A Semiparametric Fractile Approach to Dichotomous 50% Dose and Willingness to Pay Analysis

Dichotomous 50% doses and the willingness to pay have been modelled using discrete choice models in which the variable that identifies the treatment or price (x) is regressed on a discrete outcome of a physical process or a choice p(y=1). The majority of the literature that applies this method employs linear models in x and are specified of the form G(p) = γ +βx where G(p) is usually specified as either the logit or probit function. The critical value of x is defined when p(y=1) = .5 and thus G(p) = 0. In this case the critical value is –γ/β. A typical application would include other covariate variables as well which does not alter the assumption of a linear impact of x. The consideration of more general specifications has been considered by some authors however these proposals often require specialized methods for estimation.In this paper we present a technique to define the critical value and the corresponding confidence interval using a fractile semiparametric model of the form G(p) = γ + β(x). The application considered here uses a Chebyshev polynomial in the fractile of the variable x to allow for a flexible specification and constructs the confidence interval using the inverse of a test that G(p) = 0. In the simple linear case these intervals are equivalent to the Fieller interval. The definition of the confidence intervals from the inverse of the hypothesis test allows the consideration of a much more flexible functional form. To demonstrate this technique a classic WTP specification used model the survival from the Titanic as a function of the fare paid for passage. The code for the Stata computer program to perform this estimation is provided in an appendix