Bivariate random-effects models are currently widely used to synthesize pairs of test sensitivity and specificity across studies. Inferences drawn based on these models may be distorted in the presence of outlying or influential studies. Currently, subjective methods such as inspection of forest plots are used to identify outlying studies in meta-analysis of diagnostic test accuracy studies. We proposed objective methods based on solid statistical reasoning for identifying outlying and/or influential studies. The proposed methods have been validated using simulation study and illustrated on two published meta-analysis data. Our methods outperform and neglect the subjectivity of the currently used ad hoc methods. The proposed methods can be used as a sensitivity analysis tool concurrently with the current bivariate random-effects models or as a preliminary analysis tool for robust models that accommodate outlying and/or influential studies in meta-analysis of diagnostic test accuracy studies.