Tapan K. Nayak

Statistical Significance: Rationale, Validity and Utility

A Litany of Criticisms of NHSTP The Null-Hypothesis Significance-Test Procedure (NHSTP) A Phenomenon and its Quartets of Hypotheses Evidential Support for a Theory and NHSTP Effect Size and Related Issues A Critical Look at Statistical Power Bayesianism A Case for NHSTP

Estimation of location and scale parameters using generalized Pitman nearness criterion

Abstract A generalization of the Pitman nearness criterion for comparing estimators when the criterion for accuracy is based on a loss function is developed. The concept is applicable to multi-parameter problems. The generalized criterion is then used with invariance to find best estimators of a location or scale parameter. These best estimators do not depend on the form of the loss function provided that it satisfies some reasonable conditions. Thus, the generalized criterion is useful for finding a best estimator when the exact loss function is not known. Some examples are given which establish an additional optimality property of some commonly used estimators.

ON RANDOMIZED RESPONSE SURVEYS FOR ESTIMATING A PROPORTION

In studies about sensitive characteristics, randomized response (RR) methods are useful for generating reliable data, protecting respondents’ privacy. It is shown that all RR surveys for estimating a proportion can be encompassed in a common model and some general results for statistical inferences can be used for any given survey. The concepts of design and scheme are introduced for characterizing RR surveys. Some consequences of comparing RR designs based on statistical measures of efficiency and respondent’ protection are discussed. In particular, such comparisons lead to the designs that may not be suitable in practice. It is suggested that one should consider other criteria and the scheme parameters for planning a RR survey.

Calculating and Describing Uncertainty in Risk Assessment: The Bayesian Approach

Quantification of uncertainty associated with risk estimates is an important part of risk assessment. In recent years, use of second-order distributions, and two-dimensional simulations have been suggested for quantifying both variability and uncertainty. These approaches are better interpreted within the Bayesian framework. To help practitioners better use such methods and interpret the results, in this article, we describe propagation and interpretation of uncertainty in the Bayesian paradigm. We consider both the estimation problem where some summary measures of the risk distribution (e.g., mean, variance, or selected percentiles) are to be estimated, and the prediction problem, where the risk values for some specific individuals are to be predicted. We discuss some connections and differences between uncertainties in estimation and prediction problems, and present an interpretation of a decomposition of total variability/uncertainty into variability and uncertainty in terms of expected squared error of prediction and its reduction from perfect information. We also discuss the role of Monte Carlo methods in characterizing uncertainty. We explain the basic ideas using a simple example, and demonstrate Monte Carlo calculations using another example from the literature.

Large Sample Theory for the Bounds on the Gini and Related Indices of Inequality Estimated from Grouped Data

The joint asymptotic distribution of the upper and lower bounds for the Gini index derived by Gastwirth for grouped data are obtained. From them a conservative asymptotically distribution-free confidence interval for the population Gini index is presented. The methods also yield similar results for other indices of inequality (e.g., Theils and Atkinsons).

On best unbiased prediction and its relationships to unbiased estimation

Let Y be a random vector and Z be a random variable with joint density f(y,z|θ), where θ∈Θ is a vector of unknown parameters. This paper discusses minimum mean squared error (MSE) unbiased prediction of Z based on Y, and its relationships to minimum variance unbiased estimation of ψ(θ)=E[Z|θ], the expected value of Z. A Rao–Cramer type lower bound for the MSE of an unbiased predictor is presented and a characterization of uniformly minimum MSE unbiased predictors (UMMSEUP) is discussed. When Y and Z are independent given θ, the UMMSEUP of Z and the uniformly minimum variance unbiased estimator (UMVUE) of ψ(θ) are shown to be identical. If the marginal model {f(y|θ),θ∈Θ} admits a complete sufficient statistic T(Y), we prove that (a) the UMMSEUP of Z exists if and only if Z admits an unbiased predictor and there exist two functions k and h such that E[Z|y,θ]=k(y)+h(T(y),θ) with probability 1 for all θ∈Θ, and (b) the UMMSEUP of Z and the UMVUE of ψ(θ) are the same if and only if E[Z|y,θ] depends on y only through T(y) with probability 1. We also discuss optimum predictions when the bias and MSE are defined conditionally on y and z, respectively. The results are applied to u–v method of estimation, prediction in mixed linear models, and estimation of the mean of a finite population under a super population model.

