15  Network Meta-Analysis: Models and Identifiability

15.1 Introduction

The previous chapter built an indirect comparison from a single triangle: a study of \(A\) against a common comparator \(C\), a study of \(B\) against the same \(C\), and the subtraction \(\hat d_{AB}=\hat d_{AC}-\hat d_{BC}\) that the algebra of contrasts licenses under transitivity (Theorem 14.1). A health technology assessment is rarely so tidy. It must usually weigh a dozen treatments compared in a scatter of trials, some two-armed and some multi-armed, some sharing comparators and some not, with the same pairwise contrast occasionally informed by both a direct head-to-head trial and one or more indirect paths. Network meta-analysis (NMA) is the apparatus that pools all of this evidence at once into a single coherent set of relative effects, estimating every pairwise contrast in the network, including contrasts never studied head to head, while honoring the correlations that multi-arm trials and shared comparators impose.

Why pool everything into one model instead of running a separate Bucher comparison for each pair of treatments? Three reasons, each made precise in this chapter. First, pooling borrows strength: a contrast with no direct head-to-head trial is still estimated, informed by every indirect path through the network, and a contrast carrying both direct and indirect evidence combines the two efficiently rather than discarding either. Second, pooling respects structure: a three-arm trial is one randomized experiment, not three independent two-arm trials, and a single model preserves the correlations that such shared-comparator designs impose, correlations that separate pairwise calculations would silently throw away. Third, pooling makes coherence testable: isolated pairwise comparisons cannot notice that the whole web of relative effects must fit together, whereas the network can check whether the direct and indirect routes to a contrast agree, and flag them when they do not. Chapter 14 gave the single triangle; this chapter gives the whole web, together with the tools to decide which of its contrasts a reader may trust.

This chapter develops the standard NMA model as a piece of mathematics. The organizing idea is that an NMA is a generalized linear model (Definition 7.4): each trial arm contributes one observation, the link-scale arm means are a linear function \(\boldsymbol\theta=\mathbf X\boldsymbol\beta\) of a parameter vector \(\boldsymbol\beta\) through a design matrix \(\mathbf X\), and the entire analytic content of the method, its estimability and its assumptions, is read off from the structure of \(\mathbf X\). Bucher’s estimator is then visible as the special case in which \(\mathbf X\) encodes one triangle and two studies, and the question “which contrasts can the network estimate?” becomes the linear-algebra question “which columns of \(\mathbf X\) are independent?”

We prove, in full, the results that make this picture rigorous. We give the two standard parameterizations of the arm means, the baseline-shift form (Definition 15.2) that names a separate contrast inside each study and the reference-treatment form (Definition 15.3) that names a single network-wide contrast per treatment, and we prove they describe the same model exactly on the subspace where the consistency equations hold (Theorem 15.2). We prove that the network contrasts are identifiable if and only if the treatment graph is connected, by computing the rank of the design matrix through the rank of a graph incidence matrix (Theorem 15.3). We derive the characteristic one-half correlation between two contrasts that share an arm in a multi-arm trial, and we show that the one-half is not an arbitrary modeling choice but the unique value that makes the heterogeneity variance independent of which arm is called the baseline (Theorem 15.4). We prove the algebraic closure of the consistency equations (Theorem 15.1), which says that the \(\binom{K}{2}\) pairwise contrasts of a \(K\)-treatment network are generated by \(K-1\) basis contrasts and constrained by exactly \(\binom{K-1}{2}\) loop identities, the very identities that node-splitting (Definition 15.4) tests for inconsistency. We extend the model to individual patient data and prove the identifiability of the network meta-regression (Theorem 15.5), the springboard for the population-adjustment methods of Parts IV and V. We close the theory with the goodness-of-fit machinery the applied literature reports for these models, the residual deviance (Definition 15.5) and the deviance information criterion (Definition 15.6). A fully worked triangle-plus-extra-arm network, carried from a design matrix to fitted contrasts, standard errors, a node-split, and a residual deviance, makes every formula concrete.

Throughout we use the notation of Notation: \(j\) indexes studies, \(k\) indexes treatments and arms, \(\theta_{jk}\) is the link-scale mean of arm \(k\) in study \(j\), \(\mu_j\) is a study baseline, \(d_{ab}\) is the network contrast of treatment \(b\) versus treatment \(a\), and \(\delta_{j,\,t_1 t_k}\) is a study-level contrast. The chapter corresponds to results D.5 through D.14 of the proof catalogue in proofs/D_network_meta_analysis.md, and it builds directly on the Bucher development of Chapter 14 (Definition 14.3, Definition 14.2, Theorem 14.1, Theorem 14.2) and the pairwise meta-analysis of Chapter 13 (Theorem 13.1, Theorem 13.3, Definition 13.3).

15.2 A running example and a reader’s toolkit

The formal development below moves quickly into design matrices, graphs, and rank conditions. To keep it anchored, we fix one small clinical network now and carry it through every section. A reader who holds this picture in mind can translate each abstract step back into trials and drugs.

Example 15.1 (Three treatments, four trials) A health technology assessment compares three treatments for a single condition:

• treatment \(1\) = placebo (the network reference),
• treatment \(2\) = Drug A,
• treatment \(3\) = Drug B.

Four randomized trials are available. Each reports a relative effect on the log-odds scale (a log odds ratio); a positive value means the second-named treatment raises the odds of a good outcome.

Study \(j\)	Arms	Design	Comparison(s) it reports
1	placebo, Drug A	two-arm	Drug A vs placebo
2	placebo, Drug B	two-arm	Drug B vs placebo
3	Drug A, Drug B	two-arm (head-to-head)	Drug B vs Drug A
4	placebo, Drug A, Drug B	three-arm	Drug A vs placebo, and Drug B vs placebo

Drawing a dot for each treatment and a line for each within-study comparison gives the treatment-comparison graph:

                  (2) Drug A
                  /         \
       studies 1, 4          study 3
                /             \
   (1) placebo  --------------  (3) Drug B
                studies 2, 4

Three facts can be read straight off the picture, and the rest of the chapter turns each one into a theorem.

• Direct versus indirect evidence. Every pair is compared directly by at least one trial: placebo vs Drug A (studies 1 and 4), placebo vs Drug B (studies 2 and 4), Drug A vs Drug B (study 3). Drug A vs Drug B is also estimable indirectly, by chaining Drug A vs placebo with placebo vs Drug B.
• Connectedness. One can travel from any treatment to any other along the lines. This is what lets the network estimate every relative effect, including ones with no direct trial; Theorem 15.3 makes it precise.
• A loop. The three edges \(1\)-\(2\), \(2\)-\(3\), \(1\)-\(3\) close a triangle. That loop is what makes it possible to test whether the direct and indirect routes to Drug A vs Drug B agree; Theorem 15.1 and Section 15.10 build the test. A tree-shaped network with no loop offers nothing to test.

Study \(4\), the three-arm trial, is the source of most of the subtlety in this chapter. Its full numerical analysis is completed in Example 15.2. \(\blacktriangleleft\)

The picture uses standard graph words. Here is the whole vocabulary in trial language; keep it nearby.

Graph term	What it means for the evidence
vertex (node)	a treatment
edge	one within-study comparison (a single randomized contrast)
path	a chain of comparisons linking two treatments
connected	every treatment is reachable from every other by comparisons
connected component	a maximal cluster of treatments joined by comparisons
loop (cycle)	a closed chain: both a direct and an indirect route between two treatments
direct evidence (on \(a\) vs \(b\))	a trial that randomized patients between \(a\) and \(b\)
indirect evidence (on \(a\) vs \(b\))	an estimate of \(a\) vs \(b\) obtained through a path via other treatments

Four quantities recur throughout, and beginners routinely blend them. The next table previews them with plain meanings and a running-example instance; the precise definitions follow in Section 15.3 and Section 15.4, so nothing here is used before it is defined later.

Symbol	Plain-language meaning	In the running example
\(\theta_{jk}\)	outcome level of arm \(k\) in study \(j\), on the link scale	log-odds of a good outcome on the Drug A arm of study \(1\)
\(\mu_j\)	study \(j\) baseline: its reference arm’s level	log-odds on the placebo arm of study \(1\)
\(\delta_{j,\,t_{j,1}k}\)	a study-specific contrast: treatment \(k\) vs that study’s own baseline arm	Drug A vs placebo as measured inside study \(1\)
\(d_k\)	the network-wide contrast of treatment \(k\) vs the reference \(1\) (one number for the whole network)	Drug A vs placebo, pooled across all trials
\(d_{ab}=d_b-d_a\)	the network contrast of treatment \(b\) vs treatment \(a\), recovered by subtraction	Drug B vs Drug A, pooled
\(t_{j,1}\)	the arm chosen as study \(j\)’s baseline	placebo, in study \(1\)
\(\tilde{\mathbf d}=(d_1,\dots,d_K)\)	the network contrasts padded with \(d_1=0\) (length \(K\)); the unpadded \(\mathbf d=(d_2,\dots,d_K)\) has length \(K-1\)	\(\tilde{\mathbf d}=(0,d_2,d_3)\)

The single most common conceptual slip is to confuse a study-specific contrast \(\delta_{j,\,t_{j,1}k}\) (one study’s answer) with a network contrast \(d_k\) or \(d_{ab}\) (the pooled answer). The whole point of an NMA is to build the network contrasts out of the study-specific ones.

Remark. Reading the sign of a contrast. The convention \(d_{ab}=d_b-d_a\) means “treatment \(b\) versus treatment \(a\)”, with \(b\) in the favored-if-positive direction. On the logit scale a contrast is a log odds ratio: \(d_{ab}>0\) says \(b\) raises the odds of the outcome relative to \(a\), and the odds ratio is \(\mathrm{OR}_{ab}=e^{d_{ab}}\). Reversing the order flips the sign and inverts the odds ratio: \(d_{ba}=-d_{ab}\) and \(\mathrm{OR}_{ba}=1/\mathrm{OR}_{ab}\). For instance a log-odds contrast of \(0.5\) is an odds ratio \(e^{0.5}\approx1.65\); the same comparison written the other way is \(-0.5\), an odds ratio \(e^{-0.5}\approx0.61\). Applied papers are inconsistent about direction, so always check which treatment is named first. \(\blacktriangleleft\)

15.3 The network as a generalized linear model

A network of evidence is a finite collection of randomized studies. Study \(j\in\{1,\dots,J\}\) randomizes patients among a set of arms \(\mathcal A_j\subseteq\{1,\dots,K\}\), where \(\{1,\dots,K\}\) enumerates the treatments under comparison and treatment \(1\) is a fixed network reference (often placebo or standard of care). We write \(A_j=|\mathcal A_j|\ge 2\) for the number of arms in study \(j\), and we designate one arm \(t_{j,1}\in\mathcal A_j\) as the study baseline; nothing of substance depends on which arm is chosen as \(t_{j,1}\), a freedom we exploit in Section 15.7. The total number of arms in the network is \[ N \;=\; \sum_{j=1}^{J} A_j , \tag{15.1}\] and the total number of within-study contrasts, one fewer than the arm count in each study, is \[ C \;=\; \sum_{j=1}^{J} (A_j-1) \;=\; N - J . \tag{15.2}\]

The treatments that are compared, directly or by sharing a trial, form a graph.

Definition 15.1 (The arm-based network model) The treatment-comparison graph \(G=(\mathcal V,\mathcal E)\) has vertex set \(\mathcal V=\{1,\dots,K\}\), one vertex per treatment, and one edge for each within-study contrast: study \(j\) with baseline \(t_{j,1}\) contributes the \(A_j-1\) edges \(\{t_{j,1},k\}\) for \(k\in\mathcal A_j\setminus\{t_{j,1}\}\). Parallel edges from different studies are kept distinct, so \(G\) is in general a multigraph with \(|\mathcal E|=C\) edges.

For a chosen link function \(g\) (Definition 7.2), let \(\theta_{jk}=g(\bar\mu_{jk})\) be the link-scale mean outcome of arm \(k\) in study \(j\), where \(\bar\mu_{jk}\) is the marginal mean response in that arm. The arm-based network model writes the vector \(\boldsymbol\theta\in\mathbb R^{N}\) of all arm means as a linear predictor \[ \boldsymbol\theta \;=\; \mathbf X\,\boldsymbol\beta , \tag{15.3}\] for a design matrix \(\mathbf X\in\mathbb R^{N\times p}\) and a parameter vector \(\boldsymbol\beta\in\mathbb R^{p}\). Coupled with a response distribution from an exponential family (Definition 7.1), for instance \(r_{jk}\sim\mathrm{Bin}(n_{jk},g^{-1}(\theta_{jk}))\) for binary outcomes under the logit link, Equation 15.3 makes the network a generalized linear model. \(\blacktriangleleft\)

Remark. Reader’s translation of the arm model. Stack all the trial arms into one tall column \(\boldsymbol\theta\), one entry per arm. In the running example the arms are placebo and Drug A (study \(1\)), placebo and Drug B (study \(2\)), Drug A and Drug B (study \(3\)), and placebo, Drug A, Drug B (study \(4\)), so \(\boldsymbol\theta\) has \(N=9\) entries; for example \(\theta_{4,2}\) is the log-odds of a good outcome on the Drug A arm of the three-arm study \(4\). The model Equation 15.3 says every one of these arm levels is assembled from a handful of shared ingredients, the study baselines and the treatment effects, through the table \(\mathbf X\). The two appearances of the link, \(\theta_{jk}=g(\bar\mu_{jk})\) in the definition and \(g^{-1}(\theta_{jk})\) in the binomial example, are the same map read in opposite directions: \(\theta\) is the arm’s outcome level moved onto the additive scale where treatment effects add up, and \(g^{-1}\) moves it back to a probability. With the logit link, \(\theta_{jk}=0.5\) corresponds to a response probability \(g^{-1}(0.5)=1/(1+e^{-0.5})\approx0.62\). The treatment-comparison graph is nothing more than the picture of Example 15.1: a dot per treatment, a line per within-study comparison. \(\blacktriangleleft\)

Remark. Multi-arm trials: clinical pairs versus contrast rows. A three-arm trial such as study \(4\) compares all three treatment pairs within one randomized experiment, so clinically it carries direct evidence on placebo vs Drug A, placebo vs Drug B, and Drug A vs Drug B. The graph definition, however, gives it only \(A_j-1=2\) edges: after we name one arm the study baseline (here placebo), we keep the \(2\) contrasts to that baseline and treat the third pairwise comparison, Drug A vs Drug B, as their difference. This is no loss of information. The omitted contrast is recovered exactly by subtraction, \(\delta_{\text{A vs B}}=\delta_{\text{placebo vs B}}-\delta_{\text{placebo vs A}}\), and it is correlated with the two kept contrasts because all three share the placebo arm. Keeping \(A_j-1\) independent contrasts rather than all \(\binom{A_j}{2}\) pairwise comparisons is what keeps the rows of the design linearly independent; carrying their correlation, computed in Theorem 15.4, is what keeps the standard errors honest. \(\blacktriangleleft\)

Equation Equation 15.3 is the entire formal commitment of an NMA: a single linear predictor across all studies, tying the arms together through shared treatment parameters. Two specifications of \((\mathbf X,\boldsymbol\beta)\) are in standard use, and the bulk of this chapter studies them and their consequences. Before naming them we record the sense in which the chapter generalizes the last one.

