Steven F. Arnold

Nonparametric Hypotheses and Rank Statistics for Unbalanced Factorial Designs

Abstract Factorial designs are studied with independent observations, fixed number of levels, and possibly unequal number of observations per factor level combination. In this context, the nonparametric null hypotheses introduced by Akritas and Arnold are considered. New rank statistics are derived for testing the nonparametric hypotheses of no main effects, no interaction, and no factor effects in unbalanced crossed classifications. The formulation of all results includes tied observations. Extensions of these procedures to higher-way layouts are given, and the efficacies of the test statistics against nonparametric alternatives are derived. A modification of the test statistics and approximations to their finite-sample distributions are also given. The small-sample performance of the procedures for two factors is examined in a simulation study. As an illustration, a real dataset with ordinal data is analyzed.

Fully Nonparametric Hypotheses for Factorial Designs I: Multivariate Repeated Measures Designs

Abstract We introduce nonparametric versions for many of the hypotheses tested in analysis of variance and repeated measures models, such as the hypotheses of no main effects, no interaction effects, and no factor effects. These natural extensions of the nonparametric hypothesis of equality of the k distributions in the k sample problem have appealing practical interpretations. We concentrate on multivariate repeated measures designs and obtain simple rank statistics for testing these hypotheses. These statistics are the rank transform (RT) versions of the classical statistics for testing hypotheses in repeated measures designs. We emphasize that even though recent research has demonstrated the inappropriateness of the RT method for many parametric hypotheses, the RT procedure is always valid for testing our nonparametric hypotheses. We show that the rank statistics converge in distribution to central chi-squared distributions under their respective nonparametric null hypotheses. The noncentrality paramet...

Asymptotics for Analysis of Variance When the Number of Levels is Large

Abstract We study asymptotic results for F tests in analysis of variance models as the number of factor levels goes to ∞ but the number of observations for each factor combination is fixed. Asymptotic derivations of the type discussed in this article would be relevant whenever both the numerator and denominator degrees of freedom go to ∞ (at the same rate). We consider null and alternative distributions of F, the usual F statistic, for fixed-effects and random-effects, balanced and unbalanced, one-way and two-way, and normal and nonnormal analysis of variance (ANOVA) models. The results may be most relevant for random-effects and mixed models. For example, we may have an agricultural experiment in which the number of cows is quite large but the number of measurements on each cow is small. The results would also be relevant for fixed-effects models in which there are many factor levels but not many observations for each factor level.

Linear Models with Exchangeably Distributed Errors

Abstract A generalization of the general linear model is considered. Let Y i = α + T i y + e i, i = 1, …, n, where the Y i are 1 × p observed random variables, the T i are 1 × (r − 1) constant vectors, and α and γ are unobserved constants. The exchangeable linear model (EGLM) occurs when we assume that the e i are unobserved and exchangeably, jointly normally distributed. The classical general linear model (CGLM) occurs when the e i are independent and identically distributed (iid). Optimal procedures for testing hypotheses about γ for the CGLM are also optimal for the EGLM. Optimal methods for testing hypotheses about α for the CGLM have size 1 for the EGLM. There is no sensible test that the errors are iid in the EGLM. Estimation is considered.

Asymptotic Validity of F Tests for the Ordinary Linear Model and the Multiple Correlation Model

Abstract In this article we establish under fairly general conditions that the F tests used in the linear model and the correlation model are asymptotically valid in the presence of nonnormality, in that their sizes are unaffected, asymptotically, by this nonnormality. Similar results could be derived for Scheffe-type simultaneous confidence intervals as well as the one-sided t tests used in these models. Finally, we find the asymptotic distribution of the sample variance and show why the size of a χ2 test about the variance for the linear model is not asymptotically valid in the presence of nonnormal errors.

Statistical Analysis for Use with the Schruben and Margolin Correlation Induction Strategy

In 1978, L. W. Schruben and B. H. Margolin recommended a correlation induction strategy for a special class of multipopulation simulation experiments. In this paper we present methods for conducting statistical analysis when using their strategy. We give formulas for parameter estimation, hypothesis testing and confidence interval estimation. An example demonstrates the mechanics of the formulas.

Sample size selection for improved Nelder-Mead performance

The Nelder-Mead (1965) simplex algorithm has been used for sequential optimization of simulation response functions. The rescaling operations of this algorithm can lead to inappropriate termination at non-optimal points. We have used the probabilistic characterization of this behaviour to develop special rules for determining the number of replications to take for each experimental design point. Computational experiments indicate that the quality of the solution is often improved.

Control Variates for Multipopulation Simulation Experiments

Abstract In this paper the application of control variates for multipopulation simulation experiments is discussed. Procedures for statistical analysis, when these variates are observed, are presented, and results on the efficiency of employing control variates are established.

Inference about Multivariate Means for a Nonstationary Autoregressive Model

Abstract Inference about multivariate means is investigated for the case in which the covariance matrix is assumed to follow an antedependence model, which is a general first-order autoregressive model. We present two test statistics that offer alternatives to Hotellings T 2 test for testing whether a vector mean is equal to some specified value. These antedependence statistics have simple representations in terms of Hotellings T 2 statistics computed for certain bivariate and univariate variables. Extensions to the multivariate linear model provide easy-to-compute antedependence analogs to the usual Wilks lambda, Lawley-Hotelling trace, and Pillais trace statistics.

Variable order ante-dependence models

Ante-dependence models can be used to model the covariance structure in problems involving repeated measures through time. They are conditional regression models which generalize Gabriel’s constant-order ante-dependence model. Likelihood-based procedures are presented, together with simple expressions for likelihood ratio test statistics in terms of sum of squares from appropriate analysis of covariance. The estimation of the orders is approached as a model selection problem, and penalized likelihood criteria are suggested. Extensions of all procedures discussed here to situations with a monotone pattern of missing data are presented.