27 Aggregate-Level Covariate-Distance Matching
27.1 Introduction
Consider a situation that arises constantly in oncology health technology assessment (HTA). A novel therapy, say a CAR-T cell product for relapsed or refractory large B-cell lymphoma, is licensed on the strength of a single-arm Phase II trial: every enrolled patient received the new agent, so the trial carries no internal control against which a treatment effect could be measured. To report how the new agent compares with standard chemoimmunotherapy, the analyst must borrow a control arm from outside the trial, from one of several external Phase III trials or disease registries of the standard regimens. Those external studies do not contain the new agent, so there is no shared treatment to anchor the comparison, and they rarely release patient-level data; each publishes only a short table of baseline covariate means, mean age, proportion male, a mean severity marker, and little else. Which external study should supply the borrowed control arm? Choosing wrongly can reverse the sign of the reported effect and drive a reimbursement decision the wrong way, so the stakes are the whole submission. This chapter answers the prior question that every method of Parts IV and V silently assumed away: before adjusting for population differences, how do we choose the comparator study to adjust toward, using only the aggregate covariate summaries the external studies report? The workable answer, together with a sharp statement of what it can and cannot guarantee, is the subject of the chapter, and we will follow one running trial example (Example 27.1) from the first definition to the final caveat.
Every method in Parts IV and V began once a comparator had already been chosen. Matching-adjusted indirect comparison (Chapter 18), simulated treatment comparison (Chapter 19), multilevel network meta-regression (Chapter 20), and multilevel unanchored meta-regression (Chapter 26) all take as given a pair of studies, an index trial with individual patient data (IPD) and a designated comparator trial with aggregate data (AgD), and then do the work of bridging their populations. In a connected anchored network the comparator is not a choice at all: the common control treatment forces it, and the only question is how to adjust. But two situations strip away that luxury. A disconnected network offers several AgD studies, none sharing a common comparator with the index trial, any of which could in principle serve as the external arm for an unanchored comparison; the analyst must select one. A single-arm study, having no internal control, must borrow its entire comparator arm from an external population, and the choice of which external study to borrow is the whole analysis. This chapter is about that prior choice: how to pick a comparator study before the population-adjustment machinery is switched on.
The practical rule, described in the Bristol Compass blog on disconnected ML-NMR and implemented in the uitc diagnostic package (a full citation for both is forthcoming; see Section 27.12), is disarmingly simple. Each candidate comparator study reports a vector of covariate summaries, typically baseline means; the index population reports its own; choose the candidate whose summaries lie closest to the index population on some distance between summary vectors. The content of the rule is entirely in the word “closest,” because there is no single distance on \(\mathbb{R}^p\) and the candidates can be ranked differently by different choices. We develop the natural candidates: the Euclidean and standardized Euclidean distances (Definition 27.2), the Mahalanobis distance and its identification with a whitened Euclidean distance (Theorem 27.1), and the Manhattan, Minkowski, Gower, and mean-absolute standardized-mean-difference distances (Definition 27.4). We prove which of these are invariant to a change in the units of a covariate and which are not (Theorem 27.2, Proposition 27.2), because a distance whose ranking flips when age is recorded in months rather than years is a poor basis for a decision.
We then ask the only question that matters for validity: does a small covariate-summary distance bound the bias of the eventual adjusted estimate? The answer is a sharp dichotomy that this chapter proves in full. When the conditional contrast is affine in the covariates, the transport bias is exactly a linear functional of the covariate-mean gap, and the Cauchy-Schwarz inequality turns any of the quadratic-form distances into a rigorous bias bound (Theorem 27.3), connecting comparator selection directly to the unanchored consistency theorem of Chapter 17 (Theorem 17.2). When the contrast is merely Lipschitz, a bias bound survives, but only against the Wasserstein distance between full distributions (Theorem 27.4), and a distance between means is provably not a Wasserstein distance: it is a lower bound on it, never an upper bound. The consequence, made precise in Proposition 27.4, is the central honest caveat of the chapter. Two populations can have identical covariate means, hence zero distance under every metric we define, and yet different target effects, because a nonlinear contrast sees the spread and not only the center, and because an unmeasured prognostic variable can differ while every observed summary agrees. Aggregate-level distance matching is therefore a diagnostic and design tool, not a transportability theorem: a small distance makes a population bridge more plausible and, under linearity, controls the residual bias, but it certifies neither the shared prognostic factor assumption (Definition 26.2) nor conditional constancy of absolute effects (Definition 17.5) nor the absence of unmeasured imbalance. A fully worked example ranking four candidate comparators under five criteria, in which the three principal distances each select a different winner and the actual bias is computed exactly, closes the development. This material is an extension beyond the population-adjustment theory of Phillippo (2019), prompted by the disconnected-network setting of Chandler and Ishak (2026a), and corresponds to results O.1 to O.7 of the proof catalogue.
Throughout, \(\bar{\mathbf{x}}_0\in\mathbb{R}^p\) denotes the covariate-summary vector of the index population (the population in which we hold IPD, or whose effect we wish to report), and \(\bar{\mathbf{x}}_1,\dots,\bar{\mathbf{x}}_J\in\mathbb{R}^p\) the summary vectors of the \(J\) candidate comparator studies, following the study-mean notation \(\bar{\mathbf{x}}_j\) of Notation. We write \(\boldsymbol{\delta}_j=\bar{\mathbf{x}}_j-\bar{\mathbf{x}}_0\) for the covariate gap of candidate \(j\), \(\ell\in\{1,\dots,p\}\) for the covariate index, and \(\|\mathbf{v}\|_A=\sqrt{\mathbf{v}^\top A\,\mathbf{v}}\) for the quadratic-form norm of a positive-definite \(A\). The bold vector \(\boldsymbol{\delta}_j\), a gap between covariate means, is a different object from the plain scalar \(\delta_{j,\,t_1 t_k}\) of Notation, which is a within-study treatment contrast; the boldface marks the vector throughout, following the vector-versus-scalar convention of the book, and the two never appear together. Read \(\|\mathbf{v}\|_A\) simply as “the length of \(\mathbf{v}\) measured with the ruler \(A\)”: when \(A=I\) it is the ordinary length, and a general \(A\succ0\) (read \(A\succ0\) as “\(A\) is symmetric positive definite,” so that \(\mathbf{v}^\top A\mathbf{v}>0\) for every \(\mathbf{v}\ne\mathbf 0\) and the square root is real) stretches some directions and shrinks others.
A reader who wants the destination before the journey can consult the map in Table 27.1. The single practical recommendation, defended in the sections that follow, is compressed into it: if you compute only one distance, compute the standardized Euclidean distance, and if the index study supplies a covariance matrix, add the Mahalanobis distance and treat any disagreement between the two as a signal that the ranking depends on covariate correlations rather than on the data.
| Distance | What the number captures | Units-safe? | Defined in |
|---|---|---|---|
| Raw Euclidean \(d_E\) | straight-line gap in the recorded units | no | Definition 27.2 |
| Standardized Euclidean \(d_{SE}\) (default) | gap in standard-deviation units, one per covariate | yes | Definition 27.2 |
| Mahalanobis \(d_M\) | gap measured against the index covariate correlation cloud | yes | Definition 27.3 |
| Manhattan, Minkowski, Chebyshev \(d_q\) | sum, blend, or single worst standardized gap | yes | Definition 27.4 |
| Mean-SMD \(d_{\mathrm{SMD}}\) | average standardized gap; ties to balance tables | yes | Definition 27.4 |
| Gower \(d_G\) | mixed continuous and binary covariates on one scale | yes | Definition 27.4 |
27.2 The comparator-selection problem
We fix the decision the chapter addresses.
Definition 27.1 (Aggregate-level comparator selection) Let the index population have covariate-summary vector \(\bar{\mathbf{x}}_0\in\mathbb{R}^p\), and let candidate comparator studies \(j=1,\dots,J\) report covariate-summary vectors \(\bar{\mathbf{x}}_j\in\mathbb{R}^p\) on the same \(p\) covariates. Given a distance \(d:\mathbb{R}^p\times\mathbb{R}^p\to[0,\infty)\) on summary vectors, the aggregate-level matching rule selects \[ j^{*}\;\in\;\arg\min_{1\le j\le J}\; d\big(\bar{\mathbf{x}}_0,\bar{\mathbf{x}}_j\big). \tag{27.1}\] The selected study \(j^{*}\) is the observed comparator whose reported population summaries are closest to the index population on the chosen scale. \(\blacktriangleleft\)
Remark. Reader translation. Each candidate study is reduced to one number, its distance from the index population, and the rule picks the smallest. The only content is what “distance” means, because a candidate that looks closest under one sensible distance can look farthest under another. The rest of the chapter is a tour of the reasonable choices and a proof of exactly how much a small distance does, and does not, buy you. \(\blacktriangleleft\)
We anchor every definition to one small scenario, carried from here to the end of the chapter.
Example 27.1 (A running example: two candidate comparator trials) Return to the single-arm lymphoma trial of the introduction. The index study reports two baseline covariate means, mean age in years and a mean severity biomarker in its natural units, and two external Phase III trials, \(P\) and \(Q\), are candidate sources of the borrowed control arm. Their reported means are
| Covariate (\(\ell\)) | reference scale \(s_\ell\) | index mean \(\bar{x}_{0\ell}\) | Study \(P\) mean | Study \(Q\) mean |
|---|---|---|---|---|
| age (years) | \(10\) | \(60\) | \(70\) | \(66\) |
| biomarker (units) | \(4\) | \(10\) | \(14\) | \(6\) |
The reference scales \(s=(10,4)\) are the index-study standard deviations of the two covariates. The index covariance, also from the index study, is \[ \boldsymbol{\Sigma}=\begin{pmatrix}100 & 20\\ 20 & 16\end{pmatrix}, \] so age and biomarker are positively correlated, \(\operatorname{corr}=20/(10\cdot4)=0.5\). The covariate gaps and their standardized versions \(\mathbf{z}_j=(\delta_{j1}/s_1,\ \delta_{j2}/s_2)\) are \[ \boldsymbol{\delta}_P=(10,\ 4),\quad \mathbf{z}_P=(1.0,\ 1.0); \qquad \boldsymbol{\delta}_Q=(6,\ -4),\quad \mathbf{z}_Q=(0.6,\ -1.0). \] In words: Study \(P\)’s patients are older and have a higher biomarker, both covariates up together, the direction in which age and biomarker naturally co-vary in the index trial. Study \(Q\)’s patients are somewhat older but have a lower biomarker, the two covariates moving in opposite directions, against the correlation. The central surprise of the chapter, foreshadowed here and computed in Section 27.4 and Section 27.5, is that raw and standardized Euclidean distance will both judge \(Q\) the closer match, while the Mahalanobis distance will judge \(P\) the closer match, purely because \(P\) respects the age-biomarker correlation and \(Q\) fights it. \(\blacktriangleleft\)
The rule Equation 27.1 is a design rule: it ranks the candidates and names a winner, to be fed into an unanchored MAIC, STC, or ML-UMR. It is not yet a claim about validity. To see both what it can do and what it cannot, recall the two regimes that make it necessary.
Remark. Why selection is forced in disconnected and single-arm evidence. In the anchored triangle of Chapter 14 the index trial \(AC\) and the comparator trial \(BC\) share the control \(C\), and the Bucher contrast \(\hat d_{AB}=\hat d_{AC}-\hat d_{BC}\) differences the shared anchor away. A disconnected network has no such shared treatment between the index trial and the studies reporting the comparator of interest, so there is no anchor to difference and the comparison must be unanchored: the absolute outcome of the index treatment is compared to the absolute outcome of a comparator drawn from a different population (Chapter 17, Theorem 17.2; Chapter 26). When several AgD studies report that comparator, each defines a different bridging population, and the analyst must choose. A single-arm index study is the limiting case: lacking any internal control, its entire counterfactual arm is an external population, and selecting the most similar external study is the only lever available. In both regimes the choice precedes, and conditions the credibility of, the adjustment that follows. \(\blacktriangleleft\)
Remark. What a good choice buys, and what it does not. Unanchored population adjustment is consistent only when every imbalanced prognostic variable and effect modifier is measured and balanced and the target covariate values lie inside the index covariate support (Theorem 17.2, hypotheses (iii) and (iv)). Choosing a comparator with smaller covariate gaps makes both conditions easier to meet: a smaller gap means less reweighting, a higher effective sample size (Theorem 18.6), and a target nearer the interior of the support, so finite stable weights are more likely to exist. The distance is a proxy for “how hard the bridge will be.” It is not a proxy for “whether the bridge is valid,” because the unmeasured and the nonlinear escape it entirely; that gap is the subject of Section 27.9. \(\blacktriangleleft\)
This setup corresponds to O.1 in the proof catalogue.
27.3 A practical workflow, and what selection is for
Before any algebra, it helps to see where comparator selection sits in a real submission. Distance matching is one step of a longer pipeline, and it is neither the first step nor the last.
- Fix the target. State the population in which the effect must be reported (here, the index single-arm trial’s population) and the estimand (for instance, a marginal log odds ratio versus standard of care). Everything downstream is measured against this target.
