NONPARAMETRIC RELIABILITY ESTIMATION BASED ON A FEW ORDERED OBSERVATIONS

The usual one-sided Kolmogorov-Smirnov distance is generalized to obtain an improved lower confidence region for the extreme left tail of the reliability function based on k observations in a “k out of n censored” plan. Finite sample and asymptotic critical values necessary for implementation are given. The two numerical comparisons with existing parametric procedures for the case of complete or censored samples demonstrate the applicability of the proposed nonparametric procedure.     

An extension of the Anderson–Darling k-sample test to arbitrary sample space partition sizes

In this paper we first show that the k-sample Anderson–Darling test is basically an average of Pearson statistics in 2?×?k contingency tables that are induced by observation-based partitions of the sample space. As an extension, we construct a family of rank test statistics, indexed by c?∈??, which is based on similarly constructed c?×?k partitions. An extensive simulation study, in which we compare the new test with others, suggests that generally very high powers are obtained with the new tests. Finally we propose a decomposition of the test statistic in interpretable components.     

Rank-based analysis of repeated measures block designs

A rank-based inference is developed for repeated measures balanced incomplete block and randomized complete block designs using a suitable dispersion function. Asymptotic distributions of rank estimators are developed after establishing approximate linearity of the gradient vector of the dispersion function. Unlike available nonparametric procedures for those designs, estimation and testing are tied together. Three different test statistics are developed for testing the linear hypotheses. Friedman's (1937) statistic and Durbin's (1951) statistic are particular cases of one of the three proposed statistics. An estimate of a scale parameter which appears in the ARE expression as well as as in the variences and covariances of the rank estimators is discussed.     

Statistical Inference of Adaptive Designs with Binary Responses

Adaptive designs of clinical trials are ethical alternatives when the traditional randomization becomes ethically infeasible in desperate medical situations. However, such a design creates a dependency among trial data and its statistical analysis becomes more complex than the analysis for traditional randomized clinical trials. In this article, we examine adaptive designs with dichotomous responses from two treatments and extend some commonly used statistical methods for independent data. Under a regularity condition, the estimated odds ratio and its logarithm are shown to follow asymptotically normal distributions. Moreover, the ordinary goodness-of-fit test statistic for two-by-two contingency tables with dependent data is shown to be asymptotically chi-square distributed. We also discuss the consistency of maximum likelihood estimators of the unknown parameters for a wide class of adaptive designs.     

Bayesian Analysis of Contingency Tables

ABSTRACT

The display of the data by means of contingency tables is used in different approaches to statistical inference, for example, to broach the test of homogeneity of independent multinomial distributions. We develop a Bayesian procedure to test simple null hypotheses versus bilateral alternatives in contingency tables. Given independent samples of two binomial distributions and taking a mixed prior distribution, we calculate the posterior probability that the proportion of successes in the first population is the same as in the second. This posterior probability is compared with the p-value of the classical method, obtaining a reconciliation between both results, classical and Bayesian. The obtained results are generalized for r × s tables.     

Testing for or against a trend in odds ratios

Consider data arranged into k × 2 × 2 contingency tables. The principal result of this paper is the derivation of the likelihood ratio test and its asymptotic distribution for testing for or against an order restriction placed upon the odds ratios. We will show that the limiting distributions are of chi-bar square type and give the expression of the weighting values.     

Inference with overlapping 2×2 tables

Exact probabilities (under a Markov-like assumption) are calculated for overlapping 2×2 contingency tables. These contingency tables often arise in biological and legal situations that yield dichotomous responses, including the evaluation of clinical trials and the determination of prima facie evidence of employer discrimination. In this note, probabilities, means and variances are derived and comparisons are made with results when assuming no overlap exists.     