Remark. Bucher as a two-study, one-triangle network. Recall the Bucher triangle of Chapter 14: two active treatments each compared against a common comparator. Relabel that comparator as the network reference, treatment \(1\), and the two actives as treatments \(2\) and \(3\). With two two-arm studies, one on \(\{1,2\}\) and one on \(\{1,3\}\), we have \(K=3\), \(N=4\), \(J=2\), and \(C=(2-1)+(2-1)=2\) within-study contrasts; the graph \(G\) is the path \(2-1-3\). The arm-based model has one baseline and one contrast per study, and eliminating the two baselines leaves exactly the contrasts \(d_{12}\) and \(d_{13}\), whose difference \(d_{13}-d_{12}=d_{23}\) is the Bucher estimator (Theorem 14.1). The graph is connected but contains no cycle, so there is direct evidence on \(2\) versus \(1\) and on \(3\) versus \(1\) but only indirect evidence on \(3\) versus \(2\); there is nothing to test for inconsistency. Adding a third study on \(\{2,3\}\) closes the triangle into a cycle and creates the direct-versus-indirect tension that node-splitting (Definition 15.4) adjudicates. The running example of Example 15.1 is this closed triangle with one further multi-arm trial, and everything in this chapter is the systematic version of this enlargement. \(\blacktriangleleft\)

Remark. Arm-based and contrast-based likelihoods. The model Equation 15.3 is arm-based: the observations are the \(N\) arms. An equivalent contrast-based form, used for estimation and in the worked example of Section 15.13, takes as data the \(C\) within-study contrast estimates \(y_{j,\,t_{j,1}k}=\hat\theta_{jk}-\hat\theta_{j,t_{j,1}}\). In large samples each contrast estimate is approximately normal about the true contrast by the delta method (Theorem 5.3), and two contrasts that share the baseline arm of a multi-arm study are correlated because the shared arm’s sampling error enters both; Theorem 15.4 computes that correlation. Passing from the arm-based to the contrast-based likelihood is exactly the elimination of the per-study baselines \(\mu_j\), which appear in each study’s arms alike and cancel in within-study differences. Concretely, in a two-arm study with baseline arm \(1\), the two arm means are \(\theta_{j1}=\mu_j\) and \(\theta_{j2}=\mu_j+\delta_{j,12}\); subtracting the baseline arm removes \(\mu_j\) and leaves the contrast \(\delta_{j,12}=\theta_{j2}-\theta_{j1}\). The recipe is the same in general: (i) start from the \(N\) arms; (ii) in each study pick a baseline arm \(t_{j,1}\); (iii) subtract that arm’s estimate from the others; (iv) keep the resulting \(A_j-1\) within-study contrasts; (v) carry their covariance, which within a multi-arm study has the shared-arm correlation of Theorem 15.4. Steps (i) to (iv) turn \(N\) arm observations into \(C\) contrast observations by discarding the \(J\) baselines; step (v) is the price of that convenience. We develop both views and prove the identifiability theory on the arm-based design, where the baselines are explicit. \(\blacktriangleleft\)

15.4 Two parameterizations of the arm means

The arm means \(\boldsymbol\theta\) carry two layers of information: the level of each study’s outcomes, which is a nuisance reflecting that study’s population and baseline risk, and the contrasts between treatments, which are the quantities of interest. The two standard parameterizations split these layers in two different ways.

Definition 15.2 (Baseline-shift parameterization) The baseline-shift parameterization (corresponds to D.5 in the proof catalogue) writes each arm mean as the study baseline plus a study-specific contrast to that baseline: \[ \theta_{jk} \;=\; \mu_j^{\mathrm{BS}} \;+\; \delta_{j,\,t_{j,1}k}, \qquad k\in\mathcal A_j, \tag{15.4}\] with the convention \(\delta_{j,\,t_{j,1}t_{j,1}}=0\) for the baseline arm, so that \(\mu_j^{\mathrm{BS}}=\theta_{j,t_{j,1}}\). The free parameters are the \(J\) baselines \(\mu_j^{\mathrm{BS}}\) and the \(C=N-J\) within-study contrasts \(\delta_{j,\,t_{j,1}k}\) for the non-baseline arms \(k\ne t_{j,1}\), a total of \(J+C=N\) parameters. \(\blacktriangleleft\)

In words, the baseline-shift form gives every study its own private answer: it names a separate contrast \(\delta_{j,t_{j,1}k}\) inside every study. It places no restriction on the arm means at all: there are as many parameters, \(N\), as arms, and Equation 15.4 is simply a relabeling of the \(N\) numbers \(\theta_{jk}\) into \(J\) baselines and \(C\) within-study differences. Its virtue is that each \(\delta_{j,t_{j,1}k}\) is the natural target of a random effect in the random-effects model of Section 15.9. Its defect is that it does not, by itself, tie studies together: the contrast \(\delta_{j,t_{j,1}k}\) of study \(j\) and the contrast \(\delta_{j',t_{j',1}k}\) of study \(j'\) for the very same pair of treatments are, a priori, distinct parameters. Tying them together is the content of the consistency assumption, which the second parameterization builds in from the start.

Definition 15.3 (Reference-treatment parameterization) Fix the network reference treatment \(1\) and a vector of network contrasts \(\mathbf d=(d_2,\dots,d_K)^\top\in\mathbb R^{K-1}\), where \(d_k\) is the relative effect of treatment \(k\) versus the reference on the link scale, with the convention \(d_1=0\). The reference-treatment parameterization (corresponds to D.6 in the proof catalogue) writes \[ \theta_{jk} \;=\; \mu_j \;+\; \bigl(d_k - d_{t_{j,1}}\bigr), \qquad k\in\mathcal A_j, \tag{15.5}\] where \(\mu_j=\theta_{j,t_{j,1}}\) is the study-\(j\) baseline. The free parameters are the \(J\) baselines \(\mu_j\) and the \(K-1\) network contrasts \(d_2,\dots,d_K\), a total of \(J+K-1\) parameters. \(\blacktriangleleft\)

In the reference-treatment form a treatment carries a single contrast \(d_k\) for the whole network. In words, this form gives each treatment one number, its effect versus the reference, good for the entire network; every study comparing two treatments must then show the same gap between them. The reference treatment \(1\) is only a coordinate origin, like the choice of where to put zero on a thermometer: it is usually placebo for readability, but any treatment could serve, and the network’s answers (any \(d_{ab}\)) do not depend on the choice. The within-study contrast of treatment \(k\) versus the study baseline is forced to be \[ \delta_{j,\,t_{j,1}k} \;=\; \theta_{jk}-\theta_{j,t_{j,1}} \;=\; d_k - d_{t_{j,1}} , \tag{15.6}\] the same expression in every study that compares those two treatments. Equation Equation 15.6 is the consistency constraint: it is automatic in the reference-treatment parameterization and is the formal statement that the relative effects are constant across the network, the population-level constancy of Definition 14.2 rendered as an algebraic identity among parameters. A caution on the word: consistency here is a property of a whole network of relative effects (direct and indirect routes agree), and it is unrelated to the SUTVA consistency of Chapter 9, which links an observed outcome to a potential outcome. The two ideas share only a name. The two parameterizations therefore differ in their parameter counts, \(N\) for baseline-shift against \(J+K-1\) for reference-treatment, and they agree precisely when the within-study contrasts obey Equation 15.6. Making that statement exact is the work of Section 15.7; first we study the algebra of the network contrasts themselves.

15.5 Consistency equations and their algebraic closure

The reference-treatment parameterization defines a contrast only against the reference. The contrast between any two treatments is recovered by subtraction, and the resulting family of \(\binom{K}{2}\) pairwise contrasts is highly redundant. The next theorem makes the redundancy precise: the family is generated by \(K-1\) free numbers and is pinned down by a definite number of loop constraints. These constraints are what an inconsistency analysis tests.

Before the general statement, here is the identity in the running example. Suppose the pooled effects come out as Drug A vs placebo \(d_{12}=0.5\) and Drug B vs placebo \(d_{13}=1.0\) on the log-odds scale. Then consistency forces the Drug B vs Drug A contrast to be \(d_{23}=d_{13}-d_{12}=0.5\): it is not free to take any other value. If a head-to-head trial instead reported Drug B vs Drug A as \(0.1\), the gap between the forced \(0.5\) and the observed \(0.1\) is exactly the disagreement that node-splitting (Section 15.10) measures. The next theorem shows that this triangle identity, and its \(K\)-treatment generalizations, are the only constraints the network imposes on its pairwise effects.

Theorem 15.1 (Closure of the consistency equations) Define, from a network-contrast vector \(\mathbf d=(d_1,\dots,d_K)\) with \(d_1=0\), the full family of pairwise network contrasts \[ d_{ab} \;:=\; d_b - d_a, \qquad a,b\in\{1,\dots,K\}. \tag{15.7}\] Then the following hold (corresponds to D.10 in the proof catalogue).

1. Antisymmetry. \(d_{ab}=-d_{ba}\) for all \(a,b\), and \(d_{aa}=0\).

2. Additivity (the triangle, or loop, identity). For all \(a,b,c\), \[ d_{ab} + d_{bc} \;=\; d_{ac}, \qquad\text{equivalently}\qquad d_{ac}-d_{bc}=d_{ab}. \tag{15.8}\]

3. Generation. Every pairwise contrast is determined by the \(K-1\) basis contrasts \(d_2,\dots,d_K\) through \(d_{ab}=d_{1b}-d_{1a}=d_b-d_a\).

4. Converse and uniqueness. Conversely, let \((e_{ab})_{a,b}\) be any family of real numbers satisfying the additivity identity Equation 15.8 for all triples. Then \(e\) is antisymmetric, and there is a unique vector \(\mathbf d\) with \(d_1=0\) such that \(e_{ab}=d_b-d_a\) for all \(a,b\), namely \(d_k=e_{1k}\).

5. Count. The set of consistent contrast families, those of the form Equation 15.7, is a linear subspace of dimension \(K-1\) inside the \(\binom{K}{2}\)-dimensional space of antisymmetric families. The additivity identities therefore impose exactly \[ \binom{K}{2}-(K-1)=\binom{K-1}{2} \tag{15.9}\] independent constraints.

Proof. Parts 1 and 2 are immediate from Equation 15.7 and reproduce, for the network contrasts, the antisymmetry and additivity of relative effects proved within a single population in Chapter 14 (Lemma 14.1). For antisymmetry, \(d_{ab}=d_b-d_a=-(d_a-d_b)=-d_{ba}\), and \(d_{aa}=0\). For additivity, \(d_{ab}+d_{bc}=(d_b-d_a)+(d_c-d_b)=d_c-d_a=d_{ac}\); the middle term \(d_b\) cancels, and subtracting \(d_{bc}\) from both sides rearranges to \(d_{ac}-d_{bc}=d_{ab}\). Part 3 is the definition Equation 15.7 read with \(a\) or \(b\) equal to \(1\): \(d_{1b}=d_b-d_1=d_b\) and \(d_{ab}=d_b-d_a=d_{1b}-d_{1a}\).

For part 4, suppose \((e_{ab})\) satisfies Equation 15.8 for all triples. Because the identity holds for every triple, we are free to choose convenient ones. Plugging the same index into all three slots, \(a=b=c\), gives \(e_{aa}+e_{aa}=e_{aa}\), so \(e_{aa}=0\): the diagonal vanishes. Choosing \(c=a\) gives \(e_{ab}+e_{ba}=e_{aa}=0\), so \(e_{ba}=-e_{ab}\) and \(e\) is antisymmetric. Define \(d_k:=e_{1k}\); then \(d_1=e_{11}=0\). Apply additivity to the triple \((a,1,b)\): \(e_{a1}+e_{1b}=e_{ab}\), and using antisymmetry \(e_{a1}=-e_{1a}=-d_a\), so \(e_{ab}=-d_a+d_b=d_b-d_a\), as claimed. Uniqueness: if \(e_{ab}=d_b-d_a=d_b'-d_a'\) for all \(a,b\) with \(d_1=d_1'=0\), then taking \(a=1\) gives \(d_b=e_{1b}=d_b'\) for every \(b\).

For part 5, let \(\mathcal A\cong\mathbb R^{\binom{K}{2}}\) be the vector space of antisymmetric families, coordinatized by the values \(e_{ab}\) for \(a<b\), and let \(\mathcal D_{\mathrm{cons}}\subseteq\mathcal A\) be the subset satisfying all additivity identities, the consistent families. The map \(T:\mathbb R^{K-1}\to\mathcal A\) sending \((d_2,\dots,d_K)\mapsto (e_{ab})\) with \(e_{ab}=d_b-d_a\) (and \(d_1=0\)) is linear. It is injective: if \(d_b-d_a=0\) for all \(a,b\), then taking \(a=1\) gives \(d_b=0\) for all \(b\). By parts 3 and 4 its image is exactly \(\mathcal D_{\mathrm{cons}}\). Hence \(\mathcal D_{\mathrm{cons}}\) is a linear subspace with \(\dim\mathcal D_{\mathrm{cons}}=\dim\mathbb R^{K-1}=K-1\), and its codimension in \(\mathcal A\) is \(\binom{K}{2}-(K-1)\). The arithmetic \(\binom{K}{2}-(K-1)=\tfrac{K(K-1)}{2}-(K-1)=(K-1)\tfrac{K-2}{2}=\binom{K-1}{2}\) gives Equation 15.9. \(\square\)

Remark. Why this matters, and how loops carry the constraints. In evidence-synthesis terms: of all the pairwise relative effects in a network, only \(K-1\) are genuinely free; every other is forced by the ones already chosen, and the forcing identities are the loop equations. Those loop equations are the only places inconsistency can surface, so a network can be checked for coherence exactly where it contains a loop, and nowhere else. The count Equation 15.9 has a matching graph-theoretic reading. On the complete graph \(K_K\) a spanning tree has \(K-1\) edges and contains no cycle; the contrasts along the tree edges are free and generate all others, which is part 3. Each of the remaining \(\binom{K}{2}-(K-1)=\binom{K-1}{2}\) edges closes exactly one independent cycle, and that cycle is one additivity identity, an equation of the form \(d_{a_1a_2}+d_{a_2a_3}+\cdots+d_{a_m a_1}=0\). The number of independent constraints equals the cycle rank of the complete graph, and a contrast is testable for inconsistency precisely when the evidence contains an independent cycle, that is, both a direct edge and an indirect path between its endpoints. A tree-shaped network, such as the Bucher triangle-as-path of Section 15.3, has no cycle and nothing to test. \(\blacktriangleleft\)

15.6 The incidence matrix and a rank toolkit

Both the equivalence of the parameterizations (Section 15.7) and the identifiability of the contrasts (Section 15.8) reduce to one number: the rank of the design matrix \(\mathbf X\). That rank is in turn controlled by a single combinatorial object, the incidence matrix of the treatment graph. This short section isolates the two linear-algebra facts we need. Lemma 15.1 computes the rank of a graph’s incidence matrix, which will pin down the contrast columns of \(\mathbf X\); Lemma 15.2 says that a full-rank \(\mathbf X\) yields an invertible weighted normal matrix, so that the generalized least squares estimator exists and is unique. A reader in a hurry may take both lemmas on faith and skip to Section 15.7; the payoff they deliver is that the question “which contrasts can the network estimate?” becomes the purely graph-theoretic “is the network connected?”

