Lord's Paradox and two network meta-analysis models

The contrast-based model (CBM) is the most popular network meta-analysis (NMA) method, although alternative approaches, e.g., the baseline model (BM), have been proposed but seldom used. This article aims to illuminate the difference between the CBM and BM and explores when they produce different results. These models differ in key assumptions: The CBM assumes treatment contrasts are exchangeable across trials and models the reference (baseline) treatment's outcome levels as fixed effects, while the BM further assumes that the baseline treatment's outcome levels are exchangeable across trials and treats them as random effects. We show algebraically and graphically that the difference between the CBM and BM is analogous to the difference between the two analyses in a statistical conundrum called Lord's Paradox, in which the t-test and analysis of covariance (ANCOVA) yield conflicting conclusions about the group difference in weight gain. We show that this conflict arises because the t-test compares the observed weight change, whereas ANCOVA compares an adjusted weight change. In NMA, analogously, the CBM compares observed treatment contrasts, while the BM compares adjusted treatment contrasts. We demonstrate how the difference in modeling baseline effects can cause the CBM and BM to give different results. The analogy of Lord's Paradox provides insights into the different assumptions of the CBM and BM regarding the relationship between baseline effects and treatment contrasts. When these two models produce substantially different results, it may indicate a violation of the transitivity assumption. Therefore, we should be cautious in interpreting the results from either model.