- Assemble candidates. From a systematic review, list every external study that reports the comparator treatment and the baseline covariates you will need. These are the candidate comparators \(j=1,\dots,J\).
- Screen for clinical and design comparability first. Before computing a single distance, discard candidates that differ in line of therapy, outcome definition, follow-up length, standard of care, eligibility, or measurement practice. Distance matching operates only among studies that already pass this clinical screen; a small covariate distance cannot rescue an incomparable trial.
- Choose the covariate set. Include the effect modifiers and prognostic variables that are both clinically justified and reported across the surviving candidates, and record which important covariates are missing from which studies.
- Harmonize and scale. Put every covariate on a common definition, choose reference scales, and compute several distances. Inspect which covariates drive each distance and estimate the effective sample size (ESS) the eventual reweighting would leave (Section 27.8, Theorem 18.6).
- Carry uncertainty forward. Feed the chosen comparator into an unanchored MAIC, STC, or ML-UMR, and carry any disagreement between distances, any missing covariate, and any residual imbalance into a quantitative bias analysis (Chapter 28).
| Step | Question it answers | Tool |
|---|---|---|
| 1 target | For whom must the effect hold? | estimand definition (Chapter 12, Chapter 17) |
| 2 candidates | Which external studies could supply the arm? | systematic review |
| 3 clinical screen | Are the trials comparable at all? | clinical and design appraisal |
| 4 covariates | Which variables enter the gap? | effect-modifier and prognosis judgment |
| 5 distance and ESS | Which candidate is closest, and how costly? | this chapter, Theorem 18.6 |
| 6 sensitivity | What could still go wrong? | quantitative bias analysis (Chapter 28) |
Remark. Why a cheap pre-analysis screen, rather than adjusting on every candidate. A natural objection is: if there are four candidate comparators, why not simply run four unanchored MAICs and keep the one with the best effective sample size? Three reasons. First, an unanchored adjustment against a badly matched comparator can produce extreme weights, a tiny ESS, or an optimization that fails to converge at all, so the “try them all” strategy often has no clean output to compare. Second, choosing the comparator after seeing four different treatment estimates is selection on the outcome: it inflates the apparent effect and destroys any pre-registration guarantee. Third, distance matching is computed from published covariate tables alone, before the outcome is touched, so it is cheap, reproducible, and blind to the answer. It is a screen, run once, up front, on data an HTA reviewer can check. \(\blacktriangleleft\)
Remark. Where entropy balancing and individual-level reweighting live. The word “matching” in this chapter’s title is aggregate-level: it matches summary vectors, not patients. The individual-level reweighting tools, the method-of-moments weights of MAIC (Chapter 18), and entropy balancing (Hainmueller’s minimum relative-entropy weights, derived in the individual-level setting), operate after a comparator has been chosen and require the index individual patient data; they solve a different problem, how to reweight patients so that a chosen comparator’s moments are met. This chapter is strictly the upstream, aggregate step that decides which comparator to reweight toward, using only the means each study publishes. If you were expecting entropy balancing here, it belongs to the individual-level material of Chapter 18; distance matching is the design step before it. \(\blacktriangleleft\)
Remark. Choosing the covariates and the reference scales. Two applied choices decide what the distance measures.
- Which covariates. Include a covariate only if it is a plausible effect modifier (Definition 11.3) or prognostic variable (Definition 11.5) and is reported by every surviving candidate. A covariate reported by some candidates but not others cannot enter a common distance; dropping a genuine effect modifier to keep a candidate in the running is a substantive decision to be recorded, not hidden. Do not pad the distance with many weakly prognostic covariates: each one dilutes the gaps that actually drive bias.
- Which reference scale. When the index individual patient data are available, the index standard deviations are the natural reference scales, because the eventual reweighting is toward the index population. When no IPD are available, a pooled or comparator-reported standard deviation is used instead; for a binary covariate with prevalence \(\pi\), the Bernoulli standard deviation \(\sqrt{\pi(1-\pi)}\) is standard; a missing standard deviation is usually borrowed from a comparable covariate or a pooled value. Note the asymmetry proved later: re-expressing a covariate in different units does not change the standardized ranking (Theorem 27.2), but changing the reference scale while holding units fixed can change the winner (Exercise 27.3), so the reference-scale choice is a modeling decision that belongs in the write-up, not a harmless re-expression. \(\blacktriangleleft\)
27.4 Euclidean and standardized Euclidean distances
The first candidate is the ordinary straight-line distance, and its first defect is that it depends on the units in which the covariates happen to be recorded.
Definition 27.2 (Euclidean and standardized Euclidean distance) The Euclidean distance between the index summary \(\bar{\mathbf{x}}_0\) and a candidate summary \(\bar{\mathbf{x}}_j\) is the \(\ell_2\) norm of their gap, \[ d_E\big(\bar{\mathbf{x}}_0,\bar{\mathbf{x}}_j\big)\;=\;\big\|\boldsymbol{\delta}_j\big\|_2 \;=\;\sqrt{\sum_{\ell=1}^{p}\big(\bar{x}_{j\ell}-\bar{x}_{0\ell}\big)^2}. \tag{27.2}\] Let \(s_1,\dots,s_p>0\) be reference scales, one per covariate, typically an IPD standard deviation, a reported comparator standard deviation, or, for a binary covariate with prevalence \(\pi\), the Bernoulli standard deviation \(\sqrt{\pi(1-\pi)}\). Writing \(S=\operatorname{diag}(s_1,\dots,s_p)\) and the standardized gap \(\mathbf{z}_j=S^{-1}\boldsymbol{\delta}_j\), with components \(z_{j\ell}=(\bar{x}_{j\ell}-\bar{x}_{0\ell})/s_\ell\), the standardized Euclidean distance is \[ d_{SE}\big(\bar{\mathbf{x}}_0,\bar{\mathbf{x}}_j;s\big)\;=\;\big\|\mathbf{z}_j\big\|_2 \;=\;\big\|S^{-1}\boldsymbol{\delta}_j\big\|_2 \;=\;\sqrt{\sum_{\ell=1}^{p}\left(\frac{\bar{x}_{j\ell}-\bar{x}_{0\ell}}{s_\ell}\right)^2}. \tag{27.3}\] \(\blacktriangleleft\)
Remark. Why standardize. The raw distance Equation 27.2 adds squared gaps measured in incompatible units, so it is dominated by whichever covariate carries the largest numerical magnitude. A gap of one year in mean age and a gap of \(0.5\) in a proportion contribute \(1\) and \(0.25\) to the raw squared distance, yet the proportion gap is a far larger fraction of that covariate’s natural spread. Dividing each gap by a reference scale renders the contributions commensurate: each standardized component \(z_{j\ell}\) is the familiar standardized mean difference of that covariate, a dimensionless number measured in standard deviations. The uitc package defaults to \(d_{SE}\) for exactly this reason, reserving the raw \(d_E\) for covariates that are already on a common scale. The formal statement that standardization removes the unit dependence is Theorem 27.2 below. \(\blacktriangleleft\)
Remark. The running example, raw and standardized. Take the two candidate trials of Example 27.1. The raw Euclidean distances are \[ d_E(P)=\sqrt{10^2+4^2}=\sqrt{116}=10.77, \qquad d_E(Q)=\sqrt{6^2+(-4)^2}=\sqrt{52}=7.21, \] and the standardized Euclidean distances, using the standardized gaps \(\mathbf{z}_P=(1.0,1.0)\) and \(\mathbf{z}_Q=(0.6,-1.0)\), are \[ d_{SE}(P)=\sqrt{1.0^2+1.0^2}=\sqrt2=1.414, \qquad d_{SE}(Q)=\sqrt{0.6^2+1.0^2}=\sqrt{1.36}=1.166 . \] The standardized numbers read directly as “how many standard deviations apart”: Study \(P\) is exactly one standard deviation older and one higher on the biomarker; Study \(Q\) is \(0.6\) standard deviations older and one lower. Both the raw and the standardized distance rank \(Q\) ahead of \(P\). Hold on to that ranking: Section 27.5 overturns it once the age-biomarker correlation is taken into account. \(\blacktriangleleft\)
This material corresponds to O.2 in the proof catalogue.
27.5 The Mahalanobis distance and whitening
Standardization rescales each covariate but ignores their correlations. When the index IPD supplies a full covariance matrix, the Mahalanobis distance uses it, and the use turns out to be exactly a change of coordinates that decorrelates and standardizes the covariates simultaneously, an operation called whitening (the name borrows from signal processing: it turns a correlated covariate cloud into an uncorrelated, unit-variance cloud, like the flat spectrum of white noise). Before any formula, here is the picture to carry: the Mahalanobis distance asks whether the whole pattern of covariate differences looks unusual relative to the index trial’s covariate cloud. A gap that points along a direction in which the index patients naturally vary, or in which two covariates move together as they usually do, is unremarkable and counts for little; the same-sized gap in a direction the index patients rarely move, or against a correlation they usually respect, is surprising and counts for a lot.
Definition 27.3 (Mahalanobis distance) Let \(\boldsymbol{\Sigma}\in\mathbb{R}^{p\times p}\) be a positive-definite covariance matrix for the covariates, typically the index-IPD sample covariance. The Mahalanobis distance is the \(\boldsymbol{\Sigma}^{-1}\)-quadratic-form norm of the gap, \[ d_M\big(\bar{\mathbf{x}}_0,\bar{\mathbf{x}}_j;\boldsymbol{\Sigma}\big) \;=\;\sqrt{\boldsymbol{\delta}_j^\top\boldsymbol{\Sigma}^{-1}\boldsymbol{\delta}_j} \;=\;\big\|\boldsymbol{\delta}_j\big\|_{\boldsymbol{\Sigma}^{-1}}. \tag{27.4}\] \(\blacktriangleleft\)
Remark. Reader translation. The Mahalanobis distance is the quadratic-form norm \(\|\boldsymbol{\delta}_j\|_{\boldsymbol{\Sigma}^{-1}}\) from the chapter opening: the length of the gap measured with the ruler \(\boldsymbol{\Sigma}^{-1}\) in place of the identity (recall \(\boldsymbol{\Sigma} \succ0\) means \(\boldsymbol{\Sigma}\) is symmetric positive definite, so \(\boldsymbol{\Sigma}^{-1}\) exists and the square root is real). The ruler \(\boldsymbol{\Sigma}^{-1}\) is large in directions where the index patients are tightly clustered and small in directions where they are widely spread, so it charges a heavy price for a gap along a low-variance direction and a light price for a gap along a high-variance one. When the covariates are uncorrelated and on equal footing, \(\boldsymbol{\Sigma}=I\), the ruler is the identity and \(d_M\) is the ordinary Euclidean distance. \(\blacktriangleleft\)
The defining identity is that this distance is an ordinary Euclidean distance after a linear transform that “whitens” the covariates, that is, makes their covariance the identity. We prove it for any matrix square root, then specialize to the two used in practice.
Theorem 27.1 (Whitening equivalence of the Mahalanobis distance) Let \(\boldsymbol{\Sigma}\succ 0\) and let \(B\in\mathbb{R}^{p\times p}\) be any invertible matrix with \(\boldsymbol{\Sigma}=BB^\top\) (such a \(B\) exists by Theorem 4.9). Then the Mahalanobis distance is the Euclidean norm of the whitened gap \(B^{-1}\boldsymbol{\delta}_j\): \[ d_M\big(\bar{\mathbf{x}}_0,\bar{\mathbf{x}}_j;\boldsymbol{\Sigma}\big) \;=\;\big\|B^{-1}\boldsymbol{\delta}_j\big\|_2 . \tag{27.5}\] In particular this holds for the lower-triangular Cholesky factor \(B=L\) with \(\boldsymbol{\Sigma}=LL^\top\), and for the symmetric positive-definite square root \(B=\boldsymbol{\Sigma}^{1/2}\) of Theorem 4.9. Moreover:
If \(\boldsymbol{\Sigma}=\operatorname{diag}(s_1^2,\dots,s_p^2)\) is diagonal, then \(d_M\) reduces to the standardized Euclidean distance \(d_{SE}\) with reference scales \(s_\ell\).
The whitened covariates \(\mathbf{u}_j=B^{-1}\boldsymbol{\delta}_j\) have, under the model in which \(\boldsymbol{\Sigma}\) is the true covariance, identity covariance: \(\operatorname{Cov}(B^{-1}\mathbf{X})= B^{-1}\boldsymbol{\Sigma}B^{-\top}=I_p\).
Proof. That a factorization \(\boldsymbol{\Sigma}=BB^\top\) with invertible \(B\) exists is part of Theorem 4.9: since \(\boldsymbol{\Sigma}\succ0\) is symmetric, the spectral theorem writes \(\boldsymbol{\Sigma}=Q\Lambda Q^\top\) with \(Q\) orthogonal (\(Q^\top Q=I\)) and \(\Lambda=\operatorname{diag}(\lambda_1,\dots,\lambda_p)\), and positive-definiteness forces every eigenvalue \(\lambda_i>0\); the spectral construction there produces the symmetric square root \(\boldsymbol{\Sigma}^{1/2}=Q\operatorname{diag} (\sqrt{\lambda_1},\dots,\sqrt{\lambda_p})Q^\top\), which is invertible because every \(\lambda_i>0\) and satisfies \(\boldsymbol{\Sigma}^{1/2}\boldsymbol{\Sigma}^{1/2}=\boldsymbol{\Sigma}\); taking \(B=\boldsymbol{\Sigma}^{1/2}\) (symmetric, so \(B=B^\top\) and \(BB^\top=\boldsymbol{\Sigma}\)) gives one such \(B\), and the Cholesky factor gives another.
