Overview

The ANOVA designs discussed in Chapters 23 through 26 examined the effect of one or more independent variables on a single dependent variable. Because such designs focus on a single dependent variable, they are labeled as univariate ANOVA designs. The present chapter addresses designs in which two or more dependent variables are analyzed simultaneously; such designs are known as multivariate analysis of variance (MANOVA) designs. We limit our discussion to the simplest illustration of such a design: a two-group one-way between-subjects design. More information about MANOVA can be found in Meyers et al. (2006), Stevens (2002), and Warner (2008).

Univariate and multivariate ANOVA

Univariate ANOVA designs are extremely useful but their focus is on a single outcome measure. For example, in evaluating a new curriculum designed to teach children to read more quickly, a natural variable to measure is reading speed. But at the same time that the reading speed of the children was improving (assuming a successful curriculum), there might potentially be other variables changing in synchrony, such as reading comprehension, enhanced levels of self-confidence, and feelings of mastery. Perhaps improvements in other academic subjects might be observed as well. All of these related (correlated) effects could serve as potential dependent variables. To focus only on one of these variables, reading speed for example, narrows the focus of the study perhaps to an unnecessary extent.

The present book, Data Analysis Using SAS Enterprise Guide, provides readers with an overview of Enterprise Guide, the newest point-and-click interface from SAS. SAS Enterprise Guide is a graphical user (point-and-click) interface to the main SAS application, having relatively recently replaced the Analyst interface, which itself had replaced the original Assist interface. Enterprise Guide makes it easier than ever to access many SAS statistical analyses without learning to write the SAS code underlying its procedures.

We have written this book for readers who have little or no knowledge of SAS Enterprise Guide but who may wish to employ it for statistical analysis. Some of these readers will be students in an introductory statistics or data-analysis course; other readers will have taken an introductory statistics course and possibly a research methods course at some time in their past; and still other readers may have had several statistics and research design courses as a part of their background. We have therefore included in this book a relatively wide range of statistical procedures to meet the needs of various readers. There are chapters devoted to the more basic procedures such as descriptive statistics, correlation and simple linear regression, t tests, and one-way chi-square analysis. In addition, we have also included statistical procedures at a somewhat higher level; these include data transformations and other types of computations, multiple linear regression, logistic regression, and some analysis of variance designs.

Overview

Factor analysis refers to a set of procedures whose goal is to organize a relatively large set of variables into a few sets of interrelated variables. One common application of the technique is to identify the subscale structure of a paper-and-pencil inventory. Specifically, we can use factor analysis to organize a set of items on an inventory into relatively homogeneous subsets. Items that are relatively strongly related to a factor may be combined together in the scoring system to yield a subscale score.

Some history

We generally ascribe the origin of factor analysis to Charles Spearman (1904a), who, according to Harman (1962), developed this statistical tool in the process of constructing his theoretical model of intelligence. Spearman's intent was to determine the conceptual dimensions underlying a series of mostly perceptual and memory testing modules (e.g., sensory discrimination of tones, just noticeable difference measurements for weights, memory span) that he had administered to over 100 English school children that were presumed to measure aspects of intelligence. However, Harman also tells us that one of the bases for Spearman's mathematical treatment of the data was published in an earlier paper by Karl Pearson (1901), who should therefore share a small portion of the credit for the development of the procedure.

Overview

Analysis of variance (ANOVA) is a family of research and statistical designs allowing us to determine if the means of two or more distributions are significantly different. Each of the next four chapters focuses on a separate ANOVA design.

Naming of ANOVA designs

There are three important pieces of information that are contained in the name of each ANOVA design: the number of independent variables in the design, the number of levels contained in each independent variable, and an indication of the type of independent variables that are included in the design.

The number of independent variables

It is possible to have any number of independent variables in an ANOVA design, although each additional variable that is added substantially escalates the logistics of the data collection. In this chapter and in Chapter 25, we discuss designs containing one independent variable; in Chapters 24 and 26, we discuss designs containing two independent variables.

We communicate the number of independent variables by speaking of n-way designs where n is the count of independent variables. For example, a one-way design contains a single independent variable and a two-way design contains two independent variables.

The number of levels of the independent variables

It is also especially useful when there is more than one independent variable to include an indication of the number of levels of each.

Overview

In a one-way within-subjects design, sometimes referred to as a repeated-measures design, each case is measured on or contributes a data point to every level of the independent variable. Because of this, subjects function in the design as their own controls; this in turn enhances the power of the statistical design. If the drawbacks to this design can be overcome (e.g., carry-over effects; see Gamst et al., 2008), it often becomes the design of choice for a one-way design.

Numerical example

The Automobile Manufacturers Association wished to study the effects of alcohol consumption on driving different types of vehicles. This hypothetical study called for drivers to consume the equivalent of three alcoholic drinks and then drive a complex prescribed closed-track course in one of four kinds of vehicles. Because of the considerable individual differences in drinking and driving that were expected, and because it was believed that the carry-over effects from the different conditions could be largely negated by knowledge of the track, this was designed as a within-subjects study. The organization recruited 14 college students from a local university who were 21 years of age and familiarized them with the track layout. Students were then scheduled for 4 days over the next 2 weeks to drive the course. On each test day, each student was to drive a different vehicle (determined randomly for each student) around the course.

