Meta-analysis is a popular statistical method that synthesizes evidence from multiple studies. Conventionally, both the hierarchical and bivariate models for meta-analysis of diagnostic test accuracy (DTA) studies assume that the random-effects follow the bivariate normal distribution. However, this assumption is restrictive, and inferences could be misleading when it is violated. On the other hand, subjective methods such as inspection of forest plots are used to identify outlying studies in a meta-analysis of DTA studies. Moreover, inferences made using the well-established bivariate random-effects models, when outlying or influential studies are present, may lead to misleading conclusions. Thus, the aim of this thesis is to address these issues by introducing alternative and robust statistical methods. First, we extend the current bivariate linear mixed model (LMM) by assuming a flexible bivariate skew-normal distribution for the random-effects. The marginal distribution of the proposed model is analytically derived so that parameter estimation can be performed using standard likelihood methods. Overall, the proposed model performs better in terms of confidence interval width of the overall sensitivity and specificity, and with regards to bias and root mean squared error of the between-study (co)variances than the traditional bivariate LMM. Second, we propose objective methods based on solid statistical reasoning for identifying outlying and/or influential studies in a meta-analysis of DTA studies. The performances of the proposed methods are evaluated using a simulation study. The proposed methods outperform and avoid the subjectivity of the currently used ad hoc approaches. Finally, we develop a new robust bivariate random-effects model which accommodates outlying and influential observations and leads to a robust statistical inference by down-weighting the effect of outlying and influential studies. The proposed model produces robust point estimates of sensitivity and specificity compared to the standard models, and also generates a similar point and interval estimates of sensitivity and specificity as the standard models in the absence of outlying or influential studies. / Thesis / Doctor of Philosophy (PhD) / Diagnostic tests vary from the noninvasive rapid strep test used to identify whether a patient has a bacterial sore throat to the much complex and invasive biopsy test used to examine the presence, cause, and extent of a severe condition, say cancer. Meta-analysis is a widely used statistical method that synthesizes evidence from several studies. In this thesis, we develop novel statistical methods extending the traditional methods for meta-analysis of diagnostic test accuracy studies. Our proposed methods address the issue of modelling asymmetrical data, identifying outlier studies, and optimally accommodating these outlying studies in a meta-analysis of diagnostic test accuracy studies. Using both real-life and simulated datasets, we show that our proposed methods perform better than conventional methods in a wide range of scenarios. %Therefore, we believe that our proposed methods are essential for methodologists, clinicians and health policy professionals in the process of making a correct judgment to using the appropriate diagnostic test to diagnose patients.