Numerical approximations are important research areas for dealing with complicated functional forms. Techniques for developing accurate and efficient calculation of combined likelihood functions in meta-analysis are studied. The first part of the thesis introduces a B-spline approximation for making a parsimonious model in the simplest case(2-dimensional case) of correlation structure. Inference about the correlation between vitamin C intake & vitamin C serum level is developed by using likelihood intervals and the MLE, along with comparison with conventional methods. The second part studies a multivariate numerical integration method for developing a better approximation of the likelihood for correlation matrices. Analyses for (1) intercorrelations among Math, Spatial and Verbal scores in an SAT exam and (2) intercorrelations among Cognitive Anxiety, Somatic Anxiety and Self Confidence from Competitive State Anxiety Inventory (CSAI-2) are explored. Algorithms to evaluate likelihood and to find the MLE is developed. Comparison with two conventional methods (joint asymptotic weighted average method & marginal asymptotic weighted average method) is shown
