  We propose simple formulas of confidence intervals for the Wald statistic,
likelihood ratio statistic, and score statistic for a network meta-analysis. In
addition, we consider resolutions for concerns that network meta-analyses with
a small number of studies cannot hold a nominal confidence level. For a bias
adjustment in analyses with a small number of studies, a Bartlett-type
adjustment is a well-known method. Many Bartlett-type adjustment-type methods
are based on maximum likelihood estimators. However, the network meta-analysis
often uses the restricted maximum likelihood estimators that have not been
extensively discussed in Bartlett-type adjustment. In this paper, 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. In addition, we propose a higher-order adjustment by applying the
bootstrap method to the Bartlett-type adjusted statistics. Using a computer
simulation, we confirmed that the adjusted confidence intervals maintained a
nominal confidence level. In addition, we confirmed that the confidence
interval of the likelihood ratio test based on the restricted maximum
likelihood method performs well without further bootstrap adjustment. Finally,
we demonstrated that confidence intervals were adjusted for actual network
meta-analysis.