Lemma 15.1 (Rank of a graph incidence matrix) Let \(G=(\mathcal V,\mathcal E)\) be a multigraph on \(K=|\mathcal V|\) vertices, with each edge \(e\) assigned an orientation from a tail \(a(e)\) to a head \(b(e)\). Its signed incidence matrix \(\mathbf B\in\mathbb R^{|\mathcal E|\times K}\) has, in the row for edge \(e\), the entry \(-1\) in column \(a(e)\), the entry \(+1\) in column \(b(e)\), and \(0\) elsewhere. If \(G\) has \(q\) connected components, then \[ \operatorname{rank}(\mathbf B) \;=\; K-q . \tag{15.10}\] In particular \(\operatorname{rank}(\mathbf B)=K-1\) if and only if \(G\) is connected. (We write \(q\) for the number of components to keep it distinct from the within-study contrast count \(C\) of Equation 15.2.)

Proof. View \(\mathbf B\) as the linear map \(\mathbf v\mapsto \mathbf B\mathbf v\) from \(\mathbb R^K\) to \(\mathbb R^{|\mathcal E|}\). By construction the \(e\)-th coordinate of \(\mathbf B\mathbf v\) is \((\mathbf B\mathbf v)_e=v_{b(e)}-v_{a(e)}\). In words, \(\mathbf B\) turns a labeling of the vertices, one number \(v_i\) per treatment, into the list of its differences across the edges: the row for edge \(e\) reports \(v\) at the head minus \(v\) at the tail. The whole proof is then the observation that the labelings which are flat across every edge are exactly the labelings that are constant on each connected piece. Hence \(\mathbf v\in\mathcal N(\mathbf B)\) if and only if \(v_{a(e)}=v_{b(e)}\) for every edge \(e\), that is, if and only if \(\mathbf v\) takes equal values on the two endpoints of every edge.

We claim \(\mathcal N(\mathbf B)\) is the set of vectors that are constant on each connected component of \(G\). If \(\mathbf v\) is constant on components and \(e\) is an edge, its endpoints lie in the same component, so \(v_{a(e)}=v_{b(e)}\) and \(\mathbf v\in\mathcal N(\mathbf B)\). Conversely, suppose \(\mathbf v\) equalizes across every edge, and let \(u,w\) lie in the same component. By definition of a component there is a path \(u=x_0,x_1,\dots,x_m=w\) with each \(\{x_{i-1},x_i\}\in\mathcal E\); equalizing across each edge of the path gives \(v_{x_0}=v_{x_1}=\cdots=v_{x_m}\), so \(v_u=v_w\). Thus \(\mathbf v\) is constant on each component.

Let \(\mathcal V=\mathcal V_1\sqcup\cdots\sqcup\mathcal V_q\) be the partition into components and let \(\mathbf 1_{\mathcal V_i}\in\mathbb R^K\) be the indicator vector of \(\mathcal V_i\). A vector is constant on each component if and only if it is a linear combination \(\sum_i \alpha_i\mathbf 1_{\mathcal V_i}\), so these indicators span \(\mathcal N(\mathbf B)\). They are linearly independent: if \(\sum_i\alpha_i\mathbf 1_{\mathcal V_i}=\mathbf 0\), then evaluating at any vertex of \(\mathcal V_i\), which is nonempty, gives \(\alpha_i=0\). Hence \(\dim\mathcal N(\mathbf B)=q\). By the rank-nullity theorem (Theorem 1.3) applied to the map on \(\mathbb R^K\), \(\operatorname{rank}(\mathbf B)=K-\dim\mathcal N(\mathbf B)=K-q\), which is Equation 15.10. The final claim is the case \(q=1\). \(\square\)

Remark. What this buys us. The incidence matrix is pure bookkeeping for which treatments are joined by randomized comparisons. Its rank, \(K\) minus the number of disconnected pieces \(q\), is the number of independent relative effects those comparisons can pin down. A connected network (\(q=1\)) pins down all \(K-1\) contrasts; each additional disconnected piece costs exactly one estimable contrast. This single fact drives both theorems that follow, the equivalence of the two parameterizations and the identifiability of the network. \(\blacktriangleleft\)

Lemma 15.2 (Full column rank and the weighted Gram matrix) Let \(\mathbf X\in\mathbb R^{n\times p}\) and let \(\mathbf W\in\mathbb R^{n\times n}\) be symmetric positive definite. Then \[ \mathcal N(\mathbf X^\top\mathbf W\mathbf X)=\mathcal N(\mathbf X). \tag{15.11}\] Consequently \(\mathbf X^\top\mathbf W\mathbf X\) is invertible if and only if \(\mathbf X\) has full column rank \(p\), and in that case \(\mathbf X^\top\mathbf W\mathbf X\) is positive definite.

Proof. The inclusion \(\mathcal N(\mathbf X)\subseteq\mathcal N(\mathbf X^\top\mathbf W\mathbf X)\) is immediate: \(\mathbf X\mathbf v=\mathbf 0\) gives \(\mathbf X^\top\mathbf W\mathbf X\mathbf v=\mathbf 0\). For the reverse inclusion, suppose \(\mathbf X^\top\mathbf W\mathbf X\mathbf v=\mathbf 0\). Then \[ 0=\mathbf v^\top\mathbf X^\top\mathbf W\mathbf X\mathbf v=(\mathbf X\mathbf v)^\top\mathbf W(\mathbf X\mathbf v). \] Since \(\mathbf W\) is positive definite (Definition 3.7), \(\mathbf u^\top\mathbf W\mathbf u=0\) forces \(\mathbf u=\mathbf 0\); applying this with \(\mathbf u=\mathbf X\mathbf v\) gives \(\mathbf X\mathbf v=\mathbf 0\), so \(\mathbf v\in\mathcal N(\mathbf X)\). This proves Equation 15.11. The matrix \(\mathbf X^\top\mathbf W\mathbf X\) is \(p\times p\), hence invertible if and only if its null space is trivial; by Equation 15.11 this happens if and only if \(\mathcal N(\mathbf X)=\{\mathbf 0\}\), that is, if and only if \(\mathbf X\) has full column rank (Lemma 1.10). When \(\mathbf X\) has full column rank, every \(\mathbf v\ne\mathbf 0\) has \(\mathbf X\mathbf v\ne\mathbf 0\) and therefore \(\mathbf v^\top\mathbf X^\top\mathbf W\mathbf X\mathbf v=(\mathbf X\mathbf v)^\top\mathbf W(\mathbf X\mathbf v)>0\), so \(\mathbf X^\top\mathbf W\mathbf X\) is positive definite. \(\square\)

Remark. Reader’s translation. This is the workhorse behind every “the estimator exists and is unique” claim in the chapter. If the columns of \(\mathbf X\) are linearly independent (full column rank), then no matter what positive-definite weighting \(\mathbf W\) the data supply, the weighted normal matrix \(\mathbf X^\top\mathbf W\mathbf X\) can be inverted, so the generalized least squares formula returns one and only one answer. Full column rank of \(\mathbf X\) is thus the algebraic form of “the parameters are estimable.” \(\blacktriangleleft\)

We now assemble the reference-treatment design matrix explicitly, so that Lemma 15.1 can be applied to it. Order the \(N\) arms arbitrarily and let \(\boldsymbol\beta=(\boldsymbol\mu^\top,\mathbf d^\top)^\top\in\mathbb R^{J+(K-1)}\) collect the baselines and the network contrasts. By Equation 15.5 the arm-mean vector is \(\boldsymbol\theta=\mathbf X_{\mathrm{RT}}\boldsymbol\beta\) with the block structure \[ \mathbf X_{\mathrm{RT}}=\bigl[\ \mathbf X_\mu \ \big|\ \mathbf X_d\ \bigr], \qquad \mathbf X_\mu\in\mathbb R^{N\times J},\quad \mathbf X_d\in\mathbb R^{N\times (K-1)} . \tag{15.12}\] The baseline block \(\mathbf X_\mu\) has a single \(1\) in each row, in the column of the study to which that arm belongs. The contrast block \(\mathbf X_d\) has, in the row for arm \((j,k)\), a \(+1\) in column \(k\) and a \(-1\) in column \(t_{j,1}\) (omitting any entry in the dropped column \(1\), since \(d_1=0\)); for the baseline arm \(k=t_{j,1}\) the row of \(\mathbf X_d\) is zero. Thus the rows of \(\mathbf X_d\) for the \(C\) non-baseline arms are precisely the rows of the signed incidence matrix \(\mathbf B\) of \(G\) with column \(1\) deleted, each non-baseline arm \((j,k)\) corresponding to the oriented edge \(t_{j,1}\to k\). The running example makes this concrete: its contrast design \(\mathbf X\) is written out, one row per within-study contrast, in Step 1 of Example 15.2.

15.7 Equivalence of the two parameterizations

We can now state in what sense the baseline-shift and reference-treatment forms describe the same model. The baseline-shift parameterization is a saturated relabeling of the arm means, meaning it has exactly as many free parameters as there are arms, so it can reproduce any pattern of arm means whatsoever and ranges over all of \(\mathbb R^N\); the reference-treatment parameterization ranges over only a subspace, the image of the design map Equation 15.12. The theorem identifies that subspace and shows the two forms coincide on it.

Why prove this at all? We want to be certain that the tidy reference-treatment form, which bakes in consistency by giving each treatment a single effect \(d_k\), does not secretly discard any believable pattern of arm means. The theorem gives that reassurance and locates the discarded directions precisely: the reference-treatment form captures exactly the consistent arm-mean patterns, and the surplus parameters that the baseline-shift form carries are precisely the directions in which real data can disagree with consistency. That surplus is where inconsistency lives. The following table lines up the two coordinate systems.

Quantity	Baseline-shift name	Reference-treatment name
level of study \(j\)	\(\mu_j^{\mathrm{BS}}\) (free)	\(\mu_j\) (free)
study-\(j\) contrast of \(k\) vs its baseline	\(\delta_{j,\,t_{j,1}k}\) (free)	\(d_k-d_{t_{j,1}}\) (forced by Equation 15.6)
free parameters in \(\mathbb R^N\)	\(N\)	\(J+K-1\)
surplus, where inconsistency can appear	\(C-(K-1)\)	\(0\)

Theorem 15.2 (Equivalence on the consistency-constrained subspace) Consider a connected network and the reference-treatment design map \(\Phi:\mathbb R^{J}\times\mathbb R^{K-1}\to\mathbb R^{N}\), \(\Phi(\boldsymbol\mu,\mathbf d)=\mathbf X_{\mathrm{RT}}(\boldsymbol\mu,\mathbf d)\), of Equation 15.12. Define the consistency-constrained subspace \(\mathcal S_{\mathrm{cons}}=\operatorname{image}(\Phi) =\mathcal C(\mathbf X_{\mathrm{RT}})\subseteq\mathbb R^{N}\). Then (corresponds to D.7 in the proof catalogue):

1. Rank and bijection. \(\operatorname{rank}(\mathbf X_{\mathrm{RT}})=J+K-1\), so \(\Phi\) is injective, and \(\Phi:\mathbb R^{J+K-1}\to\mathcal S_{\mathrm{cons}}\) is a linear isomorphism. In particular \(\dim\mathcal S_{\mathrm{cons}}=J+K-1\).

2. Characterization of the subspace. An arm-mean vector \(\boldsymbol\theta\) lies in \(\mathcal S_{\mathrm{cons}}\) if and only if its baseline-shift contrasts satisfy the consistency constraint Equation 15.6, that is, if and only if there is a vector \(\mathbf d\) with \(\delta_{j,\,t_{j,1}k}=d_k-d_{t_{j,1}}\) for every non-baseline arm \((j,k)\). On \(\mathcal S_{\mathrm{cons}}\) the correspondence between reference-treatment coordinates \((\boldsymbol\mu,\mathbf d)\) and baseline-shift coordinates \((\boldsymbol\mu^{\mathrm{BS}}, \boldsymbol\delta)\) is a bijection.

3. Likelihood identity. Because the data enter the model only through the linear predictor \(\boldsymbol\theta\) (Definition 7.4), any two parameterizations that produce the same \(\boldsymbol\theta\) produce the same likelihood. Hence on \(\mathcal S_{\mathrm{cons}}\) the baseline-shift and reference-treatment models have identical likelihoods.

The two parameterizations have different free-parameter counts in \(\mathbb R^N\), namely \(N\) for baseline-shift against \(J+K-1\) for reference-treatment; they agree exactly on the \((J+K-1)\)-dimensional subspace \(\mathcal S_{\mathrm{cons}}\), where the consistency constraints remove the surplus \(N-(J+K-1)=C-(K-1)\) degrees of freedom.

Proof. Rank and injectivity (part 1). We compute \(\mathcal N(\mathbf X_{\mathrm{RT}})\). Suppose \(\Phi(\boldsymbol\mu,\mathbf d)=\mathbf 0\), that is, \(\mu_j+(d_k-d_{t_{j,1}})=0\) for every arm \((j,k)\). Apply this to the baseline arm \(k=t_{j,1}\) of each study, where \(d_k-d_{t_{j,1}}=0\); it gives \(\mu_j=0\) for every \(j\). With all baselines zero, the equation for the non-baseline arms reads \(d_k-d_{t_{j,1}}=0\) for every within-study edge \(t_{j,1}\to k\), that is, \(\mathbf B\tilde{\mathbf d}=\mathbf 0\) where \(\tilde{\mathbf d}=(d_1,\dots,d_K)\) is the padded contrast vector that restores the dropped reference coordinate \(d_1=0\) (the toolkit table of Section 15.2 records this convention; the unpadded \(\mathbf d=(d_2,\dots,d_K)\) has length \(K-1\)), and \(\mathbf B\) is the full signed incidence matrix of \(G\). By Lemma 15.1, \(\tilde{\mathbf d}\in\mathcal N(\mathbf B)\) means \(\tilde{\mathbf d}\) is constant on each connected component of \(G\). Since \(G\) is connected, \(\tilde{\mathbf d}\) is constant on all of \(\mathcal V\), and the constant equals \(d_1=0\), so \(\tilde{\mathbf d}=\mathbf 0\) and hence \(\mathbf d=\mathbf 0\). Thus \(\mathcal N(\mathbf X_{\mathrm{RT}})=\{\mathbf 0\}\), \(\Phi\) is injective, and \(\operatorname{rank}(\mathbf X_{\mathrm{RT}})=J+K-1\). An injective linear map onto its image is an isomorphism, so \(\Phi:\mathbb R^{J+K-1}\to\mathcal S_{\mathrm{cons}}\) is a linear isomorphism and \(\dim\mathcal S_{\mathrm{cons}}=J+K-1\).

