Network meta-analysis and random walks

Network meta-analysis (NMA) is a central tool for evidence synthesis in clinical research. The results of an NMA depend critically on the quality of evidence being pooled. In assessing the validity of an NMA, it is therefore important to know the proportion contributions of each direct treatment comparison to each network treatment effect. The construction of proportion contributions is based on the observation that each row of the hat matrix represents a so-called 'evidence flow network' for each treatment comparison. However, the existing algorithm used to calculate these values is associated with ambiguity according to the selection of paths. In this work we present a novel analogy between NMA and random walks. We use this analogy to derive closed-form expressions for the proportion contributions. A random walk on a graph is a stochastic process that describes a succession of random 'hops' between vertices which are connected by an edge. The weight of an edge relates to the probability that the walker moves along that edge. We use the graph representation of NMA to construct the transition matrix for a random walk on the network of evidence. We show that the net number of times a walker crosses each edge of the network is related to the evidence flow network. By then defining a random walk on the directed evidence flow network, we derive analytically the matrix of proportion contributions. The random-walk approach, in addition to being computationally more efficient, has none of the associated ambiguity of the existing algorithm.

Robustness of Central Composite Design and Modified Central Composite Design to a Missing Observation for Non-Standard Models

Missing observations in an experimental design may lead to ambiguity in decision making thereby bringing an experiment to disrepute. Robustness, therefore, enables a process, not to break down in the presence of missing observations. This work constructed a modified central composite design (MCCD) from a four-variable central composite design (CCD) augmented with four center points using the leverage of a hat-matrix. The robustness of the CCD and MCCD were assessed when a design point is missing at the factorial, axial, and center points of the experiment, for a non-standard model, using the loss criterion, D-optimality, D-efficiency, and relative D-efficiency. When the designs are complete the MCCD shows higher D-efficiency and D-optimality for the non-standard model when compared to the CCD. In the absence of an observation from any of the designs, the CCD is found to be a more robust and efficient design compared to the MCCD as it has overall lower loss values at all the factors levels.

The statistical importance of a study for a network meta-analysis estimate

Background: In pairwise meta-analysis, the contribution of each study to the pooled estimate is given by its weight, which is based on the inverse variance of the estimate from that study. For network meta-analysis (NMA), the contribution of direct (and indirect) evidence is easily obtained from the diagonal elements of a hat matrix. It is, however, not fully clear how to generalize this to the percentage contribution of each study to a NMA estimate. Methods: We define the importance of each study for a NMA estimate by the reduction of the estimate's variance when adding the given study to the others. An equivalent interpretation is the relative loss in precision when the study is left out. Importances are values between 0 and 1. An importance of 1 means that the study is an essential link of the pathway in the network connecting one of the treatments with another. Results: Importances can be defined for two-stage and one-stage NMA. These numbers in general do not add to one and thus cannot be interpreted as `percentage contributions'. After briefly discussing other available approaches, we question whether it is possible to obtain unique percentage contributions for NMA. Conclusions: Importances generalize the concept of weights in pairwise meta-analysis in a natural way. Moreover, they are uniquely defined, easily calculated, and have an intuitive interpretation. We give some real examples for illustration.

The statistical importance of a study for a network meta-analysis estimate

Background In pairwise meta-analysis, the contribution of each study to the pooled estimate is given by its weight, which is based on the inverse variance of the estimate from that study. For network meta-analysis (NMA), the contribution of direct (and indirect) evidence is easily obtained from the diagonal elements of a hat matrix. It is, however, not fully clear how to generalize this to the percentage contribution of each study to a NMA estimate.MethodsWe define the importance of each study for a NMA estimate by the reduction of the estimate's variance when adding the given study to the others. An equivalent interpretation is the relative loss in precision when the study is left out. Importances are values between 0 and 1. An importance of 1 means that the study is an essential link of the pathway in the network connecting one of the treatments with another. ResultsImportances can be defined for two-stage and one-stage NMA. These numbers in general do not add to one and thus cannot be interpreted as `percentage contributions'. After briefly discussing other available approaches, we question whether it is possible to obtain unique percentage contributions for NMA. ConclusionsImportances generalize the concept of weights in pairwise meta-analysis in a natural way. Moreover, they are uniquely defined, easily calculated, and have an intuitive interpretation. We give some real examples for illustration.

The statistical importance of a study for a network meta-analysis estimate

Background: In pairwise meta-analysis, the contribution of each study to the pooled estimate is given by its weight, which is defined based on the inverse variance of the estimate from that study. For network meta-analysis (NMA), the contribution of direct (and indirect) evidence is easily obtained from the diagonal elements of the hat matrix. It is, however, not fully clear how to generalize this to the percentage contribution of each study to a NMA estimate. Methods: After briefly discussing available approaches, we want to question whether it is possible to obtain unique percentage contributions and discuss another approach. We define the importance of each study for a NMA estimate by the reduction of the estimate's variance when adding the given study to the others. An equivalent interpretation is the relative loss in precision when the study is left out. Results: Importances are values between 0 and 1. An importance of 1 means that the study is an essential link of the pathway in the network connecting one of the treatments with the other. These numbers in general do not add to one and thus cannot be interpreted as `percentage contributions'. Conclusions: Importances generalize the concept of weights in pairwise meta-analysis in a natural way. Moreover, they are uniquely defined, easily calculated, and have an intuitive interpretation. We give some real examples for illustration.

