14  Indirect Comparison and Bucher’s Theorem

14.1 Introduction

A health technology assessment routinely faces a question that the available trials never asked. A regulator must decide whether a new drug \(A\) is better than an established drug \(B\), but the two have never been compared head to head. To keep a concrete picture in view throughout the chapter, let \(A\) be a new statin and \(B\) an established statin, each tested in its own randomized trial against the same comparator \(C\) (a placebo or standard of care), with an unfavorable cardiovascular event as the outcome. What does exist is a trial of \(A\) against \(C\), and a separate trial of \(B\) against the same comparator \(C\). The clinical question is about \(A\) versus \(B\); the evidence speaks only about \(A\) versus \(C\) and \(B\) versus \(C\). The task of this chapter is to make precise the simplest device that bridges the gap: the anchored indirect comparison, and the theorem of Bucher et al. (1997) that governs it.

The idea is disarmingly simple. If the effect of \(A\) relative to \(C\) is \(d_{AC}\) and the effect of \(B\) relative to \(C\) is \(d_{BC}\), both measured on a scale where contrasts add, then the effect of \(A\) relative to \(B\) ought to be the difference \(d_{AC}-d_{BC}\), because the common comparator \(C\) cancels. Replacing the unknown true effects by their trial estimates gives the Bucher estimator \(\hat d_{AB}=\hat d_{AC}-\hat d_{BC}\). The algebra is the algebra of a triangle. What makes the chapter a piece of mathematics rather than a piece of arithmetic is the question of when the algebra is licensed. The estimator is unbiased only if the true effect of \(A\) versus \(C\) is the same quantity in the \(A\) versus \(C\) trial as it would be in the (hypothetical) population where we want to report \(A\) versus \(B\), and likewise for \(B\) versus \(C\). That requirement is the transitivity assumption, equivalently the constancy of relative effects. It is the load-bearing hypothesis of every method in this book, and the rest of the book is, in a precise sense, the study of what to do when it fails.

We prove four things in full. First, that the relative effects on a fixed scale obey an antisymmetry law and an additivity law within any single population, so that \(d_{AC}-d_{BC}=d_{AB}\) is a tautology inside one population (Lemma 14.1). Second, that under transitivity the same identity holds across the three trial populations, which is the content of constancy (Proposition 14.2). Third, that the Bucher estimator is then unbiased for the cross-population contrast (Theorem 14.1), with variance equal to the sum of the two direct variances under independence (Theorem 14.2). Fourth, that the common comparator is exactly what cancels the study-specific baselines, so that the anchored comparison is protected against a bias that destroys the naive single-arm comparison (Proposition 14.4, Proposition 14.5). A fully worked example on the log odds-ratio scale, computed from raw two-by-two tables, closes the development.

Throughout we use the relative-effect notation of Notation: \(d_{ab(\mathcal P)}\) is the marginal effect of treatment \(b\) versus treatment \(a\) in population \(\mathcal P\), on a stated scale, and \(\hat d_{ab(j)}\) is its estimator from study \(j\). The potential-outcome machinery of Chapter 9 (Definition 9.1, Definition 9.4, Theorem 9.3) supplies the meaning of a single trial’s relative effect; the inference tools of Chapter 5 (Theorem 5.3, Theorem 5.6) supply its standard error. This chapter corresponds to results D.1 to D.4 of the proof catalogue.

14.2 A running example and a map of the argument

Before any formalism, fix the picture we will return to at every turn. Two randomized trials share the comparator \(C\) (placebo):

  • Trial \(AC\) compares the new statin \(A\) with placebo \(C\) and reports an odds ratio of the cardiovascular event of \(2\) for \(C\) versus \(A\); on the log scale, \(\hat d_{AC}=\log 2\approx 0.69\). In words, placebo patients had twice the odds of the event, so \(A\) is protective.
  • Trial \(BC\) compares the established statin \(B\) with placebo \(C\) and reports an odds ratio of \(1.5\) for \(C\) versus \(B\); on the log scale, \(\hat d_{BC}=\log 1.5\approx 0.41\). Placebo patients had one and a half times the odds, so \(B\) is protective too, but less so.

We will compute the full example from raw counts in Section 14.11; the two trials there produce exactly these numbers.

What “population” means here. By a population we do not mean the specific people enrolled, but the distribution of their baseline characteristics: the mix of ages, disease severities, and risk factors a trial would draw from. Formally it is a probability distribution over the covariate space \(\mathcal X\). The two trials draw from two different such distributions, \(\mathcal P_{AC}\) and \(\mathcal P_{BC}\), and we will also speak of a third, \(\mathcal P_{AB}\): the population in which we actually want the \(A\)-versus-\(B\) answer, typically the patients who would receive the drug in routine care. No trial was ever run in \(\mathcal P_{AB}\) for \(A\) against \(B\); that absence is the entire reason an indirect comparison is needed.

Two recurring words. A contrast is a difference of two positions on the chosen scale, such as \(\eta_B-\eta_A\); it is the mathematical form of the phrase “relative effect.” The link scale is the transformed scale (log-odds for a probability, log for a rate or hazard, or the untransformed mean for a continuous outcome) on which we agree to do all subtraction, chosen precisely so that contrasts add along a path.

The available and missing comparisons form a triangle:

              A  - - - - - - - - -  B        target contrast d_AB
               \                   /            (dashed: NO A-vs-B trial)
                \                 /
          d_AC   \               /   d_BC
       (trial AC) \             / (trial BC)
                   \           /
                    \         /
                     \       /
                        C                     common comparator (the anchor)

The two solid legs \(A\!-\!C\) and \(B\!-\!C\) are the trials we have; the dashed edge \(A\!-\!B\) is the comparison we want. Bucher walks the triangle: go from \(A\) down to \(C\), then from \(C\) up to \(B\), and the detour through \(C\) recovers the missing edge.

Remark. The object, stated once, up front. Everything below serves one formula. Writing \(\hat d_{AC}\) and \(\hat d_{BC}\) for the two trials’ estimated log-scale effects against the shared comparator \(C\), the Bucher estimator of the \(A\)-versus-\(B\) effect is the single subtraction \[ \hat d_{AB}=\hat d_{AC}-\hat d_{BC}, \] with standard error \(\sqrt{\mathrm{SE}_{AC}^2+\mathrm{SE}_{BC}^2}\) when the two trials are independent. In the running example \(\hat d_{AC}=\log 2\) and \(\hat d_{BC}=\log 1.5\), so \(\hat d_{AB}=\log 2-\log 1.5=\log\tfrac43\approx 0.29\): the indirect odds ratio of the event for \(B\) versus \(A\) is \(\tfrac43\). The rest of the chapter answers one question: when is this subtraction allowed? \(\blacktriangleleft\)

Orientation, once and for all. The single most common source of confusion in indirect comparison is sign. We follow Notation: \(d_{ab}=\eta_b-\eta_a\) places the reference treatment first and the active treatment second. Because the outcome is unfavorable, a positive contrast flags higher, hence worse, odds for the treatment in the second slot. The following table translates between the applied phrasing and this chapter’s symbols; keep it at hand.

Applied phrasing This chapter’s symbol Equals A positive value means
effect of \(C\) relative to \(A\) \(d_{AC}\) \(\eta_C-\eta_A\) \(C\) has higher (worse) event odds than \(A\), so \(A\) is protective
“\(A\) versus \(C\)” (common applied phrasing) \(-d_{AC}\) \(\eta_A-\eta_C\) \(A\) has higher (worse) event odds than \(C\)
effect of \(B\) relative to \(A\) (the target) \(d_{AB}\) \(\eta_B-\eta_A\) \(B\) has higher (worse) event odds than \(A\), so \(A\) is the better statin
the Bucher combination \(d_{AB}=d_{AC}-d_{BC}\) \((\eta_C-\eta_A)-(\eta_C-\eta_B)\) the shared \(\eta_C\) cancels

In the running example \(\hat d_{AC}=\log 2>0\) says placebo is worse than \(A\); \(\hat d_{AB}=\log\tfrac43>0\) says \(B\) has slightly higher event odds than \(A\), so on this evidence \(A\) is the better statin.

Map of the argument. The chapter has one destination (the subtraction is licensed and its variance is known) reached through five results. This table is a scaffold; each entry is proved below.

Result Plain statement Its job in the indirect comparison
Lemma 14.1 contrasts reverse sign and add along a path makes \(d_{AC}-d_{BC}=d_{AB}\) arithmetic inside one population
Definition 14.3, Proposition 14.2 the relevant effects are the same in every trial population lets that subtraction carry across the three populations
Theorem 14.1 the Bucher estimator has the right expectation the indirect estimate is on target
Theorem 14.2 its variance is the sum of the two direct variances indirect evidence is less precise than direct
Proposition 14.4, Proposition 14.5 the anchor cancels the study baselines anchored comparison beats the naive single-arm one

14.3 The relative effect and its scale

The goal of this section is modest but easy to lose sight of: we want to put every treatment arm on a single additive scale, a number \(\eta_k\) we can subtract, so that “effect” becomes “difference.” Everything in the section is machinery for doing that honestly when the outcome is binary or a survival time rather than a plain average.

Fix three treatments \(A\), \(B\), \(C\) and a strictly increasing, differentiable link function \(g\) (Definition 7.2) that maps a mean outcome to a “linear-predictor” scale on which effects are to be reported: the logit for binary outcomes (giving log odds ratios), the log for rates and hazards (giving log rate or hazard ratios), and the identity for continuous outcomes (giving mean differences). For a treatment \(k\in\{A,B,C\}\) write \(\mu_k(x)=\mathbb E[Y(k)\mid X=x]\) for the conditional mean of the potential outcome under \(k\) (Definition 9.1), as a function of the baseline covariates \(x\in\mathcal X\).

