Sample size determination for the parallel model in a survey with sensitive questions

Recently, a new non-randomized parallel design is proposed by Tian (2013) for surveys with sensitive topics. However, the sample size formulae associated with testing hypotheses for the parallel model are not yet available. As a crucial component in surveys, the sample size formulae with the parallel design are developed in this paper by using the power analysis method for both the one- and two-sample problems. We consider both the one- and two-sample problems. The asymptotic power functions and the corresponding sample size formulae for both the one- and two-sided tests based on the large-sample normal approximation are derived. The performance is assessed through comparing the asymptotic power with the exact power and reporting the ratio of the sample sizes with the parallel model and the design of direct questioning. We numerically compare the sample sizes needed for the parallel design with those required for the crosswise and triangular models. Two theoretical justifications are also provided. An example from a survey on ‘sexual practices’ in San Francisco, Las Vegas and Portland is used to illustrate the proposed methods.