Calibrated-Two Optional Randomized Response Techniques (C-TORRT) for the estimation of quantitative sensitive variable information

Accurate estimation of sensitive quantitative variables remains a challenge in survey research due to respondents' reluctance to disclose truthful information. While existing randomized response techniques (RRT) offer privacy protection, many suffer from inefficiencies and limited robustness. This study addresses this critical gap by proposing new classes of Calibrated-Two Optional Randomized Response Techniques (C-TORRT), developed through calibration methods that incorporate auxiliary information to enhance estimation accuracy and respondent privacy. The theoretical framework of the proposed models demonstrates unbiasedness, reduced variance, higher privacy protection, and a superior combined metric of efficiency and privacy. Empirical studies based on real-life and simulated data showed that the proposed C-TORRT models consistently outperformed existing RRT models. For instance, under Population I, the proposed model achieved a variance of 21.76704, privacy level of 222.4369, and a percentage relative efficiency (PRE) of 608.93, compared to the Azeem et al. model with variance 142.4927 and PRE 93.02. Similarly, under Population II, the C-TORRT model reduced the variance to 4.3098 and raised the PRE to 499.60, a significant improvement over Gjestvang and Singh's variance of 21.5316 and PRE of 100. Real-life data application using academic records confirmed these findings, where the C-TORRT estimators yielded lower variance (0.8084), higher privacy levels (817.28), and smaller combined efficiency-privacy metrics (0.000991) compared to existing models. These results underscore the superior efficiency, precision, and privacy protection of the proposed C-TORRT models, making them robust alternatives for sensitive quantitative data collection.