Remark. Reader translation. The phrase “link-scale position” is jargon coined here; take it to mean “the arm’s adjusted location on the report scale, before we subtract two arms.” The object \(\eta_{k(\mathcal P)}\) is a two-step recipe. Step one: in population \(\mathcal P\), work out the average outcome you would see if everyone took treatment \(k\); this is the inner integral \(\bar\mu_{k(\mathcal P)}\), an ordinary population mean. Step two: push that single average through the link \(g\) to land on the additive scale (the logit of a probability, the log of a rate). The relative effect \(d_{ab(\mathcal P)}\) is then just the gap between two such numbers. One notational caution: \(\eta_{k(\mathcal P)}\) is a population-level (marginal) quantity indexed by a treatment and a population; it is not the individual-level linear predictor \(\eta_{ijk}\) of Notation, which carries a person, study, and arm subscript. Same Greek letter, a coarser object. \(\blacktriangleleft\)

Two further points of interpretation deserve emphasis before we use Equation 14.2.

Remark. Marginal, not conditional. Equation Equation 14.2 marginalizes the conditional means first (the inner integral) and applies the link second. This is the effect a trial reports when it compares arm-level summaries: the marginal odds ratio, the marginal hazard ratio. It is in general not equal to the conditional effect \(g(\mu_b(x))-g(\mu_a(x))\) averaged over \(x\), because a nonlinear \(g\) does not commute with averaging. Here is the phenomenon in one line of arithmetic: if half a population has event probability \(0.9\) and half has \(0.1\), the average probability is \(0.5\) with logit \(0\), whereas the average of the two logits is \(\tfrac12\operatorname{logit}(0.9)+\tfrac12\operatorname{logit}(0.1)=0\) only by the symmetry of this toy case; shift the split to \(0.8\) and \(0.1\) and \(g\) of the average (\(\operatorname{logit}0.45=-0.20\)) no longer equals the average of the \(g\)’s (\(\tfrac12\operatorname{logit}0.8+\tfrac12\operatorname{logit}0.1\approx-0.40\)). For the identity link the two always coincide; for the logit and log links they differ, the phenomenon of non-collapsibility (Definition 7.7, Definition 11.6, Theorem 11.2). The Bucher construction operates on whatever marginal effects the trials report, and the transitivity assumption below is a statement about those marginal effects. We return to the interaction between non-collapsibility and indirect comparison in Chapter 16. \(\blacktriangleleft\)

Remark. Why a scale on which contrasts add. The whole construction lives on the link scale precisely because on that scale a relative effect is a difference of two positions, and differences subtract cleanly. On the natural outcome scale the same information is multiplicative: an odds ratio of \(A\) versus \(C\) times the odds ratio of \(C\) versus \(B\) gives the odds ratio of \(A\) versus \(B\). Taking logarithms turns the product into the sum Equation 14.2. The risk difference is the one common effect measure that adds on the identity scale directly. We make the additive structure a lemma in Section 14.4. \(\blacktriangleleft\)

14.3.1 The single trial as a source of an unbiased contrast

A randomized trial comparing two of the treatments in a population \(\mathcal P\) delivers an (approximately) unbiased estimator of the corresponding marginal relative effect Equation 14.2. We record this as the input to everything that follows; the proof is the identification argument of Chapter 9. The reader who wants only the upshot may read the proposition, skip its proof, and note the plain-language remark that follows it.

Proposition 14.1 (A randomized trial estimates its own marginal contrast) Let study \(j\) randomize patients drawn from population \(\mathcal P_j\) between arms \(a\) and \(b\), with randomization probability \(P(T=b\mid X=x)=\pi\in(0,1)\) independent of \(x\). Then the arms satisfy ignorability and positivity (Definition 9.7, Definition 9.8) by design, the arm means identify the marginal potential-outcome means \(\bar\mu_{a(\mathcal P_j)}\) and \(\bar\mu_{b(\mathcal P_j)}\) (Theorem 9.3), and any estimator \(\hat d_{ab(j)}\) formed by applying \(g\) to consistent arm-mean estimators is consistent for \(d_{ab(\mathcal P_j)}\), with \(\mathbb E[\hat d_{ab(j)}]\to d_{ab(\mathcal P_j)}\).

Proof. Randomization makes the treatment indicator \(T\) independent of the potential outcomes \((Y(a),Y(b))\) and of the covariates \(X\); in particular \(T\perp (Y(a),Y(b))\mid X\) holds with the conditioning on \(X\) vacuous, which is ignorability (Definition 9.7), and \(\pi\in(0,1)\) is positivity (Definition 9.8). By the identification theorem (Theorem 9.3) applied within \(\mathcal P_j\), the observed mean in arm \(a\), namely \(\mathbb E[Y\mid T=a]\), equals \(\mathbb E[Y(a)]=\bar\mu_{a(\mathcal P_j)}\), and likewise for \(b\). An estimator that plugs consistent arm-mean estimators \(\widehat{\bar\mu}_{a(j)},\widehat{\bar\mu}_{b(j)}\) into \(g\) and differences them is, by the continuous mapping theorem and the differentiability of \(g\), consistent for \(g(\bar\mu_{b(\mathcal P_j)})-g(\bar\mu_{a(\mathcal P_j)})=d_{ab(\mathcal P_j)}\); its bias vanishes in the limit, and to first order in the delta-method expansion of Section 14.10 it is unbiased. \(\square\)

Remark. What the proof said, in plain words. Ignorability means that, thanks to randomization, which treatment a patient received tells you nothing about how they would have fared under either treatment; the two arms are exchangeable. Because the coin flip is independent of everything about the patient, the conditioning on \(X\) is “vacuous,” that is, unnecessary. Positivity is just \(\pi\in(0,1)\): everyone had a genuine chance of either arm. Together these are what a randomized trial buys you by design, so the arm averages estimate the population averages you would have seen if everyone had taken that treatment. \(\blacktriangleleft\)

Remark. Consistent, and unbiased to first order. A subtlety worth stating plainly: the empirical log odds-ratio is not exactly unbiased in a finite sample. What Proposition 14.1 delivers is (i) consistency, meaning that as the arm sizes grow, \(\hat d_{ab(j)}\) converges in probability to the true contrast, and (ii) first-order (asymptotic) unbiasedness, meaning the leading term of its bias vanishes as the sample grows. Every use of the word “unbiased” in this chapter, including in the statement of Bucher’s theorem below, is meant in this large-sample sense, which is the scale on which all Bucher calculations operate. For trials of realistic size the distinction is harmless; we flag it only so the word is not over-read. A separate warning about the word consistent: in Proposition 14.1 it is the statistical sense (an estimator converging to its target); in Equation 14.6 below it is the network-meta-analysis sense (direct and indirect estimates agreeing). The two senses are unrelated; context disambiguates. \(\blacktriangleleft\)

The phrase “unbiased estimator of \(d_{AC}\)” in Bucher’s theorem is shorthand for the conclusion of Proposition 14.1: each trial estimates its own population’s marginal contrast without (first-order) bias. The entire subtlety of indirect comparison is that the trial estimates the contrast in its population \(\mathcal P_j\), whereas we wish to combine the trials into a statement about a single contrast. Bridging the populations is the work of the transitivity assumption.

14.4 The algebra of contrasts

Inside a single population the relative effects Equation 14.2 are differences of three numbers \(\eta_{A},\eta_{B},\eta_{C}\) (suppressing the population subscript), and therefore satisfy two elementary laws. They are the entire algebraic content of Bucher’s identity; the statistics enters only through Proposition 14.1 and the variance bookkeeping of Section 14.8.

It is fair to ask why a book states a theorem for something so elementary. The reason is that this trivial algebra is the entire mathematical skeleton of the Bucher estimator: the variance result (Theorem 14.2), the cancellation of baselines (Proposition 14.4), and the unbiasedness theorem (Theorem 14.1) all reduce to the two laws below applied on the right scale. Naming them now lets those later proofs cite them by name rather than re-derive them.

Lemma 14.1 (Antisymmetry and additivity of contrasts) Fix a population \(\mathcal P\) and let \(d_{ab}=\eta_{b}-\eta_{a}\) for \(a,b\in\{A,B,C\}\) as in Equation 14.2. Then for all treatments \(a,b,c\):

  1. Antisymmetry (reversal): \(d_{ab}=-d_{ba}\), and in particular \(d_{aa}=0\).

  2. Additivity (the triangle identity): \[ d_{ac} \;=\; d_{ab}+d_{bc}, \qquad\text{equivalently}\qquad d_{ab}=d_{ac}-d_{bc}. \tag{14.3}\]

Proof. Both laws are immediate from the representation \(d_{ab}=\eta_b-\eta_a\). For antisymmetry, \(d_{ab}=\eta_b-\eta_a=-(\eta_a-\eta_b)=-d_{ba}\), and setting \(b=a\) gives \(d_{aa}=\eta_a-\eta_a=0\). For additivity, \[ d_{ab}+d_{bc}=(\eta_b-\eta_a)+(\eta_c-\eta_b)=\eta_c-\eta_a=d_{ac}. \] Adding \(-d_{bc}\) to both sides of \(d_{ac}=d_{ab}+d_{bc}\) and using antisymmetry rearranges this to \(d_{ab}=d_{ac}-d_{bc}\). The middle term \(\eta_b\) cancels because it enters \(d_{ab}\) with a plus sign and \(d_{bc}\) with a minus sign; this cancellation of the intermediate treatment’s link-scale position is the algebraic seed of the indirect comparison. \(\square\)

Remark. Why this matters. Read the triangle identity with the roles of the running example: setting \(a=A,b=C,c=B\) in \(d_{ab}=d_{ac}-d_{bc}\) gives exactly \(d_{AB}=d_{AC}-d_{BC}\), the Bucher combination. The lemma says this is a tautology whenever all three contrasts live in one population; the whole difficulty, taken up next, is that our three contrasts come from three different populations. \(\blacktriangleleft\)

Remark. A note on index order and the clinical convention. With the convention of Notation, \(d_{ab}=\eta_b-\eta_a\) places the reference treatment first and the active treatment second. The common comparator of an anchored comparison is \(C\), so the two trial contrasts that involve \(C\) are \(d_{AC}=\eta_C-\eta_A\) and \(d_{BC}=\eta_C-\eta_B\), in which \(C\) occupies the second slot; the target contrast between the two new treatments is \(d_{AB}=\eta_B-\eta_A\). Much applied writing uses the opposite order, “\(A\) versus \(C\),” and reports \(-d_{AC}=\eta_A-\eta_C\). By the antisymmetry law Lemma 14.1(1) the two orderings differ only by an overall sign and carry identical information; the indirect identity \(d_{AB}=d_{AC}-d_{BC}\) holds verbatim in either bookkeeping, because reversing all three contrasts negates both sides. The practical rule to remember, tabulated in Section 14.2: with an unfavorable outcome, a positive \(d_{ab}\) means the second-slot treatment has the higher (worse) odds. We keep the Notation order and let the reader translate by the reversal law where a “\(A\) versus \(C\)” phrasing is more natural. \(\blacktriangleleft\)

The additivity law Equation 14.3 is a tautology when all three contrasts live in one population, where the \(\eta\)’s are three fixed numbers. The trials, however, live in three different populations \(\mathcal P_{AB}\), \(\mathcal P_{AC}\), \(\mathcal P_{BC}\), each with its own \(\eta\)’s. To carry Equation 14.3 across populations we need exactly the assumption that the relevant contrasts do not depend on which population we evaluate them in.

