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Background Data on single-nucleotide polymorphisms (SNPs) have been found to be useful in predicting phenotypes ranging from an individual's class membership to his/her risk of developing a disease. In multi-class classification scenarios, clinical samples are often limited due to cost constraints, making it necessary to determine the sample size needed to build an accurate classifier based on SNPs. The performance of such classifiers can be assessed using the Area Under the Receiver Operating Characteristic (ROC) Curve (AUC) for two classes and the Volume Under the ROC hyper-Surface (V US) for three or more classes. Sample size determination based on AUC or V US would not only guarantee an overall correct classification rate, but also make studies more cost-effective. Results For coded SNP data from D(⩾ 2) classes, we derive an optimal Bayes classifier and a linear classifier, and obtain a normal approximation to the probability of correct classification for each classifier. These approximations are then used to evaluate the associated AUCs or V USs, whose accuracies are validated usingMonte Carlo simulations. We give a sample size determination method, which ensures that the difference between the two approximate AUCs (or V USs) is below a pre-specified threshold. The performance of our sample size determination method is then illustrated via simulations. For the HapMap data with three and four populations, a linear classifier is built using 92 independent SNPs and the required total sample sizes are determined for a continuum of threshold values. In all, four different sample size determination studies are conducted with the HapMap data, covering cases involving well-separated populations to poorly-separated ones. Conclusion For multi-classes, we have developed a sample size determination methodology and illustrated its usefulness in obtaining a required sample size from the estimated learning curve. For classification scenarios, this methodology will help scientists determine whether a sample at hand is adequate or more samples are required to achieve a pre-specified accuracy. A PDF manual for R package "SampleSizeSNP" is given in Additional file 1, and a ZIP file of the R package "SampleSizeSNP" is given in Additional file 2. [ABSTRACT FROM AUTHOR]

In this article, we propose a new generalized index to recover relationships between two sets of random vectors by finding the vector projections that minimize an distance between each projected vector and an unknown function of the other. The unknown functions are estimated using the Nadaraya-Watson smoother. Extensions to multiple sets and groups of multiple sets are also discussed, and a bootstrap procedure is developed to detect the number of significant relationships. All the proposed methods are assessed through extensive simulations and real data analyses. In particular, for environmental data from Los Angeles County, we apply our multiple-set methodology to study relationships between mortality, weather, and pollutants vectors. Here, we detect existence of both linear and nonlinear relationships between the dimension-reduced vectors, which are then used to build nonlinear time-series regression models for the dimension-reduced mortality vector. These findings also illustrate potential use of our method in many other applications. A comprehensive assessment of our methodologies along with their theoretical properties are given in a Web Appendix. [ABSTRACT FROM AUTHOR]

We propose a notion of central mean dimension reduction subspace for time series {x1} which does not require specification of a model but seeks to find a p x d matrix Φd, d ≤ p, so that the d x 1 vector where ΦdTXt-1 where Xt-1= (xt-1,…Xt-p)T for some p ≥ 1, includes all the information about xt that is available from E(xt∣Xt-1). For known p and d, we estimate the mean central subspace through the Nadaraya-Watson kernel smoother and establish the strong consistency of our estimator. In addition, we propose estimation of d and p using a modified Schwarz Bayesian criterion, if either of d and p is unknown. Finally, we examine the performance of all the estimators extensively through a variety of simulations and provide a new analysis of the well-known Canadian lynx data. Supplemental materials for this article are available online. [ABSTRACT FROM AUTHOR]

In many applications, it is important to find the mixture with fewest number of components, known as the mixture complexity, that provides a satisfactory fit to the data. This article focuses on developing an estimator of mixture complexity that is consistent when the form of component densities are unknown but are postulated to be members of some parametric family and is simultaneously robust against model misspecification. We treat the estimation of mixture complexity as a model selection problem and construct an estimator of mixture complexity as a byproduct of minimizing a Hellinger information criterion. This estimator is shown to be consistent for any parametric family of mixtures. When the model is correctly specified, Monte Carlo simulations for a wide variety of normal mixtures show that our estimator is very competitive with several others in the literature in correctly identifying the true mixture complexity. The basic construction, being firmly rooted in the minimum Hellinger distance approach, enables our estimator to naturally inherit the property of robustness, which is examined, through simulations, under symmetric departures from postulated component normality. In terms of correctly identifying the mixture complexity under model misspecification, our estimator performs much better than an estimator based on the Kullback--Leibler distance due to James, Priebe, and Marchette. An example concerning hypertension is revisited to further illustrate the performance of our estimator. [ABSTRACT FROM AUTHOR]

