A parametric bootstrap approach for ANOVA with unequal variances: Fixed and random models

This article is about testing the equality of several normal means when the variances are unknown and arbitrary, i.e., the set up of the one-way ANOVA. Even though several tests are available in the literature, none of them perform well in terms of Type I error probability under various sample size and parameter combinations. In fact, Type I errors can be highly inflated for some of the commonly used tests; a serious issue that appears to have been overlooked. We propose a parametric bootstrap (PB) approach and compare it with three existing location-scale invariant tests—the Welch test, the James test and the generalized F (GF) test. The Type I error rates and powers of the tests are evaluated using Monte Carlo simulation. Our studies show that the PB test is the best among the four tests with respect to Type I error rates. The PB test performs very satisfactorily even for small samples while the Welch test and the GF test exhibit poor Type I error properties when the sample sizes are small and/or the number of means to be compared is moderate to large. The James test performs better than the Welch test and the GF test. It is also noted that the same tests can be used to test the significance of the random effect variance component in a one-way random model under unequal error variances. Such models are widely used to analyze data from inter-laboratory studies. The methods are illustrated using some examples.     

Bootstrap approach to the multi-sample test of means with imprecise data

A bootstrap approach to the multi-sample test of means for imprecisely valued sample data is introduced. For this purpose imprecise data are modelled in terms of fuzzy values. Populations are identified with fuzzy-valued random elements, often referred to in the literature as fuzzy random variables. An example illustrates the use of the suggested method. Finally, the adequacy of the bootstrap approach to test the multi-sample hypothesis of means is discussed through a simulation comparative study.     

A studentized range test for the equivalency of normal means under heteroscedasticity

A studentized range test using a two-stage and a one-stage sampling procedures, respectively, is proposed for testing the null hypothesis that the average deviation of the normal means is falling into a practical indifference zone. Both the level and the power of the proposed test associated with the hypotheses are controllable and they are completely independent of the unknown variances. The two-stage procedure is a design-oriented procedure that satisfies certain probability requirements and simultaneously determines the required sample sizes for an experiment while the one-stage procedure is a data-analysis procedure after the data have been collected, which can supplement the two-stage procedure when the later has to end its experiment sooner than its required experimental process is completed. Tables needed for implementing these procedures are given.     

Bayesian analysis of semiparametric reproductive dispersion mixed-effects models

Semiparametric reproductive dispersion mixed-effects model (SPRDMM) is an extension of the reproductive dispersion model and the semiparametric mixed model, and it includes many commonly encountered models as its special cases. A Bayesian procedure is developed for analyzing SPRDMMs on the basis of P-spline estimates of nonparametric components. A hybrid algorithm combining the Gibbs sampler and the Metropolis-Hastings algorithm is used to simultaneously obtain the Bayesian estimates of unknown parameters, smoothing function and random effects, as well as their standard error estimates. The Bayes factor for model comparison is employed to select better approximation of the smoothing function via path sampling. Several simulation studies and a real example are used to illustrate the proposed methodologies.     

Optimal designs for conjoint experiments

In conjoint experiments, each respondent receives a set of profiles to rate. Sometimes, the profiles are expensive prototypes that respondents have to test before rating them. Designing these experiments involves determining how many and which profiles each respondent has to rate and how many respondents are needed. To that end, the set of profiles offered to a respondent is treated as a separate block in the design and a random respondent effect is used in the model because profile ratings from the same respondent are correlated. Optimal conjoint designs are then obtained by means of an adapted version of an algorithm for finding D-optimal split-plot designs. A key feature of the design construction algorithm is that it returns the optimal number of respondents and the optimal number of profiles each respondent has to evaluate for a given number of profiles. The properties of the optimal designs are described in detail and some practical recommendations are given.     