14.5 Constancy of relative effects and transitivity

Before the formal definitions, the assumption in plain terms, because it is the most important idea in the chapter. The two trials were run in different populations. Anchoring (Section Section 14.9) will handle any difference in baseline risk for free. What it cannot handle is a difference in the size of a treatment effect between the populations. A covariate that changes the size of a treatment effect is an effect modifier (Definition 11.3); a covariate that shifts outcomes but leaves treatment effects unchanged is a prognostic variable (Definition 11.5). Transitivity is the assumption that the trial populations are alike in their effect modifiers, and it allows them to differ freely in everything else.

Question about the trial populations Answer under transitivity
What must be similar? The distribution of any effect modifier of the relevant relative effects, that is, any covariate that changes the size of a treatment effect.
What is allowed to differ? Baseline risk and purely prognostic factors, which shift outcomes but not treatment effects; the anchor cancels these (Proposition 14.4).
What cannot be repaired by more data? Bias from an imbalance in effect modifiers. It is a bias, not a variance, so a larger trial does not shrink it.

Remark. A clinical instance, before the algebra. Return to the two statin trials. Suppose age modifies the effect of statin \(B\) versus placebo \(C\): the reduction in cardiovascular events is larger in older patients. If the \(BC\) trial enrolled a substantially younger population than the one in which we want to report \(A\) versus \(B\), then \(\hat d_{BC}\) estimates \(B\)’s effect in a younger population, and subtracting it from \(\hat d_{AC}\) silently mixes two different populations; transitivity fails. If instead age only predicts cardiovascular risk but does not change the size of the statin effect, then age is a prognostic variable, not an effect modifier: it shifts a study’s baseline but not its treatment effect, and anchoring absorbs the difference at no cost. Transitivity is thus a statement about effect modifiers, not about baseline comparability. \(\blacktriangleleft\)

Definition 14.2 (Constancy of relative effects) Let a network of studies be given, study \(j\) having population \(\mathcal P_j\) with covariate density \(f_j\). The marginal relative effect of \(b\) versus \(a\) in \(\mathcal P_j\) on the scale \(g\) is, as in Equation 14.2, \[ d_{ab(\mathcal P_j)}=g\!\left(\int \mu_b(x)\,f_j(x)\,dx\right)-g\!\left(\int \mu_a(x)\,f_j(x)\,dx\right). \] The constancy of relative effects assumption asserts that this quantity does not depend on the population: for every ordered pair of treatments \((a,b)\) and all studies \(j,j'\), \[ d_{ab(\mathcal P_j)} \;=\; d_{ab(\mathcal P_{j'})} \;=:\; d_{ab}. \tag{14.4}\] The common value \(d_{ab}\) is then a population-free network contrast. \(\blacktriangleleft\)

Remark. Reader translation. Constancy says every relative effect in the network is a single number that does not care which study you look at, so “the effect of \(b\) versus \(a\)” is well defined without naming a population. It is the general network assumption. Transitivity, defined next, is its restriction to the one triangle \(\{A,B,C\}\) we need for a single Bucher comparison: constancy for the whole network, transitivity for the triangle. \(\blacktriangleleft\)

Constancy Equation 14.4 is a property of every pair of treatments across every study. For a single indirect comparison we need only its restriction to the triangle \(\{A,B,C\}\) and the three populations in play. That restriction is the transitivity assumption.

Definition 14.3 (Transitivity (for the anchored triangle)) Let \(A\), \(B\), \(C\) be three treatments studied in populations \(\mathcal P_{AB}\), \(\mathcal P_{AC}\), \(\mathcal P_{BC}\) (the population of a hypothetical direct \(A\)-versus-\(B\) trial, and the populations of the available \(A\)-versus-\(C\) and \(B\)-versus-\(C\) trials). The transitivity assumption asserts that each of the three relative effects takes the same value in every one of the three populations: \[ d_{AB(\mathcal P_{AB})}=d_{AB(\mathcal P_{AC})}=d_{AB(\mathcal P_{BC})}=:d_{AB}, \tag{14.5}\] and likewise \(d_{AC(\mathcal P_{AB})}=d_{AC(\mathcal P_{AC})}=d_{AC(\mathcal P_{BC})}=:d_{AC}\) and \(d_{BC(\mathcal P_{AB})}=d_{BC(\mathcal P_{AC})}=d_{BC(\mathcal P_{BC})}=:d_{BC}\). Equivalently, transitivity is the restriction of constancy Equation 14.4 to the treatment triangle \(\{A,B,C\}\). \(\blacktriangleleft\)

[machine-checked: theories/Bucher.v, Definition transitivity]

Remark. Reader translation. Transitivity says: the number you would call “the effect of \(B\) versus \(C\)” is the same number whether you measure it in the \(A\)-vs-\(C\) trial’s population, the \(B\)-vs-\(C\) trial’s population, or the target population where you actually want the answer, and likewise for the other two contrasts. It does not require the populations to be alike in every way; it requires them to be alike in the ingredients that change treatment effects, the effect modifiers. It is the population-level cousin of “no unmeasured confounding”: ignorability (Definition 9.7) lets one study impersonate a randomized experiment, and transitivity lets three studies impersonate one connected experiment run in a single population. In practice a reviewer checks it not from the data but from the trial reports, by asking whether the trials enrolled comparable patients with respect to known effect modifiers of the outcome; it is a judgment about clinical similarity, informed by subject-matter knowledge, that the data alone cannot confirm. \(\blacktriangleleft\)

The next proposition makes the bridge precise: under transitivity the additivity law Equation 14.3, a tautology inside one population, becomes a genuine relation among the three well-defined network contrasts.

Proposition 14.2 (Transitivity yields the population consistency equation) Under transitivity (Definition 14.3), the network contrasts satisfy the consistency equation \[ d_{AB} \;=\; d_{AC}-d_{BC}. \tag{14.6}\] Conversely, if the relative effects in each population obey the within-population additivity law Equation 14.3 and the consistency equation Equation 14.6 holds for the common values, then the three contrasts are mutually consistent in the sense required by Definition 14.3.

[machine-checked: theories/Bucher.v, Lemma transitivity_subtract, Lemma subtract_transitivity]

Proof. By Definition 14.3 the three symbols \(d_{AB},d_{AC},d_{BC}\) are well defined, each being the common value of a relative effect across the three populations. Evaluate the within-population additivity law Lemma 14.1(2) in, say, the population \(\mathcal P_{AC}\): \[ d_{AB(\mathcal P_{AC})}=d_{AC(\mathcal P_{AC})}-d_{BC(\mathcal P_{AC})}. \] Transitivity replaces each population-specific effect on the right and the left by its common value, \(d_{AB(\mathcal P_{AC})}=d_{AB}\), \(d_{AC(\mathcal P_{AC})}=d_{AC}\), \(d_{BC(\mathcal P_{AC})}=d_{BC}\), giving \(d_{AB}=d_{AC}-d_{BC}\), which is Equation 14.6. The same conclusion follows from evaluating the additivity law in any of the three populations, since transitivity forces them to agree.

For the converse, suppose each population obeys Equation 14.3 and the common values obey Equation 14.6. Then in any population \(\mathcal P_j\) the additivity law reads \(d_{AB(\mathcal P_j)}=d_{AC(\mathcal P_j)}-d_{BC(\mathcal P_j)}\); if two of the three contrasts are constant across populations the third is forced to equal \(d_{AC}-d_{BC}=d_{AB}\), the common value, in every population, which is Definition 14.3. \(\square\)

Remark. Reading the two directions. The forward direction is the one an analyst uses; the table records both.

Direction Assume Conclude Who uses it
Forward transitivity (Definition 14.3) consistency equation Equation 14.6 Theorem 14.1, the whole method
Converse within-population additivity plus consistency of the common values transitivity bookkeeping only

The converse is a consistency check, not something a data analyst invokes: it says the three contrasts cannot be defined inconsistently. “If two of the three are constant across populations, the third is forced” because inside each population the three already satisfy \(d_{AB}=d_{AC}-d_{BC}\) by pure algebra (Lemma 14.1), so fixing any two of them across populations pins the third by subtraction. Which two does not matter; the algebra is symmetric. A first-time reader can take the forward direction and move on. \(\blacktriangleleft\)

Remark. The Coq development encodes transitivity in the equivalent additive form \(d_{AC}=d_{AB}+d_{BC}\) (Definition transitivity) and proves the two implications \(d_{AC}=d_{AB}+d_{BC}\Rightarrow d_{AB}=d_{AC}-d_{BC}\) (Lemma transitivity_subtract) and its converse (Lemma subtract_transitivity); these are precisely the two directions of Proposition 14.2, stripped to their field algebra. \(\blacktriangleleft\)

14.6 The anchored indirect comparison

We can now name the estimator and the contrasting naive alternative it improves upon.

