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2.7

SECTION 2.7 TRANSFORMATIONS

ASSUMPTION OF ADDITIVITY OF EFFECTS

A basic assumption of the experimental designs described in this
book is that a score is the sum of the effects in the linear model. If the
assumptien of additivity of effects is not tenable, it may be possible to
achieve additivity by a suitable transformation of the scores.

References that provide additional discussion of the assumptions
in analysis of variance may be found in the papers by Eisenhart (1947) and
Cochran (1947).

TRANSFORMATIONS

A transformation is any systematic alteration in a set of scores
whereby certain characteristics of the set are changed and other character-
istics remain unchanged. Three major reasons for using transformations
in analysis of variance are

1. To achieve homogeneity of error variance.
2. To achieve normality of treatment-level distributions (or within-cell
distributions).

3. To obtain additivity of treatment effects.

Because the F distribution is relatively unaffected by lack of normality
and heterogeneity of variance, the first two reasons for performing a
transformation are less compelling than the third. Obtaining additivity of
effects is particularly important in designs such as a randomized block
design in which a residual mean sguare (abbreviated MS,,,) is used as an
estimate of experimental error. For example, if treatment levels and blocks
are not additive, the expected value of the residual mean sguare is

E(MS,,) < o; -£ o,
instead of

E(MS,.,) — o;,

where ož, refers to the interaction of treatment levels and blocks. Inter-
action in this context is said to be present when the dependent variable that
is measured under the k treatment levels behaves differently for different
blocks of subjects. The expected value of the treatment mean sguare for a
fixed-effects model is

E(MS,) < o? -£ nož.

If the null hypothesis is true, then, according to Section 2.5, the numerator
and denominator of the ratio .