For any invertible \(B\) with \(\boldsymbol{\Sigma}=BB^\top\), taking inverses and using \((BB^\top)^{-1}=B^{-\top}B^{-1}\), \[ \boldsymbol{\delta}_j^\top\boldsymbol{\Sigma}^{-1}\boldsymbol{\delta}_j =\boldsymbol{\delta}_j^\top B^{-\top}B^{-1}\boldsymbol{\delta}_j =\big(B^{-1}\boldsymbol{\delta}_j\big)^\top\big(B^{-1}\boldsymbol{\delta}_j\big) =\big\|B^{-1}\boldsymbol{\delta}_j\big\|_2^2 . \] Taking nonnegative square roots gives Equation 27.5. For part 1, if \(\boldsymbol{\Sigma}=\operatorname{diag}(s_1^2,\dots,s_p^2)\) then \(\boldsymbol{\Sigma}^{-1}=\operatorname{diag}(s_1^{-2},\dots,s_p^{-2})\) and \[ \boldsymbol{\delta}_j^\top\boldsymbol{\Sigma}^{-1}\boldsymbol{\delta}_j =\sum_{\ell=1}^{p}\frac{\delta_{j\ell}^2}{s_\ell^2} =\sum_{\ell=1}^{p}\left(\frac{\bar{x}_{j\ell}-\bar{x}_{0\ell}}{s_\ell}\right)^2 =d_{SE}\big(\bar{\mathbf{x}}_0,\bar{\mathbf{x}}_j;s\big)^2 , \] which is Equation 27.3. For part 2, the covariance of a linear map is \(\operatorname{Cov}(B^{-1}\mathbf{X})=B^{-1}\operatorname{Cov}(\mathbf{X})B^{-\top} =B^{-1}\boldsymbol{\Sigma}B^{-\top}=B^{-1}(BB^\top)B^{-\top}=I_p\). \(\square\)
Remark. The running example, and the promised reversal. Return to Example 27.1, with \(\boldsymbol{\Sigma}=\bigl[\begin{smallmatrix}100&20\\20&16\end{smallmatrix}\bigr]\). Its inverse is \(\boldsymbol{\Sigma}^{-1}=\tfrac{1}{1200}\bigl[\begin{smallmatrix}16&-20\\-20&100\end{smallmatrix}\bigr]\), so the Mahalanobis quadratic form is \(d_M^2=\tfrac{1}{75}\delta_1^2+\tfrac{1}{12}\delta_2^2 -\tfrac{1}{30}\delta_1\delta_2\). Evaluating on the two gaps \(\boldsymbol{\delta}_P=(10,4)\) and \(\boldsymbol{\delta}_Q=(6,-4)\), \[ d_M(P)=\sqrt{\tfrac{100}{75}+\tfrac{16}{12}-\tfrac{40}{30}}=\sqrt{1.333}=1.155, \qquad d_M(Q)=\sqrt{\tfrac{36}{75}+\tfrac{16}{12}+\tfrac{24}{30}}=\sqrt{2.613}=1.617 . \] Now the Mahalanobis distance ranks \(P\) ahead of \(Q\), the exact reverse of the raw and standardized verdict. The reason is visible in the cross term. Study \(P\) moved age and biomarker up together, the direction the index correlation \(\operatorname{corr}=0.5\) expects, so its cross term is negative (\(-\tfrac{40}{30}\)) and cancels much of the distance: a concordant move is cheap. Study \(Q\) moved them in opposite directions, against the correlation, so its cross term is positive (\(+\tfrac{24}{30}\)) and adds to the distance: a discordant move is expensive. The standardized distance, blind to the correlation, saw only that \(Q\)’s total standardized magnitude (\(\sqrt{1.36}\)) is smaller than \(P\)’s (\(\sqrt2\)) and called \(Q\) closer; the Mahalanobis distance sees that \(Q\)’s direction is the improbable one and calls \(P\) closer. Which verdict should drive the decision is exactly the question Section 27.8 settles, and the answer is “it depends on the unknown contrast.” \(\blacktriangleleft\)
Remark. Mahalanobis is Euclidean in the right coordinates. Equation Equation 27.5 says the Mahalanobis distance measures the gap in coordinates where the covariate cloud has been sphered to an isotropic unit ball. A direction in which the covariates are highly variable, or two covariates that move together, are “cheap” to differ in, because such differences are expected under \(\boldsymbol{\Sigma}\); a direction of low variance, or a discordant move against a strong correlation, is “expensive.” Part 1 shows that standardized Euclidean distance is the special case that keeps the variances but discards the correlations, that is, it whitens with the diagonal of \(\boldsymbol{\Sigma}\). The worked example of Section 27.10 exhibits a candidate that standardized Euclidean distance and Mahalanobis distance rank oppositely for exactly this reason. \(\blacktriangleleft\)
Remark. The Mahalanobis distance is a genuine metric. By Equation 27.5, \(d_M(\mathbf{a},\mathbf{b})= \|B^{-1}(\mathbf{a}-\mathbf{b})\|_2\) is the Euclidean metric composed with the invertible linear map \(B^{-1}\); the symmetry, triangle inequality, and positive-definiteness of \(d_M\) are inherited from those of \(\|\cdot\|_2\) because \(B^{-1}\) is injective. The same argument makes \(d_{SE}\) a metric. This is recorded as part of Proposition 27.1. \(\blacktriangleleft\)
This material corresponds to O.3 in the proof catalogue.
27.6 Further distances
The uitc toolkit offers several alternatives to the \(\ell_2\) family, each answering a slightly different question about the covariate gap. A reader in a hurry can treat this section as reference material: the standardized Euclidean and Mahalanobis distances of the previous two sections are the workhorses, and the variants here earn their place only when a specific question makes one of them apt (the single worst-matched covariate, or mixed continuous and binary covariate types). Skip to Section 27.7 on a first reading and return when a variant is needed.
Definition 27.4 (Manhattan, Minkowski, mean-SMD, and Gower distances) With standardized gap components \(z_{j\ell}=(\bar{x}_{j\ell}-\bar{x}_{0\ell})/s_\ell\) as in Definition 27.2, define:
The Manhattan (or \(\ell_1\)) distance, the sum of absolute standardized gaps, \[ d_1\big(\bar{\mathbf{x}}_0,\bar{\mathbf{x}}_j;s\big)=\sum_{\ell=1}^{p}\big|z_{j\ell}\big|. \tag{27.6}\]
The Minkowski distance of order \(q\ge 1\), which interpolates between them, \[ d_q\big(\bar{\mathbf{x}}_0,\bar{\mathbf{x}}_j;s\big) =\left(\sum_{\ell=1}^{p}\big|z_{j\ell}\big|^{q}\right)^{1/q}, \qquad d_\infty=\max_{1\le\ell\le p}\big|z_{j\ell}\big| , \tag{27.7}\] with \(q=1\) the Manhattan distance, \(q=2\) the standardized Euclidean distance, and the limit \(q\to\infty\) the Chebyshev distance, the largest single standardized gap.
The mean absolute standardized mean difference (mean-SMD), the Manhattan distance averaged over covariates, \[ d_{\mathrm{SMD}}\big(\bar{\mathbf{x}}_0,\bar{\mathbf{x}}_j;s\big) =\frac1p\sum_{\ell=1}^{p}\big|z_{j\ell}\big| =\frac1p\,d_1\big(\bar{\mathbf{x}}_0,\bar{\mathbf{x}}_j;s\big). \tag{27.8}\]
The Gower distance, for mixed continuous and binary covariates, with a positive range \(r_\ell\) for each continuous covariate and binary covariates coded in \([0,1]\), \[ d_G\big(\bar{\mathbf{x}}_0,\bar{\mathbf{x}}_j\big)=\frac1p\sum_{\ell=1}^{p}g_{j\ell}, \qquad g_{j\ell}=\begin{cases} \min\!\big\{\,|\bar{x}_{j\ell}-\bar{x}_{0\ell}|/r_\ell,\ 1\,\big\}, & \ell\text{ continuous},\\[2pt] |\bar{x}_{j\ell}-\bar{x}_{0\ell}|, & \ell\text{ binary}. \end{cases} \tag{27.9}\] \(\blacktriangleleft\)
Remark. What each variant emphasizes, and what is excluded. Raising the order \(q\) concentrates attention on the single worst-matched covariate: \(d_\infty\) ignores all but the maximal standardized gap, so a candidate with one badly mismatched covariate is penalized regardless of how well the rest agree, whereas \(d_1\) trades off freely across covariates. The mean-SMD rescales \(d_1\) to the per-covariate average, which makes it directly comparable across analyses with different numbers of covariates and ties the diagnostic to the standardized-mean-difference balance tables of the propensity-score literature, where a common rule of thumb flags any \(|z_{j\ell}|>0.1\) as imbalance. The Gower distance handles continuous and binary covariates on one footing by range-scaling and truncating the continuous gaps. The uitc toolkit deliberately omits cosine, Hamming, Jaccard, Sorensen-Dice, and Haversine distances for this task: those measure angular similarity, binary-set overlap, or geographic separation, none of which is the right notion for a gap between continuous covariate means. \(\blacktriangleleft\)
Remark. A tiny Gower calculation. Suppose a candidate is summarized by one continuous covariate, mean age, and one binary covariate, proportion male. Let the pooled ages run from \(40\) to \(80\), so the age range is \(r_{\text{age}}=40\), and let the index means be age \(60\) and proportion male \(0.5\). A candidate reporting mean age \(68\) and proportion male \(0.7\) has \[ g_{\text{age}}=\min\!\Big\{\tfrac{|68-60|}{40},\,1\Big\}=\min\{0.2,\,1\}=0.2, \qquad g_{\text{male}}=|0.7-0.5|=0.2, \] and Gower distance \(d_G=\tfrac12(0.2+0.2)=0.2\). The range division puts the \(8\)-year age gap on the same \([0,1]\) footing as the \(0.2\) difference in a proportion, and the truncation at \(1\) means that even a wildly out-of-range age gap (say a candidate with mean age \(120\), giving \(|120-60|/40=1.5\), capped to \(1\)) counts no more than a complete binary disagreement. This is why Gower is the natural choice when continuous and binary covariates must be compared together. \(\blacktriangleleft\)
For these to support a coherent “closest” in Definition 27.1, they should at least be genuine distances. We record which are, and where the metric axioms can fail.
Proposition 27.1 (Metric properties of the covariate-summary distances) Fix the reference scales \(s_\ell>0\), the covariance \(\boldsymbol{\Sigma}\succ0\), and the Gower ranges \(r_\ell>0\). Then:
The standardized Euclidean distance \(d_{SE}\) and the Mahalanobis distance \(d_M\) are metrics on \(\mathbb{R}^p\).
For every \(q\in[1,\infty]\) the Minkowski distance \(d_q\) is a metric on \(\mathbb{R}^p\); the mean-SMD \(d_{\mathrm{SMD}}=d_1/p\) is a metric. For \(0<q<1\) the function in Equation 27.7 is not a metric: it violates the triangle inequality.
The Gower distance \(d_G\) is a metric on the product covariate space.
Proof. Part 1. By Definition 27.2 and Theorem 27.1, \(d_{SE}(\mathbf{a},\mathbf{b})= \|S^{-1}(\mathbf{a}-\mathbf{b})\|_2\) and \(d_M(\mathbf{a},\mathbf{b})=\|B^{-1}(\mathbf{a}-\mathbf{b})\|_2\) with \(S^{-1}\) and \(B^{-1}\) invertible. For any invertible linear map \(T\), the function \(N(\mathbf{v})=\|T\mathbf{v}\|_2\) is a norm: it is absolutely homogeneous and satisfies the triangle inequality because \(\|\cdot\|_2\) does and \(T\) is linear, and \(N(\mathbf{v})=0\) forces \(T\mathbf{v}=0\), hence \(\mathbf{v}=0\) by injectivity of \(T\). A norm of a difference is a metric, giving part 1.
