldsymbol\xi}-\boldsymbol\xi_0)=\mathbf M^{-1}\,\frac1{\sqrt n}\mathbf s_n(\boldsymbol\xi_0)+o_p(1)\ \xrightarrow{d}\ \mathcal N\!\bigl(\mathbf 0,\ \mathbf M^{-1}\mathbf M\,\mathbf M^{-1}\bigr)=\mathcal N\!\bigl(\mathbf 0,\ \mathbf M^{-1}\bigr), \] which is Equation 19.6. The sandwich form \(\mathbf M^{-1}\mathbf M\mathbf M^{-1}\) collapses to \(\mathbf M^{-1}\) precisely because correct specification forces the meat to equal the bread. \(\square\)

Why this matters. Every STC estimator in this chapter, the plug-in, the marginal estimator, and their delta-method and Bayesian variances, is a function of \(\hat{\boldsymbol\xi}\). Theorem 19.1 is what lets us carry convergence and a limiting distribution through those functions: consistency of \(\hat{\boldsymbol\xi}\) becomes consistency of the transported effect (Theorem 19.3), and the Gaussian limit becomes the standard error we report. It is the same relationship regression has to everything built on it: pin down the coefficients first, then read off whatever functional the application needs.

Remark. Misspecification and the sandwich. Drop hypothesis 1. The estimating function Equation 19.5 still has a well-defined zero \(\boldsymbol\xi^\ast\), the pseudo-true parameter, solving \(\mathbb E[\mathbf s_i(\boldsymbol\xi^\ast)]=\mathbf 0\); it minimizes the Kullback-Leibler divergence from the true law to the model family (Vaart (1998), Section 5.5). The same Taylor argument then yields \(\sqrt n(\hat{\boldsymbol\xi}-\boldsymbol\xi^\ast)\xrightarrow{d}\mathcal N(\mathbf 0,\ \mathbf A^{-1}\mathbf B\mathbf A^{-1})\), the sandwich of Definition 5.11, with bread \(\mathbf A=\mathbb E[-\nabla\mathbf s_i(\boldsymbol\xi^\ast)]\) and meat \(\mathbf B=\mathbb E[\mathbf s_i(\boldsymbol\xi^\ast)\mathbf s_i(\boldsymbol\xi^\ast)^\top]\), which no longer coincide. The estimator is then consistent for \(\boldsymbol\xi^\ast\), not \(\boldsymbol\xi_0\), and the STC contrasts built from it inherit a bias equal to the discrepancy between the pseudo-true surface and the truth. This is the formal statement of STC’s central vulnerability: it is only as good as its model. \(\blacktriangleleft\)

Remark. A misspecification example. Suppose the true event probability rises with the square of a covariate, \(\operatorname{logit}\mu_C(x)=-1+x^2\), but the analyst fits the straight-line model \(\operatorname{logit}\mu_C(x)=\alpha+\beta_1 x\). Inside the index covariate range the straight line can track the curve tolerably, so the fit looks fine on the data at hand. But STC’s selling point is that it extrapolates into the comparator population, and where that population places patients at larger \(x\) than the index trial ever observed, the straight line and the true parabola diverge without bound. STC then transports a confident prediction that is simply wrong: the pseudo-true surface \(\mu_k(\cdot;\boldsymbol\xi^\ast)\) is not the truth off the index support, and the transported contrast inherits exactly that error. This is why “no overlap required” (Section 19.5 below) is a statement about mathematical definability, not a license to extrapolate blindly. \(\blacktriangleleft\)

19.4 Conditional simulated treatment comparison and its plug-in bias

The fitted model can be queried in two essentially different ways, and the difference is the subject of the chapter. The first, historically the original STC of Caro and Ishak (2010), evaluates the fitted contrast at a single synthetic patient whose covariates equal the comparator-population mean.

Definition 19.2 (Conditional (plug-in) STC) Given the fitted model \(\hat{\boldsymbol\xi}\) and the comparator covariate mean \(\bar{\mathbf x}_{BC}\), the conditional STC estimator of the \(A\)-versus-\(C\) relative effect is the fitted conditional relative effect Equation 19.3 evaluated at \(\bar{\mathbf x}_{BC}\), \[ \hat d_{AC}^{\,\mathrm{cond}} \;:=\; g\!\bigl(\mu_C(\bar{\mathbf x}_{BC};\hat{\boldsymbol\xi})\bigr)-g\!\bigl(\mu_A(\bar{\mathbf x}_{BC};\hat{\boldsymbol\xi})\bigr) \;=\;\eta_C(\bar{\mathbf x}_{BC};\hat{\boldsymbol\xi})-\eta_A(\bar{\mathbf x}_{BC};\hat{\boldsymbol\xi}) \;=\;-\bigl(\hat\gamma+\bar{\mathbf x}_{BC}^\top\hat{\boldsymbol\beta}_2\bigr), \tag{19.7}\] in the reference-first orientation of Notation. It is the relative effect predicted for an individual whose covariate vector equals the comparator mean. \(\blacktriangleleft\)

Reader’s translation. Conditional STC builds one synthetic “average patient,” whose covariates are the target-population means, and reports the treatment effect the model predicts for that single patient. It was the original proposal of Caro and Ishak (2010) for a blunt practical reason: it needs only the published covariate means \(\bar{\mathbf x}_{BC}\), which every trial reports, and never the full covariate distribution, which most trials do not. The rest of this section shows what that convenience costs.

The middle equality in Equation 19.7 is exact and not an approximation: applying the link \(g\) to a fitted conditional mean returns the linear predictor, \(g(\mu_k(\mathbf x;\hat{\boldsymbol\xi}))=\eta_k(\mathbf x;\hat{\boldsymbol\xi})\), because \(\mu_k=g^{-1}\circ\eta_k\). The conditional estimator is therefore a difference of linear predictors, and it inherits no curvature. That is exactly the trouble: the marginal estimand it is meant to approximate is a difference of links of averaged means, which does carry curvature. We isolate the target.

Definition 19.3 (The marginal and conditional-at-the-mean estimands) Let \(\mathcal P\) be a population with covariate density \(f_{\mathcal P}\), and write the arm marginal mean and its link-scale position, as in Equation 14.1 of Chapter 14, \[ \bar\mu_{k(\mathcal P)}=\int \mu_k(\mathbf x)\,f_{\mathcal P}(\mathbf x)\,d\mathbf x, \qquad \eta_{k(\mathcal P)}=g\!\bigl(\bar\mu_{k(\mathcal P)}\bigr). \tag{19.8}\] The marginal relative effect in \(\mathcal P\) is the contrast of link positions (Definition 11.2), \[ d_{AC(\mathcal P)}^{\,\mathrm{marg}}\;=\;\eta_{C(\mathcal P)}-\eta_{A(\mathcal P)}\;=\;g\!\bigl(\bar\mu_{C(\mathcal P)}\bigr)-g\!\bigl(\bar\mu_{A(\mathcal P)}\bigr), \tag{19.9}\] and the conditional-at-the-mean effect is the link contrast at the mean covariate (Definition 11.1), \[ d_{AC(\mathcal P)}^{\,\mathrm{cond}}\;=\;g\!\bigl(\mu_C(\bar{\mathbf x}_{\mathcal P})\bigr)-g\!\bigl(\mu_A(\bar{\mathbf x}_{\mathcal P})\bigr)\;=\;\eta_C(\bar{\mathbf x}_{\mathcal P})-\eta_A(\bar{\mathbf x}_{\mathcal P}). \tag{19.10}\] \(\blacktriangleleft\)

Reader’s translation: the average of outcomes versus the outcome of the average. The marginal effect Equation 19.9 averages the outcomes and then contrasts (what a trial run in \(\mathcal P\) would report); the conditional-at-the-mean effect Equation 19.10 contrasts at the single average patient. A straight-line model makes the two equal; a curved link splits them, and a population-level decision wants the first.

The marginal effect Equation 19.9 is the quantity a randomized trial in \(\mathcal P\) would report from its arm summaries, and the quantity Bucher’s identity requires (Definition 14.2). The conditional-at-the-mean effect Equation 19.10 is the effect for a single hypothetical average patient. The conditional STC estimator Equation 19.7 is consistent for Equation 19.10, by Theorem 19.1 and the continuity of \(g\circ g^{-1}\), but for the marginal estimand it carries a bias that we now compute exactly. The result is catalogue G.2.

The intuition is one sentence: a curved link makes the prediction at the mean covariate differ from the mean of the predictions. Bend a curve, mark the midpoint of two inputs, and the curve’s height at that midpoint is not the average of its two output heights; the gap grows with how much the curve bends and how spread out the inputs are. The next theorem turns that sentence into an exact formula: the arm-level gap is, to second order, half the variance of the linear predictor times the curvature of the inverse link (Equation 19.12).

Theorem 19.2 (Plug-in bias of conditional STC under a nonlinear link) Assume the outcome model Definition 19.1 is correctly specified and that \(g^{-1}\) is twice differentiable on the range of the linear predictors. Then:

  1. What it estimates. \(\hat d_{AC}^{\,\mathrm{cond}}\xrightarrow{p} d_{AC(\mathcal P_{BC})}^{\,\mathrm{cond}}\), the conditional-at-the-mean effect Equation 19.10, not the marginal effect.

  2. Arm-level Jensen gap. For each arm \(k\) the link of the marginal mean differs from the linear predictor at the mean by \[ \eta_{k(\mathcal P_{BC})}-\eta_k(\bar{\mathbf x}_{BC}) = g\!\bigl(\mathbb E_{f_{BC}}[\mu_k(X)]\bigr)-g\!\bigl(\mu_k(\bar{\mathbf x}_{BC})\bigr), \tag{19.11}\] and, writing \(b_k:=\bar\mu_{k(\mathcal P_{BC})}-\mu_k(\bar{\mathbf x}_{BC})=\mathbb E_{f_{BC}}[\mu_k(X)]-\mu_k(\bar{\mathbf x}_{BC})\) for the response-scale gap, \[ b_k=\tfrac12\,\mathbb E_{f_{BC}}\!\Bigl[\bigl(\boldsymbol\beta_{(k)}^\top(X-\bar{\mathbf x}_{BC})\bigr)^2\,(g^{-1})''(\xi_{k,X})\Bigr], \qquad \boldsymbol\beta_{(C)}=\boldsymbol\beta_1,\ \boldsymbol\beta_{(A)}=\boldsymbol\beta_1+\boldsymbol\beta_2, \tag{19.12}\] for some \(\xi_{k,X}\) between \(\eta_k(X)\) and \(\eta_k(\bar{\mathbf x}_{BC})\). The gap \(b_k\) is nonzero whenever \((g^{-1})''\not\equiv0\) on the relevant range and \(\mathrm{Var}_{f_{BC}}(\boldsymbol\beta_{(k)}^\top X)>0\).

