Analysis of Discrete Data under Order Restrictions

Strategies for the analysis of discrete data under order restrictions are discussed. We consider inference for sequences of binomial populations, and the corresponding risk difference, relative risk and odds ratios. These concepts are extended to deal with independent multinomial populations. Natural orderings such as stochastic ordering and cumulative ratio probability ordering are discussed. Methods are developed for the estimation and testing of differences between binomial as well as multinomial populations under order restrictions. In particular, inference for sequences of ordered binomial probabilities and cumulative probability ratios in multinomial populations are presented. Closed-form estimates of the multinomial parameters under order restrictions and test procedures for testing equality of two multinomial populations against the notion of cumulative probability ratio ordering which is stronger than stochastic ordering of the distributions are presented. Numerical examples are given to illustrate the techniques developed

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 or nonnegative. 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 noncarriers, and reveals a significant difference between the distribution in compound heterozygous carriers compared to noncarriers, but not between heterozygous carriers and noncarriers.Comment: Published in at http://dx.doi.org/10.1214/14-AOAS730 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

Evaluation of Tranche in Securitization and Long-range Ising Model

This econophysics work studies the long-range Ising model of a finite system with $N$ spins and the exchange interaction $\frac{J}{N}$ and the external field $H$ as a modely for homogeneous credit portfolio of assets with default probability $P_{d}$ and default correlation $\rho_{d}$. Based on the discussion on the $(J,H)$ phase diagram, we develop a perturbative calculation method for the model and obtain explicit expressions for $P_{d},\rho_{d}$ and the normalization factor $Z$ in terms of the model parameters $N$ and $J,H$. The effect of the default correlation $\rho_{d}$ on the probabilities $P(N_{d},\rho_{d})$ for $N_{d}$ defaults and on the cumulative distribution function $D(i,\rho_{d})$ are discussed. The latter means the average loss rate of the``tranche'' (layered structure) of the securities (e.g. CDO), which are synthesized from a pool of many assets. We show that the expected loss rate of the subordinated tranche decreases with $\rho_{d}$ and that of the senior tranche increases linearly, which are important in their pricing and ratings.Comment: 21 pages, 9 figure

The Flow Permanence Index: A Statistical Assessment of Flow Regime in Austin Streams

The report makes two explicit mentions of Waller Creek, briefly mentioning flow patterns and how they are highest near its mouth. In addition, the report contains valuable information concerning the importance of baseflow in the sustainability of a creek ecosystem.Flow permanence or the reliability of baseflow in a stream is an important metric in determining the potential of local streams to support aquatic life and can be used to provide an indication of future ecological changes. This report looks at quantifying the probabilities associated with permanent flow at all streams monitored for the City of Austin Environmental Integrity Index. Spatial patterns in flow permanence were examined, as well as the contributions of rainfall to flow permanence. Among the principal results is an index and ranking of streams with the most and least consistently flowing monitoring sites and a heuristic to calculate the probability of flow in a stream given the cumulative rainfall in the previous three months.Waller Creek Working Grou

Nonparametric survival analysis of epidemic data

This paper develops nonparametric methods for the survival analysis of epidemic data based on contact intervals. The contact interval from person i to person j is the time between the onset of infectiousness in i and infectious contact from i to j, where we define infectious contact as a contact sufficient to infect a susceptible individual. We show that the Nelson-Aalen estimator produces an unbiased estimate of the contact interval cumulative hazard function when who-infects-whom is observed. When who-infects-whom is not observed, we average the Nelson-Aalen estimates from all transmission networks consistent with the observed data using an EM algorithm. This converges to a nonparametric MLE of the contact interval cumulative hazard function that we call the marginal Nelson-Aalen estimate. We study the behavior of these methods in simulations and use them to analyze household surveillance data from the 2009 influenza A(H1N1) pandemic. In an appendix, we show that these methods extend chain-binomial models to continuous time.Comment: 30 pages, 6 figure

Clopper-Pearson Bounds from HEP Data Cuts

For the measurement of $N_s$ signals in $N$ events rigorous confidence bounds on the true signal probability $p_{\rm exact}$ were established in a classical paper by Clopper and Pearson [Biometrica 26, 404 (1934)]. Here, their bounds are generalized to the HEP situation where cuts on the data tag signals with probability $P_s$ and background data with likelihood $P_b<P_s$. The Fortran program which, on input of $P_s$, $P_b$, the number of tagged data $N^Y$ and the total number of data $N$, returns the requested confidence bounds as well as bounds on the entire cumulative signal distribution function, is available on the web. In particular, the method is of interest in connection with the statistical analysis part of the ongoing Higgs search at the LEP experiments