Definition 14.4 (Anchored indirect comparison) Suppose study \(AC\) provides an estimator \(\hat d_{AC}\) of the \(A\)-versus-\(C\) network contrast and study \(BC\) provides an estimator \(\hat d_{BC}\) of the \(B\)-versus-\(C\) network contrast, the two studies sharing the common comparator \(C\). The anchored indirect comparison of \(A\) and \(B\), or Bucher estimator, is \[ \hat d_{AB}^{\,\mathrm{Bucher}} \;:=\; \hat d_{AC}-\hat d_{BC}. \tag{14.7}\] The comparison is called anchored because the within-study contrasts to the shared comparator \(C\) (the anchor) are differenced, rather than the bare arm-level outcomes. \(\blacktriangleleft\)

Remark. Reader translation and orientation. The estimator subtracts the two “versus \(C\)” log-effects. In the chapter’s orientation the result estimates \(d_{AB}=\eta_B-\eta_A\), so a positive Bucher estimate means \(B\) has the higher (with an unfavorable outcome, worse) odds, hence \(A\) is the better treatment. In the running example \(\hat d_{AB}^{\,\mathrm{Bucher}}=\log 2-\log 1.5=\log\tfrac43>0\): \(B\) carries slightly higher event odds than \(A\). Always restate the direction when reporting: “\(B\) versus \(A\), higher is worse.” \(\blacktriangleleft\)

We also name the wrong thing to do, because seeing why it is wrong is the fastest route to seeing why anchoring works. The following naive comparison is a foil: it is what you get if you ignore the common comparator and simply compare the two active arms across studies. It is used in practice only as a last resort, for disconnected networks (Chapter 26); here it is the cautionary contrast that makes the value of the anchor visible.

Definition 14.5 (Unanchored (naive single-arm) comparison) With the same two studies, the unanchored comparison discards the comparator arms and contrasts the two active arms directly across studies: \[ \hat d_{AB}^{\,\mathrm{naive}} \;:=\; \hat\eta_{A(\mathcal P_{AC})}-\hat\eta_{B(\mathcal P_{BC})}, \tag{14.8}\] where \(\hat\eta_{A(\mathcal P_{AC})}\) is the estimated link-scale outcome of arm \(A\) in study \(AC\), and \(\hat\eta_{B(\mathcal P_{BC})}\) that of arm \(B\) in study \(BC\). This comparison ignores the anchor \(C\) entirely. \(\blacktriangleleft\)

Remark. Reader translation. The unanchored estimator throws away the placebo arms and directly compares the new statin’s event rate in one trial to the established statin’s event rate in the other. That is exactly what you would do if you had two single-arm studies with no common comparator. The comparison looks tempting because it seems to use the arms you care about, but it silently assumes the two trials had the same baseline risk, which they almost never do; Proposition 14.5 makes the error precise and Example 14.1 shows it as a number. \(\blacktriangleleft\)

Section Section 14.9 shows that the unanchored comparison carries a bias equal to the difference of the two study baselines, a bias that the anchored comparison removes by construction. The distinction between anchored and unanchored comparison organizes the whole of population-adjusted indirect comparison: anchored methods (Chapters 17 to 20) repair residual bias in Equation 14.7, while unanchored methods (Chapter 26) confront the baseline bias of Equation 14.8 head on.

14.7 Bucher’s theorem: unbiasedness

The next two results look almost the same, and it helps to say up front how they differ. First we prove the deterministic identity: if the trials reported the true contrasts with no sampling error at all, the subtraction lands exactly on the target (Proposition 14.3). Then we reintroduce sampling error and show that the subtraction is still right on average (Theorem 14.1). The first is arithmetic; the second is arithmetic plus one line of expectation.

Proposition 14.3 (Bucher’s identity (exact estimates)) If \(\hat d_{AC}=d_{AC}\) and \(\hat d_{BC}=d_{BC}\) hold exactly and transitivity (Definition 14.3) holds, then \(\hat d_{AB}^{\,\mathrm{Bucher}}=d_{AB}\).

Proof. Substitute the exact values into the estimator Equation 14.7 and apply the consistency equation Equation 14.6 of Proposition 14.2: \[ \hat d_{AB}^{\,\mathrm{Bucher}}=\hat d_{AC}-\hat d_{BC}=d_{AC}-d_{BC}=d_{AB}. \qquad\square \]

[machine-checked: theories/Bucher.v, Theorem bucher_identity]

The stochastic version replaces exact equality by unbiasedness and the substitution by linearity of expectation. This is the theorem of Bucher et al. (1997).

Theorem 14.1 (Bucher’s theorem (unbiasedness)) Suppose two studies provide estimators of pairwise relative effects of the form \[ \hat d_{AC}=d_{AC(\mathcal P_{AC})}+\varepsilon_{AC},\qquad \hat d_{BC}=d_{BC(\mathcal P_{BC})}+\varepsilon_{BC}, \] with mean-zero errors \(\mathbb E[\varepsilon_{AC}]=\mathbb E[\varepsilon_{BC}]=0\), so that \(\hat d_{AC}\) and \(\hat d_{BC}\) are unbiased for the relative effects in their own trial populations (Proposition 14.1). Assume transitivity (Definition 14.3). Then the Bucher estimator Equation 14.7 is unbiased for the network contrast \(d_{AB}\): \[ \mathbb E\!\left[\hat d_{AB}^{\,\mathrm{Bucher}}\right]=d_{AB}. \]

[machine-checked: theories/Bucher.v, Theorem bucher_unbiased]

Remark. Why this matters. This is the license for the whole method: provided transitivity holds, the one-line subtraction \(\hat d_{AC}-\hat d_{BC}\) targets the right quantity, so an indirect estimate is a legitimate answer to the \(A\)-versus-\(B\) question. The theorem localizes all the risk in a single assumption, which the remainder of the chapter, and much of the book, learns to inspect. \(\blacktriangleleft\)

Proof. Linearity of expectation applied to Equation 14.7 gives \[ \mathbb E\!\left[\hat d_{AB}^{\,\mathrm{Bucher}}\right] =\mathbb E[\hat d_{AC}]-\mathbb E[\hat d_{BC}]. \] By the unbiasedness hypothesis the two expectations equal the trial-population relative effects, \[ \mathbb E[\hat d_{AC}]=d_{AC(\mathcal P_{AC})},\qquad \mathbb E[\hat d_{BC}]=d_{BC(\mathcal P_{BC})}. \] Transitivity (Definition 14.3) identifies each trial-population effect with the corresponding network contrast, \(d_{AC(\mathcal P_{AC})}=d_{AC}\) and \(d_{BC(\mathcal P_{BC})}=d_{BC}\), so \[ \mathbb E\!\left[\hat d_{AB}^{\,\mathrm{Bucher}}\right]=d_{AC}-d_{BC}. \] Finally the consistency equation Equation 14.6, which Proposition 14.2 derives from transitivity, gives \(d_{AC}-d_{BC}=d_{AB}\). Chaining the three displays yields \(\mathbb E[\hat d_{AB}^{\,\mathrm{Bucher}}]=d_{AB}\). \(\square\)

Remark. Where each hypothesis is spent. The proof uses each assumption exactly once and at a distinct step. Unbiasedness of the trial estimators converts expectations into trial-population effects; this is a property of randomization within each trial and uses no cross-trial information. Transitivity converts trial-population effects into network contrasts; this is the only cross-population assumption and is the one that can fail. The consistency equation converts the difference of network contrasts into the target; this is pure algebra (Lemma 14.1) and never fails. The theorem is therefore as strong as its weakest hypothesis, and the weakest is invariably transitivity. \(\blacktriangleleft\)

Remark. The estimator is trivial; the assumption is not. Equation Equation 14.7 is one subtraction, and one might wonder why it warrants a theorem. The answer is that the subtraction is licensed only on the consistency-constrained set where Equation 14.6 holds, and membership in that set is the substantive scientific claim. Every population-adjusted method in Parts IV and V is, in the end, a device for making Equation 14.6 hold after a correction, or for quantifying the error when it does not. The simplicity of Bucher’s formula is exactly why the assumption that underwrites it must be stated with care. \(\blacktriangleleft\)

14.8 The Bucher variance

We already know from Theorem 14.1 that the indirect estimate is centered correctly. The question now is how precise it is, and the headline is simple: an indirect estimate is built from two uncertain inputs, so its uncertainty is larger than either. The estimator Equation 14.7 is a linear combination of two trial summaries, so its variance is a two-line consequence of the bilinearity of covariance. We first record the general identity, valid whether or not the trials are independent, because the general form is what later chapters need when a multi-arm trial makes the two contrasts dependent.

Lemma 14.2 (Variance of a difference) For any two square-integrable random variables \(U,V\) (that is, random variables with finite variance), \[ \mathrm{Var}(U-V)=\mathrm{Var}(U)+\mathrm{Var}(V)-2\,\mathrm{Cov}(U,V). \tag{14.9}\]

Proof. Write \(\mathrm{Var}(U-V)=\mathbb E\big[(U-V-\mathbb E[U-V])^2\big]\) and put \(\tilde U=U-\mathbb E[U]\), \(\tilde V=V-\mathbb E[V]\), so \(U-V-\mathbb E[U-V]=\tilde U-\tilde V\). Expanding the square and using linearity of expectation, \[ \mathbb E\big[(\tilde U-\tilde V)^2\big] =\mathbb E[\tilde U^2]-2\,\mathbb E[\tilde U\tilde V]+\mathbb E[\tilde V^2] =\mathrm{Var}(U)-2\,\mathrm{Cov}(U,V)+\mathrm{Var}(V), \] which is Equation 14.9. The covariance is a symmetric bilinear form on centered variables, the same form whose positive semidefiniteness is recorded in Theorem 4.9. \(\square\)

Theorem 14.2 (The Bucher variance) Let \(\hat d_{AC}\) and \(\hat d_{BC}\) have variances \(v_{AC}=\mathrm{Var}(\hat d_{AC})\) and \(v_{BC}=\mathrm{Var}(\hat d_{BC})\). If the two estimators are uncorrelated, \(\mathrm{Cov}(\hat d_{AC},\hat d_{BC})=0\) (as holds when they come from disjoint sets of trials), then the Bucher estimator Equation 14.7 has variance \[ \mathrm{Var}\!\left(\hat d_{AB}^{\,\mathrm{Bucher}}\right)=v_{AC}+v_{BC}. \tag{14.10}\] In general, without the independence assumption, \(\mathrm{Var}(\hat d_{AB}^{\,\mathrm{Bucher}})=v_{AC}+v_{BC}-2\,\mathrm{Cov}(\hat d_{AC},\hat d_{BC})\).