  3. Contrast bias. The asymptotic bias of conditional STC for the marginal effect is the difference of the two arm-level link gaps Equation 19.11, \[ d_{AC(\mathcal P_{BC})}^{\,\mathrm{cond}}-d_{AC(\mathcal P_{BC})}^{\,\mathrm{marg}} =\bigl[\eta_A(\bar{\mathbf x}_{BC})-\eta_{A(\mathcal P_{BC})}\bigr]-\bigl[\eta_C(\bar{\mathbf x}_{BC})-\eta_{C(\mathcal P_{BC})}\bigr], \tag{19.13}\] which vanishes for the identity link and is generally nonzero otherwise.

Proof. Part 1 (what the plug-in converges to). By Theorem 19.1, \(\hat{\boldsymbol\xi}\xrightarrow{p}\boldsymbol\xi_0\), and the maps \(\boldsymbol\xi\mapsto\eta_k(\bar{\mathbf x}_{BC};\boldsymbol\xi)\) are continuous (indeed affine), so by the continuous mapping theorem\(\hat d_{AC}^{\,\mathrm{cond}}=\eta_C(\bar{\mathbf x}_{BC};\hat{\boldsymbol\xi})-\eta_A(\bar{\mathbf x}_{BC};\hat{\boldsymbol\xi})\xrightarrow{p}\eta_C(\bar{\mathbf x}_{BC})-\eta_A(\bar{\mathbf x}_{BC})=d_{AC(\mathcal P_{BC})}^{\,\mathrm{cond}}\), which is Equation 19.10 by definition.

Part 2 (the per-arm Jensen gap). Fix an arm \(k\) and abbreviate \(\eta_k(\mathbf x)=\eta_k(\mathbf x;\boldsymbol\xi_0)\), \(m(\mathbf x)=\mu_k(\mathbf x)=g^{-1}(\eta_k(\mathbf x))\), \(\bar{\mathbf x}=\bar{\mathbf x}_{BC}\). Because \(\eta_k\) is affine, \(\eta_k(X)-\eta_k(\bar{\mathbf x})=\boldsymbol\beta_{(k)}^\top(X-\bar{\mathbf x})\) with \(\boldsymbol\beta_{(k)}\) the covariate slope of arm \(k\), namely \(\boldsymbol\beta_1\) for \(C\) and \(\boldsymbol\beta_1+\boldsymbol\beta_2\) for \(A\). A second-order Taylor expansion of \(m=g^{-1}\circ\eta_k\) about \(\bar{\mathbf x}\) with Lagrange remainder gives, for each \(\mathbf x\), \[ m(\mathbf x)=m(\bar{\mathbf x})+(g^{-1})'(\eta_k(\bar{\mathbf x}))\,\boldsymbol\beta_{(k)}^\top(\mathbf x-\bar{\mathbf x})+\tfrac12(g^{-1})''(\xi_{k,\mathbf x})\,\bigl(\boldsymbol\beta_{(k)}^\top(\mathbf x-\bar{\mathbf x})\bigr)^2, \] with \(\xi_{k,\mathbf x}\) between \(\eta_k(\mathbf x)\) and \(\eta_k(\bar{\mathbf x})\). Take the expectation over \(X\sim f_{BC}\). The first term is constant; the second term vanishes because \(\mathbb E_{f_{BC}}[X-\bar{\mathbf x}]=\mathbf 0\) and the coefficient is constant; the third term gives Equation 19.12: \[ b_k=\mathbb E_{f_{BC}}[m(X)]-m(\bar{\mathbf x})=\tfrac12\,\mathbb E_{f_{BC}}\!\Bigl[(g^{-1})''(\xi_{k,X})\bigl(\boldsymbol\beta_{(k)}^\top(X-\bar{\mathbf x})\bigr)^2\Bigr]. \] The integrand is a product of \((g^{-1})''\) and a nonnegative quadratic with positive expectation when the slope contrast has positive variance; if moreover \((g^{-1})''\) has a constant sign on the range (true for the logit on either side of zero and for the log everywhere), the gap \(b_k\) is strictly of that sign, hence nonzero. Applying the increasing function \(g\) to the two means in Equation 19.8 and subtracting gives the link-scale gap Equation 19.11, which is a strictly monotone transform of \(b_k\) and therefore vanishes iff \(b_k=0\).

Part 3 (the contrast bias is the difference of the two gaps). Subtract the marginal estimand Equation 19.9 from the conditional-at-the-mean estimand Equation 19.10 and regroup by arm: \[ \begin{aligned} d_{AC(\mathcal P_{BC})}^{\,\mathrm{cond}}-d_{AC(\mathcal P_{BC})}^{\,\mathrm{marg}} &=\bigl[\eta_C(\bar{\mathbf x}_{BC})-\eta_A(\bar{\mathbf x}_{BC})\bigr]-\bigl[\eta_{C(\mathcal P_{BC})}-\eta_{A(\mathcal P_{BC})}\bigr]\\ &=\bigl[\eta_A(\bar{\mathbf x}_{BC})-\eta_{A(\mathcal P_{BC})}\bigr]-\bigl[\eta_C(\bar{\mathbf x}_{BC})-\eta_{C(\mathcal P_{BC})}\bigr], \end{aligned} \] which is Equation 19.13. For the identity link \(g^{-1}(\eta)=\eta\), so \((g^{-1})''\equiv0\) and each \(b_k=0\) by Equation 19.12, hence each bracket in Equation 19.13 vanishes and the contrast bias is zero. For a nonlinear link the two brackets are the arm-level gaps Equation 19.11, which are individually nonzero and need not cancel, so the contrast bias is generally nonzero. \(\square\)

Why this matters. A cost-effectiveness model, the engine of a health technology assessment, is handed a single marginal effect together with a target population, and it prices a technology for that whole population. Feed it the conditional-at-the-mean effect instead and you have silently evaluated the drug for one hypothetical average patient, not for the population the funding decision actually covers. Theorem 19.2 says the substitution is not a rounding error: under any nonlinear link it is a systematic bias, invisible without the covariate distribution and unshrinkable by collecting more data (it is a bias, not a variance). This is why STC, and its descendant ML-NMR (Chapter 20), integrate on the natural outcome scale rather than plug in a mean.

The decomposition Equation 19.13 is the precise content of the slogan “the link does not commute with averaging.” It also shows exactly when conditional STC happens to be unbiased for the marginal effect: not only under the identity link, but whenever the two arm-level Jensen gaps coincide. The log link makes this boundary explicit and connects it to collapsibility.

Remark. A legitimate estimand for the wrong question. The conditional-at-the-mean effect Equation 19.10 is not a mistake of definition; it is a well-defined quantity, the effect a clinician would quote for a patient whose covariates sit at the population average. Its defect is one of fitness for purpose. A health technology assessment compares two treatments at the level of a whole population, and the population-level contrast is the marginal effect Equation 19.9, the average of outcomes, not the outcome of the average. Reporting Equation 19.10 where Equation 19.9 is wanted substitutes one estimand for another and, by Theorem 19.2, does so with a bias that is invisible without the covariate distribution. The defect is also not removed by a larger sample: it is a bias, not a variance. \(\blacktriangleleft\)

19.5 Marginal simulated treatment comparison

The cure is to integrate before linking. Marginal STC averages the fitted conditional means over the comparator covariate distribution on the natural outcome scale, and applies the link only to the resulting marginal means.

Definition 19.4 (Marginal STC) With the fitted model \(\hat{\boldsymbol\xi}\) and a sample (or density) representing the comparator covariate distribution \(f_{BC}\), the marginal STC predicted means under the two arms are \[ \hat\mu_A^{\mathrm{STC}}=\int \mu_A(\mathbf x;\hat{\boldsymbol\xi})\,f_{BC}(\mathbf x)\,d\mathbf x, \qquad \hat\mu_C^{\mathrm{STC}}=\int \mu_C(\mathbf x;\hat{\boldsymbol\xi})\,f_{BC}(\mathbf x)\,d\mathbf x, \tag{19.14}\] and the marginal STC estimator of the relative effect is their link contrast \[ \hat d_{AC}^{\,\mathrm{marg}}\;=\;g\!\bigl(\hat\mu_C^{\mathrm{STC}}\bigr)-g\!\bigl(\hat\mu_A^{\mathrm{STC}}\bigr). \tag{19.15}\] In practice the integrals Equation 19.14 are Monte Carlo averages over \(M\) covariate vectors \(\{\mathbf x_s^\ast\}_{s=1}^M\) drawn from \(f_{BC}\), or over the observed AgD patient covariates when those are available, \[ \hat\mu_k^{\mathrm{STC}}\;\approx\;\frac1M\sum_{s=1}^M \mu_k(\mathbf x_s^\ast;\hat{\boldsymbol\xi}). \tag{19.16}\] When only marginal covariate summaries are published, \(f_{BC}\) is reconstructed from a parametric model, typically a Gaussian copula (Chapter 22), before sampling. \(\blacktriangleleft\)

Reader’s translation: this is where the “simulation” happens. The integral Equation 19.14 is almost never available in closed form for a nonlinear model, so in practice we simulate: draw \(M\) synthetic patients from the target covariate distribution \(f_{BC}\), push each through the fitted model to get their predicted event probabilities under \(A\) and \(C\), and average Equation 19.16. That averaging is the “simulated” in simulated treatment comparison. We are not simulating outcomes, nor a missing trial arm; we are numerically averaging fitted predictions over a target case mix.

