Showing papers by "T. W. Anderson published in 1996"

R. A. Fisher and multivariate analysis

Abstract: This paper reviews R. A. Fisher's many fundamental contri- butions to multivariate statistical analysis-from the derivation of the distribution of the sample correlation coefficient to discriminant analysis. The emphasis here is on the conceptual and mathematical development. All of his papers on multivariate analysis will be included in this survey.

Some inequalities for symmetric convex sets with applications

Abstract: Under appropriate conditions the probability of a convex symmetric set decreases as the spread or scatter of the distribution increases. This paper studies the conditions when the random vector has a symmetric unimodal distribution.

Adequacy of asymptotic theory for goodness-of-fit criteria for spectral distributions

Abstract: . Any of the Cramer-von Mises, Anderson-Darling, and Kolmogorov-Smirnov statistics can be used to test the null hypothesis that the standardized spectral distribution of a stationary stochastic process is a specified one. The asymptotic distributions of the criteria have been characterized (Anderson, 1993). They are the same as for probability distributions if the observations are independent (all autocorrelations zero), but are different when there is dependence. In this paper simulation with 10000 replications has been used to determine the distributions of the criteria for samples of size 6, 10, 30 and 100 when the observations are independent. These empirical distributions have been compared with the asymptotic distributions in order to ascertain the sample sizes necessary for using the asymptotic tables. For practical purposes they are 30 for the Cramer-von Mises and Kolmogorov statistics and over 100 for Anderson-Darling.

Measures of Location

Abstract: After a set of data has been collected, it must be organized and condensed or categorized for purposes of analysis In addition to graphical summaries, numerical indices can be computed that summarize the primary features of the data set One is an indicator of location or central tendency that specifies where the set of measurements is “located” on the number line; it is a single number that designates the center of a set of measurements In this chapter we consider several indices of location and show how each of them tells us about a central point in the data

Simple Regression Analysis

Abstract: In this chapter we return to the statistical relationship between two quantitative variables. In Chapter 5 the correlation coefficient is described as a symmetrie index of strength of association. In this chapter we examine the directional relationship of two variables. In many instances one variable may have a direct effect on the other or may be used to predict the other. For example, sodium intake may affect blood pressure; rainfall influences crop yield; SAT scores may predict college grade averages; and parents’ heights may predict offsprings’ heights.

Organization of Data

Abstract: In Chapter 2 we start with the statistical information as it is obtained by the investigator; this information might be an instructor’s list of students and their grades, a record of the tax rates of counties in Florida, or the prices of Grade A large eggs in each of ten Chicago grocery stores averaged over the past 36 months. We refer to such statistical information as data, the recorded results of observation. After the collection of data, the next step in a statistical study is to organize the information in meaningful ways, often in the form of tables and graphs or charts. These displays or summaries are descriptions which help the investigator, as well as the eventual reader of the study, to understand the implications of the collected information. In later chapters we shall develop numerical descriptions that are more succinct than these tables and charts.

Inference on Categorical Data

Abstract: In this chapter we present some methods for treatment of categorical data. The methods involve the comparison of a set of observed frequencies with frequencies specified by some hypothesis to be tested. In Section 14.1 the hypothesis is that one categorical variable has a specific distribution. A test of such a hypothesis is called a test of goodness of fit.

Summarizing Multivariate Data: Association between Categorical Variables

Abstract: Statistical data are often used to answer questions about relationships between variables. Chapters 2 through 4 of this book describe ways to summarize data on a single variable. In Chapters 5 and 6 methods are described for summarizing the. relationship or association between 2 or among 3 or more variables. Chapter 5 considers association among variables measured on numerical scales; Chapter 6 discusses two or more categorical variables.

Differences Between Populations

Abstract: Frequently an investigator wishes to compare or contrast two populations—sets of individuals or objects. This may be done on the basis of a sample from each of the two populations, as when average incomes in two groups, average driving skills of males and females, or average attendance rates in two school districts are compared. The polio vaccine trial compared the incidence rate of polio in the hypothetical population of children who might be inoculated with the vaccine and the rate in the hypothetical population of those who might not be inoculated; the two groups of children observed were considered as samples from these respective (hypothetical) populations. This example illustrates an experiment in which a group receiving an experimental treatment is compared with a “control” group. Ideally, the control group is similar to the experimental group in every way except that its members are not given the treatment.

Measures of Variability

Abstract: Although for some purposes an average may be a sufficient description of a set of data, usually more information about the data is needed. An important feature of statistical data is their variability—how much the measurements differ from individual to individual. In this chapter we discuss the numerical evaluation of variability. A synonym for variability is dispersion, and other terms are sometimes used for the same concept including “spread” or “scatter.”