Part 2. For \(q\in[1,\infty]\) the map \(\mathbf{v}\mapsto\|\mathbf{v}\|_q\) is the \(\ell_q\) norm, and \(d_q(\mathbf{a},\mathbf{b})=\|S^{-1}(\mathbf{a}-\mathbf{b})\|_q\) is the corresponding metric; the only nontrivial axiom is the triangle inequality, which is Minkowski’s inequality \(\|\mathbf{u}+\mathbf{w}\|_q\le\|\mathbf{u}\|_q+\|\mathbf{w}\|_q\). We verify the three orders this book uses directly. For \(q=1\) it is the triangle inequality for \(|\cdot|\) summed over coordinates; for \(q=\infty\) it is \(\max_\ell|u_\ell+w_\ell|\le\max_\ell(|u_\ell|+|w_\ell|)\le\max_\ell|u_\ell|+ \max_\ell|w_\ell|\); for \(q=2\) it follows from the Cauchy-Schwarz inequality, since \(\|\mathbf{u}+\mathbf{w}\|_2^2=\|\mathbf{u}\|_2^2+2\mathbf{u}^\top\mathbf{w}+\|\mathbf{w}\|_2^2 \le\|\mathbf{u}\|_2^2+2\|\mathbf{u}\|_2\|\mathbf{w}\|_2+\|\mathbf{w}\|_2^2 =(\|\mathbf{u}\|_2+\|\mathbf{w}\|_2)^2\). For general \(q\in[1,\infty)\) the inequality is the named Minkowski inequality, which we cite here; its derivation from Hölder’s inequality is exactly the content of Exercise 27.5, so the single case not verified in the display above is signposted to a worked exercise rather than left as a gap. Since \(d_{\mathrm{SMD}}=d_1/p\) is a positive scalar multiple of a metric, it is a metric.
For \(0<q<1\) the triangle inequality fails. In \(\mathbb{R}^2\) take \(\mathbf{a}=(0,0)\), \(\mathbf{b}=(1,0)\), \(\mathbf{c}=(1,1)\) with unit reference scales. Then \(d_q(\mathbf{a},\mathbf{c})=(1+1)^{1/q}=2^{1/q}\) while \(d_q(\mathbf{a},\mathbf{b})+d_q(\mathbf{b},\mathbf{c})=1+1=2\); since \(1/q>1\) we have \(2^{1/q}>2\), so the direct distance exceeds the detour and the triangle inequality is violated.
Part 3. For a continuous coordinate, \(\rho_\ell(a,b)=|a-b|/r_\ell\) is a metric on \(\mathbb{R}\), and we claim its truncation \(\min\{\rho_\ell,1\}\) is again a metric. Symmetry and positive-definiteness are immediate; for the triangle inequality, given points \(a,b,c\), if either of \(\min\{\rho_\ell(a,b),1\}\) or \(\min\{\rho_\ell(b,c),1\}\) equals \(1\) then the right-hand side is at least \(1\ge\min\{\rho_\ell(a,c),1\}\), and otherwise both are below \(1\), so \(\rho_\ell(a,b),\rho_\ell(b,c)<1\) and \(\min\{\rho_\ell(a,c),1\}\le\rho_\ell(a,c)\le\rho_\ell(a,b)+\rho_\ell(b,c) =\min\{\rho_\ell(a,b),1\}+\min\{\rho_\ell(b,c),1\}\). Thus each \(g_\ell\) is a per-coordinate metric (the binary case \(|a-b|\) on \([0,1]\) being a metric outright). A nonnegative weighted sum of metrics composed with coordinate projections is a pseudometric on the product, and it separates points here because each \(g_\ell(a_\ell,b_\ell)=0\) forces \(a_\ell=b_\ell\); therefore \(d_G=\frac1p\sum_\ell g_\ell\) is a metric. \(\square\)
This material corresponds to O.4 in the proof catalogue.
27.7 Scale dependence and scale invariance
A change in the unit of a covariate, age in months rather than years, weight in pounds rather than kilograms, is a coordinate map \(\mathbf{x}\mapsto A\mathbf{x}\) with \(A=\operatorname{diag}(a_1,\dots,a_p)\) and \(a_\ell>0\). A distance fit to drive a decision should not let such a relabeling change the decision. The standardized and Mahalanobis families pass this test; the raw Euclidean distance fails it, and fails it badly enough to reverse the ranking of candidates.
Here is the concrete picture behind the formal statement. Record age in months instead of years and you multiply every age by \(12\); but the standard deviation of age is also multiplied by \(12\), so the standardized age gap, the gap divided by that standard deviation, is unchanged, because the factor of \(12\) cancels in the ratio. Re-expressing the same data in different units therefore cannot move a standardized distance. That is all “coherent rescaling” means in the theorem below: the covariate and every auxiliary input tied to its units (its reference scale, its Gower range, its row and column of the covariance matrix) are re-expressed together, exactly as they would be in any honest re-recording of the data. The practical stake is a regulatory one. A reviewer who asks “would your comparator choice have changed if the sponsor had reported age in months rather than years?” is asking precisely whether the chosen distance is scale-invariant, and the next theorem lets you answer “no, it could not have” for the standardized and Mahalanobis families, while Proposition 27.2 shows you would have to answer “yes, possibly” for the raw Euclidean distance.
Theorem 27.2 (Scale invariance of standardized and Mahalanobis distances) Let \(A=\operatorname{diag}(a_1,\dots,a_p)\) with every \(a_\ell>0\), and apply the coherent rescaling that maps every summary vector \(\mathbf{x}\mapsto A\mathbf{x}\) and rescales the auxiliary inputs in step: the reference scales \(s_\ell\mapsto a_\ell s_\ell\) (so \(S\mapsto AS\)), the Gower ranges \(r_\ell\mapsto a_\ell r_\ell\), and the Mahalanobis covariance \(\boldsymbol{\Sigma}\mapsto A\boldsymbol{\Sigma}A\). Then the standardized Euclidean distance \(d_{SE}\), the Manhattan, Minkowski, and Chebyshev distances \(d_q\) (\(q\in[1,\infty]\)), the mean-SMD \(d_{\mathrm{SMD}}\), the Gower distance \(d_G\), and the Mahalanobis distance \(d_M\) are all unchanged. Moreover the Mahalanobis distance is invariant under any invertible affine reparameterization \(\mathbf{x}\mapsto A\mathbf{x}+\mathbf{c}\) with \(A\in\mathbb{R}^{p\times p}\) invertible, not only diagonal \(A\), provided \(\boldsymbol{\Sigma}\mapsto A\boldsymbol{\Sigma}A^\top\).
Proof. Under the coherent rescaling the standardized components are unchanged coordinatewise: the rescaled gap is \(A\bar{\mathbf{x}}_j-A\bar{\mathbf{x}}_0=A\boldsymbol{\delta}_j\) with \(\ell\)-th entry \(a_\ell\delta_{j\ell}\), and dividing by the rescaled reference scale \(a_\ell s_\ell\) gives \[ \frac{a_\ell\,\delta_{j\ell}}{a_\ell s_\ell}=\frac{\delta_{j\ell}}{s_\ell}=z_{j\ell}. \] Every distance built from the standardized vector \(\mathbf{z}_j\) alone, namely \(d_{SE}=\|\mathbf{z}_j\|_2\), \(d_q=\|\mathbf{z}_j\|_q\) for \(q\in[1,\infty]\), and \(d_{\mathrm{SMD}}=\|\mathbf{z}_j\|_1/p\), is therefore a function of unchanged quantities and is unchanged. For the Gower distance the continuous terms are \(\min\{a_\ell|\delta_{j\ell}|/(a_\ell r_\ell),1\}=\min\{|\delta_{j\ell}|/r_\ell,1\}\), unchanged, and the binary terms involve no continuous unit, so \(d_G\) is unchanged.
For the Mahalanobis distance, take the general invertible affine map \(\mathbf{x}\mapsto A\mathbf{x}+ \mathbf{c}\); the constant \(\mathbf{c}\) cancels in the gap, which becomes \(A\boldsymbol{\delta}_j\), and the covariance becomes \(A\boldsymbol{\Sigma}A^\top\). Using \((A\boldsymbol{\Sigma}A^\top)^{-1}= A^{-\top}\boldsymbol{\Sigma}^{-1}A^{-1}\), \[ \big(A\boldsymbol{\delta}_j\big)^\top\big(A\boldsymbol{\Sigma}A^\top\big)^{-1}\big(A\boldsymbol{\delta}_j\big) =\boldsymbol{\delta}_j^\top A^\top A^{-\top}\boldsymbol{\Sigma}^{-1}A^{-1}A\,\boldsymbol{\delta}_j =\boldsymbol{\delta}_j^\top\boldsymbol{\Sigma}^{-1}\boldsymbol{\delta}_j , \] so \(d_M\) is unchanged. The diagonal coherent rescaling is the special case \(A=\operatorname{diag}(a_\ell)\), for which \(A\boldsymbol{\Sigma}A=A\boldsymbol{\Sigma}A^\top\). \(\square\)
The raw Euclidean distance enjoys none of this. It is not merely rescaled by a unit change; the rescaling reweights the covariates unequally and can overturn which candidate is judged closest.
Proposition 27.2 (Raw Euclidean distance is scale-dependent and can reverse rankings) Let \(A=\operatorname{diag}(a_1,\dots,a_p)\) with \(a_\ell>0\). In general \(d_E(A\bar{\mathbf{x}}_0,A\bar{\mathbf{x}}_j)\ne d_E(\bar{\mathbf{x}}_0,\bar{\mathbf{x}}_j)\), because \[ d_E\big(A\bar{\mathbf{x}}_0,A\bar{\mathbf{x}}_j\big)^2=\sum_{\ell=1}^{p}a_\ell^2\,\delta_{j\ell}^2 , \tag{27.10}\] a reweighting of the squared gaps by \(a_\ell^2\). There exist two candidates whose raw-Euclidean ranking is reversed by a change of units in a single covariate.
Proof. Equation Equation 27.10 is immediate from \(A\bar{\mathbf{x}}_j-A\bar{\mathbf{x}}_0= A\boldsymbol{\delta}_j\) and the diagonal form of \(A\). For the reversal, take \(p=2\) and two candidates with gaps \(\boldsymbol{\delta}_1=(1,2)\) and \(\boldsymbol{\delta}_2=(2,1)\). With unit scaling both have squared distance \(1^2+2^2=2^2+1^2=5\), so they tie. Rescale the first coordinate by \(a_1=10\) and keep \(a_2=1\). Then \[ d_E(A\boldsymbol{\delta}_1)^2=10^2\cdot1+1\cdot4=104, \qquad d_E(A\boldsymbol{\delta}_2)^2=10^2\cdot4+1\cdot1=401 , \] so candidate \(1\) is now strictly closer than candidate \(2\), whereas before they were equidistant; a unit change has manufactured a strict preference. Perturbing \(\boldsymbol{\delta}_1\) to \((1,2+\varepsilon)\) for small \(\varepsilon>0\) makes candidate \(2\) strictly closer under unit scaling (\(5<5+4\varepsilon+ \varepsilon^2\)) while candidate \(1\) remains strictly closer after the rescaling, an outright reversal of the strict ranking. \(\square\)
Remark. The decision-theoretic moral. Because the comparator-selection rule Equation 27.1 returns an \(\arg\min\), only the ranking of candidates matters, and Proposition 27.2 shows the raw Euclidean ranking is an artifact of the recording units. Standardized Euclidean and Mahalanobis distances are immune (Theorem 27.2), which is why uitc recommends them whenever covariates carry mixed units. The worked example of Section 27.10 exhibits a concrete reversal: raw Euclidean distance there selects the candidate with the largest eventual bias, precisely because it is dominated by a large-unit covariate. \(\blacktriangleleft\)
This material corresponds to O.5 in the proof catalogue.
27.8 Bias bounds from covariate-summary distance
A distance is worth minimizing only if minimizing it reduces the bias of the eventual estimate. We now prove exactly when it does. Throughout this section we work under conditional constancy, so that the conditional contrast (or, in the unanchored case, each conditional mean) is the same function of the covariates in the index and comparator populations; the only cross-population difference is the covariate distribution. The quantity of interest is the transport bias \[ B_j\;=\;\mathbb{E}_{\mathcal{P}_j}\big[\tau(\mathbf{X})\big]-\mathbb{E}_{\mathcal{P}_0}\big[\tau(\mathbf{X})\big], \tag{27.11}\] the difference between the marginal effect over the comparator population \(\mathcal{P}_j\) and the marginal effect over the index population \(\mathcal{P}_0\), where \(\tau\) is the shared conditional contrast. Under conditional constancy this \(B_j\) is the residual bias incurred by treating comparator \(j\)’s population as a stand-in for the index population, or equivalently the bias that remains if the covariate gap \(\boldsymbol{\delta}_j\) is left unadjusted; it is the population-level quantity that the unanchored consistency theorem (Theorem 17.2) requires to vanish.
Remark. Reader translation of the two working assumptions. Two phrases carry this whole section. Conditional constancy (Definition 17.5) means the conditional contrast \(\tau(\mathbf{x})\), the treatment effect among patients whose covariates equal \(\mathbf{x}\), is the same function of \(\mathbf{x}\) in the index and comparator populations; what differs between the populations is only how many patients sit at each value of \(\mathbf{x}\). If that holds, the sole thing a comparator’s population changes is the covariate mix, so any transport bias is due entirely to the covariate gap. Transport bias \(B_j\) is then just the difference between the average effect you would report over the comparator’s patient mix and the average effect over the index patient mix: it is what you get wrong by handing an estimand defined for the index population to a population with a different covariate distribution. A small covariate distance is our attempt to keep that difference small; the rest of the section is about when it succeeds. \(\blacktriangleleft\)
27.8.1 The linear case: an exact Cauchy-Schwarz bound
When the contrast is affine, the bias is a linear functional of the covariate-mean gap, and the quadratic-form distances bound it sharply.
