Original Article Received 25 April 2012,

Revised 6 August 2012,

Accepted 10 August 2012

Published online in Wiley Online Library

(wileyonlinelibrary.com) DOI: 10.1002/jrsm.1058

Network meta-analysis, electrical networks and graph theory Gerta Rücker*† Network meta-analysis is an active field of research in clinical biostatistics. It aims to combine information from all randomized comparisons among a set of treatments for a given medical condition. We show how graph-theoretical methods can be applied to network meta-analysis. A meta-analytic graph consists of vertices (treatments) and edges (randomized comparisons). We illustrate the correspondence between meta-analytic networks and electrical networks, where variance corresponds to resistance, treatment effects to voltage, and weighted treatment effects to current flows. Based thereon, we then show that graph-theoretical methods that have been routinely applied to electrical networks also work well in network meta-analysis. In more detail, the resulting consistent treatment effects induced in the edges can be estimated via the Moore–Penrose pseudoinverse of the Laplacian matrix. Moreover, the variances of the treatment effects are estimated in analogy to electrical effective resistances. It is shown that this method, being computationally simple, leads to the usual fixed effect model estimate when applied to pairwise meta-analysis and is consistent with published results when applied to network meta-analysis examples from the literature. Moreover, problems of heterogeneity and inconsistency, random effects modeling and including multi-armed trials are addressed. Copyright © 2012 John Wiley & Sons, Ltd. Keywords:

network meta-analysis; electrical network; graph theory; Laplacian matrix; Moore–Penrose pseudoinverse

1. Meta-analytic networks and electrical networks Network meta-analysis is a statistical method attracting great interest in medicine (Nietert et al., 2012), and it is a very active field of research in clinical biostatistics (Salanti et al., 2008; Salanti, 2012). It aims to combine information from all randomized comparisons among a set of treatments for a given medical condition. In this article, we show that graph-theoretical methods that have been routinely applied to electrical networks also work well in network meta-analysis. To this aim, we study electrical networks and meta-analytic networks as graphs and elucidate the correspondence between meta-analytic networks and electrical networks. In general, a simple undirected graph is a pair (V,E) where V denotes a set of n vertices and E is a subset of V  V containing those pairs of vertices that are connected by an edge. More generally, edges may be associated with non-negative weights. A weight function is a function w : E ! Rþ where w = w(e) ≥ 0 is the weight of the edge e 2 E. An electrical network consists of nodes and edges, where the edges are associated with resistances R(e) > 0. It is convenient to take as edge weights the inverse resistances 1/R(e), called conductances. The larger is the conductance, the smaller is the resistance. For a meta-analytic network, the vertices (nodes) correspond to treatments and the edges to existing comparisons between treatments, based on studies (ideally randomized controlled trials, RCTs). Each two-arm study contributes a direct comparison, represented by an edge joining the two treatments compared. A reasonable weight function is given by the inverse of the observed variance of the treatment effect, called the precision. This consideration suggests that the variance of a comparison corresponds to the resistance of the edge in the network, whereas the precision corresponds to the conductance. This correspondence will be elucidated in detail in the following.

Institute of Medical Biometry and Medical Informatics, University Medical Center Freiburg, Freiburg, Germany *Correspondence to: Gerta Rücker, Institute of Medical Biometry and Medical Informatics, University Medical Center Freiburg, Stefan-Meier-Straße 26, D-79104 Freiburg, Germany. † E-mail: [email protected]

Copyright © 2012 John Wiley & Sons, Ltd.

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1.1. Parallel connection For a parallel connection of edges between two nodes in an electrical network, the effective resistance is given by the inverse of the sum of the inverse resistances 1 X 1 ¼ R Rk k In the meta-analytic context, a parallel connection of a number of studies between two treatments corresponds to a pairwise meta-analysis. We denote the observed effects by xk and their empirical variances by Vk. For the fixed effect model with inverse variance weighting, the weights are given by the inverse variances wk = 1/Vk. As common in meta-analysis, we interpret the variances and thus the weights as fixed. Although xk, Vk, wk are estimated quantities, we omit hats in order to keep notation simple. The variance estimate V ðx Þ of the pooled effect X x ¼ Xk

wk x k

k

(1)

wk

is given by the inverse of the sum of the inverse weights, 1 V ðx Þ ¼ X k

wk ;

so that its inverse, the total precision, is the sum of the precisions m X 1 1 ¼ : V ðx Þ k¼1 Vk

