The issues studied in this thesis are: the assumption of local independence (LI) in item response theory (IRT) models in the first three chapters, and differential item functioning (DIF) in IRT in the last chapter. A new term "conditional trivariance" for three items is defined, which is a measurement of joint local dependence among item triples. This is extended to any J items. Relevant theorems are stated and proved, which provide necessary and sufficient conditions involving conditional co-variances, trivariances, etc. for local independence to hold. These theorems lead us to a method to test LI that goes beyond conditional covariance exploration. We realize this procedure by using a kernel smoothing technique. Asymptotic normality of the test statistic is given and proved, which in turn justifies our procedure. Simulation studies show the procedure works well for the case of large examinee sample sizes. Some Graduate Record Examination (GRE) verbal test data is analyzed.
