Abstract
Meta-analysis of diagnostic studies experiences the common problem that different studies might not be comparable since they have been using a different cut-off value for the continuous or ordered categorical diagnostic test value defining different regions for which the diagnostic test is defined to be positive. Hence specificities and sensitivities arising from different studies might vary just because the underlying cut-off value had been different. To cope with the cut-off value problem, interest is usually directed towards the receiver operating characteristic (ROC) curve which consists of pairs of sensitivities and false positive rate (1–specificity). In the context of meta-analysis, one pair represents one study and the associated diagram is called SROC curve where the S stands for ‘summary’. The paper will consider—as a novel approach—modelling SROC curves with the Lehmann family that assumes log-sensitivity is proportional to the log-false positive rate across studies. The approach allows for study-specific false positive rates which are treated as (infinitely many) nuisance parameters and eliminated by means of the profile likelihood. The adjusted profile likelihood turns out to have a simple univariate Gaussian structure which is ultimately used for building inference for the parameter of the Lehmann family. The Lehmann model is further extended by allowing the constant of proportionality to vary across studies to cope with unobserved heterogeneity. The simple Gaussian form of the adjusted profile likelihood allows this extension easily as a form of a mixed model in which unobserved heterogeneity is incorporated by means of a normal random effect. Some meta-analytic applications on diagnostic studies including brain natriuretic peptides for heart failure, alcohol use disorder identification test (AUDIT) and the consumption part of AUDIT for detection of unhealthy alcohol use as well as the mini-mental state examination for cognitive disorders are discussed to illustrate the methodology.