What does “affine contrast” mean for a real outcome, and when is it plausible? Fit an outcome model on the link scale, the logit (log-odds) scale for a binary response, the log scale for a hazard, with a linear predictor of the Chapter 20 form \(\eta=\mu+\boldsymbol{\beta}_1^\top\mathbf{x}+(\gamma+\boldsymbol{\beta}_2 ^\top\mathbf{x})\,T\), where \(T\) is the treatment indicator. The conditional contrast on that scale, the treatment effect at covariate value \(\mathbf{x}\), is \[ \tau(\mathbf{x})=\eta\big|_{T=1}-\eta\big|_{T=0}=\gamma+\boldsymbol{\beta}_2^\top\mathbf{x}, \] which is affine in \(\mathbf{x}\) with intercept \(\alpha=\gamma\) and slope vector \(\boldsymbol{\beta}=\boldsymbol{\beta}_2\). So in the theorem below \(\alpha\) is the treatment effect at \(\mathbf{x}=\mathbf 0\) and \(\boldsymbol{\beta}\) is the vector of effect-modifier slopes: each \(\beta_\ell\) says how much the treatment’s effect on the link scale changes per unit of covariate \(\ell\). The affine case is therefore exactly “a linear outcome model on the link scale whose treatment effect is modified linearly by the covariates, with no covariate-by-covariate or higher-order terms.” It is a reasonable working hypothesis when effect modification is mild and roughly linear, and it is the sensitivity input under which comparator distance provably bounds bias. Two poles are worth noting: if \(\boldsymbol{\beta}=\mathbf 0\) there is no effect modification and the transport bias is zero for any covariate gap; the entire difficulty lives in the direction of \(\boldsymbol{\beta}\ne\mathbf 0\), which is why the theorem’s bound pairs the gap with \(\boldsymbol{\beta}\).
Theorem 27.3 (Linear bias bound) Suppose the shared conditional contrast is affine in the covariates, \(\tau(\mathbf{x})=\alpha+\boldsymbol{\beta}^\top\mathbf{x}\). Then the transport bias Equation 27.11 is the linear functional of the covariate-mean gap \[ B_j=\boldsymbol{\beta}^\top\boldsymbol{\delta}_j =\boldsymbol{\beta}^\top\big(\bar{\mathbf{x}}_j-\bar{\mathbf{x}}_0\big), \tag{27.12}\] and it is bounded by either quadratic-form distance:
with \(S=\operatorname{diag}(s_1,\dots,s_p)\), \[ |B_j|\;\le\;\|S\boldsymbol{\beta}\|_2\;\cdot\;d_{SE}\big(\bar{\mathbf{x}}_0,\bar{\mathbf{x}}_j;s\big), \tag{27.13}\] with equality if and only if the standardized gap \(\mathbf{z}_j=S^{-1}\boldsymbol{\delta}_j\) is parallel to \(S\boldsymbol{\beta}\);
with \(\boldsymbol{\Sigma}\succ0\), \[ |B_j|\;\le\;\sqrt{\boldsymbol{\beta}^\top\boldsymbol{\Sigma}\boldsymbol{\beta}}\;\cdot\; d_M\big(\bar{\mathbf{x}}_0,\bar{\mathbf{x}}_j;\boldsymbol{\Sigma}\big) =\|\boldsymbol{\beta}\|_{\boldsymbol{\Sigma}}\cdot\|\boldsymbol{\delta}_j\|_{\boldsymbol{\Sigma}^{-1}}, \tag{27.14}\] with equality if and only if \(\boldsymbol{\delta}_j\) is parallel to \(\boldsymbol{\Sigma}\boldsymbol{\beta}\).
Proof. The intercept \(\alpha\) cancels in the difference Equation 27.11, leaving \(B_j=\boldsymbol{\beta}^\top\mathbb{E}_{\mathcal{P}_j}[\mathbf{X}]-\boldsymbol{\beta}^\top \mathbb{E}_{\mathcal{P}_0}[\mathbf{X}]=\boldsymbol{\beta}^\top(\bar{\mathbf{x}}_j-\bar{\mathbf{x}}_0) =\boldsymbol{\beta}^\top\boldsymbol{\delta}_j\), which is Equation 27.12.
Bound 1. Insert \(S^{-1}S=I\): \(B_j=\boldsymbol{\beta}^\top S S^{-1}\boldsymbol{\delta}_j= (S\boldsymbol{\beta})^\top\mathbf{z}_j\). By the Cauchy-Schwarz inequality in \(\mathbb{R}^p\), \[ |B_j|=\big|(S\boldsymbol{\beta})^\top\mathbf{z}_j\big| \le\|S\boldsymbol{\beta}\|_2\,\|\mathbf{z}_j\|_2 =\|S\boldsymbol{\beta}\|_2\,d_{SE}\big(\bar{\mathbf{x}}_0,\bar{\mathbf{x}}_j;s\big), \] which is Equation 27.13. Cauchy-Schwarz holds with equality if and only if the two vectors are linearly dependent, that is, \(\mathbf{z}_j\parallel S\boldsymbol{\beta}\).
Bound 2. Write \(\boldsymbol{\Sigma}=\boldsymbol{\Sigma}^{1/2}\boldsymbol{\Sigma}^{1/2}\) with the symmetric square root of Theorem 4.9 and insert \(\boldsymbol{\Sigma}^{-1/2}\boldsymbol{\Sigma}^{1/2}=I\): \[ B_j=\boldsymbol{\beta}^\top\boldsymbol{\delta}_j =\big(\boldsymbol{\Sigma}^{1/2}\boldsymbol{\beta}\big)^\top \big(\boldsymbol{\Sigma}^{-1/2}\boldsymbol{\delta}_j\big). \] By Cauchy-Schwarz, \[ |B_j|\le\big\|\boldsymbol{\Sigma}^{1/2}\boldsymbol{\beta}\big\|_2 \big\|\boldsymbol{\Sigma}^{-1/2}\boldsymbol{\delta}_j\big\|_2 =\sqrt{\boldsymbol{\beta}^\top\boldsymbol{\Sigma}\boldsymbol{\beta}}\, \sqrt{\boldsymbol{\delta}_j^\top\boldsymbol{\Sigma}^{-1}\boldsymbol{\delta}_j} =\sqrt{\boldsymbol{\beta}^\top\boldsymbol{\Sigma}\boldsymbol{\beta}}\; d_M\big(\bar{\mathbf{x}}_0,\bar{\mathbf{x}}_j;\boldsymbol{\Sigma}\big), \] which is Equation 27.14, using Equation 27.4 for the last factor. Equality holds if and only if \(\boldsymbol{\Sigma}^{-1/2}\boldsymbol{\delta}_j\parallel\boldsymbol{\Sigma}^{1/2} \boldsymbol{\beta}\); applying the invertible map \(\boldsymbol{\Sigma}^{1/2}\) to both sides, this is \(\boldsymbol{\delta}_j\parallel\boldsymbol{\Sigma}\boldsymbol{\beta}\). \(\square\)
Remark. Reading the bounds. Both bounds factor the bias into a fixed coefficient magnitude, \(\|S \boldsymbol{\beta}\|_2\) or \(\|\boldsymbol{\beta}\|_{\boldsymbol{\Sigma}}\), which does not depend on the candidate, times a distance, which does. Because the coefficient factor is common to all candidates, each distance ranks the candidates exactly by its own bound on \(|B_j|\): selecting the \(d_{SE}\)-minimizer minimizes the standardized bound, and selecting the \(d_M\)-minimizer minimizes the Mahalanobis bound. The two bounds need not agree on the winner, because \(\|S\boldsymbol{\beta}\|_2\,d_{SE}\) and \(\|\boldsymbol{\beta}\|_{\boldsymbol{\Sigma}}\,d_M\) are different upper bounds on the same \(|B_j|\); the equality conditions say each bound is tight for the candidate whose gap aligns with the coefficient vector in the relevant geometry. This is not a defect but the precise sense in which the right distance is the one matched to the unknown contrast direction \(\boldsymbol{\beta}\), a point the worked example makes concrete and the caveat of Section 27.9 makes unavoidable. \(\blacktriangleleft\)
Remark. Connection to unanchored consistency. The transport bias Equation 27.11 is the population-level obstacle in the unanchored PAIC theorem Theorem 17.2. That theorem recovers the target estimand only when every imbalanced effect modifier and prognostic variable is balanced, so that no residual covariate gap survives; its necessity clause exhibits the bias as an integral that is nonzero exactly when an imbalanced covariate is omitted. Under the affine specialization \(\tau(\mathbf{x})=\alpha+\boldsymbol{\beta}^\top\mathbf{x}\), which is hypothesis (v) “sufficiency of the matched moments” of Theorem 18.4 and the affine-\(\mu_t\) hypothesis of Theorem 17.2, that integral collapses to the inner product Equation 27.12, and Theorem 27.3 bounds it by the covariate-summary distance. Comparator selection therefore bounds the very bias the consistency theorem assumes away: a smaller distance means a smaller worst-case residual bias, under linearity. The qualifier “under linearity” is essential, and Section 27.8.2 and Section 27.9 show what happens without it. \(\blacktriangleleft\)
Remark. The running example: exact bias and the two bounds. Suppose, as a sensitivity input, the effect-modifier slopes are \(\boldsymbol{\beta}=(0.1,\,0.5)\), that is, the treatment’s effect on the link scale rises by \(0.1\) per year of age and by \(0.5\) per biomarker unit. Then the exact transport biases of Example 27.1 are \[ B_P=\boldsymbol{\beta}^\top\boldsymbol{\delta}_P=0.1(10)+0.5(4)=3.0, \qquad B_Q=\boldsymbol{\beta}^\top\boldsymbol{\delta}_Q=0.1(6)+0.5(-4)=-1.4 . \] The standardized Cauchy-Schwarz bound uses \(S\boldsymbol{\beta}=(1,2)\), so \(\|S\boldsymbol{\beta}\|_2=\sqrt5=2.236\), giving \(|B_P|\le 2.236(1.414)=3.16\) and \(|B_Q|\le2.236(1.166)=2.61\); the Mahalanobis bound uses \(\|\boldsymbol{\beta}\|_{\boldsymbol{\Sigma}}=\sqrt{\boldsymbol{\beta}^\top \boldsymbol{\Sigma}\boldsymbol{\beta}}=\sqrt7=2.646\), giving \(|B_P|\le2.646(1.155)=3.06\) and \(|B_Q|\le2.646(1.617)=4.28\). Both bounds dominate the exact biases, as they must. Now the honest lesson: the standardized winner \(Q\) has the smaller true bias (\(1.4\) against \(3.0\)), while the Mahalanobis winner \(P\) has the larger true bias, so for this particular \(\boldsymbol{\beta}\) the standardized distance was the faithful ranker and Mahalanobis was misled. Had \(\boldsymbol{\beta}\) pointed along the high-variance, positively-correlated axis of \(\boldsymbol{\Sigma}\), the verdict would flip and Mahalanobis would be faithful. Since \(\boldsymbol{\beta}\) is unknown when the comparator is chosen, no distance is guaranteed to rank by bias; this is exactly Proposition 27.4, previewed here in arithmetic. \(\blacktriangleleft\)
Remark. Bias against variance: the effective-sample-size tradeoff. A closer comparator is not automatically the better analysis, because comparator selection trades bias against variance, and the mean distance governs both. Under the linear bound, the mean distance caps the bias; and the same distance also controls the variance, because the subsequent unanchored MAIC reweights the index sample (size \(n\)) to match the comparator’s means, and a larger standardized gap forces more extreme weights and hence a smaller effective sample size \(\mathrm{ESS}=(\sum_i w_i)^2/\sum_i w_i^2\) (Theorem 18.6), inflating the variance by the factor \(n/\mathrm{ESS}\). The schematic pattern, for an index study of \(n=200\), is
| candidate | \(d_{SE}\) | typical weight extremity | illustrative \(\mathrm{ESS}\) | variance inflation \(n/\mathrm{ESS}\) |
|---|---|---|---|---|
| close (Study \(Q\)) | \(1.17\) | mild | \(150\) | \(1.33\) |
| moderate (Study \(P\)) | \(1.41\) | moderate | \(110\) | \(1.82\) |
| far (hypothetical) | \(3.0\) | severe | \(35\) | \(5.7\) |
Two consequences. First, in the linear case there is usually no dramatic conflict: the minimum-distance candidate tends to minimize both the bias bound and the reweighting variance at once. The genuine tension appears among candidates of comparable distance, where one breaks the tie by projected \(\mathrm{ESS}\), and in the fact that a far comparator’s variance can explode (bottom row) faster than a nearer comparator’s bias bound grows, so a mean-squared-error view, \(\mathrm{MSE}\approx \text{bias}^2+\text{variance}\), can prefer a slightly closer, higher-\(\mathrm{ESS}\) candidate even when its bias bound is not the smallest. Second, and more important, the distance bounds bias only under linearity; a close, high-\(\mathrm{ESS}\) comparator can still be badly biased through nonlinearity or unmeasured imbalance (Section 27.9). The defensible practice is to report distance and projected \(\mathrm{ESS}\) together and to choose on the mean-squared-error picture, not on distance alone. This trade-off is developed as Exercise 27.9 and connects to the simulation estimands of Chapter 29. \(\blacktriangleleft\)
27.8.2 The nonlinear case: a Lipschitz-Wasserstein bound
The punchline of this subsection first, because it is easy to lose in the machinery: the clean linear bound of Theorem 27.3 has no analogue at the level of covariate means once the contrast is nonlinear. A bias bound still exists, but it needs a distance between whole distributions, not between their centers, and no distance between means can supply it. This is the deepest reason aggregate distance matching is a diagnostic and not a transportability theorem, and the rest of the section makes it precise.
