Empirical likelihood-based tests for stochastic ordering

Testing against umbrella or tree orderings for binomial proportions with an adaptation of an insect resistance case

Alternative hypotheses for order restrictions, such as umbrella or inverse umbrella (a.k.a tree) orderings, have been studied extensively in the literature, although less so when the studied response for each individual is the presence or absence of the event of interest. Two families of test statistics for solving the problem of testing against an umbrella or a tree ordering when the responses are binomial proportions are studied in this work and their asymptotic distributions are derived. A simulation study is conducted to compare the empirical power of some members of the derived families of test statistics with competing approaches. The methodology developed here was driven by an applied problem arising in stored products research where despite universal mortality in the case of doses of 1000 ppm of the insecticide phosphine, unexpected survival was noted at higher doses.

A Study of Precipitation Patterns through Stochastic Ordering

The study of spatial and temporal changes in precipitation patterns is important to agriculture and natural ecosystems. These changes can be described by some climate change indices. Because these indices often have skewed probability distributions, some common statistical procedures become either inappropriate or less powerful when they are applied to the indices. A nonparametric approach based on stochastic ordering is proposed, which does not make any assumption on the shape of the distribution. This approach is applied to the average length of the period between two adjacent precipitation days, which is called the average number of consecutive dry days (ACDD). This approach is shown to be able to reveal some patterns in precipitation that other approaches do not. Using daily precipitations at 756 stations in China from 1960 to 2015, this work compares the ACDDs in three periods, 1960–1965, 1985–1990, and 2010–2015 for each province in China. The results show that ACDD increases stochastically from the period 1960–1965 to either the period 1985–1990 or the period 2010–2015, or from the period 1985–1990 to the period 2010–2015 in all but three provinces in China.

NONPARAMETRIC TESTING FOR MULTIPLE SURVIVAL FUNCTIONS WITH NON-INFERIORITY MARGINS

New nonparametric tests for the ordering of multiple survival functions are developed with the possibility of right censorship taken into account. The motivation comes from non-inferiority trials with multiple treatments. The proposed tests are based on nonparametric likelihood ratio statistics, which are known to provide more powerful tests than Wald-type procedures, but in this setting have only been studied for pairs of survival functions or in the absence of censoring. We introduce a novel type of pool adjacent violator algorithm that leads to a complete solution of the problem. The limit distributions can be expressed as weighted sums of squares involving projections of certain Gaussian processes onto the given ordered alternative. A simulation study shows that the new procedures have superior power to a competing combined-pairwise Cox model approach. We illustrate the proposed methods using data from a three-arm non-inferiority trial.

Testing for positive quadrant dependence

We develop an empirical likelihood approach to test independence of two univariate random variables X and Y versus the alternative that X and Y are strictly positive quadrant dependent (PQD). Establishing this type of ordering between X and Y is of interest in many applications, including finance, insurance, engineering, and other areas. Adopting the framework in Einmahl and McKeague (2003, Bernoulli), we create a distribution-free test statistic that integrates a localized empirical likelihood ratio test statistic with respect to the empirical joint distribution of X and Y. When compared to well known existing tests and distance-based tests we develop by using copula functions, simulation results show the EL testing procedure performs well in a variety of scenarios when X and Y are strictly PQD. We use three data sets for illustration and provide an online R resource practitioners can use to implement the methods in this article.

Tests for stochastic ordering under biased sampling

In two-sample comparison problems it is often of interest to examine whether one distribution function majorizes the other, i.e., for the presence of stochastic ordering. This paper develops a nonparametric test for stochastic ordering from size-biased data, allowing the pattern of the size bias to differ between the two samples. The test is formulated in terms of a maximally-selected local empirical likelihood statistic. A Gaussian multiplier bootstrap is devised to calibrate the test. Simulation results show that the proposed test outperforms an analogous Wald-type test, and that it provides substantially greater power over ignoring the size bias. The approach is illustrated using data on blood alcohol concentration of drivers involved in car accidents, where the size bias is due to drunker drivers being more likely to be involved in accidents. Further, younger drivers tend to be more affected by alcohol, so in making comparisons with older drivers the analysis is adjusted for differences in the patterns of size bias.

Empirical likelihood based tests for stochastic ordering under right censorship

This paper develops an empirical likelihood approach to testing for stochastic ordering between two univariate distributions under right censorship. The proposed test is based on a maximally selected local empirical likelihood statistic. The asymptotic null distribution is expressed in terms of a Brownian bridge. The new procedure is shown via a simulation study to have superior power to the log-rank and weighted Kaplan-Meier tests under crossing hazard alternatives. The approach is illustrated using data from a randomized clinical trial involving the treatment of severe alcoholic hepatitis.

COMBINING ISOTONIC REGRESSION AND EM ALGORITHM TO PREDICT GENETIC RISK UNDER MONOTONICITY CONSTRAINT

In certain genetic studies, clinicians and genetic counselors are interested in estimating the cumulative risk of a disease for individuals with and without a rare deleterious mutation. Estimating the cumulative risk is difficult, however, when the estimates are based on family history data. Often, the genetic mutation status in many family members is unknown; instead, only estimated probabilities of a patient having a certain mutation status are available. Also, ages of disease-onset are subject to right censoring. Existing methods to estimate the cumulative risk using such family-based data only provide estimation at individual time points, and are not guaranteed to be monotonic, nor non-negative. In this paper, we develop a novel method that combines Expectation-Maximization and isotonic regression to estimate the cumulative risk across the entire support. Our estimator is monotonic, satisfies self-consistent estimating equations, and has high power in detecting differences between the cumulative risks of different populations. Application of our estimator to a Parkinson's disease (PD) study provides the age-at-onset distribution of PD in PARK2 mutation carriers and non-carriers, and reveals a significant difference between the distribution in compound heterozygous carriers compared to non-carriers, but not between heterozygous carriers and non-carriers.