Surveys on sensitive issues provide distorted prevalence estimates when participants fail to respond truthfully. The randomized-response technique (RRT) encourages more honest responding by adding random noise to responses, thereby removing any direct link between a participant's response and his or her true status with regard to a sensitive attribute. However, in spite of the increased confidentiality, some respondents still refuse to disclose sensitive attitudes or behaviors. To remedy this problem, we propose an extension of Mangat's (Journal of the Royal Statistical Society: Series B, 56, 93-95, 1994) variant of the RRT that allows for determining whether participants respond truthfully. This method offers the genuine advantage of providing undistorted prevalence estimates for sensitive attributes even if respondents fail to respond truthfully. We show how to implement the method using both closed-form equations and easily accessible free software for multinomial processing tree models. Moreover, we report the results of two survey experiments that provide evidence for the validity of our extension of Mangat's RRT approach.