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Über dieses Buch

This book is a novel exposition of the traditional workhorses of statistics: analysis of variance and regression. The key feature is that these tools are viewed in their natural mathematical setting, the geometry of finite dimensions. The Authors To introduce ourselves, Dave Saville is a practicing statistician working in agricultural research; Graham Wood is a university lecturer involved in the teaching of statistical methods. Each of us has worked for sixteen years in our current field. Features of the Book People like pictures. One picture can present a set of ideas at a glance, while a series of pictures, each building on the last, can unify a wealth of ideas. Such a series we present in this text by means of a systematic geometric approach to the presentation of the theory of basic statistical methods. This approach fills the void between the traditional extremes of the "cookbook" approach and the "matrix algebra" approach, providing an elementary but at the same time rigorous view of the subject. It combines the virtues of the traditional methods, while avoiding their vices.

Inhaltsverzeichnis

Frontmatter

Basic Ideas

Frontmatter

Chapter 1. Introduction

Abstract
In §1 of this chapter we explain why geometry is important for an understanding of statistics. We investigate a simple example in §2 to convince you of the virtues of thinking with pictures. How does this geometric method link with the more traditional approaches? The answer is presented in §3. We wrap up in §4 with a few hints on how to read the book efficiently.
David J. Saville, Graham R. Wood

Chapter 2. The Geometric Tool Kit

Abstract
This book is aboutvisualizingthe classical methods of statistical analysis. The critical link between the numbers of a data set and the picture we analyze is the notion of a vector, a visual representation of that set of numbers. This chapter introduces you to the basic ideas of the geometry of vectors. Vectors are defined in §1, then in §2 we demonstrate how to combine them. Angles between vectors are defined in §3, projections of vectors are discussed in §4, and we conclude in §5 with a discussion of Pythagoras’ Theorem. Rest assured that we discuss only those ideas which are necessary for the statistics we do later!
David J. Saville, Graham R. Wood

Chapter 3. The Statistical Tool Kit

Abstract
We complement the previous chapter by now summarizing thestatisticalconcepts and techniques which will be used in succeeding chapters. We begin in §1 with the notion of a population, a sample, a random variable and a probability distribution. Properties of combinations of random variables are dealt with in §2. We discuss estimation in §3, and finish in §4 with the definitions of theFand t distributions.
David J. Saville, Graham R. Wood

Chapter 4. Tool Kits At Work

Abstract
Here we set the scene for the remainder of the book. We begin in §1 with an overview of the scientific method, or question — research — answer process, in which statistical data analysis plays an integral part. In §2 we home in upon the novel contribution of this text, namely a consistent geometric approach to data analysis. While this is a feature which sets this book apart from other texts, we emphasize that throughout we shall view data analysis as just a part of the design and analysis of each research study.
David J. Saville, Graham R. Wood

Introduction to Analysis of Variance

Frontmatter

Chapter 5. Single Population Questions

Abstract
What is the mean of the single population which we are studying? Is it reasonable that the mean is zero? These are the types of question we deal with in this chapter.
David J. Saville, Graham R. Wood

Chapter 6. Questions About Two Populations

Abstract
In this chapter we deal with questions concerning two normal populations. The central question to be answered is “Are the two means equal?” If they are deemed unequal then typically we wish to estimate with some precision the difference between them. Situations of this type are common For example, in designing office furniture it would be of importance to know to what extent the female back is in general shorter than the male back. Furniture designed for each sex should then take into account the difference in back lengths.
David J. Saville, Graham R. Wood

Chapter 7. Questions About Several Populations

Abstract
Here we move on to discuss questions concerning an arbitrary number of populations. For simplicity we restrict our attention to the case where we sample equal numbers from each population. The unequal replications case is discussed in Appendix B.
David J. Saville, Graham R. Wood

Orthogonal Contrasts

Frontmatter

Chapter 8. Class Comparisons

Abstract
This chapter is devoted to a study of contrasts of the class comparison type. With the exception of pairwise comparisons, this is the easiest type of contrast. A class comparison contrasts the average of the population means in one class of populations, with the average for a second class of populations. For example, in §7.1 the contrast of suburban with urbanc = (µ1+ µ2)/2 –µ3was of this type. Here the class of suburban populations had two members whereas the class of urban populations had just one member.
David J. Saville, Graham R. Wood

Chapter 9. Factorial Contrasts

Abstract
This chapter will deal withfactorialcontrasts. These arise when several factors are to be simultaneously investigated in a single experiment. Factorial contrasts can be thought of as clever class comparisons. For example, when we analyze Example B we shall firstly contrast the class of “no superphosphate” treatments with the class of “superphosphate” treatments, then move on to contrast the class of “no nitrolime” treatments with the class of “nitrolime” treatments. Both of these contrasts involve all the observations in the experiment, so in effect we will have investigated two factors for the price of one. When more than two factors are investigated in the same study, even greater economies can be achieved.
David J. Saville, Graham R. Wood

