Impact of Auxiliary Information and Measurement Errors on Mean Estimation with Mixture Optional Enhanced Trust (MOET) Randomized Response Model

Randomized response technique (RRT) surveys are designed to secure honest answers to sensitive questions. In this study, we consider the important issue of measurement error (ME). While non-response, a common culprit for survey inaccuracy, is a lesser issue in RRT studies because they are conducted through face-to-face interviews, measurement error is of particular significance. RRT models are generally more complex than other survey methods, sometimes requiring that respondents follow ordered instructions, draw cards from decks, and/or perform simple mathematical calculations. All of these steps can result in measurement errors, and when such error is high, estimation efficiency will suffer. In this study, we consider the impact of measurement error on a Mixture Optional Enhanced Trust (MOET) RRT model proposed in 2024, and we propose new estimators for this model that take measurement error into account. We also study the extent to which measurement error can be tolerated before it is so large that it overwhelms and undermines the benefit that RRT was implemented to yield in the first place (the reduction in or elimination of social desirability bias-related untruthfulness). We also draw attention to a surprising finding—that the presence of measurement error inadvertently serves to provide additional scrambling, thereby leading to an increase in privacy.