That is, precision behaves like conductance. 1.2. Connection in series For a serial connection of resistors in an electrical network, the effective resistance is given by the sum of the electrical resistances of the edges R¼

X

Rk :

k

In the meta-analytic context, this corresponds to considering a serial connection (chain) of m independent studies comparing m + 1 successive treatments A0, A1, . . ., Am. Let xk(k = 1, . . ., m) be the treatment effect resulting when comparing Ak  1 with Ak. We want to determine the indirect difference of treatments A0 and Am that under P the assumption of consistent treatment comparisons can be estimated by k xk. Because studies are independent, their variances add V

X

! xk

¼

k

X

Vk :

k

1.3. The correspondence between electrical networks and meta-analytic networks The previous considerations suggest that meta-analytic networks correspond to electrical networks, where variance corresponds to resistance and precision corresponds to conductance. We now elucidate the analogs to electrical current or flow I and electrical potential U or potential difference (voltage), respectively. We start from Ohm’s law U ¼ IR or

U ¼ I; R

meaning that the voltage of an electrical element is proportional to the current, with resistance as the proportionality factor. Copyright © 2012 John Wiley & Sons, Ltd.

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Given a pairwise meta-analysis of m studies with treatment effects xk, we have by Equation (1) ! X X wk x ¼ wk xk k

k

that we can write X xk x ¼ : V ðx Þ Vk k As variance corresponds to resistance, the correspondence becomes perfect if we interpret the treatment effects xk and x as voltages (potential differences) and the ratios xk/Vk that equal the weighted treatment effects wkxk as current through the respective edges of the meta-analytic network. As treatment effects are often defined as differences between certain outcomes (e.g., proportions, or log-transformed or logit-transformed proportions, or log hazard ratios) under different treatments, these outcomes correspond to the electrical potentials at the nodes of the network. The full correspondence is outlined in Table 1.

2. Algorithm In this section, the analogy to electrical networks is successfully used for estimation of consistent direct and indirect treatment effects—in the sense of the network structure—in a meta-analytic network. First, necessary graph-theoretical notation is introduced. 2.1. Notation Let n be the number of different treatments (vertices) in a network and let m be the number of existing comparisons (edges) between the treatments. At first, we restrict our considerations to two-arm studies, so that m is the number of studies. Both the vertices (treatments) 1, . . ., n and the edges (studies) 1, . . ., m are assumed to be numbered in an arbitrary but fixed order. Based thereon, let x ¼ ðx1 ; . . . ; ; xm Þ> and w ¼ ðw1 ; . . . ; ; wm Þ> be the vectors of observed effects and their inverse sampling variances, respectively. w is the vector of weights. As common in meta-analysis, we interpret the variances and thus the weights as fixed. Moreover, we define the diagonal m  m matrix W = diag(w), containing the edge weights wk on its diagonal. Further, let Wx be the pointwise product vector of inverse variance-weighted observed effects ðwk xk Þk . We point out that xk, as they are estimated from different studies, cannot necessarily be assumed to be consistent. A network is said to be consistent if the sum of directed treatment effects over all closed circuits is zero. An example of a non-consistent network is found in Figure 1. Here, x12 = 0.5, x24 = 1.5, but x14 = 1.0 6¼ x12 + x24. To define the network structure, we introduce the edge-vertex incidence matrix B. To this aim, the edges of the graph must be arbitrarily oriented without restriction (i.e., put an arrow on each edge in an arbitrary direction). B is an m  n matrix containing for each row (corresponding to an edge), a one in the column belonging to the vertex where the edge starts and a minus one in the column belonging to the vertex where the edge ends. Thus, all row sums are zero, as each edge starts and ends in exactly one vertex.We define the n  n Laplacian matrix L by L ¼ B> WB:

(2)

The Laplacian matrix (also called Kirchhoff or admittance matrix) plays a central role in spectral graph theory (Rao and Mitra, 1971; Doyle and Snell, 1999; Bollobás, 2002; Spielman and Srivastava, 2009; Spielman, 2012).