Remark. Simulating covariates from aggregate data. When the comparator trial reports patient-level covariates, \(f_{BC}\) is those patients and step 2 is trivial. When it reports only summaries, we must reconstruct \(f_{BC}\) before we can sample it, and how much we can reconstruct depends on what was published:

  • Only proportions and means. A binary covariate is reproduced exactly from its reported prevalence; a continuous covariate is reproduced from a reported mean and standard deviation under an assumed shape (often Gaussian). With no reported correlations, the covariates are simulated as independent, which is itself an assumption.
  • Means, variances, and a correlation matrix. The joint distribution can be reconstructed with a copula (Chapter 22; see Notation), typically a Gaussian copula, which glues the reported margins together with the reported correlations.
  • Impossible combinations. A naive independent simulation can generate covariate profiles that never occur clinically (for example a pregnant male, or an implausible age-comorbidity pairing); realistic dependence, or explicit constraints, are needed to avoid predicting the fitted model into nonsense regions.

The reconstructed \(f_{BC}\) is an input, not a fact: if it is wrong, marginal STC integrates the fitted surface over the wrong measure and is biased even when the outcome model is perfect. This dependence on a reconstructed covariate distribution is a distinct source of uncertainty, listed in the checklist of Section 19.8. \(\blacktriangleleft\)

The order of operations in 1 is the whole point, and it is the order of g-computation. We make that identification precise, then prove consistency.

Proposition 19.1 (Marginal STC is transported g-computation) The marginal STC predicted means Equation 19.14 are the g-computation (standardization) functionals of Chapter 10 with the integrating measure set to the target covariate distribution \(f_{BC}\) rather than the source \(f_{AC}\). If, in addition to correct specification, the conditional outcome surface is transportable, \(\mathbb E_{\mathcal P_{BC}}[Y(k)\mid X=\mathbf x]=\mu_k(\mathbf x)\) for \(f_{BC}\)-almost every \(\mathbf x\) (Theorem 12.2), then \(\bar\mu_{k(\mathcal P_{BC})}=\mathbb E_{\mathcal P_{BC}}[Y(k)]\), the marginal potential-outcome mean of arm \(k\) in the comparator population.

Proof. The g-computation functional for arm \(k\) in a population \(\mathcal P\) is, by Theorem 10.4, \(\int\mu_k(\mathbf x)\,f_{\mathcal P}(\mathbf x)\,d\mathbf x=\mathbb E_{\mathcal P}[Y(k)]\) under consistency, ignorability, and a correct \(\mu_k\). Equation Equation 19.14 is this functional evaluated with \(f_{\mathcal P}=f_{BC}\) and the source-fitted surface \(\mu_k(\cdot;\hat{\boldsymbol\xi})\). The surface is identified from the index trial, where arm \(A\) is observed; transportability (Theorem 12.2, the population-adjustment instance of Definition 12.3) asserts that this same surface governs the conditional means in the comparator population, where arm \(A\) is unobserved. Under that assumption the tower property gives \(\mathbb E_{\mathcal P_{BC}}[Y(k)]=\mathbb E_{f_{BC}}\bigl[\mathbb E_{\mathcal P_{BC}}[Y(k)\mid X]\bigr]=\mathbb E_{f_{BC}}[\mu_k(X)]=\bar\mu_{k(\mathcal P_{BC})}\). \(\square\)

Remark. Why STC needs no overlap. The g-computation identity requires only that \(\mu_k(\cdot)\) be the correct and transportable conditional surface; it places no positivity requirement on \(f_{BC}\) relative to \(f_{AC}\). The integral Equation 19.14 can be evaluated at covariate values the index trial never observed, because a regression returns a prediction wherever its predictors are defined. This is precisely the property that inverse-probability weighting, and therefore MAIC, lacks (Theorem 18.8): a weight is undefined where the source density is zero. STC trades the overlap requirement for an extrapolation that is only as trustworthy as the model is correct off the support of \(f_{AC}\). Read those two words carefully: STC’s freedom from overlap is a statement about mathematical definability, not scientific credibility. A regression will always return a number outside the index covariate range, but that number rests entirely on the model’s assumed functional form there, which the data cannot check (recall the misspecification example after Theorem 19.1). Good practice therefore diagnoses and reports the extent of extrapolation, for instance by flagging how much of the target covariate distribution falls outside the index trial’s observed range, exactly the region where MAIC would refuse to weight and STC quietly predicts anyway. \(\blacktriangleleft\)

Theorem 19.3 (Consistency of marginal STC) Assume (i) the outcome model Definition 19.1 is correctly specified, so \(\hat{\boldsymbol\xi}\xrightarrow{p}\boldsymbol\xi_0\) by Theorem 19.1; (ii) the conditional outcome surface is transportable to the comparator population (Proposition 19.1); and (iii) the integrals Equation 19.14 are computed exactly, or by Monte Carlo Equation 19.16 with \(M\to\infty\). Then the marginal STC predicted means and the marginal STC estimator are consistent, \[ \hat\mu_k^{\mathrm{STC}}\xrightarrow{p}\bar\mu_{k(\mathcal P_{BC})}=\mathbb E_{\mathcal P_{BC}}[Y(k)], \qquad \hat d_{AC}^{\,\mathrm{marg}}\xrightarrow{p} d_{AC(\mathcal P_{BC})}^{\,\mathrm{marg}}, \tag{19.17}\] the marginal relative effect Equation 19.9 of \(A\) versus \(C\) in the comparator population. (Corresponds to G.3 in the proof catalogue.)

[machine-checked: theories/STC.v, Theorem STC_consistent]

Proof. Fix an arm \(k\). The fitted conditional mean is \(\mu_k(\mathbf x;\hat{\boldsymbol\xi})=g^{-1}(\eta_k(\mathbf x;\hat{\boldsymbol\xi}))\), a composition of the continuous \(g^{-1}\) with the affine \(\eta_k(\mathbf x;\cdot)\). Since \(\hat{\boldsymbol\xi}\xrightarrow{p}\boldsymbol\xi_0\) and the covariates are bounded (hypothesis 2 of Theorem 19.1), the convergence is uniform over the bounded covariate range: for any \(\epsilon>0\), \(\sup_{\mathbf x}\,|\mu_k(\mathbf x;\hat{\boldsymbol\xi})-\mu_k(\mathbf x;\boldsymbol\xi_0)|\xrightarrow{p}0\), because \(g^{-1}\) is uniformly continuous on the compact image of the linear predictor and the affine map is jointly continuous. Integrating against the probability measure \(f_{BC}\) preserves the uniform bound, \[ \Bigl|\,\hat\mu_k^{\mathrm{STC}}-\textstyle\int\mu_k(\mathbf x;\boldsymbol\xi_0)f_{BC}(\mathbf x)\,d\mathbf x\,\Bigr| \le \sup_{\mathbf x}\bigl|\mu_k(\mathbf x;\hat{\boldsymbol\xi})-\mu_k(\mathbf x;\boldsymbol\xi_0)\bigr|\xrightarrow{p}0, \] so \(\hat\mu_k^{\mathrm{STC}}\xrightarrow{p}\int\mu_k(\mathbf x;\boldsymbol\xi_0)f_{BC}(\mathbf x)\,d\mathbf x=\bar\mu_{k(\mathcal P_{BC})}\). By Proposition 19.1 this limit equals \(\mathbb E_{\mathcal P_{BC}}[Y(k)]\), the transported potential-outcome mean. When the integral is replaced by the Monte Carlo average Equation 19.16, the strong law of large numbers gives \(\frac1M\sum_s\mu_k(\mathbf x_s^\ast;\boldsymbol\xi)\xrightarrow{\text{a.s.}}\bar\mu_{k(\mathcal P_{BC})}\) for each fixed \(\boldsymbol\xi\) as \(M\to\infty\), and the bounded integrand makes the double limit in \((n,M)\) interchangeable, so the conclusion is unchanged. Finally, \(g\) is continuous, so by the continuous mapping theorem\[ \hat d_{AC}^{\,\mathrm{marg}}=g(\hat\mu_C^{\mathrm{STC}})-g(\hat\mu_A^{\mathrm{STC}})\xrightarrow{p} g(\bar\mu_{C(\mathcal P_{BC})})-g(\bar\mu_{A(\mathcal P_{BC})})=d_{AC(\mathcal P_{BC})}^{\,\mathrm{marg}}, \] which is Equation 19.17. \(\square\)

Why this matters. This is the theorem that licenses the whole method: integrate the fitted surface over the target covariate distribution on the natural outcome scale, and you recover the marginal effect a trial in that population would have reported, with no bias from non-collapsibility. It is the exact analogue, for transport, of the g-computation result that standardization recovers the average treatment effect (Theorem 10.4); the only change is the population you standardize to. It is also the seed of ML-NMR (Chapter 20), which puts this same natural-outcome-scale integration inside a single Bayesian network model and dispenses with the separate Bucher step.

Remark. On the machine-checked tag. The Coq development theories/STC.v defines the marginal STC predicted means mu_A_STC, mu_C_STC as the target-population means of the fitted surfaces and the estimator effect_AC_STC as their difference, and Theorem STC_consistent proves the response-scale contrast identity \(\texttt{effect\_AC\_STC}=\mathbb E_2[Y_A^{\mathrm{target}}]-\mathbb E_2[Y_C^{\mathrm{target}}]\) given the two per-arm identification premises (the transported g-computation identities of Proposition 19.1). The Coq theorem is thus the identity-scale skeleton of Equation 19.17; the link transformation by \(g\) and the uniform-convergence step are the pen-and-paper additions made above. The companion development treats the target-population expectation as an abstract linear operator, exactly as Chapter 14 treats expectation in Bucher’s theorem. \(\blacktriangleleft\)

19.6 Non-collapsibility and its collapsible boundary

Theorem 19.2 located the plug-in bias in the curvature of \(g^{-1}\). The same curvature is the source of non-collapsibility (Definition 7.7): the marginal effect is not the conditional effect even when the conditional effect is constant across patients, that is, even with no effect modification. We exhibit the gap in closed form for the odds ratio (catalogue G.4), then identify the boundary on which it disappears.

