The one-sayers model for the Extended Crosswise design

Abstract

Abstract The Extended Crosswise design is a randomized response design characterized by a sensitive and an innocuous question and two sub-samples with complementary randomization probabilities of the innocuous question. The response categories are ‘One’ with two different answers and ‘Two’ with two answers that are the same. Due to the complementary randomization probabilities, ‘One’ is the incriminating response in one sub-sample, and ‘Two’ in the other. The use of two sub-samples generates a degree of freedom to test for response biases with a goodness-of-fit test, but this test is unable to detect bias resulting from self-protective respondents giving the non-incriminating response when the incriminating response was required. This raises the question what a significant goodness-of-fit test measures? In this paper, we hypothesize that respondents are largely unaware which response is associated with the sensitive characteristic, and intuitively perceive ‘One’ as the safer response. We present empirical evidence for one-saying in six surveys among a total of 4,242 elite athletes, and present estimates of doping use corrected for it. Furthermore, logistic regression analyses are conducted to test the hypothesis that respondents who complete the survey in a short time are more likely to answer randomly, and therefore are less likely to be one-sayers.