MOST PEOPLE THINK OF MULTIPLE REGRESSIONand the analysis of variance as two totally
separate statistical techniques that answer two entirely different sets of questions. In fact,
this is not at all the case. In the first place they ask the same kind of questions, and in the
second place they return the same kind of answers, although the answers may be phrased
somewhat differently. The analysis of variance tells us that three treatments (T 1 , T 2 , and T 3 )
have different means ( ). Multiple regression tells us that means ( ) are related to treat-
ments (T 1 , T 2 , and T 3 ), which amounts to the same thing. Furthermore, the analysis of
variance produces a statistic (F) on the differences among means. The analysis of regres-
sion produces a statistic (F) on the significance of R. As we shall see shortly, these Fs are
equivalent.16.1 The General Linear Model
Just as multiple regression and the analysis of variance are concerned with the same gen-
eral type of question, so are they basically the same technique. In fact, the analysis of vari-
ance is a special case of multiple linear regression, which in turn is a special case of what
is commonly referred to as the general linear model.The fact that the analysis of variance
has its own formal set of equations can be attributed primarily to good fortune. It happens
that when certain conditions are met (as they are in the analysis of variance), the somewhat
cumbersome multiple-regression calculations are reduced to a few relatively simple equa-
tions. If it were not for this, there might not even be a separate set of procedures called the
analysis of variance.
For the student interested solely in the application of statistical techniques, a word is in
order in defense of even including a chapter on this topic. Why, you may ask, should you
study what amounts to a cumbersome way of doing what you already know how to do in
a simple way? Ignoring the cry of “intellectual curiosity,” which is something that most
people are loath to admitthat they do not possess in abundance, there are several practical
(applied) answers to such a question. First, this approach represents a relatively straightfor-
ward way of handling particular cases of unequal sample sizes, and understanding this
approach helps you make intelligent decisions about various options in statistical software.
Second, it provides us with a simple and intuitively appealing way of running, and espe-
cially of understanding, an analysis of covariance—which is a very clumsy technique when
viewed from the more traditional approach. Last, and most important, it represents a
glimpse at the direction in which statistical techniques are moving. With the greatly
extended use of powerful and fast computers, many of the traditional statistical techniques
are giving way to what were previously impractical procedures. We saw an example when
we considered the mixed models approach to repeated measures analysis of variance. Other
examples are such techniques as structural equation modeling and that old and much-
abused standby, factor analysis. Unless you understand the relationship between the analy-
sis of variance and the general linear model (as represented by multiple linear regression),
and unless you understand how the data for simple analysis of variance problems can be
cast in a multiple-regression framework, you will find yourself in the near future using more
and more techniques about which you know less and less. This is not to say that t, , F, and
so on are likely to disappear, but only that other techniques will be added, opening up
entirely new ways of looking at data. The recent rise in the use of Structural Equation Mod-
eling is a case in point, because much of what that entails builds on what you already know
about regression, and what you will learn about underlying models of processes.
In the past 25 years, several excellent and very readable papers on this general
topic have been written. The clearest presentation is still Cohen (1968). A paper byx^2Xi Yi580 Chapter 16 Analyses of Variance and Covariance as General Linear Models
general linear
model