Publisher Summary This chapter focuses on pure rank statistics in factorial designs. Pure rank statistics are invariant under any strict monotone transformation of the data and are robust against outliers. In addition, they are applicable to ordinal data such as scores in psychological tests, grading scales to describe the degree of the damage of plants or trees in ecological or environmental studies. The classical models of analysis of variance are generalized (ANOVA) in such a way that not only the assumption of normality of the error terms is relaxed but also the structure of the designs is introduced in a broader framework. In addition, the concept of treatment effects is redefined within this framework. To identify the testing problem underlying the different rank procedures, the relations between the hypotheses in the general model and in the standard linear model are investigated in the chapter. The chapter concludes with a discussion of random-factor model along with the rank procedures for heteroscedastic mixed models.

19 Nonparametric methods in design and analysis of experiments

The variability among data may have quite complex structure and the splitting of variation into its individual components is important in many areas of applications. In particular, the study of variation through a class of linear models known as random and mixed models is a central topic in statistics with wide ramifications in both theory and applications. Sometimes these models are called components of variance models since the interest in the model lies in investigating the variances of the effects of factor levels which are assumed to be randomly selected from a large population of levels. These variances are also known as variance components since the total variance can be expressed as the sum of these variances and the error variance. Knowledge of the variance components, i.e., the information on the relative importance and magnitude of the sources of variability in the data is of critical importance in many problems since the isolation of the total variance into the individual components of variance would result in the physical elimination of the sources of variability in question.

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A bibliography on variance components an introduction and an update: 1984-2002

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Locally most powerful rank tests for random effects in two-way experiments

The nonparametric problem of testing for random effects in one and two-way experiments is surveyed. The results of Govindarajulu and Deshpande (1972), Govindarajulu (1975), Shetty and Govindarajulu (1988) Clemmens and Govindarajulu (1990) and Clemmens (1988) are summarized and certain open problems are mentioned.

A brief survey of lmp rank tests for random effects in one and two-way models

We investigate the properties of the locally most powerful nonparametric criterion against logistic alternatives developed by Govindarajulu (1975) for testing one-way random effects modcls We deduce the appropriate computational forms for the test criterion T and tabulate the critical values of T for α = 01, 05 and 010, and various sample sizes Certain features of the computational methods are discussed In the tables we retain only those sample sizes beyond which the asymptotic theory is meaningful We also study the power comparison of the test for two populations with the classical F-test under a range of normal alternatives

Ann Clemmens

Papers

A certain locally most powerful test for one-way random effects model: full distribution and power considerations