Overview

The independent-groups t test is a procedure to determine if the means of exactly two independent distributions are significantly different. Because a one-way between-subjects analysis of variance (ANOVA) design is the general case of the independent-groups t test, and because t2 = F, it is common practice to defer to the ANOVA for two-group as well as multigroup designs. However, the t test is well worth covering in statistics courses, and we believe it is of sufficient importance to cover in this book as well.

Some history

William Sealy Gosset, a chemist and mathematician, was hired in 1899 by the Guinness Brewing Company. As Salsburg (2001) tells the story, in the context of monitoring the brewing of that beer, Gosset developed several statistical innovations that he wished to publish in the professional literature. However, to protect trade secrets, the company prohibited its employees from publishing their work. Gosset therefore devised a pseudonym with the help of Karl Pearson so that he could disseminate his work in Pearson's Biometrika. The pseudonym that they devised was the name Student, and in 1908 Student published an article describing a new statistical test and its distribution. The letter t was selected by Gosset and Pearson as the name of the test and distribution because it was the last letter of the word Student.

Overview

General meaning of standardizing

To standardize a variable is to transform the obtained values of a variable in such a way that we can immediately determine the following two features of any score: first, its position with respect to the mean, which is whether the score is below or above the mean of the distribution; second, the magnitude of its distance from the mean, which is how far from the mean the score falls in terms of standard deviation units (i.e., how many standard deviation units separate the score from the mean of the distribution). We do this because it is often the case that such information is not always apparent from a raw score.

Conveying direction

Direction is signified by standard scores because the value of the mean is set (transformed) to a known, fixed, arbitrary value. Three examples of commonly used standardized scores and their known or fixed means are as follows:

• z scores have a mean of 0. Negative z scores are below the mean and positive ones are above the mean.

• Linear T scores have a mean of 50. Linear T scores lower than 50 are below the mean and linear T scores higher than 50 are above the mean.

• Intelligence test scores – the Wechsler Intelligence Test for Children (WISC) is a good example – commonly have a mean of 100. Scores lower than 100 are below the mean and those higher than 100 are above the mean.

Programming involves writing relatively complex systems of instructions. There are two broad styles of programming: the imperative style (used in R, for example) involves stringing together instructions telling the computer what to do. The declarative style (used in HTML in web pages, for example) involves writing a description of the end result, without giving the details about how to get there. Within each of these broad styles, there are many subdivisions, and a given program may involve aspects of several of them. For example, R programs may be procedural (describing what steps to take to achieve a task), modular (broken up into self-contained packages), object-oriented (organized to describe operations on complex objects), functional (organized as a collection of functions which do specific calculations without having external side-effects), among other possibilities. In this book we will concentrate on the procedural aspects of programming.

As described in Chapter 1, R statements mainly consist of expressions to be evaluated. Most programs are very repetitive, but the amount of repetition depends on the input. In this chapter we start by describing several flow control statements that control how many times statements are repeated. The remainder of the chapter gives advice on how to design and debug programs.

Welcome to the world of statistical programming. This book will contain a lot of specific advice about the hows and whys of the subject. We start in this chapter by giving you an idea of what statistical programming is all about. We will also tell you what to expect as you proceed through the rest of the book. The chapter will finish with some instructions about how to download and install R, the software package and language on which we base our programming examples.

What is statistical programming?

Computer programming involves controlling computers, telling them what calculations to do, what to display, etc. Statistical programming is harder to define. One definition might be that it's the kind of computer programming statisticians do – but statisticians do all sorts of programming. Another would be that it's the kind of programming one does when one is doing statistics – but again, statistics involves a wide variety of computing tasks.

For example, statisticians are concerned with collecting and analyzing data, and some statisticians would be involved in setting up connections between computers and laboratory instruments – but we would not call that statistical programming. Statisticians often oversee data entry from questionnaires, and may set up programs to aid in detecting data entry errors. That is statistical programming, but it is quite specialized, and beyond the scope of this book.

Users of statistical computing will need to produce graphs of their data and the results of their computations. In this chapter we start with a general overview of how this is done in R, and learn how to draw some basic plots. We then discuss some of the issues involved in choosing a style of plot to draw: it is not always an easy choice, and there are plenty of bad examples in the world to lead us astray. Finally, we will go into some detail about how to customize graphs in R.

There are several different graphics systems in R. The oldest one is most directly comparable to the original S graphics, and is now known as base graphics. You can think of base graphics as analogous to drawing with ink on paper. You build up a picture by drawing fixed things on it, and once something is drawn, it is permanent, though you might be able to cover it with something else, or move to a clean sheet of paper. Since the very beginning, base graphics has been designed to allow easy production of good quality scientific plots. In this chapter we will concentrate on base graphics.

The grid package provides the basis for a newer graphics system. It also has facilities to produce good quality graphics, but the programmer has access to the individual pieces of a graph, and can modify them: a graph is more like a physical model being built and displayed, rather than just drawn.