Table 1. The correspondence between electrical networks and meta-analytic networks. Electrical network Meta-analytic network Nodes i = 1, . . ., n Edges k = 1, . . ., m Resistance Rk Conductance R1 X 1 1 R ¼ kR

Treatments A1, . . ., An Existing comparisons Variance Vk Precision wk ¼ V1

k

k

Parallel connection Total precision =

k

P R = k Rk Potential at node i Voltage at edge (i,j) Current at edge (i,j) Copyright © 2012 John Wiley & Sons, Ltd.

Connection in series

P k

wk

P V = k Vk Outcome of treatment Ai Treatment effect Ai versus Aj Weighted treatment effect Res. Syn. Meth. 2012

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4 1

1

1.5

1 0.5

3 0.5

2 Figure 1. Example of a meta-analytic network with directed effects.

Whereas B and B> depend on the chosen orientation, W and L are uniquely determined except for vertex numbering. We note that the definition given here is equivalent to the more commonly used definition of the Laplacian matrix as the difference of the weighted degree matrix and the weighted adjacency matrix of a graph (Gutman and Xiao, 2004). However, we prefer the definition given here as it is edge-based and thus naturally accounts for multiple edges between two vertices that are common in meta-analysis. It is easily seen that 1, the vector of ones, is an eigenvector of L with eigenvalue 0. There is a theorem stating that the multiplicity of the eigenvalue 0 is equal to the number of connectivity components of the graph (Spielman and Srivastava, 2009; Spielman, 2012). Thus, for a connected network, the rank of L is n  1 and L is invertible on the space orthogonal to 1. We are interested in the Moore–Penrose pseudoinverse L+ of the Laplacian matrix that can be calculated by Lþ ¼ ðL  J=nÞ1 þ J=n; where J is the n  n matrix consisting of ones (Rao and Mitra, 1971; Gutman and Xiao, 2004; Fouss et al., 2007). 2.2. Effective resistance and variance estimation Given the variances of the direct comparisons, we now want to estimate the variances based on both direct and indirect comparisons. On the basis of the correspondence between variance and resistance, the problem of variance estimation is equivalent to the problem of computing effective resistances in an electrical network, given its structure and resistors. This problem has an elegant solution based on the pseudoinverse of the Laplacian matrix. For an electrical network, the effective resistance of a pair (i,j) of vertices is given by þ þ Rij ¼ Lþ ii þ Ljj  2Lij

(Klein and Randić, 1993; Bollobás, 2002; Gutman and Xiao, 2004). The effective resistance is a metric on the set of vertices. It generalizes the formulas for the total resistance for parallel and serial connections given earlier. The effective resistance decreases with the number of paths between vertices i and j. The formula directly translates to the resulting variance in a meta-analytic network. If Vij denotes the variance of the resulting (potentially indirect) comparison of treatments i and j, we have þ þ Vij ¼ Lþ ii þ Ljj  2Lij :

(3)

The full information of the network markedly reduces the variances of all comparisons and increases precision. In Appendix A, we derive the factor to which the variance is reduced on average. We note explicitly that by Equation (3), variances are also estimated for pairs of treatments for which no direct comparison exists. 2.3. Potential differences and treatment effects In an electrical network, current can be injected into or removed from each node. Let Iext be the vector of currents injected into (if positive, otherwise removed from) each node i(i = 1, . . ., n). Further, let y be a vector of length m, containing the currents induced in the edges (in the direction of orientation). Kirchhoff’s current law then says that the sum of the currents entering a vertex is zero, that is, the amount injected is the same as the sum of currents over the incident edges (Rao and Mitra, 1971; Bollobás, 2002; Spielman and Srivastava, 2009; Spielman, 2012) B> y ¼ Iext Ohm’s law says that the current y in an edge is equal to the potential difference v across its ends times its conductance w, which we can write in vector notation Copyright © 2012 John Wiley & Sons, Ltd.

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y ¼ Wv: If we denote the vector of unknown true potentials at the n nodes by u, we have v ¼ Bu by definition. Taken together, both laws yield Iext ¼ B> y ¼ B> Wv ¼ B> WBu ¼ Lu by the definition of L. In our case of a meta-analytic network, the known quantities are the observed effects x (not necessarily consistent), the edge weights w and all matrices. The unknown quantities are the ‘true’ potentials u at the vertices and their differences, the ‘voltages’ v induced in the edges of the network. Solving the last equation for the vector u of potentials provides for a vector Iext ⊥ 1 u ¼ Lþ Iext ¼ Lþ B> y; and computing the potential differences v induced in the edges finally yields v ¼ Bu ¼ BLþ B> y: The estimation problem is solved by inserting the observed (inconsistent) currents Wx for y v ¼ Bu ¼ BLþ B> Wx:

(4)

Note that Wx is mapped to the same v as the unobserved y, as the mapping BL+B> : y ↦ v is not injective. This construction of potential differences via vertex potentials guarantees that the network obeys Kirchhoff’s potential law. This law says that the directed sum of the potential differences around any closed circuit is zero. For a metaanalytic network, this means that the treatment effects v, computed by Equation (4), guarantee consistency of the treatment effect over closed circuits, by contrast to the given observed effects x. This means that we can also extend v to all pairs of vertices. We note that Equation (4) can be written  þ v ¼ B B> WB B> Wx: This can be interpreted as the least squares solution of a regression problem that can be stated like this. Given the network structure, described by B, the weights, W, and the observed (inconsistent) effects, x, choose the potentials u at the nodes such that the effects v become consistent, but as similar as possible to the observed x in a least squares sense (Albert, 1972; Gantmacher, 1986; Senn et al., 2012). There is also a geometric interpretation (Albert, 1972) of the mapping BL+B>W : x ↦ v as the projection of x into the consistent (n  1)-dimensional subspace of Rm . In a recent article, Senn et al. (2012) present an alternative fixed effects approach using general least squares, pointing to the connection between network meta-analysis and incomplete block designs (Yates, 1940; Paterson, 1983; Bailey and Cameron, 2009). We note further that for the core relation, Equation (4), only the observed weighted effects Wx and the edge-vertex incidence matrix B must be known. Therefore, as all information is based on the edges, it is possible to consider multiple edges between the same vertices and thus to combine information of single or multiple studies joining two vertices in one step. In other words, it is possible, but not necessary to do pairwise meta-analyses first. An R function network for application of the graph-theoretical method in practice is given in Appendix B.

2.4. Measuring inconsistency and heterogeneity The observed effects are inconsistent as soon as the resulting potential differences v do not agree with the observed treatment effects on the edges, as given by x. The deviation from consistency can be measured by a quadratic form, the Q statistic Qtotal ¼ ðx  v Þ> W ðx  v Þ that under homogeneity is approximately w2 - distributed with m  1 degrees of freedom, as the edges are seen as independent. Note that in order to account for heterogeneity within each pair of treatments, data must be given at the study level. We show in subsection 3.1 that for a pairwise meta-analysis, Qtotal corresponds to the wellknown Q statistic (Higgins and Thompson, 2002). In general, a network consists of more than two treatments and more than one study per comparison. Then Qtotal can be separated into parts for each pairwise meta-analysis and a part for remaining inconsistency between Copyright © 2012 John Wiley & Sons, Ltd.

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comparisons: the first is given by summing up Q terms over pairwise meta-analyses (parallel edges between pairs of treatments), the latter by subtracting the sum of all these terms from Qtotal. In principle, there is no difference between heterogeneity (different effects of studies comparing the same treatments) and inconsistency on closed circuits over more than two studies, for example, triangles. The only difference is the length of the circuit (two or longer). 2.5. Multi-armed studies Multi-armed studies correspond to complete consistent subgraphs (in graph theory called cliques) of the network. Problems occurring here are (i) that there can be no inconsistency within a study, and (ii) that therefore pairwise effects within a study are correlated. For this reason, the assumption of additive variances, essential for our method, is violated. We account for this violation as follows. Consider a multi-armed study of p treatments. We think of its graph as a complete subgraph with effective variances already known, and back-calculate the variances that have to be entered for each two-armed substudy. To this aim, we can use a theorem by Gutman and Xiao to determine L+ from the matrix V of effective variances   1 1 1 Lþ ¼  V  ðVJ þ JV Þ þ 2 JVJ ; 2 p p þ

with J defined as in subsection 2.1 (Gutman and Xiao, 2004, Theorem 7). From L+, we obtain L ¼ ðLþ Þ . As the non-diagonal entries of L are the negative weights, the original variances can be directly derived. It is of more general interest how much the variances are reduced on average when calculating the effective variances (or, the other way round, how far the variances coming from a multi-armed study must be inflated). This question is answered by the following proposition (see Appendix A for a proof). 2.5.1. Proposition. Consider a graph with n vertices and m edges. The average fraction to which the variance is reduced when calculating the effective variances is (n  1)/m. Examples. For a complete subgraph with p vertices and p(p  1)/2 edges, the fraction is 2/p. For a pairwise meta-analysis with m studies, it is 1/m. For a circuit with n vertices, it is (n  1)/n. For a tree (a graph free of circuits), the fraction is 1, that is, without at least one circuit there is no gain.

