Methods are proposed for comparing two diagnostic tests for the same data where a threshold for positive for each test is specified. One method contrasts the diagnostic tests' estimated risks. A second method compares the two tests' kappa coefficients. When thresholds for positive test results are specified a priori, maximum likelihood estimators and their asymptotic variances are derived and test statistics are presented for both case-control and naturalistic methods of sampling. The bootstrap is proposed as a method to assess differences in risk estimators when thresholds for positive test results are chosen by scanning the data. Examples are given to illustrate the methods.