Before the algebra, see the phenomenon in a table of two patients. Take the logistic model \(\operatorname{logit}\Pr(\text{event}\mid k,x)=\alpha_k+\beta x\) with \(\alpha_C=-1\), \(\alpha_A=0.5\), \(\beta=2\) (a steeper covariate slope than the running example, chosen to enlarge the gap), and a population that is half low-risk (\(x=0\)) and half high-risk (\(x=1\)). The fitted event probabilities are \(\operatorname{expit}\) of the linear predictors:

Patient stratum \(\Pr(\text{event}\mid C)\) \(\Pr(\text{event}\mid A)\) within-stratum log odds ratio
low-risk (\(x=0\)) \(\operatorname{expit}(-1)=0.2689\) \(\operatorname{expit}(0.5)=0.6225\) \(1.5\)
high-risk (\(x=1\)) \(\operatorname{expit}(1)=0.7311\) \(\operatorname{expit}(2.5)=0.9241\) \(1.5\)
population average (50/50) \(0.5000\) \(0.7733\) \(1.227\) (marginal)

Read the last column. Within each stratum the log odds ratio of \(A\) versus \(C\) is exactly \(1.5\): the effect is constant across patients, so there is no effect modification. Yet the bottom row, formed by averaging the probabilities down each arm and then taking the odds ratio, gives \(1.227\), a different number. Averaging the probabilities and taking the odds ratio does not recover the common within-stratum odds ratio. That mismatch is non-collapsibility, and it is exactly why plug-in STC, which reports the constant conditional \(1.5\), overstates the marginal effect the population would show. The theorem below is this table written in symbols.

Example 19.2 (Two patients, one broken average) The table above is the whole idea in miniature. The plug-in reports the within-stratum effect \(1.5\); the population marginal effect is \(1.227\); the gap \(0.273\) is present with no effect modification, so it is a property of the odds-ratio scale, not of patient heterogeneity. A risk difference would show no such gap: the two within-stratum differences \(0.6225-0.2689=0.3536\) and \(0.9241-0.7311=0.1930\) average to \((0.3536+0.1930)/2=0.2733\), which is exactly the marginal difference \(0.7733-0.5000=0.2733\), because the identity link commutes with averaging. The odds ratio does not. \(\blacktriangleleft\)

Theorem 19.4 (A closed-form non-collapsibility gap for the odds ratio) Let \(g\) be the logit, so \(g^{-1}=\operatorname{expit}\), and suppose the conditional model \(\operatorname{logit}\Pr(Y=1\mid k,X=x)=\alpha_k+\beta x\) holds with a single covariate \(x\), equal slope \(\beta\ne0\) in both arms (so the conditional log odds ratio \(\Lambda=\alpha_A-\alpha_C\) is constant in \(x\), no effect modification). Then the marginal log odds ratio \[ d^{\,\mathrm{marg}}=\operatorname{logit}\bigl(\mathbb E_X[\operatorname{expit}(\alpha_A+\beta X)]\bigr)-\operatorname{logit}\bigl(\mathbb E_X[\operatorname{expit}(\alpha_C+\beta X)]\bigr) \] differs from the conditional log odds ratio \(\Lambda=\alpha_A-\alpha_C\). Concretely, take \(\alpha_C=-1\), \(\alpha_A=0.5\), \(\beta=2\), and \(X\sim\mathrm{Bernoulli}(0.5)\). Then the conditional log odds ratio is \(\Lambda=0.5-(-1)=1.5\), while \[ \mathbb E_X[\operatorname{expit}(\alpha_C+\beta X)]=\tfrac12\operatorname{expit}(-1)+\tfrac12\operatorname{expit}(1)=\tfrac12(0.268941)+\tfrac12(0.731059)=0.5, \] \[ \mathbb E_X[\operatorname{expit}(\alpha_A+\beta X)]=\tfrac12\operatorname{expit}(0.5)+\tfrac12\operatorname{expit}(2.5)=\tfrac12(0.622459)+\tfrac12(0.924142)=0.773301, \] so the marginal log odds ratio is \[ d^{\,\mathrm{marg}}=\operatorname{logit}(0.773301)-\operatorname{logit}(0.5)=\log\tfrac{0.773301}{0.226699}-0=\log(3.41122)=1.22707. \] The conditional effect \(1.5\) exceeds the marginal effect \(1.22707\) in magnitude: the gap is \(0.27293\ne0\), and conditional STC, which reports \(\Lambda=1.5\), is biased upward for the marginal estimand. (Corresponds to G.4 in the proof catalogue.)

Proof. The arithmetic is the statement. We verify the four inverse-logit values and that the gap is genuine, not rounding. With \(e=2.718281828\): \(\operatorname{expit}(-1)=1/(1+e)=0.2689414\); \(\operatorname{expit}(1)=e/(1+e)=0.7310586\), summing to \(1\) so their half-sum is exactly \(0.5\); \(\operatorname{expit}(0.5)=1/(1+e^{-0.5})=1/(1+0.6065307)=0.6224593\); \(\operatorname{expit}(2.5)=1/(1+e^{-2.5})=1/(1+0.0820850)=0.9241418\); the half-sum is \(0.7733006\). Then \(\operatorname{logit}(0.7733006)=\log(0.7733006)-\log(0.2266994)=\log(3.411221)=1.227074\), while \(\operatorname{logit}(0.5)=0\). The difference \(1.227074\) is not \(1.5\); the discrepancy \(0.272926\) is far larger than the seventh-decimal precision of the inputs, so it is a genuine gap and not a numerical artifact. That the conditional effect exceeds the marginal in magnitude is the generic direction of odds-ratio non-collapsibility established in Chapter 11 (Theorem 11.2): marginalizing a logistic model over a covariate with \(\beta\ne0\) attenuates the effect toward the null. \(\square\)

The counterexample uses no effect modification: the conditional log odds ratio is the constant \(1.5\) in both strata, yet the marginal effect is smaller. Non-collapsibility is therefore a phenomenon of the scale, not of effect heterogeneity, and it is what forces STC to choose between the conditional and marginal estimands even when Equation 19.3 is constant. The boundary on which the choice becomes vacuous is the linear, identity-link model, and there the choice provably does not matter. This is the machine-checked core of the chapter.

Proposition 19.2 (Collapsibility under a linear model) Suppose the outcome model is linear with the identity link, \(\mathbb E[Y\mid k,X=\mathbf x]=\alpha+\mathbf x^\top\boldsymbol\beta_1+\mathbf 1\{k=A\}(\gamma+\mathbf x^\top\boldsymbol\beta_2)\), so \(g=\mathrm{id}\) and \(\mu_k(\mathbf x)=\eta_k(\mathbf x)\). Then for every population \(\mathcal P\) the marginal and conditional-at-the-mean effects coincide, \[ d_{AC(\mathcal P)}^{\,\mathrm{marg}}=d_{AC(\mathcal P)}^{\,\mathrm{cond}}=-\bigl(\gamma+\bar{\mathbf x}_{\mathcal P}^\top\boldsymbol\beta_2\bigr), \tag{19.18}\] and in the absence of effect modification (\(\boldsymbol\beta_2=\mathbf 0\)) both equal the constant \(-\gamma\), independent of the covariate distribution. Marginal and conditional STC then return the same number, and the plug-in is exact.

[machine-checked: theories/STC.v, Lemma STC_collapse_linear]

Proof. With the identity link, the arm marginal mean Equation 19.8 is, by linearity of expectation, \[ \bar\mu_{k(\mathcal P)}=\mathbb E_{f_{\mathcal P}}[\eta_k(X)]=\alpha+\bar{\mathbf x}_{\mathcal P}^\top\boldsymbol\beta_1+\mathbf 1\{k=A\}\bigl(\gamma+\bar{\mathbf x}_{\mathcal P}^\top\boldsymbol\beta_2\bigr)=\eta_k(\bar{\mathbf x}_{\mathcal P}), \] so \(g(\bar\mu_{k(\mathcal P)})=\bar\mu_{k(\mathcal P)}=\eta_k(\bar{\mathbf x}_{\mathcal P})=g(\mu_k(\bar{\mathbf x}_{\mathcal P}))\), the marginal and conditional-at-the-mean arm positions coincide, and so do their contrasts: Equation 19.18. The contrast is \(\eta_C(\bar{\mathbf x}_{\mathcal P})-\eta_A(\bar{\mathbf x}_{\mathcal P})=-(\gamma+\bar{\mathbf x}_{\mathcal P}^\top\boldsymbol\beta_2)\), which reduces to \(-\gamma\) when \(\boldsymbol\beta_2=\mathbf 0\), a constant free of the covariate distribution. Equivalently, this is Equation 19.13 with every \(b_k=0\). \(\square\)

Remark. What the Coq proof isolates. The companion Lemma STC_collapse_linear proves that for a linear outcome model the marginal STC estimate equals the treatment coefficient \(\gamma\), using only linearity of the abstract target-population expectation operator and its constant rule. That single hypothesis, linearity of expectation, is exactly what a nonlinear link destroys: \(g^{-1}\) does not pass through an average, and the proof of Proposition 19.2 breaks at its first line. The machine-checked lemma is therefore the positive boundary of the non-collapsibility theorem Theorem 19.4; the counterexample itself, an arithmetic over the logit, is pen-and-paper. \(\blacktriangleleft\)

19.7 A worked simulated treatment comparison

We carry a complete logistic STC from a fitted model to a reported relative effect, computing the biased plug-in, the correct target marginal, and, for contrast, the unadjusted source marginal, so that every number can be checked by hand. The covariate is the single binary risk marker of the running example (\(x=1\) high-risk, \(x=0\) low-risk), the event is an unfavorable disease flare within six months, and a larger log odds ratio means worse.

