Network meta-analysis of rare events using penalized likelihood regression

Abstract

Network meta-analysis (NMA) of rare events has attracted little attention in the literature. Until recently, networks of interventions with rare events were analyzed using the inverse-variance NMA approach. However, when events are rare the normal approximations made by this model can be poor and effect estimates are potentially biased. Other methods for the synthesis of such data are the recent extension of the Mantel-Haenszel approach to NMA or the use of the noncentral hypergeometric distribution. In this article, we suggest a new common-effect NMA approach that can be applied even in networks of interventions with extremely low or even zero number of events without requiring study exclusion or arbitrary imputations. Our method is based on the implementation of the penalized likelihood function proposed by Firth for bias reduction of the maximum likelihood estimate to the logistic expression of the NMA model. A limitation of our method is that heterogeneity cannot be taken into account as an additive parameter as in most meta-analytical models. However, we account for heterogeneity by incorporating a multiplicative overdispersion term using a two-stage approach. We show through simulation that our method performs consistently well across all tested scenarios and most often results in smaller bias than other available methods. We also illustrate the use of our method through two clinical examples. We conclude that our "penalized likelihood NMA" approach is promising for the analysis of binary outcomes with rare events especially for networks with very few studies per comparison and very low control group risks.

Keywords: bias reduction; maximum likelihood estimates; multiple treatment meta-analysis; rare endpoints.

FIGURE 1

Simulation results in terms of mean bias for scenarios with 5, 3, and 8 treatments (T), respectively. Missing bars correspond to 0 mean bias (after rounding to two decimal places) for the respective NMA method. The Monte‐Carlo standard error across the different scenarios and methods ranges from 0.004 to 0.02 with a mean value equal to 0.01. Models marked as NR are not relevant to the figure and thus no results are plotted

FIGURE 2

Simulation results in terms of coverage probability for scenarios with 5, 3, and 8 treatments (T), respectively. The horizontal lines represent the upper and lower bounds of the 95% confidence interval which is constructed for the nominal level

FIGURE 3

Simulation results in terms of MSE for scenarios with 5, 3, and 8 treatments (T), respectively. Models marked as NR are not relevant to the figure and thus no results are plotted

FIGURE 4

Simulation results in terms of length of confidence (or credible) intervals for scenarios with 5, 3, and 8 treatments (T), respectively

FIGURE 5

Network diagram for the inhaled medications example. HH, tiotropium dry powder; ICS, inhaled corticosteroid; LABA, long‐acting β2 agonist; TIO‐TIO‐SMI, tiotropium solution

FIGURE 6

Forest plots showing the odds ratios obtained from the inhaled medications network for all comparisons against the reference

FIGURE 7

Network diagrams for the psoriasis example. Panel (A) shows the initially well‐connected network while panel (B) the resulting network after the exclusion of studies that report only zero events. The node Anti‐IL 23 is eliminated with the discarded studies. Anti‐IL, anti‐interleukin; Anti‐TNF, anti‐tumor necrosis factor

FIGURE 8

Forest plots showing the odds ratios obtained from psoriasis network for all comparisons against the reference