Detecting nonadherence without loss in efficiency: A simple extension of the crosswise model

In surveys concerning sensitive behavior or attitudes, respondents often do not answer truthfully, because of social desirability bias. To elicit more honest responding, the randomized-response (RR) technique aims at increasing perceived and actual anonymity by prompting respondents to answer with a randomly modified and thus uninformative response. In the crosswise model, as a particularly promising variant of the RR, this is achieved by adding a second, nonsensitive question and by prompting respondents to answer both questions jointly. Despite increased privacy protection and empirically higher prevalence estimates of socially undesirable behaviors, evidence also suggests that some respondents might still not adhere to the instructions, in turn leading to questionable results. Herein we propose an extension of the crosswise model (ECWM) that makes it possible to detect several types of response biases with adequate power in realistic sample sizes. Importantly, the ECWM allows for testing the validity of the model's assumptions without any loss in statistical efficiency. Finally, we provide an empirical example supporting the usefulness of the ECWM.