Example 19.3 (Conditional plug-in versus correct marginalization) The index trial of the running example (Section 19.2) compares active \(A\) against comparator \(C\) with individual data; fitting the logistic model Definition 19.1 returns the coefficients (treated here as exact for transparency) \[ \hat\alpha=-1.0,\qquad \hat{\beta}_1=1.0,\qquad \hat\gamma=1.5,\qquad \hat{\beta}_2=0, \] so \(\operatorname{logit}\mu_C(x)=-1.0+x\) and \(\operatorname{logit}\mu_A(x)=0.5+x\), and the conditional log odds ratio of \(A\) versus \(C\) is the constant \(\Lambda=\hat\gamma=1.5\) (no effect modification, \(\hat\beta_2=0\)). The fitted conditional event probabilities are \[ \mu_C(0)=\operatorname{expit}(-1)=0.26894,\quad \mu_C(1)=\operatorname{expit}(0)=0.50000, \] \[ \mu_A(0)=\operatorname{expit}(0.5)=0.62246,\quad \mu_A(1)=\operatorname{expit}(1.5)=0.81757. \] The comparator (target) population has biomarker prevalence \(\Pr_{BC}(X=1)=0.8\), hence \(\bar{x}_{BC}=0.8\); the index (source) population has \(\Pr_{AC}(X=1)=0.5\).

Step 1: conditional (plug-in) STC. Evaluate the linear predictors at the target mean \(\bar x_{BC}=0.8\): \(\eta_C(0.8)=-1.0+0.8=-0.2\) and \(\eta_A(0.8)=0.5+0.8=1.3\). The plug-in log odds ratio of \(A\) versus \(C\) is \[ \ell^{\,\mathrm{cond}}=\eta_A(0.8)-\eta_C(0.8)=1.3-(-0.2)=1.5, \] equal to \(\hat\gamma\) as Proposition 19.2 predicts on the link scale: a difference of linear predictors carries no curvature. In the reference-first convention \(\hat d_{AC}^{\,\mathrm{cond}}=-1.5\).

Step 2: marginal STC in the target population. Average the fitted probabilities over \(f_{BC}\) on the response scale Equation 19.14: \[ \hat\mu_C^{\mathrm{STC}}=0.2(0.26894)+0.8(0.50000)=0.45379,\quad \hat\mu_A^{\mathrm{STC}}=0.2(0.62246)+0.8(0.81757)=0.77855. \] Apply the logit and contrast 1: \[ \ell^{\,\mathrm{marg}}_{BC}=\operatorname{logit}(0.77855)-\operatorname{logit}(0.45379)=1.25723-(-0.18538)=1.44261. \]

Step 3: the unadjusted source marginal. The marginal effect the index trial itself reports averages over \(f_{AC}\) (\(\Pr=0.5\)): \[ \hat\mu_C^{AC}=0.5(0.26894)+0.5(0.50000)=0.38447,\quad \hat\mu_A^{AC}=0.5(0.62246)+0.5(0.81757)=0.72002, \] \[ \ell^{\,\mathrm{marg}}_{AC}=\operatorname{logit}(0.72002)-\operatorname{logit}(0.38447)=0.94454-(-0.47061)=1.41515. \]

Step 4: comparison. Collecting the three log odds ratios of \(A\) versus \(C\) and exponentiating, \[ \underbrace{\ell^{\,\mathrm{cond}}=1.50000}_{\widehat{\mathrm{OR}}=4.4817},\qquad \underbrace{\ell^{\,\mathrm{marg}}_{BC}=1.44261}_{\widehat{\mathrm{OR}}=4.2317},\qquad \underbrace{\ell^{\,\mathrm{marg}}_{AC}=1.41515}_{\widehat{\mathrm{OR}}=4.1171}. \] \(\blacktriangleleft\)

Three lessons sit in these numbers.

Remark. The plug-in bias. The conditional STC estimate \(1.50000\) overstates the target marginal effect \(1.44261\) by \(0.05739\) on the log odds-ratio scale, an odds ratio inflated from \(4.232\) to \(4.482\). This is the non-collapsibility gap Equation 19.13 of Theorem 19.2, present here even though there is no effect modification (\(\hat\beta_2=0\)): the odds ratio simply is not collapsible. By Example 19.1, had the link been the log rather than the logit, the contrast bias with \(\hat\beta_2=0\) would have been exactly zero, the arm-level Jensen gaps cancelling. \(\blacktriangleleft\)

Remark. What population adjustment moves. The correct marginal effect is not a single number; it is population-specific. In the source population it is \(1.41515\); in the target it is \(1.44261\). STC’s job is to report the target value \(1.44261\), and it does so by integrating the fitted surface over \(f_{BC}\) in Step 2 rather than over \(f_{AC}\) in Step 3. The shift \(1.41515\to1.44261\) is the entire population adjustment: it is small here because the biomarker is not an effect modifier (\(\hat\beta_2=0\)), so only the non-collapsibility interaction with the changed covariate mix moves the marginal effect. With \(\hat\beta_2\ne0\) the shift would be first-order in the covariate-mean difference and typically far larger; Exercise 19.4 works such a case. \(\blacktriangleleft\)

Remark. Closing the triangle. To reach the clinical target \(A\) versus \(B\), the STC output \(\hat d_{AC(\mathcal P_{BC})}^{\,\mathrm{marg}}=-1.44261\) (reference-first) is differenced against the comparator trial’s published \(\hat d_{BC}\) by Bucher’s identity (Theorem 14.1), \(\hat d_{AB}=\hat d_{AC(\mathcal P_{BC})}-\hat d_{BC}\), both now evaluated in the common population \(\mathcal P_{BC}\). The marginalization of Step 2 is exactly what makes the two contrasts live in the same population, repairing the transitivity failure (Definition 14.3) that the raw source effect \(1.41515\) of Step 3 would have carried into the subtraction. \(\blacktriangleleft\)

19.8 Bayesian g-computation

We now handle uncertainty properly, then (in Section 19.9) fold STC back into the Chapter 17 framework. In applied terms, Bayesian g-computation runs the five-step workflow once per posterior draw of the model coefficients: each draw is one plausible fitted surface, hence one plausible target effect, and the spread of those target effects across draws is the uncertainty interval. Because the marginal STC functional is nonlinear in \(\boldsymbol\xi\), its uncertainty is generally skewed; simulating from the posterior captures that skew automatically, whereas the delta method (Exercise 19.9) forces a symmetric Gaussian on the link scale.

Marginal STC is a smooth functional of the fitted parameter, so its uncertainty can be propagated either by the delta method (Exercise 19.9) or, more naturally for the nonlinear integration step, by simulation from a posterior. The Bayesian implementation is g-computation from the posterior and is catalogue result G.5.

Definition 19.5 (Bayesian g-computation for STC) Place a prior \(\pi(\boldsymbol\xi)\) on the outcome-model parameters and obtain the posterior \(\pi(\boldsymbol\xi\mid\mathcal D)\) from the index IPD \(\mathcal D\). Define the STC functional \[ \Psi(\boldsymbol\xi)\;=\;g\!\Bigl(\textstyle\int\mu_C(\mathbf x;\boldsymbol\xi)f_{BC}(\mathbf x)\,d\mathbf x\Bigr)-g\!\Bigl(\textstyle\int\mu_A(\mathbf x;\boldsymbol\xi)f_{BC}(\mathbf x)\,d\mathbf x\Bigr), \tag{19.19}\] which maps a parameter to the marginal STC relative effect 1 it induces. The posterior of the target effect is the pushforward \(\Psi_\#\,\pi(\cdot\mid\mathcal D)\) (the “pushforward” is simply the distribution of \(\Psi(\boldsymbol\xi)\) when \(\boldsymbol\xi\) is drawn from the posterior: feed posterior draws through \(\Psi\) and collect the outputs), approximated by drawing \(\boldsymbol\xi^{(1)},\dots,\boldsymbol\xi^{(S)}\sim\pi(\boldsymbol\xi\mid\mathcal D)\) and forming \(\{\Psi(\boldsymbol\xi^{(s)})\}_{s=1}^S\), with each integral in Equation 19.19 itself a Monte Carlo average Equation 19.16 over \(f_{BC}\). \(\blacktriangleleft\)

Theorem 19.5 (Validity of Bayesian g-computation) Let \(\Psi\) be the STC functional Equation 19.19 with \(g\) and \(g^{-1}\) continuous. Then:

  1. Pushforward. \(\Psi\) is a measurable (indeed continuous) function of \(\boldsymbol\xi\), so the posterior distribution of the target effect is well defined as \(\Psi_\#\,\pi(\cdot\mid\mathcal D)\), and the empirical measure of \(\{\Psi(\boldsymbol\xi^{(s)})\}_{s=1}^S\) converges weakly to it as \(S\to\infty\), with the posterior mean and any posterior quantile of \(\Psi\) estimated at the Monte Carlo rate \(O_p(S^{-1/2})\).

  2. Frequentist calibration. Under the regularity conditions of Theorem 19.1 and a prior with a continuous positive density at \(\boldsymbol\xi_0\), the Bernstein-von Mises theorem (Vaart (1998), Chapter 10) gives \(\|\pi(\cdot\mid\mathcal D)-\mathcal N(\hat{\boldsymbol\xi},n^{-1}\mathcal I(\boldsymbol\xi_0)^{-1})\|_{\mathrm{TV}}\xrightarrow{p}0\). Consequently, by the delta method (Theorem 5.3) applied to the smooth \(\Psi\), the posterior of the target effect is asymptotically \(\mathcal N\bigl(\Psi(\hat{\boldsymbol\xi}),\,n^{-1}\nabla\Psi(\boldsymbol\xi_0)^\top\mathcal I(\boldsymbol\xi_0)^{-1}\nabla\Psi(\boldsymbol\xi_0)\bigr)\), the same Gaussian law as the sampling distribution of the frequentist marginal STC point estimate

    1. Equal-tailed posterior credible intervals therefore have asymptotically correct frequentist coverage. (Corresponds to G.5 in the proof catalogue.)