Characterization (part 2). The baseline-shift coordinates of any \(\boldsymbol\theta\in\mathbb R^N\) are \(\mu_j^{\mathrm{BS}}=\theta_{j,t_{j,1}}\) and \(\delta_{j,\,t_{j,1}k}=\theta_{jk}-\theta_{j,t_{j,1}}\); this map \(\Psi:\boldsymbol\theta\mapsto(\boldsymbol\mu^{\mathrm{BS}},\boldsymbol\delta)\) is a linear bijection of \(\mathbb R^N\) onto \(\mathbb R^J\times\mathbb R^C\), since \(\boldsymbol\theta\) is recovered by \(\theta_{j,t_{j,1}}=\mu_j^{\mathrm{BS}}\) and \(\theta_{jk}=\mu_j^{\mathrm{BS}}+\delta_{j,t_{j,1}k}\). If \(\boldsymbol\theta\in\mathcal S_{\mathrm{cons}}\), write \(\boldsymbol\theta=\Phi(\boldsymbol\mu,\mathbf d)\); then \(\theta_{jk}-\theta_{j,t_{j,1}}=(d_k-d_{t_{j,1}})-(d_{t_{j,1}}-d_{t_{j,1}})=d_k-d_{t_{j,1}}\), so the baseline-shift contrasts satisfy Equation 15.6. Conversely, if the baseline-shift contrasts of some \(\boldsymbol\theta\) satisfy \(\delta_{j,t_{j,1}k}=d_k-d_{t_{j,1}}\) for a vector \(\mathbf d\), set \(\mu_j:=\theta_{j,t_{j,1}}\); then \(\theta_{jk}=\theta_{j,t_{j,1}}+\delta_{j,t_{j,1}k}=\mu_j+(d_k-d_{t_{j,1}})=\Phi(\boldsymbol\mu,\mathbf d)_{jk}\), so \(\boldsymbol\theta\in\mathcal S_{\mathrm{cons}}\). The vector \(\mathbf d\) is unique by the injectivity of \(\Phi\) established in part 1, so the correspondence \((\boldsymbol\mu,\mathbf d)\leftrightarrow(\boldsymbol\mu^{\mathrm{BS}},\boldsymbol\delta)\) is a bijection on \(\mathcal S_{\mathrm{cons}}\), being the composition of the isomorphism \(\Phi\) with the bijection \(\Psi\).

Likelihood identity (part 3). In a generalized linear model the log-likelihood is a function of the data and of the mean vector \(g^{-1}(\boldsymbol\theta)\), hence of \(\boldsymbol\theta\) alone (Definition 7.4); it does not see the parameterization that produced \(\boldsymbol\theta\). If \((\boldsymbol\mu^{\mathrm{BS}},\boldsymbol\delta)\) and \((\boldsymbol\mu,\mathbf d)\) are matched coordinates of the same \(\boldsymbol\theta\in\mathcal S_{\mathrm{cons}}\), as in part 2, they yield the same \(\boldsymbol\theta\) and therefore the same likelihood value. The dimension bookkeeping is the difference of the two parameter counts established in Definition 15.2 and Definition 15.3, \(N-(J+K-1)=(N-J)-(K-1)=C-(K-1)\). \(\square\)

Remark. Why the surplus is harmless and where it bites. A naming caution first: the arm-mean subspace \(\mathcal S_{\mathrm{cons}}\subseteq\mathbb R^N\) used here and the contrast-family space \(\mathcal D_{\mathrm{cons}}\) of Theorem 15.1 both encode consistency, but they live in different spaces, one of arm means and one of pairwise contrasts; do not conflate them. When the network has only two-arm studies and no repeated comparisons, \(C=K-1\) and the two parameterizations have the same count; baseline-shift and reference-treatment are then merely two coordinate systems on the same space. The surplus \(C-(K-1)\) is strictly positive exactly when some treatments are compared more than once, either by a multi-arm study, which contributes \(A_j-1>1\) contrasts, or by several studies sharing a treatment pair. In those cases the baseline-shift form has genuinely more names than parameters, and the consistency constraints Equation 15.6, automatic under reference-treatment, become substantive equalities that the data can contradict. That possibility of contradiction is inconsistency, and Section 15.10 makes it testable. The reference-treatment parameterization is, in this sense, the minimal description of a consistent network. \(\blacktriangleleft\)

15.8 Fixed-effect identifiability

In the fixed-effect model the within-study contrasts obey the consistency constraint exactly, so the parameters are \((\boldsymbol\mu,\mathbf d)\) and the model is the reference-treatment generalized linear model with design Equation 15.12. Identifiability of the network contrasts is then a clean rank condition, and the rank is settled by Lemma 15.1. The headline is blunt and worth stating before the proof: if the network is disconnected, no statistical method can estimate a contrast between the disconnected pieces; identifiability is connectivity, full stop. Everything below makes that precise and says what to do instead.

Theorem 15.3 (Fixed-effect NMA is identifiable iff the network is connected) Consider the fixed-effect network model \(\boldsymbol\theta=\mathbf X_{\mathrm{RT}}\boldsymbol\beta\) with \(\boldsymbol\beta=(\boldsymbol\mu,\mathbf d)\), and a working covariance \(\mathbf V\succ 0\) for the arm-level (or contrast-level) data. Let \(q\) be the number of connected components of the treatment-comparison graph \(G\) (the same \(q\) as in Lemma 15.1, kept distinct from the contrast count \(C\)). Then (corresponds to D.8 in the proof catalogue):

1. \(\operatorname{rank}(\mathbf X_{\mathrm{RT}})=J+K-q\). In particular the design has full column rank \(J+K-1\) if and only if \(G\) is connected.

2. The following are equivalent: (a) \(G\) is connected; (b) \(\mathbf X_{\mathrm{RT}}\) has full column rank; (c) the weighted normal matrix \(\mathbf X_{\mathrm{RT}}^\top\mathbf V^{-1}\mathbf X_{\mathrm{RT}}\) is invertible; (d) the parameter \(\boldsymbol\beta\) is identifiable, meaning distinct \(\boldsymbol\beta\) give distinct mean vectors; (e) the generalized least squares estimator \[ \hat{\boldsymbol\beta} =\bigl(\mathbf X_{\mathrm{RT}}^\top\mathbf V^{-1}\mathbf X_{\mathrm{RT}}\bigr)^{-1} \mathbf X_{\mathrm{RT}}^\top\mathbf V^{-1}\mathbf y \tag{15.13}\] exists and is the unique minimizer of the weighted least squares objective \((\mathbf y-\mathbf X_{\mathrm{RT}}\boldsymbol\beta)^\top\mathbf V^{-1}(\mathbf y-\mathbf X_{\mathrm{RT}}\boldsymbol\beta)\), and is then the best linear unbiased estimator of \(\boldsymbol\beta\).

3. When \(G\) is disconnected, every contrast \(d_{ab}\) between treatments in different components is non-identifiable: it can be changed without changing the distribution of the data.

Proof. Part 1. From the kernel computation in the proof of Theorem 15.2, any \((\boldsymbol\mu,\mathbf d)\in\mathcal N(\mathbf X_{\mathrm{RT}})\) has \(\boldsymbol\mu=\mathbf 0\) forced by the baseline-arm rows, and \(\tilde{\mathbf d}=(0,d_2,\dots,d_K)\) lying in \(\mathcal N(\mathbf B)\). By Lemma 15.1, \(\mathcal N(\mathbf B)\) is the span of the \(q\) component indicators and has dimension \(q\); intersecting with the hyperplane \(\{d_1=0\}\) removes exactly one dimension, the one along the indicator of the component containing vertex \(1\), leaving a space of dimension \(q-1\). Hence \(\dim\mathcal N(\mathbf X_{\mathrm{RT}})=q-1\), and by rank-nullity \(\operatorname{rank}(\mathbf X_{\mathrm{RT}})=(J+K-1)-(q-1)=J+K-q\). For \(q=1\) this is the full column rank \(J+K-1\); for \(q>1\) it is strictly smaller.

Part 2. (a)\(\Leftrightarrow\)(b) is the \(q=1\) case of part 1. (b)\(\Leftrightarrow\)(c) is Lemma 15.2 with \(\mathbf W=\mathbf V^{-1}\succ 0\). For (b)\(\Rightarrow\)(d): if \(\mathbf X_{\mathrm{RT}}\) has full column rank then \(\boldsymbol\beta\mapsto\boldsymbol\theta=\mathbf X_{\mathrm{RT}}\boldsymbol\beta\) is injective, and since the mean vector \(g^{-1}(\boldsymbol\theta)\) is a coordinatewise strictly monotone, hence injective, function of \(\boldsymbol\theta\), distinct \(\boldsymbol\beta\) give distinct means and so distinct distributions; this is identifiability. For (d)\(\Rightarrow\)(b): contrapositively, if \(\mathbf X_{\mathrm{RT}}\) is column-rank-deficient there is \(\boldsymbol\beta_0\ne\mathbf 0\) with \(\mathbf X_{\mathrm{RT}}\boldsymbol\beta_0=\mathbf 0\), so \(\boldsymbol\beta\) and \(\boldsymbol\beta+\boldsymbol\beta_0\) give the same \(\boldsymbol\theta\) and the same distribution, contradicting identifiability. For (c)\(\Rightarrow\)(e): invertibility of \(\mathbf X_{\mathrm{RT}}^\top\mathbf V^{-1}\mathbf X_{\mathrm{RT}}\) makes Equation 15.13 well defined. Writing \(\mathbf V^{-1/2}\) for the symmetric positive definite square root of \(\mathbf V^{-1}\) (Proposition 3.2) and whitening, \(\tilde{\mathbf y}=\mathbf V^{-1/2}\mathbf y\), \(\tilde{\mathbf X}=\mathbf V^{-1/2}\mathbf X_{\mathrm{RT}}\), the weighted objective becomes the ordinary sum of squares \(\|\tilde{\mathbf y}-\tilde{\mathbf X}\boldsymbol\beta\|_2^2\), whose unique minimizer is the ordinary least squares estimator \((\tilde{\mathbf X}^\top\tilde{\mathbf X})^{-1}\tilde{\mathbf X}^\top\tilde{\mathbf y}=\hat{\boldsymbol \beta}\) of Equation 15.13, unique because \(\tilde{\mathbf X}=\mathbf V^{-1/2}\mathbf X_{\mathrm{RT}}\) has the same trivial kernel as \(\mathbf X_{\mathrm{RT}}\). The whitened errors have covariance \(\mathbf V^{-1/2}\mathbf V\mathbf V^{-1/2}=I\), so the Gauss-Markov theorem (Theorem 6.3) applies to the whitened model and makes \(\hat{\boldsymbol\beta}\) the best linear unbiased estimator. Finally (e)\(\Rightarrow\)(c) holds because Equation 15.13 requires the inverse to exist. This closes the cycle of equivalences.

Part 3. Suppose \(a\) and \(b\) lie in different components \(\mathcal V_i\ne\mathcal V_{i'}\) of \(G\). Construct \(\boldsymbol\beta_0=(\mathbf 0,\mathbf d_0)\) with \(\tilde{\mathbf d}_0=\mathbf 1_{\mathcal V_{i'}}\) if \(1\notin\mathcal V_{i'}\), or \(\tilde{\mathbf d}_0=\mathbf 1_{\mathcal V_i}\) if instead \(1\notin\mathcal V_i\) (at least one of the two components omits vertex \(1\)). Then \(\tilde{\mathbf d}_0\in\mathcal N(\mathbf B)\) with \(d_{0,1}=0\), so \(\boldsymbol\beta_0\in\mathcal N(\mathbf X_{\mathrm{RT}})\) by part 1, and \(\boldsymbol\beta_0\) changes \(d_{ab}=d_b-d_a\) by \(\pm 1\ne 0\) while leaving \(\boldsymbol\theta\), hence the distribution, unchanged. Thus \(d_{ab}\) is non-identifiable. \(\square\)

Remark. Plain-language takeaway. In practice: if every treatment can be reached from every other by following trial comparisons, the fixed-effect NMA can estimate all relative effects, direct or indirect. If it cannot, the missing cross-cluster contrasts are lost to statistics entirely; no amount of clever pooling recovers them. The only remedies are to find a trial that bridges the gap, or to bring in an external assumption that supplies the missing link, which is the business of the population-adjustment methods of Part V. \(\blacktriangleleft\)

Remark. The estimable contrasts of a disconnected network. For a concrete picture, suppose trials of Drug A versus placebo and Drug B versus placebo form one cluster \(\{\text{placebo},\text{A},\text{B}\}\), while a separate group of trials compares Drug C against Drug D, with no trial ever linking the two groups:

   placebo --- Drug A          Drug C --- Drug D
      |
   Drug B

Here \(q=2\) components. Within the first cluster, Drug A versus Drug B is identifiable (the two are connected through placebo); but Drug A versus Drug C is not, because one may add any constant to both Drug C’s and Drug D’s effects without changing a single observed within-study contrast. Part 3 shows that cross-component contrasts vanish from the identifiable parameters, but within each connected component the analysis goes through verbatim: restricting to a component with \(K_i\) treatments and its studies gives a connected subnetwork whose \(K_i-1\) contrasts are identifiable. A disconnected network is therefore not useless; it is several separate networks, and no indirect comparison can bridge them without an external assumption. Bridging disconnected components is exactly the task that unanchored population adjustment confronts in Part V, where the missing edge is supplied by modeling absolute outcomes rather than contrasts. \(\blacktriangleleft\)

15.9 Random-effects NMA and the one-half correlation

The single most actionable warning in this chapter lives in this section: a three-arm trial is not two independent two-arm trials. Its within-trial contrasts share an arm, so their sampling errors are correlated; treating them as independent, a genuinely common mistake in practice, produces wrong standard errors. Below we derive that correlation and show it is not an arbitrary modeling choice but a forced value.

