FUNDAMENTAL ASSUMPTIONS IN ANALYSIS OF VARIANCE

UT OO

O MS,
— MS,.s

F

should provide independent estimates of the same population error
Variance, o?. It is apparent from an examination of the expected values of
the two mean sguares that this can occur only if o), — 0. We shall return
to this point in Chapter 5.
Fortunately, a transformation that accomplishes any one of the
objectives listed above will usually accomplish the other two objectives.
In general, a transformation can be used whenever there is a relationship.
between the means and variances of the treatment levels and whenever
the form of the treatment level distributions is homogeneous. It is not
always possible to find an appropriate transformation for a set of data.
For example, if any of the following conditions are present, no trans-
formation exists that will make the data more suitable for analysis of
variance: (1) means of treatment levels are approximately egual but
variances are heterogeneous, (2) means of treatment levels vary indepen-
dently of variances, or (3) variances are homogeneous but treatment level
distributions are heterogeneous in form. If no transformation is appro-
priate, and if the departures from normality and homogeneity are gross, an
experimenter may be able to use one of the nonparametric statistics for k
treatment levels described in Chapter 13. Although these statistics reguire
less stringent assumptions than analysis of variance, they are less powerful
and provide less information concerning the outcome of an experiment. It
should also be noted that the nonparametric procedures described in
Chapter 13 provide a test of the hypothesis that k 2 2 population distribu-
tions of unspecified form are exactly alike. In order to test hypotheses con-
cerning population means, the homogeneity assumptions of analysis of
variance must be tenable. This point is discussed in Section 13.1. Another
alternative that may he available to the experimenter is to select a different
criterion measure. The choice of a dependent variable in the behavioral
sciences is often arbitrary; a different choice may fulfill the reguirements of
additivity, normality, and homogeneity.

A number of procedures exist for determining which transformation
is appropriate for a set of data. Several methods are described by Olds,
Mattson, and Odeh (1956) and by Tukey (1949b). One procedure is to
follow general rules concerning situations in which a given transformation
is often successful. This approach will be emphasized in presenting each
of the types of transformations. Alternative procedures for selecting a
transformation will be described later.

SOUARE-ROOT TRANSFORMATION

For certain types of data, treatment level means and variances
tend to be proportional, as in a Poisson distribution, where ju — o?. This
kind of distribution often results when the dependent variable is a freguency