Derek S. Young

Mixtures of regressions with predictor-dependent mixing proportions

We extend the standard mixture of linear regressions model by allowing the mixing proportions to be modeled nonparametrically as a function of the predictors. This framework allows for more flexibility in the modeling of the mixing proportions than the fully parametric mixture of experts model, which we also discuss. We present an EM-like algorithm for estimation of the new model. We also provide simulations demonstrating that our nonparametric approach can provide a better fit than the parametric approach in some instances and can serve to validate and thus reinforce the parametric approach in others. We also analyze and interpret two real data sets using the new method.

Semiparametric mixtures of regressions

We present an algorithm for estimating parameters in a mixture-of-regressions model in which the errors are assumed to be independent and identically distributed but no other assumption is made. This model is introduced as one of several recent generalizations of the standard fully parametric mixture of linear regressions in the literature. A sufficient condition for the identifiability of the parameters is stated and proved. Several different versions of the algorithm, including one that has a provable ascent property, are introduced. Numerical tests indicate the effectiveness of some of these algorithms.

Fiducial-based tolerance intervals for some discrete distributions

Fiducial quantities are proposed to construct approximate tolerance limits and intervals for functions of some discrete random variables. Using established fiducial quantities for binomial proportions, Poisson rates, and negative binomial proportions, an approach is demonstrated to handle functions of discrete random variables, whose distributions are either not available or are intractable. The construction of tolerance intervals using fiducial quantities is straightforward and, thus, amenable to numerical computation. An extensive numerical study shows that for most settings of the cases considered, the coverage probabilities are near the nominal levels. The applicability of the method is further demonstrated using four real datasets, including a discussion of the corresponding software that is available for the R programming language.

A procedure for approximate negative binomial tolerance intervals

In this article, we present a procedure for approximate negative binomial tolerance intervals. We utilize an approach that has been well-studied to approximate tolerance intervals for the binomial and Poisson settings, which is based on the confidence interval for the parameter in the respective distribution. A simulation study is performed to assess the coverage probabilities and expected widths of the tolerance intervals. The simulation study also compares eight different confidence interval approaches for the negative binomial proportions. We recommend using those in practice that perform the best based on our simulation results. The method is also illustrated using two real data examples.

Improved nonparametric tolerance intervals based on interpolated and extrapolated order statistics

The standard approach to construct nonparametric tolerance intervals is to use the appropriate order statistics, provided a minimum sample size requirement is met. However, it is well-known that this traditional approach is conservative with respect to the nominal level. One way to improve the coverage probabilities is to use interpolation. However, the extension to the case of two-sided tolerance intervals, as well as for the case when the minimum sample size requirement is not met, have not been studied. In this paper, an approach using linear interpolation is proposed for improving coverage probabilities for the two-sided setting. In the case when the minimum sample size requirement is not met, coverage probabilities are shown to improve by using linear extrapolation. A discussion about the effect on coverage probabilities and expected lengths when transforming the data is also presented. The applicability of this approach is demonstrated using three real data sets.

Mixtures of regressions with changepoints

We introduce an extension to the mixture of linear regressions model where changepoints are present. Such a model provides greater flexibility over a standard changepoint regression model if the data are believed to not only have changepoints present, but are also believed to belong to two or more unobservable categories. This model can provide additional insight into data that are already modeled using mixtures of regressions, but where the presence of changepoints has not yet been investigated. After discussing the mixture of regressions with changepoints model, we then develop an Expectation/Conditional Maximization (ECM) algorithm for maximum likelihood estimation. Two simulation studies illustrate the performance of our ECM algorithm and we analyze a real dataset.

Computing Tolerance Intervals and Regions Using R

Abstract Tolerance intervals provide bounds on a specified proportion of the sampled population ( P ) with a given confidence level ( γ ). While they are, perhaps, less known than confidence and prediction intervals, there are some applications where tolerance intervals are commonly used, such as in quality control, setting engineering tolerances, and environmental monitoring. We present a general introduction to the topic followed by overviews of how to calculate tolerance intervals for some continuous and discrete distributions, nonparametric tolerance limits, tolerance intervals for regression settings, and multivariate normal tolerance regions. Calculating tolerance intervals and regions under these as well as other settings can be accomplished using the R package tolerance ( Young, 2010 ). For the settings in our discussion, we present real-data examples and demonstrate how to calculate tolerance intervals and regions using the tolerance package.

Regression Tolerance Intervals

In this article, we discuss the utility of tolerance intervals for various regression models. We begin with a discussion of tolerance intervals for linear and nonlinear regression models. We then introduce a novel method for constructing nonparametric regression tolerance intervals by extending the well-established procedure for univariate data. Simulation results and application to real datasets are presented to help visualize regression tolerance intervals and to demonstrate that the methods we discuss have coverage probabilities very close to the specified nominal confidence level.

Increased white matter metabolic rates in autism spectrum disorder and schizophrenia

Both autism spectrum disorder (ASD) and schizophrenia are often characterized as disorders of white matter integrity. Multimodal investigations have reported elevated metabolic rates, cerebral perfusion and basal activity in various white matter regions in schizophrenia, but none of these functions has previously been studied in ASD. We used 18fluorodeoxyglucose positron emission tomography to compare white matter metabolic rates in subjects with ASD (n = 25) to those with schizophrenia (n = 41) and healthy controls (n = 55) across a wide range of stereotaxically placed regions-of-interest. Both subjects with ASD and schizophrenia showed increased metabolic rates across the white matter regions assessed, including internal capsule, corpus callosum, and white matter in the frontal and temporal lobes. These increases were more pronounced, more widespread and more asymmetrical in subjects with ASD than in those with schizophrenia. The highest metabolic increases in both disorders were seen in the prefrontal white matter and anterior limb of the internal capsule. Compared to normal controls, differences in gray matter metabolism were less prominent and differences in adjacent white matter metabolism were more prominent in subjects with ASD than in those with schizophrenia. Autism spectrum disorder and schizophrenia are associated with heightened metabolic activity throughout the white matter. Unlike in the gray matter, the vector of white matter metabolic abnormalities appears to be similar in ASD and schizophrenia, may reflect inefficient functional connectivity with compensatory hypermetabolism, and may be a common feature of neurodevelopmental disorders.

Approximate tolerance intervals for the discrete Poisson–Lindley distribution

The Poisson–Lindley distribution is a compound discrete distribution that can be used as an alternative to other discrete distributions, like the negative binomial. This paper develops approximate one-sided and equal-tailed two-sided tolerance intervals for the Poisson–Lindley distribution. Practical applications of the Poisson–Lindley distribution frequently involve large samples, thus we utilize large-sample Wald confidence intervals in the construction of our tolerance intervals. A coverage study is presented to demonstrate the efficacy of the proposed tolerance intervals. The tolerance intervals are also demonstrated using two real data sets. The R code developed for our discussion is briefly highlighted and included in the tolerance package.