One caution before we start. Random effects model heterogeneity, the scatter of study-level effects around shared network effects; they do not repair inconsistency, the disagreement of direct and indirect routes. A random-effects NMA still assumes the network is consistent in expectation. Heterogeneity is variation around the network effects; inconsistency is a conflict between them, and no heterogeneity variance can absorb it.

Fixed-effect NMA assumes the within-study contrasts equal the network contrasts exactly. The random-effects model (Definition 8.3) relaxes this: the study-level contrasts vary about the network contrasts with a heterogeneity variance \(\tau^2\), just as in pairwise random-effects meta-analysis (Theorem 13.3, Definition 13.3). In a multi-arm study the several contrasts to the shared baseline are then correlated, and the correlation takes a universal value. We first show that the value is forced by a symmetry requirement, then exhibit the construction that realizes it.

Lemma 15.3 (The one-half correlation is forced by basis invariance) Let a study compare arms \(1,2,\dots,m+1\) with arm \(1\) the baseline, and let the random within-study contrasts \(\delta_{1k}:=\delta_{j,1k}\) (\(k=2,\dots,m+1\)) have an exchangeable covariance: equal variances \(\operatorname{Var}(\delta_{1k})=v\) and equal pairwise covariances \(\operatorname{Cov}(\delta_{1k},\delta_{1l})=\rho v\) for \(k\ne l\). Suppose the heterogeneity variance is required to be the same for every contrast in the study regardless of which arm is named the baseline, that is, \(\operatorname{Var}(\delta_{1l}-\delta_{1k})=v\) for the derived contrast \(\delta_{kl}\) of arm \(l\) versus arm \(k\). Then necessarily \[ \rho=\tfrac12 . \tag{15.14}\]

Proof. The derived contrast of arm \(l\) versus arm \(k\) is \(\delta_{kl}=\delta_{1l}-\delta_{1k}\), since \((\theta_l-\theta_1)-(\theta_k-\theta_1)=\theta_l-\theta_k\). Its variance is \[ \operatorname{Var}(\delta_{1l}-\delta_{1k}) =\operatorname{Var}(\delta_{1l})+\operatorname{Var}(\delta_{1k})-2\operatorname{Cov}(\delta_{1l},\delta_{1k}) =v+v-2\rho v=2v(1-\rho). \] Setting this equal to \(v\), the common heterogeneity variance, and dividing by \(v>0\) gives \(2(1-\rho)=1\), hence \(\rho=\tfrac12\). \(\square\)

Remark. Reader’s translation. The lemma answers the question “why exactly one half, and not some other correlation?” The answer is a symmetry demand: the heterogeneity of a treatment comparison should not change if we rename which arm is called the baseline. Only the correlation \(1/2\) makes every derived contrast carry the same variance \(v\) as the originals; any other value would make the measured heterogeneity depend on an arbitrary bookkeeping choice. \(\blacktriangleleft\)

The requirement in Lemma 15.3 is natural: the heterogeneity of a treatment effect across studies should not depend on the bookkeeping choice of which treatment is written as the reference. The unique covariance compatible with that invariance is the one with all correlations equal to one half. The following construction shows that this covariance arises from the simplest possible generative assumption, exchangeable arm-level random effects, and computes it for any number of arms. One notational point to flag before the statement: the construction writes the arm-level variance as \(\tau^2/2\), not \(\tau^2\), and this is deliberate. Each contrast is a difference of two independent arm effects, so its variance is \(\tau^2/2+\tau^2/2=\tau^2\); the half on the arms is chosen precisely so that the contrast variance comes out to the network heterogeneity \(\tau^2\). Read \(\tau^2/2\) as “whatever arm-level variance makes the contrasts have variance \(\tau^2\).”

Theorem 15.4 (Contrast correlation in a multi-arm trial is one half) Model each treatment’s true effect in study \(j\) as \(\eta_{jk}=d_k+u_{jk}\), where \(d_1=0\), the network contrast of treatment \(k\) is \(d_k\), and the arm-level random effects \(u_{jk}\) are independent across treatments and studies with \(u_{jk}\sim\mathcal N(0,\tau^2/2)\). Then the within-study contrasts to the baseline arm \(1\), namely \(\delta_{j,1k}=\eta_{jk}-\eta_{j1}\) for \(k=2,\dots,A_j\), satisfy (corresponds to D.9 in the proof catalogue) \[ \mathbb E[\delta_{j,1k}]=d_k,\qquad \operatorname{Var}(\delta_{j,1k})=\tau^2,\qquad \operatorname{Cov}(\delta_{j,1k},\delta_{j,1l})=\tfrac{\tau^2}{2}\ (k\ne l), \tag{15.15}\] so the correlation between any two contrasts sharing the baseline arm is \(\tfrac12\). Stacking the \((A_j-1)\) contrasts, their covariance matrix is \[ \boldsymbol\Sigma_j =\tau^2\,\frac{I_{A_j-1}+J_{A_j-1}}{2}, \tag{15.16}\] where \(J_m=\mathbf 1_m\mathbf 1_m^\top\) is the \(m\times m\) all-ones matrix (every entry equal to \(1\)). This matrix is positive definite, with eigenvalues \(\tfrac{\tau^2}{2}(A_j)\) once and \(\tfrac{\tau^2}{2}\) with multiplicity \(A_j-2\), and determinant \(\det\boldsymbol\Sigma_j=A_j\,(\tau^2/2)^{A_j-1}\).

Proof. Since \(d_1=0\) and \(u_{j1}\sim\mathcal N(0,\tau^2/2)\), the contrast is \(\delta_{j,1k}=\eta_{jk}-\eta_{j1}=(d_k+u_{jk})-u_{j1}\), whose mean is \(d_k\). Using independence of the arm-level effects, \[ \operatorname{Var}(\delta_{j,1k})=\operatorname{Var}(u_{jk})+\operatorname{Var}(u_{j1}) =\tfrac{\tau^2}{2}+\tfrac{\tau^2}{2}=\tau^2 . \] For \(k\ne l\), both different from \(1\), bilinearity of covariance and the independence of distinct arm-level effects give \[ \operatorname{Cov}(\delta_{j,1k},\delta_{j,1l}) =\operatorname{Cov}(u_{jk}-u_{j1},\,u_{jl}-u_{j1}) =\operatorname{Cov}(u_{jk},u_{jl})-\operatorname{Cov}(u_{jk},u_{j1})-\operatorname{Cov}(u_{j1},u_{jl}) +\operatorname{Var}(u_{j1}), \] in which the first three terms vanish and the last equals \(\tau^2/2\). The correlation is \((\tau^2/2)/\tau^2=\tfrac12\), which is Equation 15.15. Writing \(m=A_j-1\), the covariance matrix has diagonal entries \(\tau^2\) and off-diagonal entries \(\tau^2/2\), that is, \(\boldsymbol\Sigma_j=\tfrac{\tau^2}{2}(I_m+J_m)\), the matrix Equation 15.16.

For the spectrum, \(J_m=\mathbf 1_m\mathbf 1_m^\top\) has eigenvalue \(m\) with eigenvector \(\mathbf 1_m\) (since \(J_m\mathbf 1_m=m\mathbf 1_m\)) and eigenvalue \(0\) with multiplicity \(m-1\) on the orthogonal complement \(\{\mathbf v:\mathbf 1_m^\top\mathbf v=0\}\). Hence \(I_m+J_m\) has eigenvalue \(1+m=A_j\) once and \(1\) with multiplicity \(m-1=A_j-2\), and \(\boldsymbol\Sigma_j=\tfrac{\tau^2}{2}(I_m+J_m)\) has eigenvalues \(\tfrac{\tau^2}{2}A_j\) and \(\tfrac{\tau^2}{2}\) as stated (Theorem 3.2). All eigenvalues are positive for \(\tau^2>0\), so \(\boldsymbol\Sigma_j\succ 0\) (Theorem 3.3); the determinant is the product of the eigenvalues, \(\det\boldsymbol\Sigma_j=\tfrac{\tau^2}{2}A_j\cdot(\tfrac{\tau^2}{2})^{A_j-2}=A_j(\tau^2/2)^{A_j-1}\). \(\square\)

Remark. The same one-half in the sampling covariance. The one-half of Theorem 15.4 governs the heterogeneity covariance of the random contrasts. An identical one-half appears in the sampling covariance of the contrast estimates from a multi-arm trial when the arms have equal precision: if \(\hat\eta_{jk}\) are independent arm estimates with common variance \(\sigma^2\), then the shared baseline arm makes \(\operatorname{Cov}(\hat\eta_{jk}-\hat\eta_{j1},\hat\eta_{jl}-\hat\eta_{j1})=\operatorname{Var}(\hat\eta_{j1})=\sigma^2\), while each contrast has variance \(2\sigma^2\), so the correlation is again \(\tfrac12\). This is the positive covariance flagged after Theorem 14.2, the reason a single three-arm trial cannot be treated as two independent two-arm trials. The worked example of Section 15.13 uses equal-precision arms, so its multi-arm sampling block has correlation exactly one half. Numerically, its study \(4\) gives two contrasts each with variance \(0.20\) and covariance \(0.10\), correlation exactly \(1/2\); their difference (the derived Drug A versus Drug B contrast) has variance \(0.20+0.20-2(0.10)=0.20\), the same as each original contrast, which is the invariance Lemma 15.3 demanded. Treating the two study-4 contrasts as independent would instead give \(0.20+0.20=0.40\), doubling the apparent variance and misstating the precision of every multi-arm comparison. \(\blacktriangleleft\)

15.10 Inconsistency and node-splitting

When a contrast is informed by both a direct comparison and one or more indirect paths, the consistency equations of Theorem 15.1 become falsifiable. Node-splitting compares the two streams of evidence at a chosen contrast and tests whether they agree. We first record the fact that lets us speak of a single indirect estimate: the network pools a contrast’s direct and indirect evidence by inverse-variance weighting.

Proposition 15.1 (A contrast is the inverse-variance blend of its direct and indirect evidence) Suppose a contrast \(d_{AB}\) is informed by two independent sources, a direct estimate \(\hat d^{\mathrm{dir}}_{AB}\) with variance \(v_{\mathrm{dir}}\) and an indirect estimate \(\hat d^{\mathrm{ind}}_{AB}\) with variance \(v_{\mathrm{ind}}\), both unbiased for \(d_{AB}\). The fixed-effect network estimate combining the two sources is the inverse-variance-weighted mean \[ \hat d^{\mathrm{net}}_{AB} =\frac{w_{\mathrm{dir}}\,\hat d^{\mathrm{dir}}_{AB}+w_{\mathrm{ind}}\,\hat d^{\mathrm{ind}}_{AB}} {w_{\mathrm{dir}}+w_{\mathrm{ind}}}, \qquad w_{\mathrm{dir}}=\frac1{v_{\mathrm{dir}}},\quad w_{\mathrm{ind}}=\frac1{v_{\mathrm{ind}}}, \tag{15.17}\] with variance \(1/(w_{\mathrm{dir}}+w_{\mathrm{ind}})\), and this is the best linear unbiased estimate of \(d_{AB}\) from the two sources.

Proof. The two sources are independent unbiased estimators of the same scalar \(d_{AB}\), so this is exactly the two-study fixed-effect meta-analysis of Chapter 13: the inverse-variance-weighted mean Equation 15.17 is the best linear unbiased combination, with the stated variance, by Theorem 13.1 (the \(k=2\) case). Independence makes the variance of Equation 15.17 equal to \((w_{\mathrm{dir}}^2v_{\mathrm{dir}}+w_{\mathrm{ind}}^2v_{\mathrm{ind}})/(w_{\mathrm{dir}}+w_{\mathrm{ind}})^2 =(w_{\mathrm{dir}}+w_{\mathrm{ind}})/(w_{\mathrm{dir}}+w_{\mathrm{ind}})^2=1/(w_{\mathrm{dir}}+w_{\mathrm{ind}})\). \(\square\)

Definition 15.4 (Node-splitting) Let \(d_{AB}\) be a contrast for which the network contains both direct evidence (one or more studies comparing \(A\) and \(B\)) and indirect evidence (a path through other treatments). Node-splitting separates the evidence on this contrast into the direct estimate \(\hat d^{\mathrm{dir}}_{AB}\), computed from the \(AB\) studies alone, and the indirect estimate \(\hat d^{\mathrm{ind}}_{AB}\), computed from the network with the direct \(AB\) evidence removed. The inconsistency statistic for the contrast is their difference, \[ \hat\omega_{AB}:=\hat d^{\mathrm{dir}}_{AB}-\hat d^{\mathrm{ind}}_{AB} \tag{15.18}\] (corresponds to D.11 in the proof catalogue). A node-split model is the network model in which the single consistency equation for \(d_{AB}\) is released, letting the direct and indirect contrasts differ by \(\omega_{AB}\). \(\blacktriangleleft\)

Remark. Reader’s translation, and the name. “Node-splitting” is a slightly misleading name: the method splits the evidence for a comparison, a contrast such as \(d_{AB}\), into its direct and indirect streams; it does not split a single treatment node. Operationally the indirect estimate is computed by refitting the whole network by generalized least squares with the direct \(AB\) rows deleted, then reading off the \(AB\) contrast; Example 15.2, Step 5, carries this out on the running example. The statistic \(\hat\omega_{AB}\) is attached to the edge \(AB\), one node-split per contrast that carries both kinds of evidence. \(\blacktriangleleft\)

Proposition 15.2 (Distribution of the inconsistency statistic) Assume the direct and indirect evidence streams are independent, \(\hat d^{\mathrm{dir}}_{AB}\) is unbiased for \(d_{AB}\) with variance \(\operatorname{Var}(\hat d^{\mathrm{dir}}_{AB})\), and the indirect estimate is formed along the path \(A\to C\to B\) by the Bucher rule \(\hat d^{\mathrm{ind}}_{AB}=\hat d_{AC}-\hat d_{BC}\) of Theorem 14.1. Under the null hypothesis \(H_0\) that the consistency equation Equation 15.8 holds for the triangle \(\{A,B,C\}\), \[ \mathbb E[\hat\omega_{AB}]=0,\qquad \operatorname{Var}(\hat\omega_{AB}) =\operatorname{Var}(\hat d^{\mathrm{dir}}_{AB})+\operatorname{Var}(\hat d_{AC})+\operatorname{Var}(\hat d_{BC}), \tag{15.19}\] and the standardized statistic \(\hat\omega_{AB}/\widehat{\operatorname{SE}}(\hat\omega_{AB})\) is approximately standard normal in large samples.

