Some methods for testing independence between the components of a random vector are discussed. The basic principle in the construction of the tests is dichotomization of each component variable. The distributions are obtained under randomization. Other applications of the tests are mentioned (Section 2). Certain limiting distributions are derived (Section 4). The exact distribution of the test statistic in a special case is tabulated (Section 5). A brief study of an alternative test is made (Section 6).