Proof. Part 1. For each fixed \(\boldsymbol\xi\) the integrand \(\mu_k(\mathbf x;\boldsymbol\xi)=g^{-1}(\eta_k(\mathbf x;\boldsymbol\xi))\) is continuous in \(\boldsymbol\xi\) (composition of continuous \(g^{-1}\) with the affine \(\eta_k\)) and bounded on the covariate range, so by dominated convergence the integrals \(\int\mu_k(\mathbf x;\boldsymbol\xi)f_{BC}(\mathbf x)\,d\mathbf x\) are continuous in \(\boldsymbol\xi\); composing with the continuous \(g\) and differencing, \(\Psi\) is continuous, hence Borel measurable. A measurable function of a random variable has a well-defined law, the pushforward \(\Psi_\#\,\pi(\cdot\mid\mathcal D)\). The draws \(\boldsymbol\xi^{(s)}\) are distributed (exactly, or in the stationary limit of the Markov chain, by Definition 19.5) according to \(\pi(\cdot\mid\mathcal D)\), so \(\Psi(\boldsymbol\xi^{(s)})\) are distributed according to the pushforward; the Glivenko-Cantelli theorem gives weak convergence of their empirical measure as \(S\to\infty\), and the central limit theorem gives the \(O_p(S^{-1/2})\) Monte Carlo rate for the posterior mean and, via the smoothness of the quantile map of the continuous pushforward, for posterior quantiles. The inner Monte Carlo over \(f_{BC}\) adds an independent \(O_p(M^{-1/2})\) error to each \(\Psi(\boldsymbol\xi^{(s)})\), controlled by taking \(M\) large.

Part 2. The Bernstein-von Mises theorem applies because Theorem 19.1 verified the local asymptotic normality conditions (smooth log-likelihood, nonsingular Fisher information \(\mathcal I(\boldsymbol\xi_0)\), consistent MLE) and the prior charges a neighborhood of \(\boldsymbol\xi_0\) with a continuous positive density; its conclusion is the stated total-variation convergence of the posterior to the Gaussian centered at the MLE with covariance \(n^{-1}\mathcal I(\boldsymbol\xi_0)^{-1}\). The functional \(\Psi\) is differentiable at \(\boldsymbol\xi_0\) with gradient \(\nabla\Psi(\boldsymbol\xi_0)\) obtained by differentiating Equation 19.19 under the integral sign (legitimate by bounded, continuous integrands), so the delta method transports the Gaussian limit through \(\Psi\): the posterior of \(\Psi(\boldsymbol\xi)\) converges to \(\mathcal N(\Psi(\hat{\boldsymbol\xi}),n^{-1}\nabla\Psi^\top\mathcal I^{-1}\nabla\Psi)\). The frequentist marginal STC estimate \(\hat d_{AC}^{\,\mathrm{marg}}=\Psi(\hat{\boldsymbol\xi})\) has, again by the delta method applied to the sampling law Equation 19.6 of \(\hat{\boldsymbol\xi}\), the identical asymptotic Gaussian distribution. Two asymptotically equal Gaussian laws produce intervals with the same coverage, so the credible interval is asymptotically a confidence interval. \(\square\)

Why this matters. Part 2 says the honest Bayesian interval and the frequentist confidence interval agree asymptotically, so a health technology assessment that runs STC in a Bayesian pipeline is not buying a different answer, only a more convenient and more faithful propagation of the nonlinearity. It is the same guarantee that lets network meta-analysis (Chapters 15 and 20) report Bayesian credible intervals to decision-makers who reason in frequentist terms.

Remark. Why simulate rather than delta. The delta-method standard error requires the gradient \(\nabla\Psi(\boldsymbol\xi_0)\), whose entries are themselves integrals of \(g^{-1}{}'\) against \(f_{BC}\) and must be approximated; the approximation is awkward and the resulting interval is symmetric on the link scale by construction. Bayesian g-computation instead propagates the full nonlinear functional through the posterior sample, so it captures asymmetry and higher-order curvature exactly within Monte Carlo error, and it composes cleanly with a Bayesian comparator summary: a posterior draw of \(\hat d_{BC}\) differenced against a posterior draw \(\Psi(\boldsymbol\xi^{(s)})\) yields, by Bucher’s identity, a posterior draw of the anchored \(\hat d_{AB}\), propagating all uncertainty coherently. This is the implementation used by the multilevel methods of Chapters 20 and 26. \(\blacktriangleleft\)

Remark. A practical uncertainty checklist. A defensible interval for the final \(A\)-versus-\(B\) effect must combine four distinct sources of uncertainty, which Bayesian g-computation can propagate jointly:

  1. Outcome-model parameters. Uncertainty in \(\hat{\boldsymbol\xi}\) from the finite index sample; this is the posterior \(\pi(\boldsymbol\xi\mid\mathcal D)\), or the sandwich variance (Definition 5.11) when misspecification is suspected.
  2. Monte Carlo integration. The \(O_p(M^{-1/2})\) error from approximating the integral Equation 19.14 by an average over \(M\) synthetic patients; drive it down by taking \(M\) large.
  3. Reconstructed covariate distribution. Uncertainty in \(f_{BC}\) itself when it was rebuilt from published summaries (the warning after Definition 19.4); the reported means and correlations are themselves estimates, and the assumed copula is a modeling choice.
  4. Published comparator effect. The standard error of \(\hat d_{BC}\) from the comparator trial, which enters at the Bucher step.

A common failure is to report only source 1 (the model fit) and to treat \(f_{BC}\) and \(\hat d_{BC}\) as fixed, understating the interval. Sources 1, 2, and 4 compose cleanly by differencing posterior draws (the “Why simulate” remark above); source 3 requires either a Bayesian model for the covariate summaries or a sensitivity analysis over plausible reconstructions. \(\blacktriangleleft\)

19.9 STC consistency under the population-adjustment conditions

Theorem 19.3 established consistency for the marginal effect in the comparator population under correct specification and transportability of the outcome surface. The population-adjustment theory of Chapter 17 expresses transportability through the conditions on effect modifiers and prognostic variables that an indirect comparison actually needs, separately for the anchored and unanchored cases. We restate STC consistency in those terms (catalogue G.6), so that STC slots into the general framework alongside MAIC.

Remark. Two data scenarios, in the running example. Anchored is the situation we have carried throughout: the index trial randomizes \(A\) against \(C\) and the comparator trial randomizes \(B\) against the same \(C\), so \(C\) is a shared anchor and Bucher’s subtraction cancels the study baselines. Here STC need only get the \(A\)-versus-\(C\) contrast right in \(\mathcal P_{BC}\), so it must include every covariate that modifies that contrast (every effect modifier). Unanchored is the harder situation where the two trials share no common arm, for instance a single-arm study of \(A\) and a single-arm study of \(B\). Now there is no baseline to cancel, so STC must get each absolute arm outcome right in \(\mathcal P_{BC}\), which means including every covariate that is merely prognostic (that shifts the outcome level) in addition to every effect modifier. Concretely: if age is prognostic for the flare but does not change the \(A\)-versus-\(C\) effect, the anchored analysis may safely omit it, but the unanchored analysis may not, because age now moves arm levels that no longer cancel. \(\blacktriangleleft\)

Theorem 19.6 (Anchored and unanchored STC consistency) Assume the outcome model Definition 19.1 is correctly specified on the chosen link scale, so \(\hat{\boldsymbol\xi}\xrightarrow{p}\boldsymbol\xi_0\) by Theorem 19.1, and the integrals are computed as in Theorem 19.3.

  1. Anchored. If conditional constancy of relative effects holds (Definition 17.4) and every effect modifier of the \(A\)-versus-\(C\) contrast that is distributed differently between the index and comparator populations is included in \(\mathbf x\) (the anchored sufficiency condition of Theorem 17.1), then the marginal STC estimator 1 is consistent for the target-population marginal relative effect \(d_{AC(\mathcal P_{BC})}^{\,\mathrm{marg}}\), and the anchored Bucher combination \(\hat d_{AC}^{\,\mathrm{marg}}-\hat d_{BC}\) is consistent for \(d_{AB(\mathcal P_{BC})}\).

  2. Unanchored. If conditional constancy of absolute effects holds (Definition 17.5) and every prognostic variable and effect modifier distributed differently between the populations is included in \(\mathbf x\) (the unanchored condition of Theorem 17.2), then the unanchored marginal STC estimator, the single-arm predicted contrast \(g(\hat\mu_C^{\mathrm{STC}})-g(\hat\mu_A^{\mathrm{STC}})\) taken without a common anchor, is consistent for the target-population effect.

(Corresponds to G.6 in the proof catalogue.)

Proof. The argument is the regression-route counterpart of the MAIC consistency theorems Theorem 18.4 and Theorem 18.5, with the weighted mean replaced by the predicted marginal mean Equation 19.14.

Anchored. By Theorem 19.3, \(\hat\mu_k^{\mathrm{STC}}\xrightarrow{p}\bar\mu_{k(\mathcal P_{BC})}\) provided the source-fitted surface \(\mu_k(\cdot)\) is the correct conditional surface in the comparator population. Conditional constancy of relative effects (Definition 17.4) states that the conditional contrast Equation 19.3 is the same function of \(\mathbf x\) in both populations; including every differentially distributed effect modifier in \(\mathbf x\) (Theorem 17.1) makes the fitted conditional means transportable in the sense of Proposition 19.1, because any covariate that would otherwise carry an unmodeled interaction is now a modeled coordinate of \(\mathbf x\). Hence \(\hat d_{AC}^{\,\mathrm{marg}}\xrightarrow{p}d_{AC(\mathcal P_{BC})}^{\,\mathrm{marg}}\). The comparator trial’s \(\hat d_{BC}\) already estimates \(d_{BC(\mathcal P_{BC})}\) in its own population, so both contrasts now live in \(\mathcal P_{BC}\), transitivity holds across them by construction, and Theorem 14.1 gives consistency of the difference for \(d_{AB(\mathcal P_{BC})}\).