Extending fuzzy and probabilistic clustering to very large data sets

Approximating clusters in very large (VL=unloadable) data sets has been considered from many angles. The proposed approach has three basic steps: (i) progressive sampling of the VL data, terminated when a sample passes a statistical goodness of fit test; (ii) clustering the sample with a literal (or exact) algorithm; and (iii) non-iterative extension of the literal clusters to the remainder of the data set. Extension accelerates clustering on all (loadable) data sets. More importantly, extension provides feasibility—a way to find (approximate) clusters—for data sets that are too large to be loaded into the primary memory of a single computer. A good generalized sampling and extension scheme should be effective for acceleration and feasibility using any extensible clustering algorithm. A general method for progressive sampling in VL sets of feature vectors is developed, and examples are given that show how to extend the literal fuzzy (c-means) and probabilistic (expectation-maximization) clustering algorithms onto VL data. The fuzzy extension is called the generalized extensible fast fuzzy c-means (geFFCM) algorithm and is illustrated using several experiments with mixtures of five-dimensional normal distributions.     

Small-sample comparisons of confidence intervals for the difference of two independent binomial proportions

This paper compares the exact small-sample achieved coverage and expected lengths of five methods for computing the confidence interval of the difference of two independent binomial proportions. We strongly recommend that one of these be used in practice. The first method we compare is an asymptotic method based on the score statistic (AS) as proposed by Miettinen and Nurminen [1985. Comparative analysis of two rates. Statist. Med. 4, 213-226.]. Newcombe [1998. Interval estimation for the difference between independent proportions: comparison of seven methods. Statist. Med. 17, 873-890.] has shown that under a certain asymptotic set-up, confidence intervals formed from the score statistic perform better than those formed from the Wald statistic (see also [Farrington, C.P., Manning, G., 1990. Test statistics and sample size formulae for comparative binomial trials with null hypothesis of non-zero risk difference or non-unity relative risk. Statist. Med. 9, 1447-1454.]). The remaining four methods compared are the exact methods of Agresti and Min (AM), Chan and Zhang (CZ), Coe and Tamhane (CT), and Santner and Yamagami (SY). We find that the CT has the best small-sample performance, followed by AM and CZ. Although AS is claimed to perform reasonably well, it performs the worst in this study; about 50% of the time it fails to achieve nominal coverage even with moderately large sample sizes from each binomial treatment.     

A goodness-of-fit test for normality based on polynomial regression

Statistical models are often based on normal distributions and procedures for testing such distributional assumption are needed. Many goodness-of-fit tests are available. However, most of them are quite insensitive in detecting non-normality when the alternative distribution is symmetric. On the other hand all the procedures are quite powerful against skewed alternatives. A new test for normality based on a polynomial regression is presented. It is very effective in detecting non-normality when the alternative distribution is symmetric. A comparison between well known tests and this new procedure is performed by simulation study. Other properties are also investigated.     

On using prototype reduction schemes to enhance the computation of volume-based inter-class overlap measures

In most pattern recognition (PR) applications, it is advantageous if the accuracy (or error rate) of the classifier can be evaluated or bounded prior to testing it in a real-life setting. It is also well known that if the two class-conditional distributions have a large overlapping volume (almost all the available work on “overlapping of classes” deals with the case when there are only two classes), the classification accuracy is poor. This is because if we intend to use the classification accuracy as a criterion for evaluating a PR system, the points within the overlapping volume tend to lead to maximal misclassification. Unfortunately, the computation of the indices which quantify the overlapping volume is expensive. In this vein, we propose a strategy of using a prototype reduction scheme (PRS) to approximately, but quickly, compute the latter. In this paper, we demonstrate, first of all, that this is an extremely expedient proposition. Indeed, we show that by completely discarding (we are not aware of any reported scheme which discards “irrelevant” sample (training) points, and which simultaneously attains to an almost-comparable accuracy) the points not included by the PRS, we can obtain a reduced set of sample points, using which, in turn, the measures for the overlapping volume can be computed. The value of the corresponding figures is comparable to those obtained with the original training set (i.e., the one which considers all the data points) even though the computations required to obtain the prototypes and the corresponding measures are significantly less. The proposed method has been rigorously tested on artificial and real-life datasets, and the results obtained are, in our opinion, quite impressive—sometimes faster by two orders of magnitude.     