Concordant integrative gene set enrichment analysis of multiple large-scale two-sample expression data sets

BackgroundGene set enrichment analysis (GSEA) is an important approach to the analysis of coordinate expression changes at a pathway level. Although many statistical and computational methods have been proposed for GSEA, the issue of a concordant integrative GSEA of multiple expression data sets has not been well addressed. Among different related data sets collected for the same or similar study purposes, it is important to identify pathways or gene sets with concordant enrichment.MethodsWe categorize the underlying true states of differential expression into three representative categories: no change, positive change and negative change. Due to data noise, what we observe from experiments may not indicate the underlying truth. Although these categories are not observed in practice, they can be considered in a mixture model framework. Then, we define the mathematical concept of concordant gene set enrichment and calculate its related probability based on a three-component multivariate normal mixture model. The related false discovery rate can be calculated and used to rank different gene sets.ResultsWe used three published lung cancer microarray gene expression data sets to illustrate our proposed method. One analysis based on the first two data sets was conducted to compare our result with a previous published result based on a GSEA conducted separately for each individual data set. This comparison illustrates the advantage of our proposed concordant integrative gene set enrichment analysis. Then, with a relatively new and larger pathway collection, we used our method to conduct an integrative analysis of the first two data sets and also all three data sets. Both results showed that many gene sets could be identified with low false discovery rates. A consistency between both results was also observed. A further exploration based on the KEGG cancer pathway collection showed that a majority of these pathways could be identified by our proposed method.ConclusionsThis study illustrates that we can improve detection power and discovery consistency through a concordant integrative analysis of multiple large-scale two-sample gene expression data sets.

The Use of Diversity Analysis to Assess the Relative Influence of Factors Affecting the Income Distribution

This article uses diversity measures based on quadratic entropy to analyze the relative effects of factors such as age, sex, and education on the income distribution. In particular, sex appears to be as important as education in explaining the overall income inequality in the data collected by the U.S. Bureau of the Census. An advantage of the diversity measures is that they possess a decomposition of the total inequality into between-group and within-group components. Moreover, analogs of partial association enable one to study the effects of several factors. By using several measures, our inferences are shown to be robust.

Estimation of upper quantiles under model and parameter uncertainty

In this paper we assess accuracy of some commonly used estimators of upper quantiles of a right skewed distribution under both parameter and model uncertainty. In particular, for each of log-normal, log-logistic, and log-double exponential distributions, we study the bias and mean squared error of the maximum likelihood estimator (MLE) of the upper quantiles under both the correct and incorrect model specifications. We also consider two data dependent or adaptive estimators. The first (tail-exponential) is based on fitting an exponential distribution to the highest 10-20 percent of the data. The second selects the best fitting likelihood-based model and uses the MLE obtained from that model. The simulation results provide some practical guidance concerning the estimation of the upper quantiles when one is uncertain about the underlying model. We found that the consequences of assuming log-normality when the true distribution is log-logistic or log-double exponential are not severe in moderate sample sizes. For extreme quantiles, no estimator was reliable in small samples. For large sample sizes the selection estimator performs fairly well. For small sample sizes the tail-exponential method is a good alternative. Presenting it and the MLE for the log-normal enables one to assess the potential effects of model uncertainty.

Statistical properties of measures of between-group income differentials

Abstract A variety of statistical measures have been used to analyze interdistributional income inequality. Two recently introduced measures indicate a much larger secular change in the black–white income differential than the currently used measures. In order to understand this phenomenon both the theoretical properties of and empiric results obtained from five measures are given. It is shown that the new measures are more sensitive to the type of change that actually occured than the usual measures which were designed to detect a shift in location.