On estimation and testing arising from order restricted balanced mixed model

The randomized complete block design is one of the most widely used experimental designs to systematically control the variability arising from known nuisance sources. The balanced mixed effects model is usually appropriate for such an experiment when the blocks used in the experiment are randomly chosen. In applications with k increasing or decreasing treatment levels, there is sometimes prior knowledge about the ordering of the treatment effects. The most commonly seen orderings include simple ordering, simple tree ordering and umbrella orderings with known or unknown peaks. A natural question is how to incorporate the prior ordering information in estimating the parameters in a balanced mixed effects model so that the estimated treatment effects are consistent with the prior information and the estimated variances of the block effects and experiment errors are nonnegative. In this paper we derive the maximum likelihood estimators of the parameters in a balanced mixed model subject to any partial ordering of the treatment effects, which includes the usual maximum likelihood estimators as a special case. An example is provided to illustrate the results.     

A Bootstrap Approach for Testing Marginal Independence Between Two Categorical Variables When Subjects Have Repeated Responses

To assess independence in two-way contingency tables, the Pearson chi-square test or Fisher’s exact test are typically used. These tests assume that each subject contributes at most one count to only one table cell (e.g., sex versus blood type). In other situations, each subject may have more than one count contributing to the table and these counts may occur in different cells of the table. One may wish to test independence, adjusting for the within-subject correlation. We provide a simple nonparametric bootstrap approach and assess its performance through simulation studies. The method is illustrated on subjects with multiple mental health presentations to Emergency Departments.     

Construction of neighbor designs of block size 4 for all values of v ≥ 5 by using the method of addition

In this article, a general method of construction of neighbor block designs is given. The designs are constructed using variation of a simple method which we refer to as the method of addition (renamed as the method of cyclic shifts). We give complete solution of neighbor balanced designs for k = 4 for any value of v. We also give many series of generalized neighbor designs (GNDs). In the last section, we have constructed GNDs in a sequential manner (as Did John 1981) for v ≤ 50 and r is multiple of k.     

A-Optimal and MV-Optimal Incomplete Block Designs for Comparing Successive Treatments

Consider an incomplete block experiment in which observations are taken from t treatments using an incomplete block design with b blocks of size k < t. Suppose the interest is in estimating the differences of effects of successive treatments. This may occur, for example, if the treatments are different dosages or concentrations of a compound. This article presents A-optimal and MV-optimal incomplete block designs for estimating the the differences of successive treatment effects. Tables of optimal designs are given for k < t ≤ 5 with b ≤ 40.     

Tables of Cyclic Incomplete Block Designs: r=k

Initial blocks and efficiencies are given for cyclic incomplete block designs in the range t= 10(1)60 and r=k= 3(1)10. The designs have been selected by search procedures using optimality criteria which maximize the efficiency factor.     

Analysis of means using ranks for the randomized complete block design

A nonparametric test procedure is proposed for the analysis of randomized complete block designs. Such a procedure may be carried out graphically in the form of a Shewhart control chart. Exact and asymptotic critical values are given for the implementation of the proposed procedure. A Monte Carlo study is made to compare the powers of the proposed procedure to those of analysis of variance, the analysis of means, and the Friedman procedures. Results of the study indicate that the proposed procedure has superior power performance when testing against slippage alternative hypotheses under heavy-tailed distributions such as the Cauchy distribution. However, when testing against symmetric alternatives under light-tailed distributions, the proposed procedure does not perform well     

Nonparametric estimation of copula-based measures of multivariate association from contingency tables

Nonparametric estimation of copula-based measures of multivariate association in a continuous random vector X=(X₁, …, X_d) is usually based on complete continuous data. In many practical applications, however, these types of data are not readily available; instead aggregated ordinal observations are given, for example, ordinal ratings based on a latent continuous scale. This article introduces a purely nonparametric and data-driven estimator of the unknown copula density and the corresponding copula based on multivariate contingency tables. Estimators for multivariate Spearman's rho and Kendall's tau are based thereon. The properties of these estimators in samples of medium and large size are evaluated in a simulation study. An increasing bias can be observed along with an increasing degree of association between the components. As it is to be expected, the bias is severely influenced by the amount of information available. Additionally, the influence of sample size is only marginal. We further give an empirical illustration based on daily returns of five German stocks.     

Multidimensional contingency tables with missing data

This paper is concerned wim ine maximum likelihood estimation and the likelihood ratio test for hierarchical loglinear models of multidimensional contingency tables with missing data. The problems of estimation and test for a high dimensional contingency table can be reduced into those for a class of low dimensional tables. In some cases, the incomplete data in the high dimensional table can become complete in the low dimensional tables through the reduction can indicate how much the incomplete data contribute to the estimation and the test.     

