Covariate Randomized Response Models

Abstract

Abstract The randomized response technique (RRT) is a survey procedure that requires respondents to decide randomly whether to respond to a sensitive question in its positive or negative form (Warner 1965) or respond to a sensitive question and an unrelated question (Greenberg, Abul-Ela, Simmons, and Horvitz 1969). Although RRT results in larger standard errors than direct questioning or anonymous questionnaires, there is empirical evidence that RRT results in less underreporting of sensitive behavior, so estimates tend to be biased to a lesser degree than when alternate techniques are used. This article develops and illustrates the application of a covariate extension of RRT that can reduce standard errors and that allows for modeling the relation between the proportion with a sensitive characteristic and a covariate. It is assumed that the covariate, X, has a known distribution. Then, the covariate RRT involves selecting a functional form for the relation between π A , the proportion with the sensitive characteristic, and X: πA|X = g(X | α), where α is a vector of parameters that generally must be estimated from the data. Because of its mathematical tractability, the two-parameter logistic function g(X | α) = [1 + exp(– α1 – α2 · X)]–1 was developed for the Warner RRT and both the πv -known and πy -unknown versions of the unrelated-question RRT. A small-scale study using contrived but realistic data situations demonstrated the relative efficiency of the covariate RRT compared with the ordinary RRT at each level of a stratified covariate. It should be noted, however, that misspecification of the relation between πA and X may result in a substantial bias component, so it is important to assess the appropriateness of a model for a particular data set. The covariate RRT was applied to data for five questions from a cheating study involving 184 university students. The covariate was estimated grade point average (GPA). The logistic function provided acceptable fit for all five questions, although the parameter estimates associated with GPA were statistically significant for only three of the five questions.