Stepwise estimation of common principal components

The standard common principal components (CPCs) may not always be useful for simultaneous dimensionality reduction in k groups. Moreover, the original FG algorithm finds the CPCs in arbitrary order, which does not reflect their importance with respect to the explained variance. A possible alternative is to find an approximate common subspace for all k groups. A new stepwise estimation procedure for obtaining CPCs is proposed, which imitates standard PCA. The stepwise CPCs facilitate simultaneous dimensionality reduction, as their variances are decreasing at least approximately in all k groups. Thus, they can be a better alternative for dimensionality reduction than the standard CPCs. The stepwise CPCs are found sequentially by a very simple algorithm, based on the well-known power method for a single covariance/correlation matrix. Numerical illustrations on well-known data are considered.     

Supersaturated designs: Are our results significant?

Two-level supersaturated designs (SSDs) are designs that examine more than n−1 factors in n runs. Although SSD literature for both construction and analysis is plentiful, the dearth of actual applications suggests that SSDs are still an unproven tool. Whether using forward selection or all-subsets regression, it is easy to select simple models from SSDs that explain a very large percentage of the total variation. Hence, naive p-values can persuade the user that included factors are indeed active. We propose the use of a global model randomization test in conjunction with all-subsets (or a shrinkage method) to more appropriately select candidate models of interest. For settings where the large number of factors makes repeated use of all-subsets expensive, we propose a short-cut approximation for the p-values. Two state-of-the-art model selection methods that have received considerable attention in recent years, Least Angle Regression and the Dantzig Selector, were likewise supplemented with the global randomization test. Finally, we propose a randomization test for reducing the number of terms in candidate models with small global p-values. Randomization tests effectively emphasize the limitations of SSDs, especially those with a large factor to run size ratio.     

Obtaining more information from conjoint experiments by best-worst choices

Conjoint choice experiments elicit individuals’ preferences for the attributes of a good by asking respondents to indicate repeatedly their most preferred alternative in a number of choice sets. However, conjoint choice experiments can be used to obtain more information than that revealed by the individuals’ single best choices. A way to obtain extra information is by means of best-worst choice experiments in which respondents are asked to indicate not only their most preferred alternative but also their least preferred one in each choice set. To create D-optimal designs for these experiments, an expression for the Fisher information matrix for the maximum-difference model is developed. Semi-Bayesian D-optimal best-worst choice designs are derived and compared with commonly used design strategies in marketing in terms of the D-optimality criterion and prediction accuracy. Finally, it is shown that best-worst choice experiments yield considerably more information than choice experiments.     

Modelling series of studies with a common structure

Consider the situation where the Structuration des Tableaux à Trois Indices de la Statistique (STATIS) methodology is applied to a series of studies, each study being represented by data and weight matrices. Relations between studies may be captured by the Hilbert-Schmidt product of these matrices. Specifically, the eigenvalues and eigenvectors of the Hilbert-Schmidt matrix S may be used to obtain a geometrical representation of the studies. The studies in a series may further be considered to have a common structure whenever their corresponding points lie along the first axis. The matrix S can be expressed as the sum of a rank 1 matrix λuu^T with an error matrix E. Therefore, the components of the vector are sufficient to locate the points associated to the studies. Former models for S where vec(E) are mathematically tractable and yet do not take into account the symmetry of the matrix S. Thus a new symmetric model is proposed as well as the corresponding tests for a common structure. It is further shown how to assess the goodness of fit of such models. An application to the human immunodeficiency virus (HIV) infection is used for assessing the proposed model.     

H∞ filtering for discrete-time systems with randomly varying sensor delays

This paper investigates an H_∞ filtering problem for discrete-time systems with randomly varying sensor delays. The stochastic variable involved is a Bernoulli distributed white sequence appearing in measured outputs. This measurement mode can be used to characterize the effect of communication delays and/or data-loss in information transmissions across limited bandwidth communication channels over a wide area. H_∞ filtering of this class of systems is used to design a filter using the measurements with random delays to ensure the mean-square stochastic stability of the filtering error system and to guarantee a prescribed H_∞ filtering performance. A sufficient condition for the existence of such a filter is presented in terms of the feasibility of a linear matrix inequality (LMI). Finally, a numerical example is given to illustrate the effectiveness of the proposed approach.     