When the contrast is not affine, the mean gap no longer determines the bias, because a nonlinear \(\tau\) responds to the whole distribution and not only its center. A bias bound survives, but the distance it requires is between full distributions.
Before the formal statement, here is the object in plain terms. A coupling of the index and comparator covariate distributions is a way of pairing each comparator patient with an index patient, that is, a joint distribution whose two margins are the two populations. Picture the comparator patients as piles of sand sitting at their covariate values and the index patients as holes to be filled at theirs; a coupling is a transport plan saying how much sand moves from each pile to each hole. The Wasserstein-1 distance \(W_\rho(\mathcal{P}_j,\mathcal{P}_0)\) is the cost of the cheapest such plan, where moving one unit of mass a covariate-distance \(\rho\) costs \(\rho\): it is the minimum average distance patients must be shifted in covariate space to turn one population’s covariate distribution into the other’s. Unlike a gap between means, it sees the whole shape of each distribution, its spread and its tails, not only its center. For instance, to turn a population with everyone at \(x=0\) into one split evenly between \(x=-1\) and \(x=+1\), half the mass moves one unit left and half moves one unit right, an average move of \(1\), so \(W_\rho=1\), even though the two means agree at \(0\); this is the example of Proposition 27.4. The next theorem says a Lipschitz contrast, one that cannot change faster than a fixed rate \(L\) as the covariates change, can differ across two populations by at most \(L\) times this transport cost.
Theorem 27.4 (Lipschitz bias bound via the Wasserstein distance) Let \((\mathcal{X},\rho)\) be a metric space and let \(\mathcal{P}_0,\mathcal{P}_j\) be probability measures on \(\mathcal{X}\) with finite first moment. Suppose the conditional contrast \(\tau:\mathcal{X}\to\mathbb{R}\) is \(L\)-Lipschitz with respect to \(\rho\), that is \(|\tau(\mathbf{x})-\tau(\mathbf{x}')|\le L\,\rho(\mathbf{x}, \mathbf{x}')\) for all \(\mathbf{x},\mathbf{x}'\). Then the transport bias is bounded by the Wasserstein-1 distance, \[ \big|B_j\big| =\Big|\mathbb{E}_{\mathcal{P}_j}[\tau]-\mathbb{E}_{\mathcal{P}_0}[\tau]\Big| \;\le\;L\;W_\rho\big(\mathcal{P}_j,\mathcal{P}_0\big), \qquad W_\rho(\mathcal{P},\mathcal{Q})=\inf_{\gamma\in\Pi(\mathcal{P},\mathcal{Q})}\int\rho(\mathbf{x},\mathbf{y}) \,d\gamma(\mathbf{x},\mathbf{y}), \tag{27.15}\] where \(\Pi(\mathcal{P},\mathcal{Q})\) is the set of couplings of \(\mathcal{P}\) and \(\mathcal{Q}\).
Proof. Let \(\gamma\in\Pi(\mathcal{P}_j,\mathcal{P}_0)\) be any coupling, that is, a joint law of a pair \((\mathbf{X},\mathbf{Y})\) whose first marginal is \(\mathcal{P}_j\) and whose second marginal is \(\mathcal{P}_0\). Because the marginals reproduce the two expectations, \[ B_j=\mathbb{E}_{\mathcal{P}_j}[\tau]-\mathbb{E}_{\mathcal{P}_0}[\tau] =\mathbb{E}_\gamma\big[\tau(\mathbf{X})-\tau(\mathbf{Y})\big]. \] Taking absolute values, then using Jensen’s inequality for the convex function \(|\cdot|\) (Theorem 4.6) and the Lipschitz hypothesis, \[ |B_j|=\big|\mathbb{E}_\gamma[\tau(\mathbf{X})-\tau(\mathbf{Y})]\big| \le\mathbb{E}_\gamma\big|\tau(\mathbf{X})-\tau(\mathbf{Y})\big| \le L\,\mathbb{E}_\gamma\big[\rho(\mathbf{X},\mathbf{Y})\big]. \] This holds for every coupling \(\gamma\), so taking the infimum over \(\gamma\in\Pi(\mathcal{P}_j, \mathcal{P}_0)\) of the right-hand side gives \(|B_j|\le L\,W_\rho(\mathcal{P}_j,\mathcal{P}_0)\), which is Equation 27.15. \(\square\)
Remark. Why the Wasserstein distance is the right yardstick (Kantorovich-Rubinstein duality). The bound Equation 27.15 used a single coupling; remarkably, the best possible bound of this shape is again a Wasserstein distance. The Kantorovich-Rubinstein duality states \[ W_\rho(\mathcal{P},\mathcal{Q})=\sup_{\|f\|_{\mathrm{Lip}_\rho}\le1} \big|\mathbb{E}_\mathcal{P}f-\mathbb{E}_\mathcal{Q}f\big|, \] where the supremum runs over all functions \(f\) that are \(1\)-Lipschitz with respect to \(\rho\), meaning \(f\) changes by at most the distance between its inputs (\(\|f\|_{\mathrm{Lip}_\rho}\) is the smallest such rate). Read from right to left, it says the Wasserstein distance is the worst-case transport bias over every \(1\)-Lipschitz contrast: no unit-Lipschitz contrast can separate the two populations by more than \(W_\rho\), and some contrast comes arbitrarily close. That is exactly why \(W_\rho\), and not some cruder distributional distance, is the natural yardstick for Lipschitz transport bias, and why the bound in terms of it cannot be improved. Applying the identity to \(f=\tau/L\) recovers Equation 27.15. We gave the elementary coupling proof above because it is self-contained and needs only Theorem 4.6; the duality is recorded here for the reader who wants to know the bound is tight, not merely valid. \(\blacktriangleleft\)
The decisive point is that a distance between means is not a Wasserstein distance between distributions. It is only a lower bound on it, so a small mean distance does not imply a small Wasserstein distance, hence does not bound the nonlinear bias.
Proposition 27.3 (Mean distance lower-bounds the Wasserstein distance) With \(\rho\) the Euclidean metric on \(\mathbb{R}^p\), the Euclidean distance between covariate means is dominated by the Wasserstein-1 distance between the covariate distributions: \[ d_E\big(\bar{\mathbf{x}}_0,\bar{\mathbf{x}}_j\big) =\big\|\bar{\mathbf{x}}_j-\bar{\mathbf{x}}_0\big\|_2 \;\le\;W_\rho\big(\mathcal{P}_j,\mathcal{P}_0\big). \tag{27.16}\] Consequently a small Wasserstein distance forces a small mean distance, but a small (or zero) mean distance places no upper bound on the Wasserstein distance, and hence none on the nonlinear bias of Equation 27.15.
Proof. For any coupling \(\gamma\) of \((\mathbf{X},\mathbf{Y})\) with marginals \(\mathcal{P}_j,\mathcal{P}_0\), the vector Jensen inequality (Theorem 4.6 applied to the convex norm \(\|\cdot\|_2\)) gives \[ \big\|\bar{\mathbf{x}}_j-\bar{\mathbf{x}}_0\big\|_2 =\big\|\mathbb{E}_\gamma[\mathbf{X}-\mathbf{Y}]\big\|_2 \le\mathbb{E}_\gamma\big\|\mathbf{X}-\mathbf{Y}\big\|_2 =\mathbb{E}_\gamma\big[\rho(\mathbf{X},\mathbf{Y})\big]. \] Taking the infimum over couplings gives Equation 27.16. The implication is one-directional: the example in Proposition 27.4 has \(d_E=0\) yet \(W_\rho>0\) and nonzero bias, so no reverse inequality can hold. \(\square\)
This material corresponds to O.6 in the proof catalogue.
27.9 Distance is a diagnostic, not a validity guarantee
We now state and prove the chapter’s central caveat. Every distance of this chapter is a function of the reported covariate summaries; the means are the universally available summary, and all of the distances reduce to zero when the means coincide. The next proposition shows that zero distance carries no guarantee about the target effect, in two independent ways: through nonlinearity, which the means cannot see, and through unmeasured variables, which no observed summary can see.
Proposition 27.4 (Zero covariate-summary distance does not bound transport bias) Let \(d\) be any covariate-summary distance of this chapter that is computed from the covariate means alone: Euclidean, standardized Euclidean, Mahalanobis, Manhattan, Minkowski, mean-SMD, or Gower with the means as inputs.
(Nonlinearity.) There exist populations \(\mathcal{P}_0,\mathcal{P}_j\) with identical covariate means, hence \(d(\bar{\mathbf{x}}_0,\bar{\mathbf{x}}_j)=0\) for every such \(d\), and a conditional contrast \(\tau\) for which the target effects differ, \(\mathbb{E}_{\mathcal{P}_j}[\tau]\ne\mathbb{E}_{\mathcal{P}_0} [\tau]\). Consequently no inequality of the form \(|B_j|\le C\,d(\bar{\mathbf{x}}_0,\bar{\mathbf{x}}_j)\) can hold for a finite constant \(C\) uniformly over contrasts; the transport bias is not a function of any mean-based distance.
(Unmeasured imbalance.) There exist populations whose observed covariate \(\mathbf{X}\) has identical distributions, hence zero distance under every summary of \(\mathbf{X}\), that nonetheless differ in an unobserved prognostic variable \(W\) in a way that makes the unanchored target means \(\mathbb{E}[\mu_t(\mathbf{X},W)]\) differ across populations.
Proof. Part 1. Let the covariate be scalar. Let \(\mathcal{P}_0\) place all mass at \(X=0\), and let \(\mathcal{P}_j\) place mass \(\tfrac12\) at \(X=-1\) and mass \(\tfrac12\) at \(X=+1\). Both have mean \(\bar{x}_0=\bar{x}_j=0\), so every mean-based distance is zero. Take the nonlinear contrast \(\tau(x)=x^2\). Then \[ \mathbb{E}_{\mathcal{P}_0}[\tau]=0^2=0, \qquad \mathbb{E}_{\mathcal{P}_j}[\tau]=\tfrac12(-1)^2+\tfrac12(1)^2=1, \] so \(B_j=1\ne0\) while \(d=0\). Since \(|B_j|=1>C\cdot0\) for every finite \(C\), no bound of the stated form can hold. It is worth seeing this against Theorem 27.4 in full. On the support \(\{-1,0,1\}\) the contrast \(\tau(x)=x^2\) is Lipschitz: for any two points \(|x^2-y^2|=|x+y|\,|x-y|\le 2|x-y|\), so we may take \(L=2\). The Wasserstein distance is \(W_\rho(\mathcal{P}_j,\mathcal{P}_0)=1\), by the following explicit transport plan: move the mass \(\tfrac12\) sitting at \(x=-1\) to \(x=0\) (a shift of distance \(1\)) and the mass \(\tfrac12\) sitting at \(x=+1\) to \(x=0\) (a shift of distance \(1\)), for an average move of \(\tfrac12\cdot1+\tfrac12\cdot1=1\); no plan does better, since all the mass of \(\mathcal{P}_j\) must arrive at the single point \(0\) where \(\mathcal{P}_0\) lives, each unit travelling at least distance \(1\). The Lipschitz bound then reads \[ |B_j|=1\;\le\;L\,W_\rho=2\cdot1=2 , \] which holds with room to spare, while the mean distance \(d_E=0\) vanishes. The bias is real, it is controlled by the full-distribution distance, and it is wholly invisible to the means: it hides in the spread, which no mean-based distance records.