[machine-checked: theories/Bucher.v, Theorem bucher_variance]

Remark. Why this matters, and a picture. Indirect evidence is real evidence, but it is more expensive: its uncertainty is the sum of the two direct uncertainties. Independent uncertainties add in quadrature, exactly as independent errors combine by the Pythagorean rule, so the indirect standard error is the hypotenuse \(\sqrt{\mathrm{SE}_{AC}^2+\mathrm{SE}_{BC}^2}\) of the two direct standard errors. A practical consequence, made concrete in Section 14.11: an indirect comparison anchored on a tiny \(B\)-versus-\(C\) trial can be uninformative even when both direct trials are individually significant, because the small trial’s large variance is added in. \(\blacktriangleleft\)

Proof. Apply Lemma 14.2 with \(U=\hat d_{AC}\) and \(V=\hat d_{BC}\): \[ \mathrm{Var}\!\left(\hat d_{AB}^{\,\mathrm{Bucher}}\right) =v_{AC}+v_{BC}-2\,\mathrm{Cov}(\hat d_{AC},\hat d_{BC}), \] which is the general formula. Under the independence hypothesis the covariance term is zero, leaving \(v_{AC}+v_{BC}\), equation Equation 14.10. \(\square\)

The independence hypothesis holds in the canonical Bucher setting, where the \(A\)-versus-\(C\) evidence and the \(B\)-versus-\(C\) evidence come from separate trials. It fails when a single multi-arm trial contributes to both contrasts through a shared comparator arm: then the shared arm’s sampling error enters both \(\hat d_{AC}\) and \(\hat d_{BC}\) with the same sign, inducing a positive covariance. Intuitively, the shared comparator’s error is partly subtracted away rather than counted twice, so a positive covariance reduces the Bucher variance below the sum Equation 14.10. The standard homoskedastic random-effects model (a model in which every contrast is given the same variance) makes that correlation exactly one half; the intuition is that two contrasts sharing one of their two arms overlap in half of their sampling error, and we compute it exactly in Theorem 15.4. Because the covariance is positive, treating correlated contrasts as independent overstates the variance, which is conservative for this comparison.

Corollary 14.1 (The price of indirectness) Under the independence hypothesis of Theorem 14.2, the standard error of the Bucher estimate strictly exceeds the standard error of either direct contrast it is built from: \[ \mathrm{SE}\!\left(\hat d_{AB}^{\,\mathrm{Bucher}}\right)=\sqrt{v_{AC}+v_{BC}} \;>\;\max\!\left(\sqrt{v_{AC}},\,\sqrt{v_{BC}}\right), \] whenever both \(v_{AC}>0\) and \(v_{BC}>0\).

Proof. Since \(v_{AC},v_{BC}>0\), we have \(v_{AC}+v_{BC}>v_{AC}\) and \(v_{AC}+v_{BC}>v_{BC}\); taking square roots, which is monotone on the nonnegative reals, gives \(\sqrt{v_{AC}+v_{BC}}>\sqrt{v_{AC}}\) and \(>\sqrt{v_{BC}}\), hence the inequality with the maximum. \(\square\)

The corollary quantifies a fact of evidence synthesis: an indirect comparison is less precise than either of the direct comparisons it rests on, because it accumulates the uncertainty of both. Indirect evidence is genuine evidence, but it is statistically more expensive than a head-to-head trial of the same size. This is one half of the case for direct trials; the other half, that indirect evidence can be biased when transitivity fails, is the subject of Section 14.12.

14.9 The common comparator cancels baseline differences

We now make precise the sense in which the anchor \(C\) protects the comparison. Decompose each arm’s link-scale position into a study-specific baseline and a treatment-specific effect. Under transitivity the treatment effects are population-free; the baselines are not, and they are exactly what the anchoring removes.

Definition 14.6 (The additive arm decomposition) Write the marginal link-scale position of arm \(k\) in study \(j\) as \[ \eta_{k(\mathcal P_j)} \;=\; \mu_j+\alpha_k, \tag{14.11}\] where \(\mu_j\) is a study-specific baseline and \(\alpha_k\) is a treatment-specific shift shared across studies, with the reference treatment normalized to \(\alpha_{\mathrm{ref}}=0\). \(\blacktriangleleft\)

Remark. Reader translation, with a picture. The decomposition splits each arm’s position into “where the study starts” plus “how far the treatment moves it.” Concretely, normalizing the comparator as reference, \(\alpha_C=0\), so \(\mu_j=\eta_{C(\mathcal P_j)}\) is literally the placebo arm’s link-scale outcome in study \(j\):

   position eta_k(P_j)  =  mu_j (study baseline)  +  alpha_k (treatment shift)

   study AC:   arm C:  mu_AC + 0            study BC:   arm C:  mu_BC + 0
               arm A:  mu_AC + alpha_A                  arm B:  mu_BC + alpha_B

   mu_AC, mu_BC differ across studies      (different baseline risk)
   alpha_A, alpha_B, alpha_C are SHARED     (constancy of relative effects)

Read \(\mu_j\) as “how risky study \(j\)’s population is under the reference treatment,” its baseline log-odds of the event; read \(\alpha_k\) as “how much treatment \(k\) moves that risk on the link scale,” the same move in every study. This \(\mu_j\) is exactly the study-\(j\) baseline (intercept) of Notation; the reuse of the symbol is deliberate, not a clash. Because both trials share \(C\), the two baselines \(\mu_{AC}\) and \(\mu_{BC}\) are the same kind of quantity, the placebo log-odds, measured in two different populations, and they will differ. \(\blacktriangleleft\)

Remark. The decomposition encodes constancy. The decomposition Equation 14.11 is precisely the fixed-effect network model in the reference-treatment parameterization (Definition 15.3). The same \(\alpha_k\) appears in every study, so the contrast \(\alpha_k-\alpha_{k'}\) is the population-free network effect \(d_{k'k}\): under Equation 14.11, \(d_{ab(\mathcal P_j)}=\eta_{b(\mathcal P_j)}-\eta_{a(\mathcal P_j)}=\alpha_b-\alpha_a\), free of \(\mu_j\). This is the model form of transitivity, and it is the bridge to the network meta-analysis of Chapter 15, where the same \(\mu_j\)-plus-\(\alpha_k\) structure is fitted to a whole network at once. \(\blacktriangleleft\)

Proposition 14.4 (Anchoring removes both the baselines and the comparator) Assume the additive decomposition Equation 14.11 with a common comparator \(C\) present in both studies. Then the Bucher estimand is free of the study baselines \(\mu_{AC},\mu_{BC}\) and of the comparator effect \(\alpha_C\): \[ d_{AC(\mathcal P_{AC})}-d_{BC(\mathcal P_{BC})}=(\alpha_C-\alpha_A)-(\alpha_C-\alpha_B)=\alpha_B-\alpha_A=d_{AB}. \] Each direct contrast already cancels its own study baseline, and the difference of contrasts then cancels the shared comparator effect \(\alpha_C\).

Proof. In study \(AC\) the two arms have positions \(\eta_{A(\mathcal P_{AC})}=\mu_{AC}+\alpha_A\) and \(\eta_{C(\mathcal P_{AC})}=\mu_{AC}+\alpha_C\) by Equation 14.11. Their contrast is \[ d_{AC(\mathcal P_{AC})}=\eta_{C(\mathcal P_{AC})}-\eta_{A(\mathcal P_{AC})}=(\mu_{AC}+\alpha_C)-(\mu_{AC}+\alpha_A)=\alpha_C-\alpha_A, \] the baseline \(\mu_{AC}\) cancelling because it is common to both arms of the same study. This is the within-study cancellation effected by randomization. Identically, study \(BC\) gives \(d_{BC(\mathcal P_{BC})}=\alpha_C-\alpha_B\). Subtracting, \[ d_{AC(\mathcal P_{AC})}-d_{BC(\mathcal P_{BC})}=(\alpha_C-\alpha_A)-(\alpha_C-\alpha_B)=\alpha_B-\alpha_A, \] the comparator effect \(\alpha_C\) cancelling because it is common to both contrasts. The result \(\alpha_B-\alpha_A\) equals \(d_{AB}\) by Equation 14.11. \(\square\)

Remark. Why this matters. This is the precise sense in which the anchor earns its keep. Differencing within each trial before differencing across trials deletes the study baselines \(\mu_j\) in two stages, so the anchored comparison is immune to the baseline-risk differences that sink the naive comparison. It does not, however, protect against differences in the treatment shifts \(\alpha_k\) across populations; that is the province of transitivity, and its failure is Section 14.12. \(\blacktriangleleft\)

The contrast with the unanchored comparison is now sharp. The naive estimator Equation 14.8 forgoes the within-study cancellation and inherits the difference of baselines as outright bias.

Proposition 14.5 (The unanchored comparison is biased by the baseline difference) Under the additive decomposition Equation 14.11, the unanchored estimand Equation 14.8 is \[ \eta_{A(\mathcal P_{AC})}-\eta_{B(\mathcal P_{BC})}=(\mu_{AC}-\mu_{BC})+(\alpha_A-\alpha_B)=(\mu_{AC}-\mu_{BC})-d_{AB}, \] so, taking \(\hat d_{AB}^{\,\mathrm{naive}}\) as an estimate of \(-d_{AB}\) (the \(A\)-versus-\(B\) contrast in the reversed orientation), its bias is the difference of study baselines: \[ \text{bias}\;=\;\mu_{AC}-\mu_{BC}. \tag{14.12}\] The bias vanishes only in the special case \(\mu_{AC}=\mu_{BC}\), that is, only when the two studies happen to share the same baseline link-scale outcome.