3. Applications 3.1. Pairwise meta-analysis revisited A pairwise meta-analysis considers only two treatments, connected by m studies. We have two vertices 1 and 2, corresponding to the treatments, and m edges from one to the other, corresponding to the studies. The observed vector of currents is Wx ¼ ðw1 x1 ; w2 x2 ; . . . ; wm xm Þ> : The edge-vertex incidence matrix is

0

1 B1 B¼B @⋮ 1

1 1 1 C C ⋮ A 1

with m rows and two columns, and the Laplacian matrix and its pseudoinverse are  P   P  1 wk P wk 1 1 þ P P L¼ ;L ¼ : wk  wk 4 wk 1 1 The vector of potential differences (dimension m) is

P wk xk 1; v ¼ BLþ B> Wx ¼ P wk

which corresponds to the usual fixed effect model in meta-analysis. For the heterogeneity statistic Q, we find P  X  wk x k 2 Qtotal ¼ ðx  v Þ> W ðx  v Þ ¼ wk xk  P ; wk which is the usual Q statistic. Copyright © 2012 John Wiley & Sons, Ltd.

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3.2. Example by Senn et al. (2012) We applied the graph-theoretical method to an example by Senn et al. (2012) who discuss fixed and random effects modeling for network meta-analysis using an example of a network of 10 diabetes treatments including 26 studies, including one 3-arm study (data given in the Appendix of the reference). They consider a continuous outcome, the HbA1c value (mean change or mean post-treatment value). We reanalyzed the data using the graph-theoretical approach and were able to reproduce their Figure 5 (Senn et al., 2012, based on the fixed effect model) exactly (Figure 2). This result confirms the equivalence between the graph-theoretical approach and the fixed effect model with inverse variance weighting. Measuring heterogeneity/inconsistency resulted in Qtotal = 96.9(df = 27). Comparing this to the sum of Q values of separate pairwise meta-analyses (74.0), we may interpret the difference (22.9) as residual inconsistency across the network. 3.3. Example by Dias et al. (2010) The second real-data example is a network of thrombolytic treatments (Dias et al., 2010). The data come from 50 trials comparing eight (originally, nine) thrombolytic drugs and angioplasty for patients with acute myocardial infarction. The endpoint was mortality, and the effect measure was the (log) odds ratio. Meta-analyses were presented in aggregate form, that is, as pairwise fixed effect meta-analyses for each existing comparison (edge) of the network. The resulting (inconsistent) effects and standard errors were listed in Dias et al. (2010, Table 2). Table 2 first contains the treatment numbers as given in the source (columns 1 and 2; note that numbers run from one to nine as number six was not part of the analysis). All edges are incident with at least one of the treatments 1, 2, or 3. In columns 3 and 4, the treatment effects with standard errors on the log odds scale are listed, based on pairwise fixed effect meta-analyses (Dias et al., 2010, Table 2, columns 5 and 6), followed by the result of the multiple treatment comparison (MTC) from Dias et al. (2010, Table 2, columns 3 and 4). These can be compared with the last two columns in Table 2 that contain our own result using the graph-theoretical method. The MTC analysis was a fixed effect Bayesian analysis, whereas our analysis corresponds to a fixed effect model, based on pairwise analyses for each direct comparison. The results of our analysis are very similar to the MTC analysis with respect to both treatment effects and standard errors. 3.4. Example by Mauri et al. (2008) Mauri et al. (2008) performed a systematic review comparing different regimens involving chemotherapy and/or targeted therapy in advanced breast cancer with respect to overall survival (Mauri et al., 2008, Tables 2 and 3). They published the results of a network of 148 comparisons, pertaining to 128 trials (26 031 patients, 22 different types of treatment). Differences between treatments were measured as hazard ratios. We reanalyzed the data applying the graph-theoretical method to the pairwise aggregate data given in Mauri et al. (2008, Table 2). The results (Table 3) are comparable, although not identical to those in the source, which is explained by our use of a fixed effect model, compared with a random effects Bayesian MTC analysis with a common heterogeneity parameter across all comparisons.

