66 FUNDAMENTAL ASSUMPTIONS iN ANALYSIS OF VARIANCE

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RECIPROCAL TRANSFORMATION

If the sguare of treatment means and standard deviations are pro-
portional, a reciprocal transformation may be appropriate. A transformed
store X' is given by

1 , 1.

X X-yYrT

The latter formula should be used if any scores are egual to zero. A recip-
rocal transformation may be useful when the dependent variable is reaction
time.

ANGULAR OR INVERSE SINE TRANSFORMATION
Xseio Ge. GE
The angular transformation is given by

A

X' < 2aresin VX, x n
where X is expressed as a proportion. It is not necessary to solve for X'
in the above formula; a table of values of X from .00L to .999 is given in
Table D.13. The transformed values in Table D.13 are in radians. Bartlett

1 1 ; — 1
(1947) suggests that zn an oč substituted for X < zero and l zn

1 . . h
or l — dn be substituted for X — 1, where n is the number of observations

on which each proportion is based. An angular transformation may be
useful when means and variances are proportional and the distribution
has a binomial form. This condition may occur when the number of trials
is fixed and X is the probability of a correct response that varies from
one treatment level to another.

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SELECTING A TRANSFORMATION

We have already described situations where particular transfor-
mations have been found to be successful. An alternative approach to
selecting a transformation uses the fact that means and variances are
unrelated for normally distributed treatment populations. The correct
transformation to use for a set of data is the one that removes the relation-
ship between the sample means and variances. This can be determined
by graphing the means and variances on the x and y axes respectively,
for each transformation and selecting the one that appears to remove the
dependency relationship best. The correciness of the selected transformation
can be verified by inspecting the transformed treatment distributions for
normality and homogeneity of variances.

An additional procedure for selecting a transformation is to apply
each of the transformations to the largest and smallest score in the treat-
ment levels. The range within each treatment level is then determined and