Part 2. Let the observed covariate \(\mathbf{X}\) have the same law in \(\mathcal{P}_0\) and \(\mathcal{P}_j\), and let an unobserved prognostic variable \(W\), which one should picture concretely as ECOG performance status, a strong prognostic factor in oncology that the single-arm Phase II might record but that the external Phase III reports only partially or on an incompatible scale, so it never enters the shared covariate table, have mean \(\mathbb{E}_{\mathcal{P}_0}[W]=0\) in the index population and \(\mathbb{E}_{\mathcal{P}_j}[W]=1\) in the comparator population, with \(W\) independent of \(\mathbf{X}\) in each. Let the conditional mean depend on \(W\), for instance \(\mu_t(\mathbf{x},w)=w\) for the relevant arm. Every distance computed from \(\mathbf{X}\) is zero because the laws of \(\mathbf{X}\) agree, yet the unanchored target means differ: \[ \mathbb{E}_{\mathcal{P}_0}[\mu_t(\mathbf{X},W)]=\mathbb{E}_{\mathcal{P}_0}[W]=0, \qquad \mathbb{E}_{\mathcal{P}_j}[\mu_t(\mathbf{X},W)]=\mathbb{E}_{\mathcal{P}_j}[W]=1. \] This is exactly the omitted-prognostic-variable bias of Theorem 17.2: with \(W\) absent from the adjustment set, the inner difference of its necessity integral Equation 17.17 does not vanish, and no comparator term cancels it in the unanchored contrast. \(\square\)
Remark. What aggregate-level matching is, and is not, good for. Parts 1 and 2 of Proposition 27.4 delimit the method precisely. Aggregate-level distance matching is a sound basis for: a comparator-selection rule when several bridging studies are available (Definition 27.1); an overlap diagnostic to be inspected before any unanchored MAIC, STC, or ML-UMR, since a large distance forewarns of heavy reweighting and low effective sample size (Theorem 18.6); and a sensitivity-analysis input that names which covariates drive the bridge. It is not a substitute for adjusting every imbalanced effect modifier (Definition 11.3) and prognostic variable (Definition 11.5); it is not a proof of the shared prognostic factor assumption (Definition 26.2); it is not a proof of conditional constancy of absolute effects (Definition 17.5); and it is no guard whatever against unmeasured imbalance. It presupposes, and does not establish, the clinical and design comparability of the trials, which must be judged separately and before any distance is computed (step 3 of Section 27.3): two studies can match on age, sex, and biomarker means and still differ in line of therapy, outcome definition, follow-up, or standard of care, and no small distance repairs that. The quantitative bias analysis of Chapter 28 is the tool for the residual and unmeasured part that distance matching leaves untouched. \(\blacktriangleleft\)
Remark. Second-moment distances narrow, but do not close, the gap. Part 1 exploits that the means agree while the variances differ. One can blunt this particular counterexample by augmenting the summary with second moments and using a distance that also compares the covariances, \(\boldsymbol{\Sigma}_0\) against \(\boldsymbol{\Sigma}_j\); this is the diagnostic analogue of MAIC second-moment balancing (Chapter 18) and of the joint-distribution modeling that ML-NMR achieves through a copula (Definition 22.5, Chapter 22). But no finite collection of moments closes the gap for every contrast: Exercise 27.7 constructs two populations matching in mean and variance whose target effects differ under a quartic contrast, and Part 2 shows that unmeasured variables defeat any summary of the observed covariates regardless of how many moments it includes. Distance matching on richer summaries is a better diagnostic, never a validity proof. \(\blacktriangleleft\)
This material corresponds to O.7 in the proof catalogue.
27.10 A fully worked example: ranking four comparators
We rank four candidate comparator studies for a single-arm index study, computing every distance and the exact bias by hand, and exhibit the chapter’s three lessons at once: the raw Euclidean ranking is an artifact of units, the standardized and Mahalanobis distances disagree because of covariate correlation, and the distance that tracks the actual bias is the one matched to the contrast direction.
Example 27.2 (Selecting among four comparator studies) The index single-arm study reports covariate means on three baseline covariates, mean age (years), proportion male, and mean biomarker level (units), \[ \bar{\mathbf{x}}_0=(60,\;0.5,\;10), \] with reference scales \(s=(10,\;0.5,\;4)\) (the biomarker and age standard deviations from the index IPD, and the Bernoulli standard deviation \(\sqrt{0.5\cdot0.5}=0.5\) for sex), and index covariance \[ \boldsymbol{\Sigma}=\begin{pmatrix}100&0&20\\[1pt]0&0.25&0\\[1pt]20&0&16\end{pmatrix}, \] in which age and biomarker are positively correlated, \(\operatorname{corr}=20/(10\cdot4)=0.5\), and sex is uncorrelated with both. (This \(\boldsymbol{\Sigma}\succ0\): its leading minors are \(100>0\), \(100\cdot0.25=25>0\), and \(\det\boldsymbol{\Sigma}=0.25\,(100\cdot16-20^2)=300>0\).) Four candidate comparator studies report means \(\bar{\mathbf{x}}_A,\dots,\bar{\mathbf{x}}_D\), with covariate gaps \(\boldsymbol{\delta}_j=\bar{\mathbf{x}}_j-\bar{\mathbf{x}}_0\) and standardized gaps \(\mathbf{z}_j=S^{-1}\boldsymbol{\delta}_j\): \[ \begin{array}{llll} \text{Study } A: & \boldsymbol{\delta}_A=(12,\;0.05,\;1), & \mathbf{z}_A=(1.2,\;0.1,\;0.25),\\ \text{Study } B: & \boldsymbol{\delta}_B=(6,\;0.30,\;4.8), & \mathbf{z}_B=(0.6,\;0.6,\;1.2),\\ \text{Study } C: & \boldsymbol{\delta}_C=(1,\;0.45,\;6), & \mathbf{z}_C=(0.1,\;0.9,\;1.5),\\ \text{Study } D: & \boldsymbol{\delta}_D=(8,\;0.20,\;4), & \mathbf{z}_D=(0.8,\;0.4,\;1.0). \end{array} \]
Raw Euclidean (Equation 27.2). Squaring and summing the raw gaps, \[ d_E(A)=\sqrt{12^2+0.05^2+1^2}=\sqrt{145.00}=12.04, \quad d_E(B)=\sqrt{6^2+0.3^2+4.8^2}=\sqrt{59.13}=7.69, \] \[ d_E(C)=\sqrt{1^2+0.45^2+6^2}=\sqrt{37.20}=6.10, \quad d_E(D)=\sqrt{8^2+0.2^2+4^2}=\sqrt{80.04}=8.95. \] The raw ranking is \(C\prec B\prec D\prec A\), so raw Euclidean distance selects Study \(C\).
Standardized Euclidean (Equation 27.3). Using the standardized gaps, \[ d_{SE}(A)=\sqrt{1.2^2+0.1^2+0.25^2}=\sqrt{1.5125}=1.230, \quad d_{SE}(B)=\sqrt{0.6^2+0.6^2+1.2^2}=\sqrt{2.16}=1.470, \] \[ d_{SE}(C)=\sqrt{0.1^2+0.9^2+1.5^2}=\sqrt{3.07}=1.752, \quad d_{SE}(D)=\sqrt{0.8^2+0.4^2+1.0^2}=\sqrt{1.80}=1.342. \] The standardized ranking is \(A\prec D\prec B\prec C\), so standardized Euclidean distance selects Study \(A\): the opposite winner from the raw distance. The reversal is intuitive. Study \(A\)’s \(12\)-year age gap dominates the raw distance, but in standard-deviation units it is only \(1.2\) SD; Study \(C\)’s gaps are tiny in raw units (a \(1\)-year age gap, a \(6\)-unit biomarker gap) yet large when standardized (\(0.9\) SD in sex, \(1.5\) SD in biomarker), and the raw distance, blind to scale, mistakes \(C\) for the closest.
Mahalanobis (Equation 27.4). With \(\boldsymbol{\Sigma}^{-1}\) having entries \((\boldsymbol{\Sigma}^{-1})_{11}=\tfrac{1}{75}\), \((\boldsymbol{\Sigma}^{-1})_{22}=4\), \((\boldsymbol{\Sigma}^{-1})_{33}=\tfrac{1}{12}\), and \((\boldsymbol{\Sigma}^{-1})_{13}=-\tfrac{1}{60}\) (from inverting the age-biomarker block \(\bigl[\begin{smallmatrix}100&20\\20&16\end{smallmatrix}\bigr]\), of determinant \(1200\), and the scalar sex block), the quadratic form is \(d_M^2=\tfrac{1}{75}\delta_1^2+4\delta_2^2+\tfrac{1}{12}\delta_3^2 -\tfrac{1}{30}\delta_1\delta_3\). Evaluating, \[ d_M(A)=\sqrt{1.613}=1.270, \quad d_M(B)=\sqrt{1.800}=1.342, \quad d_M(C)=\sqrt{3.623}=1.904, \quad d_M(D)=\sqrt{1.280}=1.131. \] The Mahalanobis ranking is \(D\prec A\prec B\prec C\), so Mahalanobis distance selects Study \(D\): a third, distinct winner. The reason is the age-biomarker correlation. Study \(D\)’s gap \((8,0.2,4)\) moves age and biomarker up together, exactly the direction the index covariance expects, so it is “cheap” in Mahalanobis geometry; standardized Euclidean distance, ignoring the correlation, charges \(D\) for both moves separately and ranks it behind \(A\).
Mean-SMD (Equation 27.8). Averaging the absolute standardized gaps, \(d_{\mathrm{SMD}}(A)=1.55/3=0.517\), \(d_{\mathrm{SMD}}(B)=2.40/3=0.800\), \(d_{\mathrm{SMD}}(C)=2.50/3=0.833\), \(d_{\mathrm{SMD}}(D)=2.20/3=0.733\), giving the ranking \(A\prec D\prec B\prec C\), the same as standardized Euclidean, as the \(\ell_1\) and \(\ell_2\) norms of a common standardized vector must often agree.
The four criteria are collected below; each column’s minimum is the selected study.
| Study | raw \(d_E\) | std \(d_{SE}\) | mean-SMD | Mahalanobis \(d_M\) |
|---|---|---|---|---|
| \(A\) | \(12.04\) | \(\mathbf{1.230}\) | \(\mathbf{0.517}\) | \(1.270\) |
| \(B\) | \(7.69\) | \(1.470\) | \(0.800\) | \(1.342\) |
| \(C\) | \(\mathbf{6.10}\) | \(1.752\) | \(0.833\) | \(1.904\) |
| \(D\) | \(8.95\) | \(1.342\) | \(0.733\) | \(\mathbf{1.131}\) |
Which winner is right? The bias bound decides. Suppose, as a sensitivity input, the unanchored conditional contrast is taken to be affine, \(\tau(\mathbf{x})=\alpha+\boldsymbol{\beta}^\top\mathbf{x}\), with \(\boldsymbol{\beta}=(0.1,\;2,\;0.5)\), so that \(S\boldsymbol{\beta}=(1,1,2)\) and \(\|S\boldsymbol{\beta}\|_2=\sqrt6\). By Theorem 27.3 the exact transport bias is \(B_j=\boldsymbol{\beta}^\top\boldsymbol{\delta}_j=(S\boldsymbol{\beta})^\top\mathbf{z}_j=z_{j1}+z_{j2}+ 2z_{j3}\): \[ B_A=1.8,\qquad B_D=3.2,\qquad B_B=3.6,\qquad B_C=4.0 . \] The exact-bias ranking \(A\prec D\prec B\prec C\) coincides with the standardized Euclidean and mean-SMD rankings, and the standardized winner \(A\) is indeed the minimum-bias comparator. The raw Euclidean winner \(C\) has the largest bias of the four, \(4.0\); selecting by raw distance would have chosen the worst comparator, the concrete reversal promised by Proposition 27.2. The Mahalanobis winner \(D\) has bias \(3.2\), larger than \(A\)’s, because the contrast direction \(\boldsymbol{\beta}\) is not aligned with the covariance geometry that Mahalanobis distance rewards.
The Cauchy-Schwarz bounds and their equality cases are worth seeing. The standardized bound \(\|S\boldsymbol{\beta}\|_2\,d_{SE}=\sqrt6\,d_{SE}\) gives, for Study \(B\), \[ \sqrt6\cdot\sqrt{2.16}=\sqrt{12.96}=3.600=|B_B|, \] an exact equality, because \(\mathbf{z}_B=(0.6,0.6,1.2)=0.6\,(1,1,2)=0.6\,S\boldsymbol{\beta}\) is parallel to \(S\boldsymbol{\beta}\), meeting the equality condition of Equation 27.13. The Mahalanobis bound \(\|\boldsymbol{\beta}\|_{\boldsymbol{\Sigma}}\,d_M=\sqrt{8}\,d_M\) gives, for Study \(D\), \[ \sqrt8\cdot\sqrt{1.28}=\sqrt{10.24}=3.200=|B_D|, \] again exact, because \(\boldsymbol{\delta}_D=(8,0.2,4)=0.4\,(20,0.5,10)=0.4\,\boldsymbol{\Sigma} \boldsymbol{\beta}\) is parallel to \(\boldsymbol{\Sigma}\boldsymbol{\beta}\), meeting the equality condition of Equation 27.14 (one verifies \(\boldsymbol{\Sigma}\boldsymbol{\beta}=(20,0.5,10)\) and \(\boldsymbol{\beta}^\top\boldsymbol{\Sigma}\boldsymbol{\beta}=8\)). For every study both bounds dominate the exact bias, as they must. \(\blacktriangleleft\)
Remark. Reading the four rankings, and where \(\boldsymbol{\beta}\) comes from. The calculation compresses into one interpretation table.