4. Discussion We have introduced a purely graph-theoretical approach to network meta-analysis that utilizes the correspondence between meta-analytic networks and electrical networks. Interpreting the observed variances as resistances and weighted effects as current, the estimated treatment effects and resulting variances can be obtained as voltages and effective resistances at the edges of the network. Contrast to placebo

95%−CI

vildagliptin sulfonylurea sitagliptin rosiglitazone placebo pioglitazone miglitol metformin benfluorex acarbose

−0.70 −0.44 −0.57 −1.20 0.00 −1.07 −0.98 −1.12 −0.91 −0.83

−1.5

−1

−0.5 0 0.5 HbA1c difference

[−0.95; −0.45] [−0.62; −0.26] [−0.82; −0.32] [−1.30; −1.11] [−1.22; −0.92] [−1.22; −0.74] [−1.24; −1.00] [−1.15; −0.66] [−1.03; −0.62]

1

Figure 2. Results for example by Senn et al. (2012, Figure 5).

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Table 2. Thrombolytic data example (Dias et al., 2010, Table 2). Pairwise meta analysis (Dias et al., MTC analysis (Dias et al., Treatments 2010) 2010)

Graph-theoretical method

i

j

Log OR

SE

Log OR

SE

Log OR

SE

1 1 1 1 1 1 2 2 2 3 3 3 3 3

2 3 5 7 8 9 7 8 9 4 5 7 8 9

0.003 0.158 0.060 0.666 0.369 0.006 0.541 0.297 0.017 0.113 0.019 0.215 0.145 1.409

0.031 0.049 0.090 0.185 0.518 0.039 0.418 0.347 0.039 0.056 0.066 0.118 0.356 0.416

0.002 0.177 0.124 0.475 0.203 0.016 0.477 0.205 0.014 0.128 0.053 0.298 0.026 0.193

0.030 0.043 0.060 0.101 0.219 0.037 0.104 0.220 0.037 0.054 0.056 0.097 0.220 0.056

0.004 0.174 0.122 0.473 0.206 0.009 0.469 0.202 0.013 0.113 0.052 0.299 0.032 0.183

0.027 0.043 0.060 0.101 0.225 0.031 0.104 0.225 0.031 0.056 0.055 0.098 0.226 0.053

Columns 1, 2: treatment numbers. Columns 3, 4: given treatment effects and standard errors. Columns 5, 6: result of the Bayesian MTC. Columns 7, 8: result using the graph-theoretical method. Treatments: 1 streptokinase, 2 alteplase, 3 accelerated alteplase, 4 streptokinase plus alteplase, 5 reteplase, 6 tenocteplase (not a part of the network), 7 per-cutaneous transluminal coronary angioplasty, 8 urokinase, 9 anistreptilase. MTC, multiple treatment comparison.

For the problem of handling p-armed studies, we used a reweighting method based on back-calculating variances using the Laplacian matrix and its pseudoinverse. This is more efficient than splitting each arm of a multi-armed study into disjoint parts belonging to different comparisons, which would lead to dividing the weight of each comparison by p  1, an approach mentioned and—with some reservation—recommended by the ‘Cochrane Handbook for Reviewers of Interventions’ (Higgins and Green, 2008, 16.5.4). As the graph-theoretical method corresponds to a fixed effect model, there are some limitations. First, it would be desirable to generalize the approach to a random effects model. To do this, a generalized estimate of the heterogeneity variance estimator may be derived in the future from the inconsistency statistic Qtotal. However, if not only random treatment-trial interactions but also random trial effects are introduced, modeling may raise problems with recovery of inter-trial information (Yates, 1940; Senn, 2000; Senn et al., 2012). Secondly, methods should be developed for including covariates into the model. Despite these limitations, the graph-theoretical approach is easy to implement, and the correspondence between electrical and meta-analytic networks seems interesting in itself and casts new light on network meta-analysis.

4.1. Software All calculations were carried out using the open source statistical environment R (R Development Core Team, 2009). We used the R package meta, written by Guido Schwarzer (2007), and developed an R function network given in Appendix B.

