Correlated or matched data is frequently collected under many study designs in applied sciences such as the social, behavioral, economic, biological, medical, epidemiologic, health, public health, and drug developmental sciences. Challenges with respect to availability and cost commonly occur with matching observational or experimental study subjects, thus researchers frequently encounter situations where the observed sample consists of a combination of correlated and uncorrelated data. This paper discusses and proposes testing procedures to handle data when partially correlated data is available. Theoretical as well as numerical investigation will be provided. The proposed testing procedures will be applied to real data. These procedures will be of special importance in meta-analysis where partially correlated data is a concern when combining results of various studies.