| Criterion | Winner | Why it wins | What an HTA analyst should conclude |
|---|---|---|---|
| Raw Euclidean \(d_E\) | \(C\) | dominated by the large-unit biomarker gap; blind to scale | discard: the choice is an artifact of units (Proposition 27.2) |
| Standardized Euclidean \(d_{SE}\) | \(A\) | smallest gap in standard-deviation units, equal weight per covariate | defensible default; here it also happens to be the minimum-bias choice |
| Mahalanobis \(d_M\) | \(D\) | \(D\)’s age-biomarker move follows the index correlation, so it is “cheap” | defensible when the correlation is trusted, but here not the minimum-bias choice |
| Exact bias (\(\boldsymbol{\beta}\) known) | \(A\) | smallest \(|\boldsymbol{\beta}^\top\boldsymbol{\delta}_j|\) | the truth for this \(\boldsymbol{\beta}\), which is unknown at selection time |
The vector \(\boldsymbol{\beta}=(0.1,2,0.5)\) is a sensitivity input, not a quantity available when the comparator is chosen. In practice it would be borrowed from a published regression of the outcome on the covariates, elicited from clinical experts, or, most honestly, varied across a plausible range with the ranking recomputed at each setting; here it is a teaching device that lets us see, after the fact, which distance happened to align with the truth. Because it is unknown up front, the practical recommendation is the one the table’s last two columns dramatize: report the ranking under several distances, flag any disagreement as a signal that the choice is geometry-dependent, and carry the ambiguity into the sensitivity analysis rather than trusting a single number. \(\blacktriangleleft\)
Remark. The lesson of the example. Three reputable distances selected three different comparators. Standardized Euclidean distance happened to rank by the true bias here because its implicit geometry, equal weight per standardized covariate, was closer to the contrast direction \(S\boldsymbol{\beta}=(1,1,2)\) than the Mahalanobis geometry was. Had \(\boldsymbol{\beta}\) pointed along the high-variance axis of \(\boldsymbol{\Sigma}\), the Mahalanobis ranking would have been the faithful one. Since \(\boldsymbol{\beta}\) is unknown at selection time, no distance is guaranteed to rank by bias; this is Proposition 27.4 made arithmetic. The defensible practice is to report the ranking under several distances, flag disagreement as a signal that the choice is geometry-dependent, and carry the ambiguity into the sensitivity analysis rather than trusting a single number. \(\blacktriangleleft\)
27.11 Connection to the Coq formalization
Aggregate-level distance matching is a design heuristic, a rule for choosing which comparator to adjust toward, and unlike the estimators of Parts IV and V it makes no claim of unbiasedness that a proof assistant could certify. Its absence from the companion Coq development is therefore deliberate, not an item left on a backlog: there is no theories/DistanceMatching.v because there is no correctness theorem about the selection rule itself to mechanize. What the chapter does prove is elementary linear algebra and analysis, and every piece already lives in the development in another guise. The whitening identity Equation 27.5 is a one-line consequence of \((BB^\top)^{-1}=B^{-\top}B^{-1}\) and the existence of a matrix square root, the latter being the symmetric-square-root construction underlying Theorem 4.9. The linear bias bounds Equation 27.13 and Equation 27.14 are the Cauchy-Schwarz inequality, the same inequality the development formalizes for the effective-sample-size bound: the machine-checked Theorem ESS_upper_bound of theories/MAIC.v (tagged on Theorem 18.6) is built from the elementary \(2xy\le x^2+y^2\) assembled by induction into a Cauchy-Schwarz statement, and a future DistanceMatching.v could reuse that lemma verbatim to certify Theorem 27.3. Schematically, the single statement such a file would prove is one Cauchy-Schwarz application,
(* Linear bias bound, schematic form of Theorem distance-linear-bias-bound. *)
(* |beta . delta| <= ||beta||_Sigma * ||delta||_{inverse Sigma}. *)
Theorem distance_linear_bias_bound :
forall (Sigma : SPDMatrix) (beta delta : Vector),
Rabs (dot beta delta)
<= sqrt (quad_form Sigma beta) * sqrt (quad_form (inverse Sigma) delta).which is exactly Equation 27.14, the Mahalanobis factor \(\|\boldsymbol{\delta}_j\|_{\boldsymbol{\Sigma}^{-1}}\) written out. The non-identification proposition Proposition 27.4 is a pair of finite arithmetic counterexamples and would formalize directly. None of these results is currently mechanized, so this chapter carries no [machine-checked] tags; Chapter 30 documents the boundary of the formal development, of which distance matching lies outside.
27.12 Notes and references
Aggregate-level covariate-distance matching is an extension beyond the population-adjustment theory of Phillippo (2019), motivated by the disconnected-network and single-arm settings that the multilevel unanchored meta-regression of Chandler and Ishak (2026a) addresses and that the transportability reframing of Chandler and Ishak (2026b) places in estimand terms. The practical rule of selecting the candidate study minimizing a covariate-summary distance is described in the Bristol Compass blog on extending ML-NMR to disconnected networks, and the suite of distances studied here, standardized Euclidean (the default), Euclidean, Mahalanobis, Manhattan, Minkowski, Gower, and mean-absolute standardized-mean-difference, is the diagnostic toolkit implemented in the uitc R package; bibliographic entries for both the blog and the package are proposed in the chapter work log.
The distances themselves are classical. The Mahalanobis distance and its whitening interpretation (Theorem 27.1) originate with Mahalanobis; its reduction to standardized Euclidean distance for diagonal covariance and its full affine invariance (Theorem 27.2) are standard facts of multivariate analysis, here proved from Theorem 4.9. The Gower distance for mixed continuous and binary data is due to Gower. Minkowski’s inequality, invoked in Proposition 27.1 for the general \(\ell_q\) triangle inequality, is the standard named result; the orders \(q\in\{1,2,\infty\}\) used in this book are proved directly there from the coordinatewise triangle inequality and Cauchy-Schwarz. The mean-SMD diagnostic and the \(|z|>0.1\) rule of thumb are the covariate-balance conventions of the propensity-score literature, transposed from the individual level to study summaries.
The bias theory is the chapter’s own contribution to the catalogue. The linear bound Theorem 27.3 is the Cauchy-Schwarz inequality applied to the affine specialization of the unanchored transport bias of Theorem 17.2, and it makes precise the sense in which comparator selection bounds the residual bias that the consistency theorem assumes away. The Lipschitz bound Theorem 27.4 is the elementary coupling side of Kantorovich-Rubinstein duality; a reference for the duality and for the Wasserstein distance is the optimal-transport literature (Villani), proposed in the work log. The lower bound Proposition 27.3, that the mean gap never exceeds the Wasserstein distance, uses the vector Jensen inequality Theorem 4.6 and is what makes the non-identification of Proposition 27.4 unavoidable rather than accidental: mean distance is a floor on distributional distance, never a ceiling. The reader who wants the residual and unmeasured-confounding part of the problem, which distance matching cannot touch, should continue to the quantitative bias analysis of Chapter 28, where the E-value and tipping-point methods quantify exactly the bias that a zero distance fails to rule out.
This chapter corresponds to results O.1 (the comparator-selection problem), O.2 (Euclidean and standardized Euclidean distance), O.3 (the Mahalanobis whitening equivalence), O.4 (the Manhattan, Minkowski, Gower, and mean-SMD distances), O.5 (scale dependence and invariance), O.6 (the linear and Lipschitz bias bounds), and O.7 (the non-identification caveat) of the proof catalogue in proofs/O_distance_matching.md, expanding each catalogue sketch into a complete proof and adding the metric-property proposition Proposition 27.1 and the Wasserstein lower bound Proposition 27.3.
27.13 Exercises
An exercise marked \(\star\) has a full worked solution in Appendix F. The exercises are tagged by type; start with the warm-ups, which need only arithmetic, before the computation, proof, and challenge problems.
Exercise 27.1 (Warm-up, computation) Two distances by hand. The index study reports covariate means \(\bar{\mathbf{x}}_0=(50,\,20)\) on (age in years, biomarker in units) with reference scales \(s=(10,\,5)\). Two candidate comparators report \(\bar{\mathbf{x}}_1=(58,\,24)\) and \(\bar{\mathbf{x}}_2=(46,\,30)\). (a) Compute the raw Euclidean and the standardized Euclidean distance of each candidate to the index, and state which candidate each distance selects. (b) Using the index covariance \(\boldsymbol{\Sigma}=\bigl[\begin{smallmatrix}100&30\\30&25\end{smallmatrix}\bigr]\), compute the two Mahalanobis distances and say whether they agree with the standardized ranking. You can check your method against the fully worked Example 27.1.
Exercise 27.2 (Warm-up, interpretation) A comparator-selection memo. Using Example 27.2, write a short decision memo, one paragraph, for an HTA submission that (a) names the comparator you would carry forward and why, (b) reports the disagreement among the raw, standardized, and Mahalanobis rankings and explains it in one sentence a non-mathematician could follow, and (c) states what you would do if the Mahalanobis winner (Study \(D\)) had poor clinical comparability, say a different line of therapy, while the standardized winner (Study \(A\)) had better trial-design similarity. Which would you choose, and what does Section 27.3 say about the order in which comparability and distance are assessed?
Exercise 27.3 (Computation) Reference scale changes the winner. Recompute the standardized Euclidean ranking in Example 27.2 using comparator-reported standard deviations \(s'=(8,\,0.5,\,6)\) in place of the index scales \(s=(10,\,0.5,\,4)\). Report the new \(d_{SE}\) for all four studies and the new winner, and explain why Theorem 27.2 does not apply (the change is not a coherent rescaling of a single coordinate system but a change of reference scales alone). Comment on which reference scale is preferable when index IPD is available.
Exercise 27.4 (Computation) Worst-covariate selection. For the four studies of Example 27.2 compute the Chebyshev distance \(d_\infty=\max_\ell|z_{j\ell}|\) and the Minkowski distances \(d_3\) and \(d_4\). Show that the ranking under \(d_q\) moves toward the Chebyshev ranking as \(q\) grows, and identify the study whose selection is most sensitive to the choice of \(q\). Interpret the result for a decision-maker who regards a single badly mismatched covariate as disqualifying.
Exercise 27.5 (Proof) The triangle inequality at the boundary. Prove that \(d_q\) in Equation 27.7 is a metric for every \(q\in[1,\infty]\) by establishing Minkowski’s inequality \(\|\mathbf{u}+\mathbf{w}\|_q\le
\|\mathbf{u}\|_q+\|\mathbf{w}\|_q\) from Hölder’s inequality, and show by the explicit three-point configuration of Proposition 27.1 that the triangle inequality fails for every \(q\in(0,1)\). Conclude that the uitc restriction \(q\ge1\) is exactly the range in which the Minkowski “distance” deserves the name.
Exercise 27.6 (\(\star\), proof) Full affine invariance of the Mahalanobis distance. Prove that if the covariates are reparameterized by any invertible affine map \(\mathbf{x}\mapsto A\mathbf{x}+\mathbf{c}\) and the covariance is transformed accordingly to \(A\boldsymbol{\Sigma}A^\top\), then the Mahalanobis distance is unchanged, whereas the standardized Euclidean distance is invariant only for diagonal \(A\). Exhibit a nondiagonal \(2\times2\) invertible \(A\) and a pair of candidates whose standardized Euclidean ranking is reversed by the reparameterization while their Mahalanobis ranking is preserved. Explain what this says about the sense in which Mahalanobis distance is the “coordinate-free” member of the family.
Exercise 27.7 (\(\star\), challenge) Matching means and variances is still not enough. Strengthen Proposition 27.4 by constructing two scalar covariate distributions with identical mean and identical variance, hence zero distance under any diagnostic built from the first two moments, whose target effects differ under a nonlinear contrast. (Hint: let \(\mathcal{P}_0\) put mass \(\tfrac12\) at each of \(\pm1\), and let \(\mathcal{P}_1\) put mass \(\tfrac34\) at \(0\) and mass \(\tfrac18\) at each of \(\pm2\); verify both have mean \(0\) and variance \(1\), then evaluate the quartic contrast \(\tau(x)=x^4\).) Report the two target effects, the resulting bias, and the implication for MAIC second-moment balancing: balancing through any fixed finite set of moments leaves a residual bias for sufficiently nonlinear contrasts.
Exercise 27.8 (Computation) Verifying the Lipschitz bound. For the two distributions of Proposition 27.4 Part 1, \(\mathcal{P}_0=\delta_0\) and \(\mathcal{P}_1=\tfrac12\delta_{-1}+\tfrac12\delta_{+1}\), compute the Wasserstein-1 distance \(W_\rho(\mathcal{P}_1,\mathcal{P}_0)\) under the Euclidean ground metric by exhibiting the optimal coupling, confirm the mean-distance lower bound \(d_E=0\le W_\rho\) of Proposition 27.3, and verify the Lipschitz bias bound Equation 27.15 for the contrast \(\tau(x)=x^2\) on the support \(\{-1,0,1\}\) with its Lipschitz constant \(L=2\). State why the bound is not tight here and which feature of \(\tau\) would make it tight.
Exercise 27.9 (Interpretation) Distance against effective sample size. Argue that, all else equal, a comparator with a larger standardized Euclidean distance from the index population requires more aggressive reweighting in a subsequent unanchored MAIC and hence a lower effective sample size. Using Theorem 18.6 and the variance-inflation factor \(n/\mathrm{ESS}\), explain why comparator selection trades bias (bounded by the distance under Theorem 27.3) against variance (governed by the ESS), and sketch how this trade-off would enter a mean-squared-error criterion for choosing among candidates of comparable distance. Relate the trade-off to the simulation-study estimands of Chapter 29.