Detecting change-points in Markov chains

Markov chains provide a flexible model for dependent random variables with applications in such disciplines as physics, environmental science and economics. In the applied study of Markov chains, it may be of interest to assess whether the transition probability matrix changes during an observed realization of the process. If such changes occur, it would be of interest to estimate the transitions where the changes take place and the probability transition matrix before and after each change. For the case when the number of changes is known, standard likelihood theory is developed to address this problem. The bootstrap is used to aid in the computation of p-values. When the number of changes is unknown, the AIC and BIC measures are used for model selection. The proposed methods are studied empirically and are applied to example sets of data.     

Vector transport for shape-from-shading

In this paper we describe a new shape-from-shading method. We show how the parallel transport of surface normals can be used to impose curvature consistency and also to iteratively update surface normal directions so as to improve the brightness error. We commence by showing how to make local estimates of the Hessian matrix from surface normal information. With the local Hessian matrix to hand, we develop an “EM-like” algorithm for updating the surface normal directions. At each image location, parallel transport is applied to the neighbouring surface normals to generate a sample of local surface orientation predictions. From this sample, a local weighted estimate of the image brightness is made. The transported surface normal which gives the brightness prediction which is closest to this value is selected as the revised estimate of surface orientation. The revised surface normals obtained in this way may in turn be used to re-estimate the Hessian matrix, and the process iterated until stability is reached. We experiment with the method on a variety of real world and synthetic data. Here we explore the properties of the fields of surface normals and the height data delivered by the method.     

Fuzzy random renewal reward process and its applications

This paper studies a renewal reward process with fuzzy random interarrival times and rewards under the ?-independence associated with any continuous Archimedean t-norm ?. The interarrival times and rewards of the renewal reward process are assumed to be positive fuzzy random variables whose fuzzy realizations are ?-independent fuzzy variables. Under these conditions, some limit theorems in mean chance measure are derived for fuzzy random renewal rewards. In the sequel, a fuzzy random renewal reward theorem is proved for the long-run expected reward per unit time of the renewal reward process. The renewal reward theorem obtained in this paper can degenerate to that of stochastic renewal theory. Finally, some application examples are provided to illustrate the utility of the result.     

Fuzzy data treated as functional data: A one-way ANOVA test approach

The use of the fuzzy scale of measurement to describe an important number of observations from real-life attributes or variables is first explored. In contrast to other well-known scales (like nominal or ordinal), a wide class of statistical measures and techniques can be properly applied to analyze fuzzy data. This fact is connected with the possibility of identifying the scale with a special subset of a functional Hilbert space. The identification can be used to develop methods for the statistical analysis of fuzzy data by considering techniques in functional data analysis and vice versa. In this respect, an approach to the FANOVA test is presented and analyzed, and it is later particularized to deal with fuzzy data. The proposed approaches are illustrated by means of a real-life case study.     

Application of a fault detection filter to structural health monitoring

A model-based fault detection filter is developed for structural health monitoring of a simply supported beam. The structural damage represented in the plant model is shown to decompose into a known fault direction vector maintaining a fixed direction, dependent on the damage location, and an arbitrary fault magnitude representing the extent of the damage. According to detection filter theory, if damage occurs, under certain circumstances the fault will be uniquely detected and identified through an associated invariance in the direction imposed on the fault detection filter residuals. The spectral algorithm used to design the detection filter is based on a left eigenstructure assignment approach which accommodates system sensitivities that are revealed as ill-conditioned matrices formed from the eigenvectors in the construction of the detection filter gains. The detection filter is applied to data from an aluminum simply supported beam with four piezoelectric sensors and one piezoelectric actuator. By exciting the structure at the first natural frequency, damage in the form of a 5 mm saw cut made to one side of the beam is detected and localized.