Appendix A: Proof of proposition

Proof. It has been shown that the effective variances given by Equation (3) are equal to the diagonal elements of the matrix G = BL+B> (which plays also a role in Equation (4)) (Spielman and Srivastava, 2009). We derive some properties of G. We see that G = GWG by GWG ¼ BLþ B> WBLþ B> ¼ BLþ LLþ B> ¼ BLþ B> ¼ G This means that G, GW, WG all have the same rank as L+, which is n  1. For an arbitrary eigenvector z of WG with eigenvalue l, we have Copyright © 2012 John Wiley & Sons, Ltd.

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Table 3. Breast cancer data (Mauri et al., 2008, Tables 2 and 3). Graph-theoretical method HR Treatment reference is old agents (95% confidence interval) Old agents (single agent) Novel non-taxane agents + taxanes Taxanes + trastuzumab Taxanes (combination regimen) Standard-dose anthracycline + trastuzumab Standard-dose anthracycline + novel non-taxane agents + taxanes Taxanes + lapatinib Standard-dose anthracycline + taxanes Standard-dose anthracycline + novel non-taxane agents Taxanes (single agent) Standard-dose anthracycline (combination regimen) Novel non-taxane agents + lapatinib Low-dose mitoxantrone (combination regimen) Low-dose anthracycline (combination regimen) Standard-dose anthracycline (single agent) Novel non-taxane agents + bevacizumab Novel non-taxane agents (combination regimen) Novel non-taxane agents (single agent) Old agents (combination regimen) Standard-dose mitoxantrone (combination regimen) Low-dose anthracycline (single agent) Standard-dose mitoxantrone (single agent)

1.00 0.53 0.53 0.60 0.57

MTC (Mauri et al., 2008, Table 3) HR (95% credible interval)

(reference) (0.43–0.67) (0.40–0.70) (0.42–0.86) (0.42–0.79)

1.00 0.49 0.51 0.53 0.55

(reference) (0.37–0.67) (0.35–0.72) (0.34–0.85) (0.36–0.84)

0.59 (0.39–0.90)

0.56 (0.34–0.94)

0.61 (0.46–0.81) 0.72 (0.60–0.86)

0.57 (0.38–0.86) 0.64 (0.53–0.78)

0.69 (0.56–0.85)

0.65 (0.49–0.85)

0.70 (0.59–0.84) 0.70 (0.61–0.80)

0.67 (0.55–0.81) 0.67 (0.57–0.78)

0.63 (0.38–1.03) 0.72 (0.46–1.11)

0.68 (0.38–1.23) 0.69 (0.41–1.18)

0.74 (0.65–0.85)

0.70 (0.61–0.81)

0.75 (0.64–0.88)

0.71 (0.60–0.84)

0.66 (0.46–0.94)

0.71 (0.45–1.12)

0.72 (0.55–0.93)

0.72 (0.53–0.99)

0.68 (0.57–0.83)

0.74 (0.60–0.91)

0.78 (0.68–0.90) 0.75 (0.63–0.89)

0.75 (0.65–0.85) 0.75 (0.62–0.90)

0.81 (0.69–0.95)

0.78 (0.64–0.94)

0.88 (0.70–1.11)

0.82 (0.64–1.06)

HR = Hazard ratio, MTC = Multiple treatment comparison. Gz ¼ GWGz ¼ GðlzÞ ¼ lGz; that is, l must be 1 or 0. As a consequence, WG has the eigenvalue 1 with multiplicity n  1, and thus its trace is n  1. Let the diagonal elements of G be numbered as in subsection 2.1 and denoted Gkk, and note that the original variances are the inverse weights, 1/wk. The ratio of the effective variance Gkk to the original variance is wkGkk. We want to find the average of these ratios, 1X wk Gkk : m k Using the properties of G and writing ek for the k-th unit vector, the sum can be written X

wk Gkk ¼

k

X k

w k e> k Gek ¼

X

e> k WGek ¼ tr ðWGÞ ¼ n  1

k

and the average reduction factor proves to be (n  1)/m, as stated. This proposition is a generalization of corollary C3 by Klein and Randić (1993). Copyright © 2012 John Wiley & Sons, Ltd.

Res. Syn. Meth. 2012

G. RÜCKER

B. Appendix: R code library(meta) # Input needed: # E list of m edges in form of a m x 2 matrix # x starting vector of m observed effect comparisons # w starting vector of m inverse variance weights # network

Network meta-analysis, electrical networks and graph theory.

Network meta-analysis is an active field of research in clinical biostatistics. It aims to combine information from all randomized comparisons among a...
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