Ordinal Dominance Curve Based Inference for Stochastically Ordered Distributions

Abstract

In a variety of applications researchers collect data that are sampled under ordered experimental conditions. In such situations it is reasonable to assume that outcomes are ordered stochastically by the level of the treatment. For example a toxicologist may want to assess the effect of a toxin on the reproductive system by studying the number of offspring produced by an animal as a function of the dose that it received. A variety of scientific questions arise in such a context. The most basic is whether a dose-response relationship is present. If so, determining the minimal toxic dose is of great interest. The strength of the ordering between groups i < j may be assessed by the probabilities P.X i X j / or monotone functions thereof. Large probabilities indicate a strong ordering. We find that such (and other) questions are naturally addressed by studying the ordinal dominance curve and the area under the curve (AUC). We develop new, non-parametric, order-restricted estimators for the AUCs in K 2 samples and investigate them in detail. We show that the restricted estimators are strongly consistent and have lower meansquared error compared with the unrestricted estimators. In the two-sample case estimators are considerably improved in those situations where the data are highly variable and/or sample sizes are small, and where the AUC is close to its boundary value. Note that these are precisely the conditions under which estimating the AUC is difficult. Simulations indicate that in the multisample case estimators are improved under a much broader set of conditions with substantial reductions in mean-squared error. These estimators provide a basis for a family of distribution-free tests for the usual stochastic order in K 2 samples. Limiting distributions are derived. An accurate and simple approximation to the tail area is given for K D 2. Simulation studies show that the new tests perform well compared with existing alternatives. The methodology proposed is illustrated by applying it to an aquatic toxicology study where it is used to determine the minimal toxic dose more accurately.

Estimating a distribution function subject to a stochastic order restriction: a comparative study

“…Note that T U is exactly the (one sided) WMW test. The statistic T P was recently investigated in detail by Davidov and Herman (2012) and the statistic T NP was proposed and studied by Lee and Wolfe (1976). The statistic T S seems to be new.…”

“…In addition, the existing studies only investigated limited types of ordering. We emphasise that the existing theoretical results, see Huang and Praestgaard (1996), Rojo and Ma (1996), El-Barmi and Mukerjee (2005) and Davidov and Herman (2012), are, due to their complexity, of limited value for comparing the different methodologies. It follows that simulation studies, as reported here, are essential.…”

“…Several estimators for the DFs, F and G, have been proposed. We will compare the usual empirical distribution function (EDF) with three estimators which explicitly account for the ordering of F and G. These are (1) Brunk, Franck, Hanson, and Hogg (1966) nonparametric maximum-likelihood estimator (the np-MLE); (2) an estimator which we refer to as the pointwise maximum-likelihood estimator (the p-MLE), see Hogg (1965), El-Barmi and Mukerjee (2005), Davidov (2006), Davidov and Herman (2009) and Davidov and Herman (2012) among others; and (3) Lo's (1987) and Rojo and Ma's (1996) estimator, which we refer to as the switch MLE (the S-MLE).…”

Nonparametric goodness-of-fit tests for uniform stochastic ordering

“…In many applications, it is of interest to compare F and G . The ordinal dominance curve (ODC), which plots ( G ( t ) , F ( t )) for −∞ ≤ t ≤∞, is a useful graphical tool that facilitates such a comparison (Bamber, 1975; Hsieh and Turnbull, 1996; Carolan and Tebbs, 2005; Davidov and Herman, 2012). The ODC can also be defined as R = FG −1 , where G −1 ( u ) = inf{ t: G ( t ) ≥ u } is the quantile function of G .…”

“…There is a substantive literature on nonparametric tests for stochastic orderings with two or more distributions; see Davidov and Herman (2012), El Barmi and McKeague (2016), and the references therein. In the two-sample case, most of this literature describes tests where the equal distribution assumption F = G is treated as the null hypothesis and the ordering (i.e., F ≤ S G , F ≤ US G , or F ≤ LR G ) is placed in the alternative.…”