Using logistic regression to estimate delay-discounting functions

The monetary choice questionnaire (MCQ) and similar computer tasks ask preference questions in order to ascertain indifference, the perceived equivalence of immediate versus larger delayed rewards. Indifference data are then fitted with a hyperbolic function, summarizing the decline in perceived value with delay time. We present a fitting method that estimates the hyperbolic parameter k directly from survey responses. Binary preferences are modeled as a function of time (X2) and a transformed reward ratio (X1), yielding logistic regression coefficients beta 2 and beta 1. The hyperbolic parameter emerges as k = beta 2/beta 1, where the logistic predicted p = .5 (the definition of indifference). The MCQ was administered to 1,073 adolescents and was scored using both standard and logistic methods. The means for In(k) were similar (standard, -4.53; logistic, -4.51), and the results were highly correlated (rho = .973). Simulated MCQ data showed that k was unbiased, except where beta 1 > or = -1, indicating a vague survey response. Jackknife standard errors provided excellent coverage.