Proof. By Equation 14.11, \(\eta_{A(\mathcal P_{AC})}=\mu_{AC}+\alpha_A\) and \(\eta_{B(\mathcal P_{BC})}=\mu_{BC}+\alpha_B\). Subtracting, \[ \eta_{A(\mathcal P_{AC})}-\eta_{B(\mathcal P_{BC})}=(\mu_{AC}-\mu_{BC})+(\alpha_A-\alpha_B). \] By Equation 14.11 again, \(\alpha_A-\alpha_B=-(\alpha_B-\alpha_A)=-d_{AB}\). Hence the estimand equals \((\mu_{AC}-\mu_{BC})-d_{AB}\), an estimate of \(-d_{AB}\) offset by \(\mu_{AC}-\mu_{BC}\), which is the claimed bias. It is zero precisely when \(\mu_{AC}=\mu_{BC}\). \(\square\)

Example 14.1 (Anchored versus unanchored, on the same data) Use the two trials of the running example (computed in full in Section 14.11). The placebo (arm \(C\)) log-odds were \(\hat\eta_C^{(AC)}=\log 0.5=-0.693\) in trial \(AC\) and \(\hat\eta_C^{(BC)}=\log 0.375=-0.981\) in trial \(BC\); normalizing \(\alpha_C=0\) makes these the two baselines, \[ \mu_{AC}=-0.693,\qquad \mu_{BC}=-0.981,\qquad \mu_{AC}-\mu_{BC}=0.288. \] The two statin arms happened to have identical event rates, \(\hat\eta_A=\hat\eta_B=\log 0.25=-1.386\), so the two analyses disagree sharply:

Quantity Anchored (Bucher) Unanchored (naive)
what is contrasted within-study contrasts \(\hat d_{AC},\hat d_{BC}\) active arms \(\hat\eta_A,\hat\eta_B\) across studies
uses the placebo arms? yes, as the anchor no, they are discarded
estimate on the log scale \(\hat d_{AB}=0.288\) \(\hat\eta_A-\hat\eta_B=0\)
target it estimates \(d_{AB}\) \(-d_{AB}\)
odds ratio reported \(e^{0.288}=\tfrac43=1.33\) \(e^{0}=1.00\)
bias none (under transitivity) \(\mu_{AC}-\mu_{BC}=0.288\)

The naive comparison concludes the two statins are identical (\(\mathrm{OR}=1\)), because their event rates look the same. But they are not comparable: statin \(B\) was tested against a placebo population with a lower event rate (\(27\%\) versus \(33\%\)), so \(B\) merely had an easier comparator. The anchored comparison corrects for this and finds \(B\) carries \(\tfrac43\) the event odds of \(A\). The entire discrepancy, \(0.288\), is precisely the gap between the two placebo baselines, exactly as Equation 14.12 predicts. \(\blacktriangleleft\)

Remark. Why anchoring matters. The baseline \(\mu_j\) absorbs everything about study \(j\)’s population that shifts the absolute outcome level: disease severity, follow-up duration, the prevalence of prognostic variables (Definition 11.5). Two trials of different drugs are run in different places at different times and essentially never share a baseline, so \(\mu_{AC}-\mu_{BC}\) is essentially never zero, and the unanchored estimator Equation 14.8 is essentially always biased, as Example 14.1 just showed. The anchored estimator Equation 14.7 removes this bias for free, by differencing within each study before differencing across studies. This is the whole reason anchored indirect comparison is the default and unanchored comparison a last resort reserved for disconnected networks (Chapter 26), where it must be propped up by strong additional assumptions to control Equation 14.12. Crucially, anchoring does not protect against differences in the treatment effects \(\alpha_k\) across populations; that is the province of transitivity, and its failure is Section 14.12. \(\blacktriangleleft\)

14.10 The variance of an empirical log-odds

To turn the worked example into honest numbers we need the standard error of a single trial’s log odds-ratio. It follows from the delta method (Theorem 5.3), and it is the standard error every applied Bucher computation uses.

Remark. You may skip the proof on a first reading. The only thing the worked example needs from this section is the boxed formula Equation 14.13: the estimated variance of a log odds-ratio is the sum of the reciprocals of the four cell counts of its two-by-two table. A reader in a hurry can take this on faith and go straight to Section 14.11; the delta-method derivation below merely justifies it, repeating the machinery of Chapter 5 (Theorem 5.3). \(\blacktriangleleft\)

Proposition 14.6 (Large-sample variance of an empirical log-odds and log odds-ratio) Let an arm enroll \(n\) patients of whom \(r\) experience the event, with \(r\sim\mathrm{Bin}(n,p)\) and \(0<p<1\). Let \(\hat p=r/n\) and let the empirical log-odds be \(\hat\eta=\operatorname{logit}(\hat p)=\log\frac{\hat p}{1-\hat p}\). Then, to first order, \[ \mathrm{Var}(\hat\eta)\approx\frac{1}{np(1-p)},\qquad \widehat{\mathrm{Var}}(\hat\eta)=\frac{1}{r}+\frac{1}{n-r}. \tag{14.13}\] Consequently the empirical log odds-ratio contrasting two independent arms, with event counts \(r_1,r_2\) out of \(n_1,n_2\), has estimated variance \(\big(\tfrac1{r_1}+\tfrac1{n_1-r_1}\big)+\big(\tfrac1{r_2}+\tfrac1{n_2-r_2}\big)\), the sum of the inverse cell counts of the underlying two-by-two table.

Proof. The logit \(g(p)=\log p-\log(1-p)\) has derivative \(g'(p)=\frac1p+\frac1{1-p}=\frac{1}{p(1-p)}\). Since \(\hat p\) is the mean of \(n\) independent Bernoulli\((p)\) variables, \(\mathbb E[\hat p]=p\) and \(\mathrm{Var}(\hat p)=p(1-p)/n\), and by the central limit theorem \(\sqrt n(\hat p-p)\xrightarrow{d}\mathcal N(0,p(1-p))\). The delta method (Theorem 5.3) applied to \(g\) gives \[ \sqrt n\big(g(\hat p)-g(p)\big)\xrightarrow{d}\mathcal N\!\big(0,\,[g'(p)]^2\,p(1-p)\big), \qquad [g'(p)]^2\,p(1-p)=\frac{p(1-p)}{p^2(1-p)^2}=\frac{1}{p(1-p)}, \] so \(\mathrm{Var}(\hat\eta)\approx\frac{1}{np(1-p)}\). Substituting the plug-in estimate \(\hat p=r/n\), \[ \widehat{\mathrm{Var}}(\hat\eta)=\frac{1}{n\hat p(1-\hat p)}=\frac{1}{n\cdot\frac{r}{n}\cdot\frac{n-r}{n}} =\frac{n}{r(n-r)}=\frac{(n-r)+r}{r(n-r)}=\frac1r+\frac1{n-r}, \] which is Equation 14.13. The two arms are independent, so by Lemma 14.2 (with zero covariance) the variance of their difference of log-odds is the sum of the two arm variances; collecting the four reciprocals gives the stated table formula. The normal approximation underlying a Wald interval is justified by the asymptotic normality of the maximum likelihood estimator (Theorem 5.6), of which \(\hat\eta\) is an instance. \(\square\)

14.11 A worked Bucher comparison on the log odds-ratio scale

We carry a complete example from raw counts to a confidence interval, so that every number can be checked by hand. The event is an unfavorable outcome (a cardiovascular event); larger log-odds means worse.

Remark. Recipe: Bucher from published trials. Most of the time you will not have raw counts but a reported estimate and interval. The five-step recipe is then:

  1. Recover each standard error from a reported \(95\%\) interval on the log scale: \(\mathrm{SE}=(\text{upper}-\text{lower})/(2\times 1.96)\), both endpoints on the log scale.
  2. Line up the orientation: express both direct effects as “versus the common comparator \(C\)” on the same log scale (log OR, log RR, or log HR), so that \(\hat d_{AC}\) and \(\hat d_{BC}\) mean \(\eta_C-\eta_A\) and \(\eta_C-\eta_B\).
  3. Subtract: \(\hat d_{AB}=\hat d_{AC}-\hat d_{BC}\).
  4. Add variances (if the trials are independent): \(\mathrm{SE}(\hat d_{AB})=\sqrt{\mathrm{SE}_{AC}^2+\mathrm{SE}_{BC}^2}\).
  5. Interval, then back-transform: form a Wald interval on the log scale, then exponentiate the estimate and both interval endpoints to report a ratio.

The example below runs this recipe starting one step earlier, from the raw tables. \(\blacktriangleleft\)

Example 14.2 (Indirect comparison of \(A\) and \(B\) through a common comparator \(C\)) Orientation check. We report on the chapter’s \(d_{ab}=\eta_b-\eta_a\) scale. The event is unfavorable, so a positive contrast flags higher, worse odds. We expect both statins to beat placebo, so we expect \(\hat d_{AC}=\eta_C-\eta_A>0\) and \(\hat d_{BC}=\eta_C-\eta_B>0\) (placebo worse than each statin), and the target is \(d_{AB}=\eta_B-\eta_A\), “\(B\) versus \(A\), higher is worse.”

Two randomized trials report the following two-by-two tables.

Study \(AC\) (treatment \(A\) versus comparator \(C\)):

Arm Events \(r\) Non-events \(n-r\) Total \(n\)
\(A\) \(30\) \(120\) \(150\)
\(C\) \(50\) \(100\) \(150\)

Study \(BC\) (treatment \(B\) versus comparator \(C\)):

Arm Events \(r\) Non-events \(n-r\) Total \(n\)
\(B\) \(40\) \(160\) \(200\)
\(C\) \(60\) \(160\) \(220\)

Step 1: arm log-odds. With \(\hat\eta=\log\frac{r}{n-r}\), \[ \hat\eta_A=\log\tfrac{30}{120}=\log 0.25=-1.38629,\qquad \hat\eta_C^{(AC)}=\log\tfrac{50}{100}=\log 0.5=-0.69315, \] \[ \hat\eta_B=\log\tfrac{40}{160}=\log 0.25=-1.38629,\qquad \hat\eta_C^{(BC)}=\log\tfrac{60}{160}=\log 0.375=-0.98083. \]

Step 2: direct contrasts. Using the Notation order \(d_{kC}=\eta_C-\eta_k\), \[ \hat d_{AC}=\hat\eta_C^{(AC)}-\hat\eta_A=-0.69315-(-1.38629)=0.69315=\log 2, \] \[ \hat d_{BC}=\hat\eta_C^{(BC)}-\hat\eta_B=-0.98083-(-1.38629)=0.40546=\log 1.5. \] Both are positive, as the orientation check anticipated. On the odds-ratio scale the comparator \(C\) has twice the event odds of \(A\) and one and a half times the event odds of \(B\); equivalently \(A\) and \(B\) are both protective relative to \(C\), with \(A\) the more so.