Proof. The direct estimate is unbiased by hypothesis, \(\mathbb E[\hat d^{\mathrm{dir}}_{AB}]=d_{AB}\). The indirect estimate is the Bucher estimator, unbiased for \(d_{AC}-d_{BC}\) by Theorem 14.1; under \(H_0\) the consistency equation Equation 15.8 gives \(d_{AC}-d_{BC}=d_{AB}\), so \(\mathbb E[\hat d^{\mathrm{ind}}_{AB}]=d_{AB}\) as well. Subtracting, \(\mathbb E[\hat\omega_{AB}]=d_{AB}-d_{AB}=0\). For the variance, the direct stream is independent of the two indirect studies, and within the indirect estimate the studies \(AC\) and \(BC\) are independent, so by the variance-of-a-difference identity (Theorem 14.2) \(\operatorname{Var}(\hat d^{\mathrm{ind}}_{AB})=\operatorname{Var}(\hat d_{AC})+\operatorname{Var}(\hat d_{BC})\), and adding the independent direct variance gives Equation 15.19. The asymptotic normality of \(\hat\omega_{AB}\) follows from that of its three independent, asymptotically normal constituents (Theorem 5.6, Theorem 5.3), and standardizing by a consistent standard-error estimate yields the standard normal reference. \(\square\)

Remark. Reading a node-split, and its limits. A small \(|\hat\omega_{AB}|\) relative to its standard error is the empirical signature of consistency on that loop; a large value signals that direct and indirect evidence disagree, which under the structural account of Chapter 14 (Section 14.12) points to an effect modifier (Definition 11.3) distributed differently across the studies in the loop, breaking transitivity (Definition 14.3). Two cautions attend the test. First, it has the variance Equation 15.19, a sum of three variances, so it is often underpowered; a non-significant node-split is weak evidence of consistency, not proof of it. Second, by Proposition 15.1 the full network estimate of \(d_{AB}\) is the inverse-variance blend Equation 15.17 of the very direct and indirect estimates the node-split compares, so a strong inconsistency means the pooled estimate averages two genuinely discordant numbers and should not be trusted without explanation. A global check compares, by the deviance information criterion of Definition 15.6, the consistency model against an inconsistency model that frees all loops, the unrelated-mean-effects model; a large drop in fit on freeing the loops is network-level inconsistency. The detailed bias calculation behind a transitivity failure, and its interaction with non-collapsibility (Definition 7.7) and ecological bias (Definition 16.2), is the subject of Chapter 16 (Theorem 16.1). \(\blacktriangleleft\)

15.11 Individual-patient-data network meta-regression

This section is a bridge to Parts IV and V, not part of the core standard-NMA toolkit; a reader meeting NMA for the first time can note the result and return to it later. Its importance is that it is the point where patient covariates, and therefore effect modifiers, formally enter the model, which is exactly what the population-adjustment methods will exploit.

When individual patient data (IPD) are available, the covariates of each patient can enter the linear predictor, and treatment-by-covariate interactions, the effect modifiers, can be estimated rather than assumed away. The resulting model is the bridge from standard NMA to the population-adjustment methods of Parts IV and V. Concretely, imagine one covariate, baseline LDL cholesterol. Then the prognostic main effect \(\boldsymbol\beta_1\) measures how LDL shifts every patient’s outcome regardless of treatment, while the interaction \(\boldsymbol\beta_{2,k}\) measures how Drug \(k\)’s benefit itself changes with LDL, that is, LDL acting as an effect modifier. Condition (iii) below, within-study covariate variation, is the demand that LDL vary among patients inside each trial, not only between trials, so that these slopes can be read off within a study.

Theorem 15.5 (The IPD network meta-regression and its identifiability) For individual \(i\) in study \(j\) on arm \(k\), with covariate vector \(\mathbf x_{ijk}\in\mathbb R^{p}\), the IPD network meta-regression (corresponds to D.12 in the proof catalogue) models the link-scale mean as \[ g\bigl(\mathbb E[Y_{ijk}]\bigr) =\mu_j+\mathbf x_{ijk}^\top\boldsymbol\beta_1+d_k+\mathbf x_{ijk}^\top\boldsymbol\beta_{2,k}, \tag{15.20}\] with study baselines \(\mu_j\), prognostic main effects \(\boldsymbol\beta_1\) (Definition 11.5), network treatment contrasts \(d_k\) (\(d_1=0\)), and treatment-by-covariate interactions \(\boldsymbol\beta_{2,k}\) (Definition 11.3; \(\boldsymbol\beta_{2,1}=\mathbf 0\)). Assume

1. shared interactions: \(\boldsymbol\beta_{2,k}\) depends on the treatment \(k\) and not on the study, so it is one vector per treatment across the network;

2. connectedness: the treatment-comparison graph \(G\) is connected;

3. within-study covariate variation: for each study \(j\) the within-study covariate covariance matrix is nonsingular, so the covariates are not collinear and not constant within a study;

4. multi-arm anchoring: each study has at least two arms.

Then the parameters \((\boldsymbol\mu,\boldsymbol\beta_1,\mathbf d,\{\boldsymbol\beta_{2,k}\}_{k\ge2})\) are identifiable: distinct parameter vectors give distinct conditional-mean functions on the observed arms and covariate supports.

Proof. Identifiability means the map from parameters to the collection of conditional-mean functions is injective. Suppose two parameter vectors, unprimed and primed, induce the same conditional mean on every observed arm \((j,k)\) and at every covariate value \(\mathbf x\) in that arm’s support. Equating Equation 15.20 for the two vectors, for each present arm \((j,k)\) and all \(\mathbf x\) in its support, \[ \bigl(\mu_j+d_k\bigr)+\mathbf x^\top\bigl(\boldsymbol\beta_1+\boldsymbol\beta_{2,k}\bigr) =\bigl(\mu_j'+d_k'\bigr)+\mathbf x^\top\bigl(\boldsymbol\beta_1'+\boldsymbol\beta_{2,k}'\bigr). \tag{15.21}\] By (iii) the covariate support within study \(j\) affinely spans \(\mathbb R^{p}\), and randomization gives each arm of study \(j\) that same support; an affine function \(\mathbf x\mapsto a+\mathbf x^\top\mathbf b\) that is identically equal to another on an affinely spanning set must agree in both its constant and its linear part. Matching Equation 15.21 therefore yields, for each present arm \((j,k)\), the slope identity and the intercept identity \[ \boldsymbol\beta_1+\boldsymbol\beta_{2,k}=\boldsymbol\beta_1'+\boldsymbol\beta_{2,k}', \qquad \mu_j+d_k=\mu_j'+d_k'. \tag{15.22}\]

Contrasts. Fix a study \(j\) and two of its arms \(k,k'\), which exist by (iv). Differencing the intercept identity across the two arms cancels \(\mu_j\) and \(\mu_j'\): \(d_k-d_{k'}=d_k'-d_{k'}'\). As \((k,k')\) ranges over the edges of \(G\) this says \(\mathbf B\,\tilde{\mathbf d}=\mathbf B\,\tilde{\mathbf d}'\), where \(\tilde{\mathbf d}=(d_1,\dots,d_K)\), so \(\tilde{\mathbf d}-\tilde{\mathbf d}'\in\mathcal N(\mathbf B)\). By (ii) and Lemma 15.1, \(\tilde{\mathbf d}-\tilde{\mathbf d}'\) is constant on the single component; since \(d_1=d_1'=0\) the constant is \(0\), so \(\mathbf d=\mathbf d'\).

Interactions. Differencing the slope identity Equation 15.22 across the same two arms \(k,k'\) of study \(j\) cancels \(\boldsymbol\beta_1\) and \(\boldsymbol\beta_1'\): \(\boldsymbol\beta_{2,k}-\boldsymbol\beta_{2,k'}=\boldsymbol\beta_{2,k}'-\boldsymbol\beta_{2,k'}'\). By (i) each \(\boldsymbol\beta_{2,k}\) is a single network-level vector, so this holds along every edge of \(G\); applying Lemma 15.1 coordinatewise (to each of the \(p\) components of the interaction vectors), connectedness (ii) and the normalization \(\boldsymbol\beta_{2,1}=\boldsymbol\beta_{2,1}'=\mathbf 0\) force \(\boldsymbol\beta_{2,k}=\boldsymbol\beta_{2,k}'\) for every \(k\).

Main effects and baselines. With the interactions equal, the slope identity Equation 15.22 for any present arm gives \(\boldsymbol\beta_1=\boldsymbol\beta_1'\). With the contrasts equal, the intercept identity gives \(\mu_j=\mu_j'\) for every study. All parameters coincide, so the parameter map is injective. \(\square\)

Remark. What each hypothesis buys, and the failure modes. The proof spends connectedness (ii) twice, once to propagate the treatment contrasts and once to propagate the interaction contrasts along the same graph; it spends within-study variation (iii) to read slopes off each arm, and the shared-interaction assumption (i) to make the interaction a single vector per treatment that connectedness can propagate. Removing any one breaks identifiability in a definite way. A disconnected network leaves cross-component contrasts and interaction contrasts unidentified, as in Theorem 15.3. If a covariate is constant within every study, varying only between studies, then within-study differencing cannot separate \(\mathbf x^\top\boldsymbol\beta_1\) from the baseline \(\mu_j\), and \(\boldsymbol\beta_1\) is identified only from study-level aggregates, confounded with the baseline; this is the seed of ecological bias (Definition 16.2). A treatment that appears in only one study with degenerate covariate variation has an unidentified interaction \(\boldsymbol\beta_{2,k}\). And if the interactions are not shared, each study carrying its own \(\boldsymbol\beta_{2,k}^{(j)}\), the parameter count grows by a factor of order \(J\) and small networks cannot support estimation; the shared-effect-modifier assumption is then a necessity, not a convenience, and is the load-bearing assumption of the multilevel network meta-regression of Part V. \(\blacktriangleleft\)

15.12 Residual deviance and the deviance information criterion

This section is a practical checklist rather than new network theory. The residual deviance is the goodness-of-fit deviance of the generalized linear model of Chapter 7, and the deviance information criterion is the Bayesian model-comparison tool of Chapter 8, each specialized to a network; a reader focused on identifiability can skim it and return when fitting a real model.

A fitted network model must be checked, and competing models, fixed against random effects, consistency against inconsistency, must be compared. Two summaries do this work in the applied literature: the residual deviance, which measures absolute fit against a saturated model, and the deviance information criterion, which trades fit against effective complexity. Both specialize the general deviance of Chapter 7 (Definition 7.6, Definition 7.5) to the network setting.

Definition 15.5 (Residual deviance) For an arm with data \(y\) and fitted mean \(\hat\mu\) under a one-parameter exponential-family likelihood, the deviance residual contribution is twice the log-likelihood ratio between the saturated model, which fits the datum exactly, and the fitted model (Definition 7.6). For a binomial arm \(y_{jk}\sim\mathrm{Bin}(n_{jk},p_{jk})\) with fitted success probability \(\hat p_{jk}\) (corresponds to D.13 in the proof catalogue), \[ D_{jk}=2\left[\,y_{jk}\log\!\frac{y_{jk}}{n_{jk}\hat p_{jk}} +(n_{jk}-y_{jk})\log\!\frac{n_{jk}-y_{jk}}{n_{jk}(1-\hat p_{jk})}\,\right], \tag{15.23}\] with the convention \(0\log 0=0\). The residual deviance of the network is the sum \(D_{\mathrm{res}}=\sum_{j,k}D_{jk}\) over all arms. \(\blacktriangleleft\)

Remark. Derivation of Equation 15.23. The binomial log-likelihood, dropping the constant binomial coefficient, is \(\ell(p;y)=y\log p+(n-y)\log(1-p)\). The saturated model maximizes \(\ell\) over \(p\) freely; setting the derivative \(y/p-(n-y)/(1-p)\) to zero gives \(\hat p^{\mathrm{sat}}=y/n\). The deviance contribution is twice the gap between saturated and fitted log-likelihoods, \[ D_{jk}=2\bigl[\ell(y_{jk}/n_{jk};y_{jk})-\ell(\hat p_{jk};y_{jk})\bigr] =2\Bigl[y_{jk}\log\tfrac{y_{jk}/n_{jk}}{\hat p_{jk}} +(n_{jk}-y_{jk})\log\tfrac{(n_{jk}-y_{jk})/n_{jk}}{1-\hat p_{jk}}\Bigr], \] which rearranges to Equation 15.23. Each term is nonnegative because the saturated model maximizes the likelihood (Proposition 7.3). Under a correctly specified model with \(N\) arms and \(p\) estimated parameters, \(D_{\mathrm{res}}\) is approximately \(\chi^2_{N-p}\) (Theorem 7.4), so the posterior mean of \(D_{\mathrm{res}}\) should be close to the residual degrees of freedom \(N-p\); a value far in excess signals lack of fit, often inconsistency or unmodeled heterogeneity. As a working rule of thumb from the applied network guidance of Dias et al. (2018), a well-fitting model has \(D_{\mathrm{res}}\) near \(N-p\) (each data point contributes about \(1\), since a \(\chi^2_{N-p}\) variable has mean \(N-p\)), whereas a \(D_{\mathrm{res}}\) several times larger than \(N-p\) is a warning; the yardstick is always the residual degrees of freedom \(N-p\), not any fixed number. For a normal likelihood with known variances the deviance contribution reduces to the squared standardized residual, and the residual deviance is the weighted residual sum of squares, as the worked example illustrates. \(\blacktriangleleft\)

Definition 15.6 (Deviance information criterion) Let \(\boldsymbol\xi\) be the model parameters with posterior \(\pi(\boldsymbol\xi\mid\mathbf y)\), and let \(D(\boldsymbol\xi)=-2\log p(\mathbf y\mid\boldsymbol\xi)\) be the deviance. Define the posterior mean deviance and the effective number of parameters \[ \bar D=\mathbb E_{\boldsymbol\xi\mid\mathbf y}[D(\boldsymbol\xi)], \qquad p_D=\bar D-D(\bar{\boldsymbol\xi}), \qquad \bar{\boldsymbol\xi}=\mathbb E[\boldsymbol\xi\mid\mathbf y]. \tag{15.24}\] The deviance information criterion (corresponds to D.14 in the proof catalogue) is \[ \mathrm{DIC}=\bar D+p_D=2\bar D-D(\bar{\boldsymbol\xi}). \tag{15.25}\] Smaller DIC indicates better posterior fit penalized for effective complexity. The conventional reading, that DIC differences below about \(3\) are not meaningful while differences above \(5\) are substantial, is a rule of thumb from the network meta-analysis guidance of Dias et al. (2018), not a theorem; it is a rough calibration against the Monte Carlo and modeling noise in a DIC, and it is no substitute for clinical judgment about whether transitivity holds. In applied use DIC most often compares a fixed-effect with a random-effects model, or a consistency with an inconsistency model. \(\blacktriangleleft\)

The name “effective number of parameters” for \(p_D\) is justified by the following identity, which shows \(p_D\) counts parameters when the prior is uninformative and counts fewer when the prior or a hierarchical structure shrinks them.