Unanchored. Without a common anchor the estimand is a contrast of absolute arm positions across populations, so the cancellation of study baselines that the anchor provides (the \(\mu_{AC}-\mu_{BC}\) cancellation of Chapter 14) is unavailable, and the prognostic structure no longer drops out. Conditional constancy of absolute effects (Definition 17.5) is then required, and every differentially distributed prognostic variable as well as every effect modifier must be in \(\mathbf x\) (Theorem 17.2), so that both fitted arm surfaces \(\mu_A(\cdot)\) and \(\mu_C(\cdot)\), not merely their contrast, are transportable. Under these stronger hypotheses Proposition 19.1 applies to each arm separately, Theorem 19.3 gives \(\hat\mu_k^{\mathrm{STC}}\xrightarrow{p}\bar\mu_{k(\mathcal P_{BC})}\), and the continuous mapping theorem delivers consistency of the unanchored predicted contrast. \(\square\)

Why this matters. This is where STC stops being a stand-alone trick and becomes a citizen of the population-adjustment framework of Chapter 17: its consistency rests on exactly the same effect-modifier and prognostic-variable conditions that govern MAIC and ML-NMR, only discharged through a fitted regression instead of weights. The practical reading is a covariate-selection rule: for an anchored comparison, model every effect modifier that differs between the trials; for an unanchored one, model every prognostic variable too. Getting that list wrong is misspecification, and by the remark after Theorem 19.1 its price is bias, not merely variance.

Remark. STC against MAIC, assumption for assumption. STC and MAIC are dual routes through the same conditional-transportability framework, and Chapter 17 shows they target the same estimand when both are valid. Their hypotheses differ in two mirror-image ways. STC is stronger in demanding a correctly specified parametric outcome model, link and functional form and included covariates; if the model is wrong, STC inherits the pseudo-true bias of the misspecification remark after Theorem 19.1, whereas MAIC asks only that the matched moments balance and is robust to functional form beyond them. STC is weaker in demanding no overlap: a regression extrapolates where weights cannot exist (Theorem 18.8, Proposition 19.1), so STC’s variance is not inflated by poor overlap the way MAIC’s is by a small effective sample size (Theorem 18.6). The companion development records this trade-off as a comparison of method profiles in theories/Comparison.v; the two methods provably agree when both sets of assumptions hold, which we make precise next. \(\blacktriangleleft\)

19.10 Equivalence of MAIC and STC under a linear model

When the outcome model is linear and only first moments are matched, the weighting route and the regression route are not merely both valid; they are asymptotically the same estimator. This is catalogue result G.7. It explains the close agreement of MAIC and STC seen in practice when the outcome surface is near-linear and overlap is good, and it locates exactly where the two methods part company.

We first record the trivial but useful fact that the two methods agree whenever both are valid, the abstract statement that has a machine-checked counterpart.

Corollary 19.1 (Agreement under joint validity) Suppose the anchored MAIC assumptions and the anchored STC assumptions both hold, so that, by Theorem 18.4 and Theorem 19.6 respectively, each estimator is consistent for the same target estimand \(d_{AC(\mathcal P_{BC})}^{\,\mathrm{marg}}\). Then the two estimators have the same probability limit, \(\hat d_{AC}^{\,\mathrm{MAIC}}-\hat d_{AC}^{\,\mathrm{marg}}\xrightarrow{p}0\).

[machine-checked: theories/Comparison.v, Theorem methods_agree_when_correct]

Proof. By hypothesis each estimator converges in probability to the common target \(d_{AC(\mathcal P_{BC})}^{\,\mathrm{marg}}\), so their difference converges in probability to \(0\) by the algebra of limits. The Coq theorem methods_agree_when_correct is this statement stripped to its logical core: from the two identification premises (MAIC’s assumptions imply its estimate equals the true effect, STC’s assumptions imply the same for its estimate) it derives equality of the two estimates. \(\square\)

The substantive theorem is sharper: under a linear model the agreement is exact in the limit even before invoking a shared target, because the two estimators are two consistent estimators of the same linear functional of the same parameters.

Theorem 19.7 (Asymptotic equivalence of MAIC and STC under a linear model) Suppose (i) the outcome model is linear with the identity link, \(\mathbb E[Y\mid k,X=\mathbf x]=\alpha_k+\mathbf x^\top\boldsymbol\beta_{(k)}\) for \(k\in\{A,C\}\); (ii) the MAIC weights are the entropy-balanced exponential weights of Theorem 18.1, with parameters chosen to satisfy the first-moment balance constraint (Definition 18.1, Definition 18.2); and (iii) only the first moments of the covariates are matched, the weighted index mean equalling \(\bar{\mathbf x}_{BC}\). Then the marginal STC estimator 1 and the MAIC weighted-difference estimator have the same probability limit, \[ \hat d_{AC}^{\,\mathrm{marg}}\ \xrightarrow{p}\ (\alpha_C-\alpha_A)+\bar{\mathbf x}_{BC}^\top(\boldsymbol\beta_{(C)}-\boldsymbol\beta_{(A)})\ \xleftarrow{p}\ \hat d_{AC}^{\,\mathrm{MAIC}}, \tag{19.20}\] and are therefore asymptotically equivalent. (Corresponds to G.7 in the proof catalogue.)

Proof. The common target. With the identity link, Proposition 19.2 gives the marginal arm position \(\bar\mu_{k(\mathcal P_{BC})}=\alpha_k+\bar{\mathbf x}_{BC}^\top\boldsymbol\beta_{(k)}\), so the marginal relative effect Equation 19.9 in the comparator population is \[ d_{AC(\mathcal P_{BC})}^{\,\mathrm{marg}}=\bar\mu_{C(\mathcal P_{BC})}-\bar\mu_{A(\mathcal P_{BC})}=(\alpha_C-\alpha_A)+\bar{\mathbf x}_{BC}^\top(\boldsymbol\beta_{(C)}-\boldsymbol\beta_{(A)}), \] the right-hand quantity in Equation 19.20.

STC side. Under the linear model the predicted marginal means Equation 19.14 are, by linearity of expectation, \(\hat\mu_k^{\mathrm{STC}}=\hat\alpha_k+\bar{\mathbf x}_{BC}^\top\hat{\boldsymbol\beta}_{(k)}\), so \(\hat d_{AC}^{\,\mathrm{marg}}=(\hat\alpha_C-\hat\alpha_A)+\bar{\mathbf x}_{BC}^\top(\hat{\boldsymbol\beta}_{(C)}-\hat{\boldsymbol\beta}_{(A)})\). By Theorem 19.1 the coefficient estimates are consistent, \((\hat\alpha_k,\hat{\boldsymbol\beta}_{(k)})\xrightarrow{p}(\alpha_k,\boldsymbol\beta_{(k)})\), so \(\hat d_{AC}^{\,\mathrm{marg}}\xrightarrow{p}d_{AC(\mathcal P_{BC})}^{\,\mathrm{marg}}\), the left convergence in Equation 19.20.

MAIC side. The MAIC estimator is the difference of weighted arm means \(\hat\mu_k^{\mathrm{MAIC}}=\sum_{i:T_i=k}w_i Y_i\big/\sum_{i:T_i=k}w_i\), with the entropy-balanced weights \(w_i=\exp(\hat\alpha_w+\hat{\boldsymbol\beta}_w^\top(\mathbf x_i-\bar{\mathbf x}_{BC}))\) of Theorem 18.1. Let \(P^\ast_k\) denote the limiting weighted law of the covariates in arm \(k\); under overlap and the law of large numbers for weighted averages, \(\hat\mu_k^{\mathrm{MAIC}}\xrightarrow{p}\mathbb E_{P^\ast_k}[Y\mid T=k]=\mathbb E_{P^\ast_k}\bigl[\mathbb E[Y\mid k,X]\bigr]=\mathbb E_{P^\ast_k}[\alpha_k+X^\top\boldsymbol\beta_{(k)}]=\alpha_k+\mathbb E_{P^\ast_k}[X]^\top\boldsymbol\beta_{(k)}\), the middle step by the tower property and the linear model, the last by linearity of expectation. The first-moment balance constraint (iii) forces \(\mathbb E_{P^\ast_k}[X]=\bar{\mathbf x}_{BC}\), so \(\hat\mu_k^{\mathrm{MAIC}}\xrightarrow{p}\alpha_k+\bar{\mathbf x}_{BC}^\top\boldsymbol\beta_{(k)}=\bar\mu_{k(\mathcal P_{BC})}\), the same arm limit as the STC side. Differencing the arms, \(\hat d_{AC}^{\,\mathrm{MAIC}}=\hat\mu_C^{\mathrm{MAIC}}-\hat\mu_A^{\mathrm{MAIC}}\xrightarrow{p}d_{AC(\mathcal P_{BC})}^{\,\mathrm{marg}}\), the right convergence in Equation 19.20.

Both estimators converge in probability to the same number, hence their difference converges to zero: they are asymptotically equivalent. \(\square\)

Remark. A one-line numeric check. Take the linear model of Exercise 19.10: \(\mathbb E[Y\mid C,x]=1+0.5x\) and \(\mathbb E[Y\mid A,x]=2+0.5x\), with target covariate mean \(\bar x_{BC}=1\). STC side: the predicted target means are \(\hat\mu_C^{\mathrm{STC}}=1+0.5(1)=1.5\) and \(\hat\mu_A^{\mathrm{STC}}=2+0.5(1)=2.5\), so the effect is \(1.5-2.5=-1\). MAIC side: entropy-balanced weights that pull the index covariate mean to \(\bar x_{BC}=1\) give weighted arm means \(\alpha_k+\beta_{(k)}\cdot 1\), again \(1.5\) and \(2.5\), so the effect is again \(-1\). The two routes land on the identical number because, under a linear model, the weighted mean of \(Y\) depends on the covariates only through their first moment, which is exactly what MAIC matches and what STC’s linear prediction uses. This is why the two methods so often agree in practice: near-linear outcome and good overlap make them two computations of the same first-moment functional. \(\blacktriangleleft\)

Remark. Where the equivalence breaks. The proof used linearity twice and first-moment balance once, and the equivalence fails as soon as any of the three is removed.

  1. Nonlinear link. If \(g\) is not the identity, \(\mu_k(\mathbf x)=g^{-1}(\eta_k(\mathbf x))\) is curved, STC’s response-scale integration carries the Jensen terms of Theorem 19.2, and MAIC’s first-moment balance no longer reproduces them; the two estimators target the same marginal effect but by different finite-sample routes and need not coincide except in the limit, and only then if both remain consistent.

