Control rate regression is a diffuse approach to account for heterogeneity
among studies in meta-analysis by including information about the outcome risk
of patients in the control condition. Correcting for the presence of
measurement error affecting risk information in the treated and in the control
group has been recognized as a necessary step to derive reliable inferential
conclusions. Within this framework, the paper considers the problem of small
sample size as an additional source of misleading inference about the slope of
the control rate regression. Likelihood procedures relying on first-order
approximations are shown to be substantially inaccurate, especially when
dealing with increasing heterogeneity and correlated measurement errors. We
suggest to address the problem by relying on higher-order asymptotics. In
particular, we derive Skovgaard's statistic as an instrument to improve the
accuracy of the approximation of the signed profile log-likelihood ratio
statistic to the standard normal distribution. The proposal is shown to provide
much more accurate results than standard likelihood solutions, with no
appreciable computational effort. The advantages of Skovgaard's statistic in
control rate regression are shown in a series of simulation experiments and
illustrated in a real data example. R code for applying first- and second-order
statistic for inference on the slope on the control rate regression is
provided
