Adjusted closed-form confidence interval formulas for network meta-analysis with a small number of studies

We derive simple formulas for closed-form confidence intervals for the Wald statistic, likelihood ratio statistic, and score statistic for network meta-analysis (NMA). Additionally, we consider resolutions of concerns that network meta-analyzes with a small number of studies cannot maintain a nominal confidence level. For bias adjustment in analyzes with a small number of studies, the Bartlett-type adjustment is a well-known method. Many Bartlett-type adjustment-type methods are based on maximum likelihood estimators (MLEs). However, NMA often uses restricted MLEs that have not been extensively discussed with respect to the Bartlett-type adjustment. In this article, we propose a Bartlett-type adjustment method for the Wald statistic, likelihood ratio statistic, and score statistic when nuisance parameters are estimated by not only the maximum likelihood method but also the restricted maximum likelihood method. We can compute closed-form confidence intervals adjusted using the Bartlett-type adjustment immediately without any numerical calculations (eg, bootstrap method). Additionally, we propose a higher-order adjustment by applying the bootstrap method to Bartlett-type adjusted statistics. Using a computer simulation, we confirmed that the adjusted confidence intervals maintained a nominal confidence level. Additionally, we confirmed that the confidence intervals of the Wald statistic, likelihood ratio statistic, and score statistic based on the restricted maximum likelihood method performed well without further bootstrap adjustment and the performances of the three adjusted confidence intervals were comparable. Finally, we demonstrated that confidence intervals were adjusted for actual NMA. In the actual NMA, the adjusted confidence intervals of the Wald statistic were wider, the adjusted confidence intervals of the likelihood ratio statistic were also wider, and the adjusted confidence intervals of the score statistic were narrower. We recommend using the likelihood ratio test statistic with the restricted maximum likelihood estimator; however, just in case, we recommend applying the Bartlett-type adjustment to remove the second order bias. From demonstrations in actual studies, we confirmed that the adjusted confidence intervals improved compared with the naive confidence intervals.