Sample size determination for the non-randomised triangular model for sensitive questions in a survey

Sample size determination is an essential component in public health survey designs on sensitive topics (e.g. drug abuse, homosexuality, induced abortions and pre or extramarital sex). Recently, non-randomised models have been shown to be an efficient and cost effective design when comparing with randomised response models. However, sample size formulae for such non-randomised designs are not yet available. In this article, we derive sample size formulae for the non-randomised triangular design based on the power analysis approach. We first consider the one-sample problem. Power functions and their corresponding sample size formulae for the one- and two-sided tests based on the large-sample normal approximation are derived. The performance of the sample size formulae is evaluated in terms of (i) the accuracy of the power values based on the estimated sample sizes and (ii) the sample size ratio of the non-randomised triangular design and the design of direct questioning (DDQ). We also numerically compare the sample sizes required for the randomised Warner design with those required for the DDQ and the non-randomised triangular design. Theoretical justification is provided. Furthermore, we extend the one-sample problem to the two-sample problem. An example based on an induced abortion study in Taiwan is presented to illustrate the proposed methods.