The statistical importance of a study for a network meta-analysis estimate

Background: In pairwise meta-analysis, the contribution of each study to the pooled estimate is given by its weight, which is defined based on the inverse variance of the estimate from that study. For network meta-analysis (NMA), the contribution of direct (and indirect) evidence is easily obtained from the diagonal elements of the hat matrix. It is, however, not fully clear how to generalize this to the percentage contribution of each study to a NMA estimate. Methods: After briefly discussing available approaches, we want to question whether it is possible to obtain unique percentage contributions and discuss another approach. We define the importance of each study for a NMA estimate by the reduction of the estimate's variance when adding the given study to the others. An equivalent interpretation is the relative loss in precision when the study is left out. Results: Importances are values between 0 and 1. An importance of 1 means that the study is an essential link of the pathway in the network connecting one of the treatments with the other. These numbers in general do not add to one and thus cannot be interpreted as `percentage contributions'. Conclusions: Importances generalize the concept of weights in pairwise meta-analysis in a natural way. Moreover, they are uniquely defined, easily calculated, and have an intuitive interpretation. We give some real examples for illustration.

Estimating the contribution of studies in network meta-analysis: paths, flows and streams

In network meta-analysis, it is important to assess the influence of the limitations or other characteristics of individual studies on the estimates obtained from the network. The proportion contribution matrix, which shows how much each direct treatment effect contributes to each treatment effect estimate from network meta-analysis, is crucial in this context. We use ideas from graph theory to derive the proportion that is contributed by each direct treatment effect. We start with the ‘projection’ matrix in a two-step network meta-analysis model, called the H matrix, which is analogous to the hat matrix in a linear regression model. We develop a method to translate H entries to proportion contributions based on the observation that the rows of H can be interpreted as flow networks, where a stream is defined as the composition of a path and its associated flow. We present an algorithm that identifies the flow of evidence in each path and decomposes it into direct comparisons. To illustrate the methodology, we use two published networks of interventions. The first compares no treatment, quinolone antibiotics, non-quinolone antibiotics and antiseptics for underlying eardrum perforations and the second compares 14 antimanic drugs. We believe that this approach is a useful and novel addition to network meta-analysis methodology, which allows the consistent derivation of the proportion contributions of direct evidence from individual studies to network treatment effects.

Estimating the contribution of studies in network meta-analysis: paths, flows and streams

In network meta-analysis, it is important to assess the influence of the limitations or other characteristics of individual studies on the estimates obtained from the network. The percentage contribution matrix, which shows how much each direct treatment effect contributes to each treatment effect estimate from network meta-analysis, is crucial in this context. We use ideas from graph theory to derive the percentage that is contributed by each direct treatment effect. We start with the ‘projection’ matrix in a two-step network meta-analysis model, called the H matrix, which is analogous to the hat matrix in a linear regression model. We develop a method to translate H entries to percentage contributions based on the observation that the rows of H can be interpreted as flow networks, where a stream is defined as the composition of a path and its associated flow. We present an algorithm that identifies the flow of evidence in each path and decomposes it into direct comparisons. To illustrate the methodology, we use two published networks of interventions. The first compares no treatment, quinolone antibiotics, non-quinolone antibiotics and antiseptics for underlying eardrum perforations and the second compares 14 antimanic drugs. We believe that this approach is a useful and novel addition to network meta-analysis methodology, which allows the consistent derivation of the percentage contributions of direct evidence from individual studies to network treatment effects.

Construction of Modified Central Composite Designs for Non-standard Models

The use of loss function in studying the reduction in determinant of information matrix due to missing observations has effectively produced designs that are robust to missing observations. Modified central composite designs are constructed for non-standard models using principles of the loss function or equivalently first compound of (I ) matrix associated with hat matrix . Although central composite designs (CCDs) are reasonably robust to model mis-specifications, efficient designs with fewer design points are more economical. By classifying the losses due to missing design points in the CCD portions, where there are multiple losses associated with specified CCD portions, the design points having less influence may be deleted from the full CCD. This leads to a possible increase in design efficiency and offers alternative designs, similar in the structure of CCDs, for non-standard models.

Estimating the contribution of studies in network meta-analysis: paths, flows and streams

In network meta-analysis, it is important to assess the influence of the limitations or other characteristics of individual studies on the estimates obtained from the network. The percentage contribution matrix, which shows how much each direct treatment effect contributes to each treatment effect estimate from network meta-analysis, is crucial in this context. We use ideas from graph theory to derive the percentage that is contributed by each direct treatment effect. We start with the ‘projection’ matrix in a two-step network meta-analysis model, called the H matrix, which is analogous to the hat matrix in a linear regression model. We develop a method to translate H entries to percentage contributions based on the observation that the rows of H can be interpreted as flow networks, where a stream is defined as the composition of a path and its associated flow. We present an algorithm that identifies the flow of evidence in each path and decomposes it into direct comparisons. To illustrate the methodology, we use two published networks of interventions. The first compares no treatment, quinolone antibiotics, non-quinolone antibiotics and antiseptics for underlying eardrum perforations and the second compares 14 antimanic drugs. We believe that this approach is a useful and novel addition to network meta-analysis methodology, which allows the consistent derivation of the percentage contributions of direct evidence from individual studies to network treatment effects.