When it is necessary to manually count a large number N s of items, significant time and energy can be saved by weighing the items. When the weights are random with unknown mean and variance, one procedure is to take a small sample of n items, weigh them, and add more items until the total weight reaches N s times the average weight of the n items. This procedure yields a batch of items. An estimation criterion used in Yu (1989) yields an optimal value n * for the original sample, but it is not useful when the coefficient of variation is unknown. To overcome this, Yu (1989) proposed a fully, sequential scheme. Here, we propose a relatively new sequential sampling scheme due to Liu (1997a). Our scheme and Yu's (1989) scheme are shown to be second-order efficient, but our scheme requires substantially fewer sampling operations. These show that a little bit of sampling using substantially fewer sampling operations can significantly reduce the effort and cost of complete counting. [ABSTRACT FROM AUTHOR]

Considerable time and energy involved in complete counting of large, but pre-specified, batch of Ns items could be saved by using weights of items. This article considers the case when the underlying distribution of weights, and its mean and variance are unknown. The problem is then reduced to that of finding an estimator of the mean and an optimal (small) sample size based on which a number [Ncirc]n of items in the batch can be determined. Using a fixed-width interval criterion, Nickerson [Nickerson, D.M. Another look at counting by weighing. Commun. Statist. Simula. 1993, 22 (2), 323–343] derived an optimal sample size, but it depends on the unknown coefficient of variation. For this case, we propose a batch-type sequential sampling scheme which requires substantially fewer sampling operations and no prior knowledge of the coefficient of variation, but performs as well as Nickerson's and other available procedures in the literature. This shows that a little bit of sampling using substantially fewer sampling operations can significantly reduce the effort of complete counting. [ABSTRACT FROM AUTHOR]

Background: Data on single-nucleotide polymorphisms (SNPs) have been found to be useful in predicting phenotypes ranging from an individual's class membership to his/her risk of developing a disease. In multi-class classification scenarios, clinical samples are often limited due to cost constraints, making it necessary to determine the sample size needed to build an accurate classifier based on SNPs. The performance of such classifiers can be assessed using the Area Under the Receiver Operating Characteristic (ROC) Curve (AUC) for two classes and the Volume Under the ROC hyper-Surface (VUS) for three or more classes. Sample size determination based on AUC or VUS would not only guarantee an overall correct classification rate, but also make studies more cost-effective., Results: For coded SNP data from D(≥2) classes, we derive an optimal Bayes classifier and a linear classifier, and obtain a normal approximation to the probability of correct classification for each classifier. These approximations are then used to evaluate the associated AUCs or VUSs, whose accuracies are validated using Monte Carlo simulations. We give a sample size determination method, which ensures that the difference between the two approximate AUCs (or VUSs) is below a pre-specified threshold. The performance of our sample size determination method is then illustrated via simulations. For the HapMap data with three and four populations, a linear classifier is built using 92 independent SNPs and the required total sample sizes are determined for a continuum of threshold values. In all, four different sample size determination studies are conducted with the HapMap data, covering cases involving well-separated populations to poorly-separated ones., Conclusion: For multi-classes, we have developed a sample size determination methodology and illustrated its usefulness in obtaining a required sample size from the estimated learning curve. For classification scenarios, this methodology will help scientists determine whether a sample at hand is adequate or more samples are required to achieve a pre-specified accuracy. A PDF manual for R package "SampleSizeSNP" is given in Additional file 1, and a ZIP file of the R package "SampleSizeSNP" is given in Additional file 2.