Step 3: direct variances by Proposition 14.6, summing inverse cell counts, \[ \widehat{\mathrm{Var}}(\hat d_{AC})=\tfrac1{30}+\tfrac1{120}+\tfrac1{50}+\tfrac1{100}=0.071667,\qquad \mathrm{SE}=0.26771, \] \[ \widehat{\mathrm{Var}}(\hat d_{BC})=\tfrac1{40}+\tfrac1{160}+\tfrac1{60}+\tfrac1{160}=0.054167,\qquad \mathrm{SE}=0.23274. \]

Step 4: the Bucher estimate. By Equation 14.7 and Equation 14.10, \[ \hat d_{AB}^{\,\mathrm{Bucher}}=\hat d_{AC}-\hat d_{BC}=0.69315-0.40546=0.28768=\log\tfrac43, \] \[ \widehat{\mathrm{Var}}\!\left(\hat d_{AB}^{\,\mathrm{Bucher}}\right)=0.071667+0.054167=0.125833,\qquad \mathrm{SE}=0.35473. \]

Step 5: interval and back-transformation. A \(95\%\) Wald interval on the log scale is \(0.28768\pm 1.95996\times 0.35473=0.28768\pm 0.69526=(-0.40758,\,0.98293)\). Exponentiating, the indirect odds ratio of the event comparing \(B\) to \(A\) is \[ \widehat{\mathrm{OR}}_{AB}=e^{0.28768}=1.3333=\tfrac43,\qquad 95\%\ \mathrm{CI}=(e^{-0.40758},\,e^{0.98293})=(0.665,\,2.672). \] \(\blacktriangleleft\)

Remark. Final interpretation, in both orientations. On this chapter’s \(d_{AB}=\eta_B-\eta_A\) scale, the event odds for statin \(B\) are estimated as \(1.33\) times those for statin \(A\); equivalently, the event odds for \(A\) are \(1/1.33=0.75\) times those for \(B\). Either way, the point estimate favors \(A\) as the better statin. But the \(95\%\) interval for the odds ratio, \((0.67,\,2.67)\), comfortably includes \(1\), so on this evidence the difference is not statistically distinguishable from no difference. And the estimate is trustworthy only under transitivity: it assumes the two trials’ populations shared the distribution of any covariate that modifies the statin effect on the log odds-ratio scale. \(\blacktriangleleft\)

Three further features of the example repay attention.

Remark. Ratios divide; logs subtract. The indirect odds ratio \(\tfrac43\) is exactly the ratio of the two reported odds ratios, \(2/1.5=\tfrac43\), which is the multiplicative face of the additive identity \(\log\tfrac43=\log 2-\log 1.5\). Had we (wrongly) subtracted the odds ratios, \(2-1.5=0.5\), we would have produced a number with no meaning. The additive scale of Lemma 14.1 is not optional bookkeeping; it is where the cancellation lives. The same arithmetic works verbatim on any additive scale: if a rheumatoid-arthritis network instead reported \(\hat d_{AC}=-0.40\) and \(\hat d_{BC}=-0.15\) as log hazard ratios of disease progression, then \(\hat d_{AB}=-0.40-(-0.15)=-0.25\), a hazard ratio \(e^{-0.25}=0.78\); only the name of the scale changed. \(\blacktriangleleft\)

Remark. The price of indirectness, concretely. The Bucher standard error \(0.355\) exceeds both direct standard errors \(0.268\) and \(0.233\), as Corollary 14.1 guarantees. The effect on inference is stark. The direct \(A\)-versus-\(C\) contrast is significant (\(z=0.69315/0.26771=2.59\)), the direct \(B\)-versus-\(C\) contrast borderline (\(z=0.40546/0.23274=1.74\)), yet the indirect \(A\)-versus-\(B\) contrast is firmly inconclusive (\(z=0.28768/0.35473=0.81\), two-sided \(p\approx 0.42\)), its \(95\%\) interval for the odds ratio running from \(0.67\) to \(2.67\) and comfortably including \(1\). Two informative direct comparisons can combine into an uninformative indirect one. \(\blacktriangleleft\)

Remark. What the number means, and what it assumes. The estimate \(\widehat{\mathrm{OR}}_{AB}=\tfrac43\) is unbiased for the true \(A\)-versus-\(B\) odds ratio only under transitivity: the populations of the two trials must share the distribution of any covariate that modifies the treatment effect on the log odds-ratio scale. If, for instance, study \(BC\) enrolled younger patients and age modifies the effect of \(B\), then \(\hat d_{BC}\) estimates the effect in a younger population than the one in which \(\hat d_{AC}\) estimates the effect of \(A\), and the subtraction mixes populations. No amount of data fixes this; it is a bias, not a variance, and it is invisible in the standard error \(0.355\). Diagnosing and repairing it is the subject of node-splitting (Definition 15.4), a diagnostic that splits a comparison into its direct and indirect parts and checks whether they agree, and of population adjustment (Parts IV and V). \(\blacktriangleleft\)

14.12 When transitivity fails

Bucher’s theorem is exact under its hypotheses, so every difficulty with indirect comparison is a difficulty with a hypothesis. The variance hypothesis (independence) is mild and is relaxed by an honest covariance term, Theorem 14.2. The unbiasedness hypothesis (randomization) holds by design in the trials that anchored comparisons are built from. The hypothesis that bears the weight is transitivity, and it fails for a definite mechanism: an effect modifier (Definition 11.3) whose distribution differs between the trial populations.

To see the mechanism, suppose a binary covariate \(Z\) modifies the \(B\)-versus-\(C\) effect, so that the conditional contrast \(d_{BC}(z)\) takes one value at \(z=0\) and another at \(z=1\). The marginal effect Equation 14.2 is then a weighted average of \(d_{BC}(0)\) and \(d_{BC}(1)\) with weights given by the prevalence of \(Z\) in the population. If study \(BC\) has a different prevalence of \(Z\) than the population in which we wish to report \(A\) versus \(B\), the marginal \(d_{BC(\mathcal P_{BC})}\) differs from \(d_{BC(\mathcal P_{AB})}\), transitivity Equation 14.5 is violated, the consistency equation Equation 14.6 no longer holds, and the Bucher estimator Equation 14.7 is biased by exactly the discrepancy \(d_{BC(\mathcal P_{BC})}-d_{BC(\mathcal P_{AB})}\).

Example 14.3 (A transitivity failure as a specific number) Let \(Z\) be presence of diabetes, an effect modifier of the statin-\(B\) effect. On the log odds-ratio scale, the conditional \(B\)-versus-\(C\) contrast is \(d_{BC}(Z=0)=0.4\) in non-diabetics and \(d_{BC}(Z=1)=1.0\) in diabetics (statin \(B\) is more protective in diabetics). Diabetes is more common in the \(BC\) trial than in the target population where we want the answer:

\(Z\) (diabetes) conditional \(d_{BC}(Z)\) prevalence in \(BC\) trial prevalence in target \(\mathcal P_{AB}\)
\(Z=0\) \(0.4\) \(0.4\) \(0.8\)
\(Z=1\) \(1.0\) \(0.6\) \(0.2\)

The marginal \(B\)-versus-\(C\) effects are the prevalence-weighted averages \[ \begin{aligned} d_{BC(\mathcal P_{BC})} &= 0.4\times 0.4 + 1.0\times 0.6 = 0.76,\\ d_{BC(\mathcal P_{AB})} &= 0.4\times 0.8 + 1.0\times 0.2 = 0.52. \end{aligned} \] Suppose the \(A\)-versus-\(C\) effect is unaffected by \(Z\) and equals \(d_{AC}=0.6\) in every population. The true target contrast is \(d_{AB}=d_{AC}-d_{BC(\mathcal P_{AB})}=0.6-0.52=0.08\). The Bucher estimand, however, subtracts the \(BC\) trial’s own marginal effect, \(d_{AC}-d_{BC(\mathcal P_{BC})}=0.6-0.76=-0.16\). The bias is \[ -0.16-0.08=-0.24=-\big(d_{BC(\mathcal P_{BC})}-d_{BC(\mathcal P_{AB})}\big), \] a log-odds-ratio error of \(-0.24\), a multiplicative error of \(e^{-0.24}\approx 0.79\) in the odds ratio, about \(21\%\). It does not shrink as the trials grow: it is a bias, not a variance, and it is invisible in any standard error. This is the mechanism the rest of the book is built to detect and repair. \(\blacktriangleleft\)

Chapter 16 makes this calculation general and names the resulting failure (Theorem 16.1), and shows how it compounds with the non-collapsibility of the odds ratio and with ecological bias (Definition 16.2), the bias that arises when a relationship estimated at the trial-average level does not hold at the individual level. The point to carry forward is structural: prognostic variables that are not effect modifiers do not break transitivity, because they shift the baselines \(\mu_j\), which Proposition 14.4 already cancels; only effect modifiers, which alter the \(\alpha_k\) across populations, can break it.

When both direct and indirect evidence exist for the same contrast, the consistency equation becomes testable by comparing them: this is node-splitting (Definition 15.4, Theorem 15.1), the diagnostic developed in Chapter 15 that splits a comparison into its direct and indirect parts and checks whether they agree. A significant discrepancy is the empirical signature of a transitivity failure and the trigger for the population-adjustment methods that occupy the remainder of the book.