Proposition 15.3 (The effective number of parameters) Suppose the deviance is quadratic in \(\boldsymbol\xi\) near its minimizer \(\hat{\boldsymbol\xi}\), \(D(\boldsymbol\xi)=(\boldsymbol\xi-\hat{\boldsymbol\xi})^\top\mathcal I(\boldsymbol\xi-\hat{\boldsymbol\xi})+c\) for a constant \(c\) and a positive definite matrix \(\mathcal I\), as holds for a normal likelihood with known variance and, asymptotically, for a regular model with \(\mathcal I\) the Fisher information. Then \[ p_D=\operatorname{tr}\!\bigl(\mathcal I\,\operatorname{Var}(\boldsymbol\xi\mid\mathbf y)\bigr). \tag{15.26}\] If in addition the prior is flat, so that the posterior is approximately \(\mathcal N(\hat{\boldsymbol\xi}, \mathcal I^{-1})\), then \(p_D\approx\operatorname{tr}(I)=\dim\boldsymbol\xi\).

Proof. We use the quadratic-form expectation identity: for a random vector \(\mathbf z\) with mean \(\boldsymbol\nu\) and covariance \(\boldsymbol\Sigma\), and a fixed symmetric matrix \(M\), \[ \mathbb E[(\mathbf z-\mathbf a)^\top M(\mathbf z-\mathbf a)] =\operatorname{tr}(M\boldsymbol\Sigma)+(\boldsymbol\nu-\mathbf a)^\top M(\boldsymbol\nu-\mathbf a), \tag{15.27}\] which follows by writing \(\mathbf z-\mathbf a=(\mathbf z-\boldsymbol\nu)+(\boldsymbol\nu-\mathbf a)\), expanding, using \(\mathbb E[\mathbf z-\boldsymbol\nu]=\mathbf 0\) to kill the cross term, and \(\mathbb E[(\mathbf z-\boldsymbol\nu)^\top M(\mathbf z-\boldsymbol\nu)] =\mathbb E\operatorname{tr}(M(\mathbf z-\boldsymbol\nu)(\mathbf z-\boldsymbol\nu)^\top) =\operatorname{tr}(M\boldsymbol\Sigma)\) by the cyclic property of trace (Lemma 3.4). Apply Equation 15.27 with \(\mathbf z=\boldsymbol\xi\mid\mathbf y\), mean \(\bar{\boldsymbol\xi}\), covariance \(\operatorname{Var}(\boldsymbol\xi\mid\mathbf y)\), matrix \(M=\mathcal I\), and \(\mathbf a=\hat{\boldsymbol\xi}\): \[ \bar D-c=\mathbb E[(\boldsymbol\xi-\hat{\boldsymbol\xi})^\top\mathcal I(\boldsymbol\xi-\hat{\boldsymbol\xi})\mid\mathbf y] =\operatorname{tr}\!\bigl(\mathcal I\operatorname{Var}(\boldsymbol\xi\mid\mathbf y)\bigr) +(\bar{\boldsymbol\xi}-\hat{\boldsymbol\xi})^\top\mathcal I(\bar{\boldsymbol\xi}-\hat{\boldsymbol\xi}). \] Also \(D(\bar{\boldsymbol\xi})-c=(\bar{\boldsymbol\xi}-\hat{\boldsymbol\xi})^\top\mathcal I(\bar{\boldsymbol\xi}-\hat{\boldsymbol\xi})\). Subtracting, \(p_D=\bar D-D(\bar{\boldsymbol\xi})=\operatorname{tr}(\mathcal I\operatorname{Var}(\boldsymbol\xi\mid\mathbf y))\), which is Equation 15.26. Under a flat prior the Bernstein-von Mises approximation makes the posterior \(\mathcal N(\hat{\boldsymbol\xi},\mathcal I^{-1})\) (Theorem 5.6, Theorem 8.3), so \(\operatorname{Var}(\boldsymbol\xi\mid\mathbf y)\approx\mathcal I^{-1}\) and \(p_D\approx\operatorname{tr}(\mathcal I\mathcal I^{-1})=\operatorname{tr}(I)=\dim\boldsymbol\xi\). \(\square\)

Remark. Why \(p_D\) is “effective.” In a hierarchical random-effects network (Definition 8.3), the study-level effects \(\delta_{j,t_{j,1}k}\) are shrunk toward the network contrasts by the heterogeneity prior, so their posterior variance is smaller than the flat-prior \(\mathcal I^{-1}\) and \(p_D<\dim\boldsymbol\xi\). The criterion thus charges a fixed-effect contrast close to one parameter and a heavily shrunk random effect close to zero, and DIC Equation 15.25 rewards a random-effects model only when the extra flexibility improves \(\bar D\) by more than the \(p_D\) it costs. This is the mechanism by which DIC adjudicates fixed against random effects, and consistency against inconsistency, in applied network meta-analysis. \(\blacktriangleleft\)

15.13 A worked triangle-plus-extra-arm network

We now complete the running example of Example 15.1, carrying a full fixed-effect analysis of its network from the design matrix to fitted contrasts, standard errors, a node-split, and a residual deviance, so that every number can be checked by hand. The network is a triangle of two-arm trials closed by a single three-arm trial: the “extra arm” that creates a multi-arm correlation block.

Example 15.2 (A three-treatment network with one multi-arm trial) The three treatments are those of Example 15.1: \(1\) = placebo (reference), \(2\) = Drug A, and \(3\) = Drug B. Four studies report log odds-ratio contrasts on the logit scale, with arm precisions equal within each study so that multi-arm contrast estimates carry the correlation \(\tfrac12\) of Theorem 15.4.

Study	Arms	Contrast(s) reported	Estimate	Variance
1	\(\{1,2\}\)	\(\hat d_{12}\)	\(0.5\)	\(0.10\)
2	\(\{1,3\}\)	\(\hat d_{13}\)	\(1.0\)	\(0.10\)
3	\(\{2,3\}\)	\(\hat d_{23}\)	\(0.1\)	\(0.20\)
4	\(\{1,2,3\}\)	\(\hat d_{12},\ \hat d_{13}\)	\(0.6,\ 0.9\)	\(0.20\) each, covariance \(0.10\)

The treatment-comparison graph \(G\) has vertices \(\{1,2,3\}\) and edges \(12,13,23\) from studies 1, 2, 3 and a further \(12,13\) from study 4; it is connected, with \(K=3\), \(J=4\), \(N=2+2+2+3=9\) arms and \(C=N-J=5\) within-study contrasts.

Step 1: the contrast-based design. Eliminating the four per-study baselines leaves the five within-study contrasts as data \(\mathbf y=(0.5,1.0,0.1,0.6,0.9)^\top\), with network-contrast parameters \(\mathbf d=(d_2,d_3)^\top\) and design rows from Equation 15.5, \[ \mathbf X=\begin{pmatrix} 1&0\\ 0&1\\ -1&1\\ 1&0\\ 0&1\end{pmatrix}, \] one row per within-study contrast. Labeling the rows makes the design transparent:

Row	Study	Contrast	Design row \((d_2,d_3)\)	Data \(y\)
1	1	\(d_{12}=d_2\)	\((1,\ 0)\)	\(0.5\)
2	2	\(d_{13}=d_3\)	\((0,\ 1)\)	\(1.0\)
3	3	\(d_{23}=d_3-d_2\)	\((-1,\ 1)\)	\(0.1\)
4	4	\(d_{12}=d_2\)	\((1,\ 0)\)	\(0.6\)
5	4	\(d_{13}=d_3\)	\((0,\ 1)\)	\(0.9\)

The contrast incidence rank is \(\operatorname{rank}(\mathbf X)=2=K-1\), so by Theorem 15.3 the contrasts are identifiable.

Step 2: the covariance matrix. The first three contrasts come from separate studies and from each other are independent; the last two come from study 4 and share its baseline arm, so by the equal-precision remark after Theorem 15.4 they have correlation \(\tfrac12\). Thus \(\boldsymbol\Sigma=\operatorname{blockdiag}(0.10,\,0.10,\,0.20,\,\boldsymbol\Sigma_4)\) with \[ \boldsymbol\Sigma_4=\begin{pmatrix}0.20&0.10\\0.10&0.20\end{pmatrix}=0.10\,(I_2+J_2), \qquad \boldsymbol\Sigma_4^{-1}=\frac{10}{3}\begin{pmatrix}2&-1\\-1&2\end{pmatrix}, \] since \((I_2+J_2)^{-1}=\tfrac13\begin{pmatrix}2&-1\\-1&2\end{pmatrix}\) and the off-diagonal correlation is \(0.10/0.20=\tfrac12\).

Step 3: the weighted normal equations. The precision is \(10\) for contrasts 1 and 2, \(5\) for contrast 3, and the block \(\boldsymbol\Sigma_4^{-1}\) for contrasts 4 and 5. Accumulating \(\mathbf A=\mathbf X^\top\boldsymbol\Sigma^{-1}\mathbf X\) and \(\mathbf b=\mathbf X^\top\boldsymbol\Sigma^{-1}\mathbf y\) row by row, the independent contrasts contribute \[ 10\!\begin{pmatrix}1&0\\0&0\end{pmatrix} +10\!\begin{pmatrix}0&0\\0&1\end{pmatrix} +5\!\begin{pmatrix}1&-1\\-1&1\end{pmatrix} =\begin{pmatrix}15&-5\\-5&15\end{pmatrix}, \] while study 4, whose two rows form the identity, contributes its precision block \(\boldsymbol\Sigma_4^{-1}=\tfrac{10}{3}\begin{pmatrix}2&-1\\-1&2\end{pmatrix}\). Hence \[ \mathbf A=\begin{pmatrix}15&-5\\-5&15\end{pmatrix}+\frac{10}{3}\begin{pmatrix}2&-1\\-1&2\end{pmatrix} =\frac{1}{3}\begin{pmatrix}65&-25\\-25&65\end{pmatrix}, \qquad \det\mathbf A=\frac{65^2-25^2}{9}=400. \] For the right-hand side, the independent contrasts give \(10(0.5)\binom{1}{0}+10(1.0)\binom{0}{1} +5(0.1)\binom{-1}{1}=\binom{4.5}{10.5}\), and study 4 gives \(\boldsymbol\Sigma_4^{-1}\binom{0.6}{0.9}=\tfrac{10}{3}\binom{2(0.6)-0.9}{-0.6+2(0.9)} =\tfrac{10}{3}\binom{0.3}{1.2}=\binom{1}{4}\), so \[ \mathbf b=\binom{4.5}{10.5}+\binom{1}{4}=\binom{5.5}{14.5}. \]

Step 4: fitted contrasts and standard errors. Inverting, \[ \mathbf A^{-1}=\frac{1}{1200}\begin{pmatrix}65&25\\25&65\end{pmatrix}, \qquad \hat{\mathbf d}=\mathbf A^{-1}\mathbf b =\frac{1}{1200}\begin{pmatrix}65&25\\25&65\end{pmatrix}\binom{5.5}{14.5} =\frac{1}{1200}\binom{720}{1080}=\binom{0.60}{0.90}. \] So \(\hat d_2=0.60\) and \(\hat d_3=0.90\), each with variance \(65/1200=0.054167\) and standard error \(0.2327\), and with covariance \(25/1200=0.020833\). The derived contrast is \(\hat d_{23}=\hat d_3-\hat d_2=0.30\), with variance \((65+65-2\cdot25)/1200=80/1200=0.066667\) and standard error \(0.2582\).

Step 5: node-split of the \(2\) versus \(3\) comparison. The direct evidence on \(d_{23}\) is study 3 alone, \(\hat d^{\mathrm{dir}}_{23}=0.1\) with variance \(0.20\). The indirect evidence is the network refitted without study 3, that is, studies 1, 2, 4. Repeating Steps 3 and 4 with the third contrast dropped gives \[ \mathbf A_{\mathrm{ind}}=\frac13\begin{pmatrix}50&-10\\-10&50\end{pmatrix},\quad \mathbf A_{\mathrm{ind}}^{-1}=\frac{1}{800}\begin{pmatrix}50&10\\10&50\end{pmatrix},\quad \mathbf b_{\mathrm{ind}}=\binom{6}{14},\quad \hat{\mathbf d}_{\mathrm{ind}}=\binom{0.55}{0.95}, \] so the indirect estimate is \(\hat d^{\mathrm{ind}}_{23}=0.95-0.55=0.40\), with variance \((50+50-2\cdot10)/800=80/800=0.10\). The inconsistency statistic is \[ \hat\omega_{23}=\hat d^{\mathrm{dir}}_{23}-\hat d^{\mathrm{ind}}_{23}=0.1-0.4=-0.30, \qquad \operatorname{Var}(\hat\omega_{23})=0.20+0.10=0.30, \] by Proposition 15.2, giving \(\widehat{\operatorname{SE}}=0.5477\) and \(z=-0.30/0.5477=-0.55\), a two-sided \(p\)-value of about \(0.58\): no evidence of inconsistency, though the test is underpowered, as the large standard error warns. As a check on Proposition 15.1, the inverse-variance blend of direct (\(0.1\), precision \(5\)) and indirect (\(0.4\), precision \(10\)) is \((5\cdot0.1+10\cdot0.4)/15=4.5/15=0.30\), exactly the full-network \(\hat d_{23}\) of Step 4.

Step 6: residual deviance. The fitted contrasts are \(\hat{\boldsymbol\theta}=\mathbf X\hat{\mathbf d}\): for the five data rows, \(\hat\theta=(0.60,\,0.90,\,0.30,\,0.60,\,0.90)\). Laying the fit beside the data row by row,

Row	Study	Contrast	Observed \(y\)	Fitted \(\hat\theta\)	Residual \(r\)	Variance	Deviance contribution
1	1	\(d_{12}\)	\(0.5\)	\(0.60\)	\(-0.10\)	\(0.10\)	\(0.10\)
2	2	\(d_{13}\)	\(1.0\)	\(0.90\)	\(0.10\)	\(0.10\)	\(0.10\)
3	3	\(d_{23}\)	\(0.1\)	\(0.30\)	\(-0.20\)	\(0.20\)	\(0.20\)
4	4	\(d_{12}\)	\(0.6\)	\(0.60\)	\(0\)	\(0.20\)	\(0\)
5	4	\(d_{13}\)	\(0.9\)	\(0.90\)	\(0\)	\(0.20\)	\(0\)

The residuals are \(\mathbf r=\mathbf y-\hat{\boldsymbol\theta}=(-0.10,\,0.10,\,-0.20,\,0,\,0)\), and for a normal likelihood the residual deviance is the quadratic form \(D_{\mathrm{res}}=\mathbf r^\top\boldsymbol\Sigma^{-1}\mathbf r\). Because the two study-\(4\) residuals are zero, that correlated block contributes nothing (in general it would contribute \(\mathbf r_4^\top\boldsymbol\Sigma_4^{-1}\mathbf r_4\), not a sum of separate squares), and \[ D_{\mathrm{res}}=\frac{(-0.10)^2}{0.10}+\frac{(0.10)^2}{0.10}+\frac{(-0.20)^2}{0.20}+0 =0.1+0.1+0.2=0.40 . \]

Degrees of freedom. We fit \(p=2\) parameters \((d_2,d_3)\) to \(N_{\mathrm{obs}}=5\) contrast data points, so the residual degrees of freedom are \(5-2=3\). This agrees with the arm-level count of Definition 15.5: the network has \(N=9\) arms and, in the reference-treatment parameterization, \(J+K-1=4+2=6\) free parameters (four baselines and two contrasts), giving \(9-6=3\) as well. The two counts match because profiling out the four baselines, that is, eliminating them from the likelihood so that only the contrasts remain (the concentration step of Section 15.3), converts \(9\) arms into \(5\) contrasts and \(6\) parameters into \(2\); the residual degrees of freedom are unchanged. Since \(D_{\mathrm{res}}=0.40\) sits well below its reference value \(3\), the model fits excellently, consistent with the small node-split statistic of Step 5.

