The analysis of variance is a comparison of different populations in studies that have several treatments or conditions.

**One-Way Analysis of Variance**

The analysis of variance is a comparison of
different populations in studies that have several treatments or conditions.
For example, we may want to compare mean scores from three or more populations
that represent three or more study conditions. Remember that we used the *Z* test or *t* test to compare two populations, as in com-paring an experimental
group with a control group. The analysis of variance will enable us to extend
the comparison to more than two groups.

In this text, we will consider only the one-way
analysis of variance (ANOVA). Typically, ANOVA is used to compare population
means (*μ*’s) that
represent inter-val- or ratio-level measurement. In the one-way analysis of
variance, there is a sin-gle factor (such as classification according to
treatment group) that differentiates the groups.

Other types of analyses of variance are also
important in statistics. ANOVA may be extended to two-way, three-way, and *N*-way designs. To illustrate, the
two-way analysis would examine the effects of two variables, such as treatment
group and age group, on an outcome variable. The *N*-way ANOVAs are used in experimental studies that have multiple
factorial designs. However, the problem of assessing the associations of
several variables with an outcome variable becomes daunting.

One common use of the two-way analysis of variance
is the randomized block design. In this design, one factor could be the
treatment and the other would be the blocks. Blocks refer to homogeneous
groupings of subsets of subjects; for example, subsets defined by race or other
demographic characteristics. These characteristics, when uncontrolled, may
increase the size of the error variance. In the randomized block design, we
look for treatment effects and block effects, both of which are called the main
effects. There is also the possibility of considering interaction effects
between the treatments and the blocks. Interaction means that certain
combi-nations of treatments and blocks may have greater or smaller impact on
the out-come than do than the sum of their main effects. As is true of
regression, the analysis of variance, which represents an important area in
applied statistics, is the subject of entire books.

Scheffe (1959) wrote the classic theoretical text
on analysis of variance. Fisher and McDonald (1978) authored a more recent
text, which provides an advanced treatment of fixed effects designs (as opposed
to random effects). Other, less ad-vanced, treatments can be found in Hocking
(1985), Dunn and Clark (1974), and Miller (1986).

In statistical computer packages, the analysis of
variance can be treated as a re-gression problem with dummy variables. A dummy
variable is a type of dichoto-mous variable created by recoding the classifications
of a categorical variable. For example, a single category of race (e.g.,
African American) would be coded as pre-sent (1) or absent (0). In the case of
a regression problem, we may regard an ANO-VA as a type of linear model. Such a
linear model (called the general linear model) can employ a mix of categorical
and continuous variables to describe a relationship between them and a response
variable. You may often see this type of analysis re-ferred to as analysis of
covariance. All these models have the decomposition of variance of the response
*Y* into proportions explained by the
predictor variables. This is the so-called ANOVA that we will describe in this
chapter.

In Chapter 12 we discussed *R*^{2}, which is a ratio of the part of the variance in the
response variable *Y* that is explained
by the regression equation divided by the total variance of the response
variable *Y*. In the ANOVA table (refer
to Appendix *A*), we will see the case
of an *F* test in which at least one of
the means of a response vari-able is different from the other means. There is a
direct mathematical relationship between this *F* statistic and *R*^{2}.

In Chapter 12, we emphasized simple linear
regression and correlation and briefly touched on multiple regression by giving
one example. Analogously, multi-way analysis of variance is similar to multiple
linear regression, in that there are two or more categorical variables in the
model to explain the response *Y*. We
will not go into the details here; the interested reader can consult some of
the texts listed in Section 13.7.

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