Chapter 10. Polynomial Contrasts

Abstract
The experiments we have considered so far have always involved treatments which arequalitativein nature: for example, variety of lupin or breed of sheep. Consequently the contrasts of interest were comparisons of one class of treatment with another (Chapter 8). In the case of factorial experiments (Chapter 9) the contrasts of interest also included the interactions between factors. Frequently in an experimental situation, however, the treatments arequantitativein nature. Such is the case when treatments correspond to the seeding rate of barley in kg/ha, or the dose rate of a medicine in a clinical trial in mg/person. In the first example, it is then natural to ask a question of the type “Does the yield of barley increase as the seeding rate increases?” In the second example, the natural question may be “Does blood pressure go down as dose rate goes up?” In this chapter we find out how to answer such questions. The mechanism will be essentially that of the previous two chapters: the effects we wish to study correspond to contrasts, and testing is carried out using the appropriate unit vector in the treatment space.
David J. Saville, Graham R. Wood

Chapter 11. Pairwise Comparisons

Abstract
The simplest type of contrast occurs when we compare just one population mean with another. This chapter is devoted to such contrasts, termedpairwise comparisons.In Example E of §7.2, which involved four fertilizer treatments for sunflowers, such contrasts occur naturally. We can assess, for example, the effect of omitting magnesium from the fertilizer solution by comparing treatment one, the complete solution, with treatment two, the solution lacking magnesium.
David J. Saville, Graham R. Wood

Introducing Blocking

Frontmatter

Chapter 12. Randomized Block Design

Abstract
In preceding chapters our measurements have largely arisen through applying experimental treatments to experimental units. For example, we measured the “bulk” of two and three day conditioned samples of wool. When the experimental units are entirely uniform, the only major influence on the measured value is the treatment. In practice, however the experimental units are seldom entirely uniform; for example, the nature of the wool may vary from sample to sample. This variation in the experimental units will in turn influence our measurements, and disguise the variation between treatments, the issue of central interest. Intelligent design, which recognizes the variation between experimental units, can help overcome this problem.
David J. Saville, Graham R. Wood

Chapter 13. Latin Square Design

Abstract
In this chapter we introduce the latin square design. This design is used to take account of two sources of variation between the experimental units, in an orthogonal fashion. In practice, the design is not widely used. However, we include it in this book since it illuminates the ideas of orthogonal blocking in a very elegant manner.
David J. Saville, Graham R. Wood

Chapter 14. Split Plot Design

Abstract
Here we introduce the simplest “hierarchical” design, the split plot design. This design has two error terms, corresponding to a subdivision of the error space into two orthogonal subspaces. Studies employing this design have (at least) two treatment factors; the effects of one factor, however, are estimated more accurately than the effects of the other factor. In practice this design is very commonly used.
David J. Saville, Graham R. Wood

Fundamentals of Regression

Frontmatter

Chapter 15. Simple Regression

Abstract
Over a century ago Sir Francis Galton investigated the manner in which a son’s height depended on that of the father. He showed that the heights of tall fathers’ sons were distributed about a value somewhat less than that of their father. The paper was entitled “Regression towards mediocrity in hereditary stature”, and the word regression has stayed with the subject. However, the word has taken on a broader meaning in the world of statistics; it now refers to more general relationships between variables, not just the special linear relationship noted by Galton.
David J. Saville, Graham R. Wood

Chapter 16. Polynomial Regression

Abstract
In Chapter 15 we made a strong assumption about the form of the relationship between thexand y variables: we assumed that y dependedlinearlyonx.In some of the exercises in that chapter such an assumption was clearly unreasonable. In this chapter we study the fitting of quadratic, cubic and higher order polynomials to sets of data, and look at the problem of how to choose the best fitting curve.
David J. Saville, Graham R. Wood

Chapter 17. Analysis of Covariance

Abstract
The analysis of covariance, or ANCOVA, technique is an amalgam of the ANOVA and regression techniques. In analysis of covariance we fit parallel straight lines to approximate the relationship between two variables, such as fatness and weight, for several groups, such as male and female sheep. This is usually done for one of two reasons. Firstly, a researcher may simply wish to describe the relationships. Alternatively, a researcher may be trying to increase the precision of comparisons between groups by explaining some of the variation in the y variable, say fatness, using a relatedxvariable, say weight, whichcovarieswithy. To rephrase this, we may be primarily interested in accounting for some of the extraneous variation which may affect the precision of a study, or bias the estimates of population means.
David J. Saville, Graham R. Wood

Chapter 18. General Summary

Abstract
In §1 of this chapter we review the method that has been used throughout this textbook, and in §2 we discuss its extension to more complicated problems using matrix methods.
David J. Saville, Graham R. Wood

Backmatter

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