  2. Higher moments needed. If the conditional mean involves \(\mathbf x\) nonlinearly, for instance a quadratic term \(\mathbf x^\top\boldsymbol\gamma\mathbf x\), then \(\mathbb E_{P^\ast_k}[\mathbb E[Y\mid k,X]]\) depends on the second moment of \(P^\ast_k\), which first-moment balance does not pin to \(\boldsymbol\Sigma_{BC}\); MAIC then estimates a different number from STC unless the higher moments are also matched.

  3. Nonlinear interactions. Even with the identity link, an interaction structure richer than the additive \(\mathbf x^\top\boldsymbol\beta_2\) makes the conditional contrast a nonlinear function of \(\mathbf x\), and first-moment balance is again insufficient.

In practice the two methods agree closely when overlap is good and the outcome surface is near-linear, and diverge under poor overlap, where MAIC’s variance inflates (Theorem 18.6) while STC extrapolates, or under misspecification, where STC is biased while MAIC remains robust to functional form beyond the matched moments. The choice between them is the choice between trusting a model and trusting an overlap. \(\blacktriangleleft\)

19.11 Notes and references

Simulated treatment comparison was introduced by Caro and Ishak (2010) under the title “No head-to-head trial? Simulate the missing arms,” as a regression-and-simulation alternative to matching; the original proposal is the conditional plug-in of Definition 19.2, evaluated at the comparator covariate means. The distinction between conditional and marginal STC, the diagnosis of the plug-in bias as non-collapsibility, and the recommendation to integrate on the response scale are due to the population-adjustment methods literature, principally the NICE technical support document Phillippo et al. (2016) and the thesis of Phillippo (2019) (Sections 2.2 to 2.4), which also frame STC and MAIC as the outcome-regression and weighting implementations of one conditional-transportability problem; the estimand-based reframing in terms of marginal versus conditional targets and the transportability of the outcome surface is developed by Chandler and Ishak (2026). The identification of marginal STC as g-computation standardized to a target population (Proposition 19.1) places it in the lineage of Hernán and Robins (2020) and Rosenbaum and Rubin (1983), and is the seed of the multilevel network meta-regression of Phillippo et al. (2020) (Chapter 20), which embeds the same response-scale integration inside a coherent Bayesian network model and removes STC’s reliance on a separate Bucher step. The comparison of MAIC and STC against the MLE asymptotics of Theorem 5.6 follows the GLM treatment of McCullagh and Nelder (1989) and the M-estimation theory of Vaart (1998), the latter also supplying the Bernstein-von Mises theorem behind Theorem 19.5.

This chapter corresponds to results G.1 through G.7 of the proof catalogue in proofs/G_stc.md: the outcome MLE asymptotics G.1 (Theorem 19.1), the conditional plug-in bias G.2 (Theorem 19.2), the marginal STC consistency G.3 (Theorem 19.3), the non-collapsibility counterexample G.4 (Theorem 19.4), Bayesian g-computation G.5 (Theorem 19.5), STC consistency under the Chapter 17 conditions G.6 (Theorem 19.6), and the MAIC-STC equivalence G.7 (Theorem 19.7).

Two results carry machine-checked tags confirmed in the companion Coq development. The marginal STC contrast identity Theorem 19.3 is Theorem STC_consistent in theories/STC.v, which proves the response-scale identity \(\texttt{effect\_AC\_STC}=\mathbb E_2[Y_A^{\mathrm{target}}]-\mathbb E_2[Y_C^{\mathrm{target}}]\) from the per-arm transported g-computation premises; the link transformation and uniform convergence are the analytic additions made here. The collapsibility boundary Proposition 19.2 is Lemma STC_collapse_linear in the same file, which proves that a linear outcome model makes the marginal STC estimate equal the treatment coefficient using only linearity of the abstract target expectation, the single property a nonlinear link destroys. The agreement-under-joint-validity corollary Corollary 19.1 is Theorem methods_agree_when_correct in theories/Comparison.v, the logical core of the equivalence. The non-collapsibility counterexample Theorem 19.4 is deliberately left as pen-and-paper arithmetic over the logit, as its content is a numerical witness rather than an algebraic identity; the companion file confirms only the collapsible boundary. As in Chapter 14, the Coq files treat the target-population expectation as an abstract linear operator, so the formalization captures the algebraic content while leaving the measure-theoretic construction of the integral Equation 19.14 to the caller. The files theories/OutcomeRegression.v (g-computation, the source-population analogue) and theories/Comparison.v (the MAIC-versus-STC method profiles) provide the surrounding structure.

19.12 Exercises

An exercise marked with a star (\(\star\)) has a full worked solution in Appendix F. The first two are warm-ups; do them before the proof-heavy exercises that follow.

Exercise 19.1 (Warm-up, computation) A full STC table by hand. An index trial fits the logistic model \(\operatorname{logit}\mu_C(x)=-0.5+0.7x\) and \(\operatorname{logit}\mu_A(x)=0.4+0.7x\) for a single binary covariate \(x\) (so there is no effect modification). The target comparator population has \(\Pr_{BC}(x=1)=0.6\). (a) Compute the four fitted event probabilities \(\mu_C(0),\mu_C(1),\mu_A(0),\mu_A(1)\) with \(\operatorname{expit}\). (b) Conditional plug-in: evaluate the linear predictors at \(\bar x_{BC}=0.6\) and report the plug-in log odds ratio of \(A\) versus \(C\). (c) Marginal STC: average the fitted probabilities over the target on the natural outcome scale, apply the logit, and report the marginal log odds ratio. (d) State which of (b) and (c) a health technology assessment should use, and by how much they differ. This is the five-step workflow of Section 19.2 with the Bucher step omitted.

Exercise 19.2 (Warm-up, interpretation) Which scale, which average? In one or two sentences each, explain to a colleague: (a) why “average then link” and “link then average” give different answers for an odds ratio but the same answer for a risk difference; (b) what “simulated” refers to in simulated treatment comparison; and (c) why STC can predict into a comparator population that MAIC cannot weight into, and what STC must assume to justify that prediction. No arithmetic required.

Exercise 19.3 Recompute with a new target. In the worked example Example 19.3, keep the fitted model (\(\hat\alpha=-1\), \(\hat\beta_1=1\), \(\hat\gamma=1.5\), \(\hat\beta_2=0\)) but change the target biomarker prevalence to \(\Pr_{BC}(X=1)=0.3\). Compute the conditional plug-in log odds ratio, the marginal STC log odds ratio in the new target, and the non-collapsibility gap between them. Confirm that the plug-in is unchanged at \(1.5\) while the marginal value moves, and explain why the plug-in ignores the covariate distribution entirely.

Exercise 19.4 With effect modification. Repeat Example 19.3 but with \(\hat\beta_2=0.8\) (the biomarker now modifies the \(A\)-versus-\(C\) effect), so \(\operatorname{logit}\mu_A(x)=0.5+1.8x\). With target prevalence \(\bar x_{BC}=0.8\) and source prevalence \(0.5\), compute the conditional plug-in, the target marginal STC, and the source marginal log odds ratios. Show that the source-to-target shift is now much larger than in the \(\hat\beta_2=0\) case, and identify how much of the conditional-versus-marginal gap is non-collapsibility and how much is effect modification interacting with the covariate shift.

Exercise 19.5 Identity link is exact. Prove directly from Definition 19.3 that for the identity link the conditional plug-in STC estimator Equation 19.7 and the marginal STC estimator 1 have the same probability limit for every covariate distribution, recovering Proposition 19.2 without the Taylor expansion. Then exhibit a continuous outcome with a log link for which they differ, and compute the gap using Example 19.1.

Exercise 19.6 No overlap required. Let the index trial observe the covariate only on \([0,1]\) but the target place mass on \([1,2]\). Explain why the marginal STC integral Equation 19.14 is still defined while the MAIC weights are not, and state precisely the assumption (correctness of \(\mu_k\) off the index support) that STC must add to justify the extrapolation. Relate this to Proposition 19.1 and Theorem 18.8.

Exercise 19.7 Direction of the gap. For the logit link with equal arm slopes (\(\boldsymbol\beta_2=\mathbf 0\)) and a covariate symmetric about its mean, use Equation 19.12 and the sign of \((\operatorname{expit})''\), which is positive for \(\eta<0\) and negative for \(\eta>0\), to argue that the conditional-at-the-mean odds ratio is generically farther from the null than the marginal odds ratio. Confirm the direction against the counterexample Theorem 19.4, where the conditional \(1.5\) exceeds the marginal \(1.227\).

Exercise 19.9 (\(\star\)) Delta-method variance of marginal STC. Let \(\Psi(\boldsymbol\xi)\) be the STC functional Equation 19.19. Using Theorem 19.1 and the delta method (Theorem 5.3), show that \(\sqrt n(\hat d_{AC}^{\,\mathrm{marg}}-d_{AC(\mathcal P_{BC})}^{\,\mathrm{marg}})\xrightarrow{d}\mathcal N(0,\nabla\Psi(\boldsymbol\xi_0)^\top\mathcal I(\boldsymbol\xi_0)^{-1}\nabla\Psi(\boldsymbol\xi_0))\), and write the gradient \(\nabla\Psi(\boldsymbol\xi_0)\) as an integral of \(g^{-1}{}'\) against \(f_{BC}\) for each coordinate of \(\boldsymbol\xi\). Explain why Theorem 19.5 reproduces this variance asymptotically, and why the Bayesian route avoids computing the gradient. Under misspecification, replace \(\mathcal I^{-1}\) by the sandwich of Definition 5.11 and state what \(\Psi(\boldsymbol\xi^\ast)\) then estimates. (A solution appears in Appendix F.)

Exercise 19.10 MAIC equals STC, numerically. Construct a linear-model dataset with one covariate: arm means \(\mathbb E[Y\mid C,x]=1+0.5x\) and \(\mathbb E[Y\mid A,x]=2+0.5x\) (so \(\boldsymbol\beta_2=0\)), source covariate mean \(0\), target covariate mean \(\bar x_{BC}=1\). Compute the marginal STC effect from Theorem 19.7, then verify that entropy-balanced MAIC weights matching the first moment give the same limit. Now add a quadratic term \(0.3x^2\) to the comparator arm only and show that first-moment MAIC and STC now disagree, matching the breakdown clause of the equivalence remark.