Rudolf J. Freund

Design of Experiments

This chapter focuses on the design of experiments, which is the process of planning and executing an experiment. It deals with the factorial experiments that are carried out within blocks, an analog of the multifactor analysis of variance (ANOVA), and classifies repeated measures designs by the number of between-subject and within-subject factors. The objective of an experimental design is to provide the maximum amount of reliable information at the minimum cost. The data resulting from the implementation of experimental designs are described by linear models and analyzed by the analysis of variance. One of the simplest and the most popular experimental design—the randomized block design—is described. In this design, the sample of experimental units is divided into groups or blocks and then treatments are randomly assigned to units in each block. The observations that come from within the same block have a natural matching mechanism. The data from a randomized block design can be described by a linear model that suggests the partitioning of the sum of squares and provides a justification for the test statistics.

The Case of the Missing Cell

Abstract The occurrence of missing data cells precludes a universally correct procedure for performing an analysis of variance. This is illustrated by the use of two computer routines to analyze a 2 × 3 factorial experiment with one missing cell. One of these routines does, however, provide information that may enhance the usefulness of the associated results.

Organizations for Statistical Consulting at Colleges and Universities

Abstract The importance of correct implementation of statistical methodology in a wide variety of research is widely acknowledged. Few universities, however, have given thought to providing a means of ensuring that their research efforts employ statistics in an appropriate manner. Specifically, many universities have not established any formal structure for the delivery of statistical consulting services. This article addresses this problem by discussing how the consultant, the consultee, and the institution fare under a number of different organizational structures. The conclusion reached is that a separately funded consulting center operated as a division of a department of statistics is the most appropriate way to provide statistical consulting and to ensure quality research, although at many universities alternate organizational structures are probably more feasible at this time.

Organization for the Conduct of Statistical Activities in Colleges and Universities

Abstract Although the importance of statistics in many phases of university activities is well-known, many universities have not given serious thought to the optimum organizational structure for statistics. This article addresses this problem in two parts: 1. Principles and/or goals of optimum administrative structures for statistics, and 2. Discussion of the most popular organizational structures now in existence together with comments on the advantages and disadvantages of each. The article concludes that a central statistics unit (department) appears the most optimal structure, although it is not without certain disadvantages and/or dangers.

The measurement of diversity: Ethnic and socioeconomic mixing in residential areas

Units of observation such as census tracts continue to be analyzed according to various modal characteristics while the variation or diversity existent in such units is often ignored. The qualitative or nominal-level indicators of diversity are examined which (1) are operative in the polytomous situation, and (2) measure within-unit diversity rather than divergences among units. Six qualitative indicators are explained and compared both theoretically and by example, with the Index of Qualitative Variation suggested as the most appropriate measure of diversity when variables representing a nominal scale are used. Quantitative or interval-level diversity also was examined with six measures analyzed, representing three operational situations. Because of the susceptibilities of five of the quantitative measures to skewness and variable sample sizes, the coefficient of variation was recommended for interval-level variables to evaluate within-unit diversity.

Inferences for Two or More Means

This chapter describes the analysis of variance (ANOVA) method for testing the equality of a set of means and examines the use of the linear model to justify this method. It presents the assumptions necessary for the validity of the results of ANOVA and discusses the remedial methods if these assumptions are not met. It also discusses the procedures for specific comparisons among selected means and provides an alternative to the analysis of variance called the “analysis of means.” The ANOVA provides a methodology for making inferences for means from any number of populations. The inferences based on data resulting from independently drawn samples from t populations are discussed. This data structure is called a “one-way classification” or “completely randomized design.” The analysis of variance is based on the comparison of the estimated variance among sample means to the estimated variance of observations within the samples. A heuristic justification for the analysis of variance procedure for the balanced case is presented.

CHAPTER 14 – Nonparametric Methods

Publisher Summary This chapter discusses rank-based nonparametric methods for one, two, and more than two independent samples, paired samples, randomized block designs, and correlation. Nonparametric methods provide alternative statistical methodology when assumptions necessary for the use of linear model-based methods fail as well as provide procedures for making inferences when the scale of measurement is ordinal or nominal. The chapter describes the use of a randomization test to assign a p value. This technique is applied to calculate p values for the standard nonparametric test statistics. Data from a randomized block design may be analyzed by a nonparametric rank-based method known as the “Friedman test.” A randomization test allows calculating a p value for the null hypothesis so that the distributions are the same, with special sensitivity to the possibility that they differ with respect to their medians.. The bootstrap is a powerful tool that allows some creativity in the estimation. Although the original intent of the bootstrap was to develop confidence intervals, it can also be used to calculate p values.

Probability and Sampling Distributions

This chapter provides some of the tools that are used in probability theory as measures of uncertainty, and particularly those tools that allow making inferences and evaluating the reliability of such inferences. It outlines the concept of a sampling distribution, which is a probability distribution that describes the way a statistic from a random sample is related to the characteristics of the population from which the sample is drawn. It presents the concept of the probability of a simple outcome of an experiment such as the probability of obtaining a head on a toss of a coin. Rules are then given for obtaining the probability of an event, which may consist of several outcomes such as obtaining no heads in the toss of five coins. These rules are used to construct probability distributions that are simply listings of probabilities of all events resulting from an experiment such as obtaining all possible number of heads in the toss of five coins. This concept is generalized to define probability distributions for the results of experiments that result in continuous numeric variables. Some of these distributions are derived from purely mathematical concepts and require the use of functions and tables to find probabilities.

Inferences for Two Populations

This chapter presents the inferential methods for making comparisons between the parameters of two populations. The comparison of two populations results in a single easily understood statistic—the difference between sample means. It discusses two distinct methods for collecting data on two populations, or equivalently, designing an experiment for comparing two populations— (1) independent samples and (2) dependent or paired samples. These two methods are illustrated with a hypothetical experiment designed to compare the effectiveness of two migraine headache remedies. The procedures for making inferences on differences in the proportions of successes using independent as well as dependent samples from two binomial populations are also provided. The assumptions underlying the various procedures for comparing two populations are also explained including a brief discussion on the detection of violations and some alternative methods. The estimate of a common variance from two independent samples is obtained by “pooling,” which is simply the weighted mean of the two individual variance estimates with the weights being the degrees of freedom for each variance.

Other Linear Models

This chapter introduces the concept of dummy or indicator variables that make an analysis of variance model look like a regression model. It discusses the use of dummy variables to analyze unbalanced factorial data sets and the use of models that contain both dummy and interval independent variables with emphasis on the analysis of covariance. It also discusses the use of covariance matrices to customize inferences on linear combinations of parameters and the use of weighted least squares when error terms show unequal variances. The chapter deals with linear models in which some parameters describe effects due to factor levels and others represent regression relationships. Such models include dummy variables representing factor levels as well as interval variables associated with regression analyses. The simplest of these models are illustrated that has parameters representing levels of a single factor and a regression coefficient for one independent interval variable. The analysis of models that include both measured and categorical independent factors are also presented.