Transitivity is also the assumption that connects this chapter to the transportability framework of Chapter 12. Transportability asks when a relative effect measured in one population may be carried to another; transitivity is exactly the special case, direct transportability (Definition 12.3, Theorem 12.1), in which the relative effect is assumed to transport unchanged from each trial population to the target. In the language of Example 14.3, transportability would have us reweight the conditional contrasts \(d_{BC}(z)\) to the target prevalence of \(Z\) before subtracting; the population-adjustment methods of Parts IV and V are precisely the machinery that performs that reweighting when it is needed. Transitivity is the assumption that it is not needed, and the bridge to Chapter 12 is the tool for the cases where it is.

14.13 Notes and references

The anchored indirect comparison and the estimator Equation 14.7 are due to Bucher et al. (1997), who introduced the difference of log relative effects and its variance Equation 14.10 as the basis for indirect comparison in meta-analysis of randomized trials. The framing in terms of constancy of relative effects and transitivity, and the embedding of the estimator in the network model with the reference-treatment decomposition Equation 14.11, follows Dias et al. (2018) and the thesis of Phillippo (2019) (Eq. 1.2 for constancy, Eq. 1.3 for the indirect estimator). The two-layer cancellation argument of Section 14.9, separating the within-study cancellation of baselines from the across-study cancellation of the comparator, is the algebraic core of why anchored comparisons are preferred to unanchored ones; the bias formula Equation 14.12 for the naive comparison is the quantity that unanchored population adjustment (Chapter 26) must control. The delta-method variance of the empirical log-odds (Proposition 14.6) is classical; the sum-of-inverse-cell-counts form is the standard error reported by every two-by-two table routine.

This chapter corresponds to results D.1 (transitivity), D.2 (Bucher’s theorem), D.3 (the Bucher variance), and D.4 (constancy of relative effects) of the proof catalogue in proofs/D_network_meta_analysis.md. The network-level material that builds on Bucher’s identity, namely the parameterizations (Definition 15.2, Definition 15.3), identifiability via the graph incidence rank (Theorem 15.3), the random-effects contrast correlation (Theorem 15.4), the consistency closure and node-splitting (Theorem 15.1, Definition 15.4), and goodness-of-fit (Definition 15.5, Definition 15.6), is developed in Chapter 15; the breakdown of transitivity under effect modification (Theorem 16.1, Definition 16.2) is developed in Chapter 16. When the two direct contrasts are themselves pooled over several trials, the pooling is the pairwise meta-analysis of Chapter 13 (Theorem 13.1 for the fixed-effect inverse-variance estimate, Theorem 13.3 for the random-effects estimate), and Bucher’s identity operates on the pooled summaries. The connection to transportability (Definition 12.3, Theorem 12.1) is developed in Chapter 12.

Three results of this chapter have machine-checked counterparts in the companion Coq development, in theories/Bucher.v. The exact identity Proposition 14.3 is Theorem bucher_identity; the unbiasedness theorem Theorem 14.1 is Theorem bucher_unbiased; the variance theorem Theorem 14.2 is Theorem bucher_variance. The transitivity definition Definition 14.3 and the equivalence Proposition 14.2 are formalized as Definition transitivity together with Lemma transitivity_subtract and Lemma subtract_transitivity. The Coq development treats expectation, variance, and covariance as abstract operators satisfying linearity and the difference identity Equation 14.9, exactly the structure used in the proofs above, so that the formalization captures the algebraic content of Bucher’s theorem while leaving the measure-theoretic construction of those operators to the caller.

14.14 Exercises

The exercises progress from warm-up to challenge, and are tagged by kind. An exercise marked \(\star\) has a full worked solution in Appendix F. The first two are gentle checks of orientation and interpretation; the later ones build toward the network theory of Chapters 15 and 16.

Exercise 14.1 Warm-up. Identify, orient, compute. A health technology assessment has two independent trials against a common comparator \(C\) (placebo). Trial \(AC\) reports the effect of a new drug on the log odds-ratio scale as \(\hat d_{AC}=\eta_C-\eta_A=0.50\) with \(\mathrm{SE}=0.20\); trial \(BC\) reports \(\hat d_{BC}=\eta_C-\eta_B=0.30\) with \(\mathrm{SE}=0.15\). The outcome is unfavorable, so higher odds is worse. (a) Name the common comparator and the target contrast. (b) State whether a positive \(\hat d_{AC}\) means the new drug \(A\) is better or worse than \(C\), and why. (c) Compute the Bucher estimate \(\hat d_{AB}\) and its standard error. (d) Back-transform to an odds ratio for \(B\) versus \(A\) and say, in one sentence, which drug the point estimate favors. (e) State the transitivity assumption you relied on, in one plain sentence.

Exercise 14.2 Interpretation. What breaks it, and what does not. In the setting of Exercise 14.1, suppose you learn that trial \(BC\) enrolled much older patients than trial \(AC\). (a) If age strongly modifies the effect of drug \(B\), is the age difference a threat to the Bucher comparison, and which assumption does it threaten? (b) Would enrolling more patients in each trial remove the threat? Explain in one sentence why or why not. (c) Now suppose instead that age predicts the outcome but does not change the size of any treatment effect. Explain why the anchored comparison is then unharmed, naming the result (Proposition 14.4) that does the work.

Exercise 14.3 Computation. A Bucher computation from summaries. Two trials report log hazard ratios on the proportional-hazards scale: \(\hat d_{AC}=-0.51\) with \(\mathrm{SE}=0.18\), and \(\hat d_{BC}=-0.22\) with \(\mathrm{SE}=0.13\), the contrasts being independent. Compute the indirect log hazard ratio of \(A\) versus \(B\), its standard error, a \(95\%\) confidence interval, and the back-transformed hazard ratio with interval. State the transitivity assumption you are relying on in words.

Exercise 14.4 Computation. From raw tables. Repeat the worked example Example 14.2 with the tables: study \(AC\) has \(A:24/120\) and \(C:48/160\); study \(BC\) has \(B:20/100\) and \(C:30/150\). Report the two direct log odds-ratios with standard errors, the Bucher estimate with standard error, and the indirect odds ratio with \(95\%\) interval. Verify that the indirect odds ratio equals the ratio of the two direct odds ratios.

Exercise 14.5 Proof. Additivity holds on the right scales. Prove that the antisymmetry and additivity laws Lemma 14.1 hold for the log risk-ratio contrast \(d_{ab}=\log\bar\mu_b-\log\bar\mu_a\) and for the mean-difference contrast \(d_{ab}=\bar\mu_b-\bar\mu_a\), and show by a numerical counterexample that they fail for the risk-ratio contrast taken on the ratio scale itself, \(\bar\mu_b/\bar\mu_a\) (that is, that \(\bar\mu_C/\bar\mu_A\ne(\bar\mu_C/\bar\mu_B)\cdot\text{(something additive)}\) unless one first takes logarithms). Conclude that Bucher’s subtraction is a statement about the link scale.

Exercise 14.6 Computation. The cost of dropping the anchor. Using the additive decomposition Equation 14.11, suppose \(\mu_{AC}=-1.2\), \(\mu_{BC}=-1.8\), \(\alpha_A=-0.7\), \(\alpha_B=-0.4\), \(\alpha_C=0\) on the logit scale. Compute the true network contrast \(d_{AB}\), the anchored estimand \(d_{AC}-d_{BC}\), and the unanchored estimand Equation 14.8, and verify that the unanchored estimand is off by exactly \(\mu_{AC}-\mu_{BC}\) as Proposition 14.5 predicts. Comment on which of the five parameters an anchored comparison is immune to.

Exercise 14.7 Proof. A longer chain. Generalize Bucher’s identity to a path of comparators \(A\to C\to D\to B\), in which trial \(AC\) estimates \(d_{AC}\), trial \(CD\) estimates \(d_{CD}\), and trial \(DB\) estimates \(d_{DB}\), all independent. Using Lemma 14.1, derive the chained indirect estimator of \(d_{AB}\) as a signed sum of the three direct contrasts, prove it unbiased under transitivity along the path, and show its variance is the sum of the three direct variances. Explain why each additional link strictly inflates the variance.

Exercise 14.8 (\(\star\), challenge) Correlated contrasts from a three-arm trial. Suppose a single three-arm trial reports both \(\hat d_{AC}\) and \(\hat d_{BC}\) (arms \(A\), \(B\), \(C\) in one study), so the shared comparator arm makes the two contrasts correlated with \(\mathrm{Cov}(\hat d_{AC},\hat d_{BC})=\rho\sqrt{v_{AC}v_{BC}}\) for some \(\rho>0\). Using the general formula in Theorem 14.2, write the variance of the Bucher estimate of \(d_{AB}\) and show it is strictly smaller than \(v_{AC}+v_{BC}\). For the homoskedastic structure in which both contrasts share a comparator arm of variance \(\sigma^2\) and have arm-level variance \(\sigma^2\) in each non-comparator arm, show \(\rho=\tfrac12\) and recover the variance of the within-study \(A\)-versus-\(B\) contrast directly; reconcile your answer with Theorem 15.4.

Exercise 14.9 (\(\star\), interpretation) Quantifying a transitivity failure. A binary effect modifier \(Z\) acts on the \(B\)-versus-\(C\) contrast so that the conditional log odds-ratios are \(d_{BC}(Z=0)=0.2\) and \(d_{BC}(Z=1)=0.8\) (on the link scale, treating these conditional contrasts as collapsible for simplicity). Study \(BC\) has \(\Pr(Z=1)=0.7\), while the target population \(\mathcal P_{AB}\) in which we wish to report \(A\) versus \(B\) has \(\Pr(Z=1)=0.3\). The \(A\)-versus-\(C\) contrast is unaffected by \(Z\) and equals \(d_{AC}=0.6\) in every population. Compute the marginal \(d_{BC(\mathcal P_{BC})}\) and \(d_{BC(\mathcal P_{AB})}\), the true target contrast \(d_{AB}\), the Bucher estimand \(d_{AC}-d_{BC(\mathcal P_{BC})}\), and hence the bias of the unadjusted indirect comparison. Identify which quantity an effect-modifier adjustment (Chapter 17 onward) must reweight to remove the bias, and explain why a prognostic variable that shifted only the baseline would have created no bias at all.