Independence in Contingency Tables Using Simplicial Geometry

Frequently, contingency tables are generated in a multinomial sampling. Multinomial probabilities are then organized in a table assigning probabilities to each cell. A probability table can be viewed as an element in the simplex. The Aitchison geometry of the simplex identifies independent probability tables as a linear subspace. An important consequence is that, given a probability table, the nearest independent table is obtained by orthogonal projection onto the independent subspace. The nearest independent table is identified as that obtained by the product of geometric marginals, which do not coincide with the standard marginals, except in the independent case. The original probability table is decomposed into orthogonal tables, the independent and the interaction tables. The underlying model is log-linear, and a procedure to test independence of a contingency table, based on a multinomial simulation, is developed. Its performance is studied on an illustrative example.     

Algorithmic and analytical construction of efficient designs in small blocks for comparing consecutive pairs of treatments

Optimal block designs in small blocks are explored under the A-, E- and D-criteria when the treatments have a natural ordering and interest lies in comparing consecutive pairs of treatments. We first formulate the problem via approximate theory which leads to a convenient multiplicative algorithm for obtaining A-optimal design measures. This, in turn, yields highly efficient exact designs, under the A-criterion, even when the number of blocks is rather small. Moreover, our approach is seen to allow nesting of such efficient exact designs which is an advantage when the resources for the experiment are available in possibly several stages. Illustrative examples are given and tables of A-optimal design measures are provided. Approximate theory is also seen to yield analytical results on E- and D-optimal design measures.     

Asymmetry models for square contingency tables: exact tests via algebraic statistics

Square contingency tables with the same row and column classification occur frequently in a wide range of statistical applications, e.g. whenever the members of a matched pair are classified on the same scale, which is usually ordinal. Such tables are analysed by choosing an appropriate loglinear model. We focus on the models of symmetry, triangular, diagonal and ordinal quasi symmetry. The fit of a specific model is tested by the chi-squared test or the likelihood-ratio test, where p-values are calculated from the asymptotic chi-square distribution of the test statistic or, if this seems unjustified, from the exact conditional distribution. Since the calculation of exact p-values is often not feasible, we propose alternatives based on algebraic statistics combined with MCMC methods.     

Nonparametric methods in factorial designs

Summary In this paper, we summarize some recent developments in the analysis of nonparametric models where the classical models of ANOVA are generalized in such a way that not only the assumption of normality is relaxed but also the structure of the designs is introduced in a broader framework and also the concept of treatment effects is redefined. The continuity of the distribution functions is not assumed so that not only data from continuous distributions but also data with ties are included in this general setup. In designs with independent observations as well as in repeated measures designs, the hypotheses are formulated by means of the distribution functions. The main results are given in a unified form. Some applications to special designs are considered, where in simple designs, some well known statistics (such as the Kruskal-Wallis statistic and the χ²-statistic for dichotomous data) come out as special cases. The general framework presented here enables the nonparametric analysis of data with continuous distribution functions as well as arbitrary discrete data such as count data, ordered categorical and dichotomous data. Received: October 13, 1999; revised version: June 26, 2000     

MULTI-FACTOR BALANCED BLOCK DESIGNS WITH COMPLETE ADJUSTED ORTHOGONALITY FOR ALL PAIRS OF TREATMENT FACTORS

A new series of multi-factor balanced block designs is introduced. Each of these designs has the following properties: (i) each of its k– 1 treatment factors is disposed in a cyclic or multi-cyclic balanced incomplete block design with parameters (v,b,r,k,Λ) = (a(k-l) + 1,a²(k-1) +a, ak, k, k) (a > 1); (ii) the incidence of any one of the treatment factors on any other is balanced; and (iii) after adjustment for blocks only, the relationship between any two of the treatment factors is that of adjusted orthogonality. The treatment factors are thus orthogonal to one another in the within-blocks stratum of the analysis of variance. The designs provide a benchmark with which other designs may be compared.