Why the study-\(4\) rows fit exactly. The zero residuals on rows \(4\) and \(5\) are a feature of these particular numbers, not a general phenomenon. Studies \(1\), \(2\), \(3\) alone already imply \((\hat d_2,\hat d_3)=(0.60,0.90)\) (solve the top-left contribution \(\begin{pmatrix}15&-5\\-5&15\end{pmatrix}\hat{\mathbf d}=\binom{4.5}{10.5}\)), and study \(4\) happens to report exactly \((0.6,0.9)\), so it agrees perfectly and adds nothing to the residuals. With generic data the two evidence streams would disagree slightly and all five residuals would be nonzero; read no structural meaning into the exact fit.

Clinical reading. On the log-odds scale the pooled estimates are \(\hat d_2=0.60\) for Drug A versus placebo and \(\hat d_3=0.90\) for Drug B versus placebo, that is, odds ratios \(e^{0.60}\approx1.82\) and \(e^{0.90}\approx2.46\): both drugs raise the odds of a good outcome, Drug B more so. The head-to-head contrast is \(\hat d_{23}=\hat d_3-\hat d_2=0.30\), an odds ratio \(e^{0.30}\approx1.35\) favoring Drug B over Drug A. Its standard error \(0.258\) gives a rough \(95\%\) interval \(0.30\pm1.96(0.258)=(-0.21,\,0.81)\) on the log-odds scale, an odds-ratio interval of about \((0.81,\,2.24)\); because this interval includes \(1\), the network does not establish that Drug B beats Drug A, even though the point estimate leans that way. This is the everyday output of an NMA: a coherent set of relative effects, each with an interval, that a decision maker can compare on one scale. \(\blacktriangleleft\)

Remark. What the example exhibits. Every structural feature of the chapter appears in the arithmetic. The design matrix \(\mathbf X\) is the contrast incidence of Equation 15.12 with the baselines profiled out; its rank \(2=K-1\) is the identifiability condition of Theorem 15.3. The block \(\boldsymbol\Sigma_4=0.10(I_2+J_2)\) is the one-half-correlation structure of Theorem 15.4, here in its sampling-covariance guise; treating study 4 as two independent trials would have used the wrong weights. The estimator Equation 15.13 is the generalized least squares solution \(\hat{\mathbf d}=\mathbf A^{-1}\mathbf b\). The node-split of Step 5 instantiates Definition 15.4 and Proposition 15.2, and the pooling identity of Proposition 15.1 is confirmed to four figures. The residual deviance of Step 6 is the normal-likelihood form of Definition 15.5, with the correlated block handled by the quadratic form \(\mathbf r^\top\boldsymbol\Sigma^{-1}\mathbf r\) rather than a sum of independent squares. \(\blacktriangleleft\)

15.14 Notes and references

The network meta-analysis model in its arm-based and contrast-based forms, the two parameterizations of Definition 15.2 and Definition 15.3, the consistency equations, the random-effects covariance with its one-half correlation, residual deviance, and the deviance information criterion are the standard apparatus of the field as codified by Dias et al. (2018) and the thesis of Phillippo (2019), building on the indirect comparison of Bucher et al. (1997) (Chapter 14) and the random-effects meta-analysis of DerSimonian and Laird (1986) (Chapter 13). The reference-treatment decomposition Equation 15.5 is Eq. (1.7c) of Phillippo (2019); the corrected statement of the parameterization equivalence (Theorem 15.2), as an isomorphism onto the consistency-constrained subspace rather than a square change of basis, follows the proof catalogue’s amendment to that footnote, and is needed precisely because multi-arm trials and repeated comparisons make the baseline-shift count \(N\) exceed the reference-treatment count \(J+K-1\).

The identifiability theorem (Theorem 15.3) is folklore in the NMA literature, usually stated as “the network must be connected”; the formulation here as the rank of the design matrix, reduced to the rank of a graph incidence matrix (Lemma 15.1), makes it a theorem of algebraic graph theory. The incidence-rank result \(\operatorname{rank}(\mathbf B)=K-q\) is classical and may be found in standard texts on algebraic graph theory; the elementary kernel proof given here uses only rank-nullity and the linear algebra of Chapter 1 (Theorem 1.3, Lemma 1.10) and the positive-definite machinery of Chapter 3 (Theorem 3.3, Proposition 3.2), and is self-contained. The reduction of generalized least squares to ordinary least squares by whitening, used in Theorem 15.3, is Aitken’s theorem, here routed through the Gauss-Markov theorem (Theorem 6.3) of Chapter 6.

The one-half contrast correlation (Theorem 15.4) is the homoscedastic multi-arm covariance of Lu and Ades; the symmetry characterization (Lemma 15.3), that one half is the unique correlation making the heterogeneity variance independent of the baseline choice, sharpens the usual presentation, which exhibits the value without showing it is forced. Node-splitting (Definition 15.4, Proposition 15.2) is due to Dias and colleagues and is presented in Dias et al. (2018); the framing as a released consistency equation, and the connection to the inverse-variance blend (Proposition 15.1), make the test’s mean-zero null and its power limitations transparent. The deviance information criterion (Definition 15.6, Proposition 15.3) is due to Spiegelhalter and colleagues; the trace identity for \(p_D\) is standard, and its use for model comparison in network meta-analysis is described in Dias et al. (2018). The individual-patient-data network meta-regression (Theorem 15.5) and its dependence on shared effect modifiers, connectedness, and within-study covariate variation is the foundation of the multilevel network meta-regression of Phillippo et al. (2020) and is discussed in the indirect-comparison guidance of Phillippo et al. (2016).

This chapter corresponds to results D.5 (baseline-shift), D.6 (reference-treatment), D.7 (equivalence), D.8 (fixed-effect identifiability), D.9 (one-half correlation), D.10 (consistency closure), D.11 (node-splitting), D.12 (IPD network meta-regression), D.13 (residual deviance), and D.14 (DIC) of the proof catalogue in proofs/D_network_meta_analysis.md. The companion Coq development formalizes only the Bucher algebra (results D.2 and D.3) in theories/Bucher.v, which underwrites the node-split null of Proposition 15.2 through Theorem 14.1 and Theorem 14.2; the network-level results of this chapter, being linear-algebra and likelihood arguments over the incidence matrix and the design matrix, are not currently in the Coq tree, and carry no machine-checked tag. The breakdown of consistency under effect modification, where a significant node-split comes from, is the subject of Chapter 16 (Theorem 16.1, Definition 16.2); the extension of the IPD model to mixed individual-and-aggregate networks is the multilevel network meta-regression of Part V.

15.15 Exercises

Exercises marked with a star (\(\star\)) have a full solution in Appendix F. The first two are warm-ups: draw the picture and read the signs before turning to the computations and proofs.

Exercise 15.1 (Warm-up.) Draw the network and name its parts. Using the four trials of Example 15.1 (placebo \(=1\), Drug A \(=2\), Drug B \(=3\)), draw the treatment-comparison graph. (a) For each of the three treatment pairs, state whether the network has direct evidence, indirect evidence, or both. (b) Is the graph connected? Name a path from Drug A to Drug B that does not use the direct head-to-head trial. (c) Identify the loop, and state in words the consistency equation it imposes on \(d_{12}\), \(d_{13}\), \(d_{23}\). (d) If study \(3\) (the head-to-head trial) were removed, would the network still be connected, and would any inconsistency remain testable?

Exercise 15.2 (Warm-up, interpretation.) Reading contrasts as odds ratios. On the log-odds scale a network reports \(d_{12}=0.5\) (Drug A versus placebo) and \(d_{13}=1.0\) (Drug B versus placebo). (a) Convert each to an odds ratio. (b) Under consistency (Theorem 15.1), what is \(d_{23}\), the Drug B versus Drug A contrast, and its odds ratio? (c) Write the same comparison in the reverse direction, \(d_{32}\), as a log-odds contrast and an odds ratio. (d) A colleague reports “the odds ratio for treatment \(2\) versus treatment \(3\) is \(0.61\).” Is this consistent with your answer, and why is naming the direction essential?

Exercise 15.3 From a network to a design matrix. A network has four treatments \(1,2,3,4\) (\(1\) the reference) and three two-arm studies: study A on \(\{1,2\}\), study B on \(\{2,3\}\), study C on \(\{3,4\}\). Write the treatment-comparison graph, the signed incidence matrix \(\mathbf B\), and the reference-treatment contrast design \(\mathbf X_d\). Compute \(\operatorname{rank}(\mathbf B)\) from Lemma 15.1 and confirm the network is connected and the three contrasts \(d_2,d_3,d_4\) are identifiable. Which pairwise contrasts have only indirect evidence?

Exercise 15.4 Disconnection and non-identifiability. Remove study B from the previous network, leaving studies A on \(\{1,2\}\) and C on \(\{3,4\}\). Show the graph has two components, compute \(\operatorname{rank}(\mathbf X_{\mathrm{RT}})\) from Theorem 15.3, and exhibit an explicit nonzero vector in the kernel of \(\mathbf X_{\mathrm{RT}}\). Which contrasts become non-identifiable, and which remain identifiable?

Exercise 15.5 A four-arm covariance. Using Theorem 15.4, write the \(3\times3\) heterogeneity covariance matrix \(\boldsymbol\Sigma_j\) of the three baseline contrasts of a four-arm trial. Verify its eigenvalues are \(2\tau^2\) once and \(\tfrac{\tau^2}{2}\) twice, and that \(\det\boldsymbol\Sigma_j=4(\tau^2/2)^3\). Confirm directly that the variance of the derived contrast \(\delta_{j,kl}=\delta_{j,1l}-\delta_{j,1k}\) equals \(\tau^2\), consistent with Lemma 15.3.

Exercise 15.6 Counting loop constraints. For a network with \(K=5\) treatments, use Theorem 15.1 to find the number of basis contrasts and the number of independent consistency (loop) constraints among the \(\binom{5}{2}\) pairwise contrasts. Verify the count two ways: by the formula \(\binom{K-1}{2}\) and by the cycle rank \(|\mathcal E|-K+1\) of the complete graph \(K_5\).

Exercise 15.7 Effective parameters under shrinkage. Consider a one-parameter normal model \(y\sim\mathcal N(\theta,\sigma^2)\) with known \(\sigma^2\) and a conjugate prior \(\theta\sim\mathcal N(0,\kappa^2)\). Using the conjugate posterior (Theorem 8.3) and Proposition 15.3, compute \(p_D=\mathcal I\operatorname{Var}(\theta\mid y)\) with \(\mathcal I=1/\sigma^2\), and show \(p_D=\kappa^2/(\kappa^2+\sigma^2)\in(0,1)\). Confirm \(p_D\to 1\) as \(\kappa^2\to\infty\) (flat prior) and \(p_D\to 0\) as \(\kappa^2\to 0\) (point-mass prior), so a shrunk parameter counts as less than one.

Exercise 15.8 Recompute the worked example with a discrepant direct study. In Example 15.2, replace the study 3 estimate \(\hat d_{23}=0.1\) by \(\hat d_{23}=1.0\), keeping its variance \(0.20\) and all other data unchanged. Recompute the full-network \(\hat d_2,\hat d_3\), the node-split statistic \(\hat\omega_{23}\) and its standard error, and the residual deviance. Comment on whether the inconsistency is now detectable and how the residual deviance reflects it.

Exercise 15.9 (\(\star\)) The network estimate is the direct-indirect blend. Prove Proposition 15.1 from the generalized least squares estimator Equation 15.13 in the special case of a contrast informed by exactly two independent sources. That is, take a \(2\times1\) design \(\mathbf X=\binom{1}{1}\) with diagonal covariance \(\operatorname{diag}(v_{\mathrm{dir}},v_{\mathrm{ind}})\) and data \((\hat d^{\mathrm{dir}}_{AB},\hat d^{\mathrm{ind}}_{AB})\), and show Equation 15.13 reduces to the inverse-variance-weighted mean Equation 15.17 with variance \(1/(w_{\mathrm{dir}}+w_{\mathrm{ind}})\). Then explain why, in a larger network, the same blend holds for any contrast whose only evidence splits into one direct and one indirect independent stream, and relate this to the node-split of Definition 15.4.

Exercise 15.10 (\(\star\)) An unidentifiable network meta-regression. In the IPD network meta-regression Equation 15.20 with a single scalar covariate \(x\), suppose \(x\) is constant within each study (it varies only between studies). Show that condition (iii) of Theorem 15.5 fails and that the prognostic main effect \(\beta_1\) is confounded with the study baselines \(\mu_j\): exhibit a one-parameter family of distinct parameter vectors \((\boldsymbol\mu,\beta_1,\dots)\) that give identical conditional means on all arms. Explain in one or two sentences why this is the individual-level seed of ecological bias (Definition 16.2), and why within-study covariate variation rescues identifiability.

Bucher, Heiner C., Gordon H. Guyatt, Lauren E. Griffith, and Stephen D. Walter. 1997. “The Results of Direct and Indirect Treatment Comparisons in Meta-Analysis of Randomized Controlled Trials.” Journal of Clinical Epidemiology 50 (6): 683–91.

DerSimonian, Rebecca, and Nan Laird. 1986. “Meta-Analysis in Clinical Trials.” Controlled Clinical Trials 7 (3): 177–88.

Dias, Sofia, A. E. Ades, Nicky J. Welton, Jeroen P. Jansen, and Alexander J. Sutton. 2018. Network Meta-Analysis for Decision-Making. Statistics in Practice. Wiley.

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

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

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