We investigate the association of a sensitive characteristic or latent variable with observed binary random variables by the randomized response (RR) technique of Warner in his publication (Warner, S.L. J. Am. Stat. Assoc. 1965, 60, 63–69) and a latent class model. First, an expectation-maximization (EM) algorithm is provided to easily estimate the parameters of the null and alternative/full models for the association between a sensitive characteristic and an observed categorical random variable under the RR design of Warner’s paper above. The likelihood ratio test (LRT) is utilized to identify observed categorical random variables that are significantly related to the sensitive trait. Another EM algorithm is then presented to estimate the parameters of a latent class model constructed through the sensitive attribute and the observed binary random variables that are obtained from dichotomizing observed categorical random variables selected from the above LRT. Finally, two classification criteria are conducted to predict an individual in the sensitive or non-sensitive group. The practicality of the proposed methodology is illustrated with an actual data set from a survey study of the sexuality of first-year students, except international students, at Feng Chia University in Taiwan in 2016.