A Markov chain approach for ranking treatments in network meta-analysis

Abstract

When interpreting the relative effects from a network meta-analysis (NMA), researchers are usually aware of the potential limitations that may render the results for some comparisons less useful or meaningless. In the presence of sufficient and appropriate data, some of these limitations (eg, risk of bias, small-study effects, publication bias) can be taken into account in the statistical analysis. Very often, though, the necessary data for applying these methods are missing and data limitations cannot be formally integrated into ranking. In addition, there are other important characteristics of the treatment comparisons that cannot be addressed within a statistical model but only through qualitative judgments; for example, the relevance of data to the research question, the plausibility of the assumptions, and so on. Here, we propose a new measure for treatment ranking called the Probability of Selecting a Treatment to Recommend (POST-R). We suggest that the order of treatments should represent the process of considering treatments for selection in clinical practice and we assign to each treatment a probability of being selected. This process can be considered as a Markov chain model that allows the end-users of NMA to select the most appropriate treatments based not only on the NMA results but also to information external to the NMA. In this way, we obtain rankings that can inform decision-making more efficiently as they represent not only the relative effects but also their potential limitations. We illustrate our approach using a NMA comparing treatments for chronic plaque psoriasis and we provide the Stata commands.

Keywords: comparative effectiveness research; multiple treatments; selection probabilities; stochastic process; treatment hierarchy.

FIGURE 1

Data and ranking results of the psoriasis network. A, The network diagram for efficacy (PASI 90). B, The treatment ranking based on P‐scores for efficacy and safety jointly. Different colors represent different clusters of treatments with respect to their ranking on both outcomes jointly. ACI = Acitretin, ADA = Adalimumab, ALEFACEPT = Alefacept, APRE = Apremilast, BRODA = Brodalumab, CERTO = Certolizumab, CICLO = Ciclopsorin, ETA = Etanercept, FUM = Fumaric acid esters, GUSEL = Guselkumab, IFX = Infliximab, ITO = Itolizumab, IXE = Ixekizumab, MTX = Methotrexate, PBO = Placebo, PONE = Ponesimod, SECU = Secukinumab, TILDRA = Tildrakizumab, TOFA = Tofacitinib, USK = Ustekinumab [Colour figure can be viewed at wileyonlinelibrary.com]

FIGURE 2

Graphical representation of the Markov Chain model for treatment selection using a hypothetical example of a three‐treatment network. Arrows represent the movement from one treatment to the other that happens when a change in the preference between the two treatments occurs

FIGURE 3

Relationship between P‐scores and POST‐R values when only the relative effects are considered (ie, z = 1). Red points correspond to drugs not presenting a good agreement between the two measures and r is the Pearson correlation coefficient

FIGURE 4

Ranking results of the psoriasis network using the POST‐R measure and considering the relative effects for efficacy in the transition probabilities and different characteristics in the initial probability distribution as indicated by the legend. Drugs have been ordered according to the ranking for efficacy and safety (green bars) [Colour figure can be viewed at wileyonlinelibrary.com]