60 FUNDAMENTAL ASSUMPTIONS IN ANALYSIS OF VARIANCE

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distribution of F and those associated with the mathematical model for a
particular design—the F ratio can be used to test the hypothesis that all
treatment population means are egual.

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2.6 EFFECTS OF FAILURE TO MEET
ASSUMPTIONS IN ANALYSIS OF
VARIANCE

The emphasis in the previous sections of this chapter has been on
the assumptions necessary for the mathematical justification of hypothesis-
testing procedures using the F distribution. What are the conseguences of
failure to meet these assumptions? Cochran and Cox (1957, 91) stated
that failure to meet the assumptions aflects both the significance level of a
test and the sensitivity of a test. For example, a test performed at the .05
level may actually be made at the .O4'or .07 level. Also, a loss in sensitivity
results when the assumptions are not fulfilled because it is often possible
to construct a more powerful test than that using the F ratio if the correct
model can be specified. Fortunately, the F distribution is very robust
with respect to violation of many of the assumptions associated with its
mathematical derivation. The effects of failure to meet certain assumptions
associated with the F distribution and the mathematical model for a
design are discussed in the following paragraphs. Cochran (1947) has
pointed out that it is impossible to be certain that all reguired assumptions
are exactly satisfied by a set of data. Thus analysis of variance must be
regarded as approximate rather than exact. However, it is generally possible,
by a careful examination of the data, to detect cases in which a standard
analysis will lead to. gross errors in interpreting the outcome of an experi-
ment.

ASSUMPTION OF NORMALLY
DISTRIBUTED POPULATION

One of the reguirements in order for an F ratio to be distributed
as the F distribution is that the numerator and denominator of the ratio
are independent. If scores are randomly sampled from a normal population,
this reguirement is satisfied.

An assumption of both the fixed-effects and random-effects models
is that the errors e;; are normally distributed for each treatment population.
Because the only source of variation within a treatment population are
the errors, the assumption of normally distributed ejj's is eguivalent to the
assumption of normally distributed scores.

A population of scores can depart from the normal distribution in
terms of either skewness or kurtosis, or in both skewness and kurtosis.