untitled

are kept separate for purposes of analysis. Red kangaroos and eastern gray kanga-

roos are now the two levels of the factor SPECIES.

In the ANOVAat the bottom of the box, the sum of squares of each source of vari-

ation is divided by the respective degrees of freedom to form a mean square. (Mean

square is just another name for variance.) The three sources of variance of interest,

those of the two factors and their interaction, are divided (in this case) by the resid-

ual mean square to form the Fratios that are our test statistics. That for SPECIESis

5.22 and we check that for significance by looking up the Ftable of Appendix 1. It

will show that an Fwith 1 degree of freedom in the numerator, and 42 (say 40) in

the denominator, will have to exceed 4.08 if the magic 5% or lower probability is to

be attained. We therefore conclude that the disparity in observed numbers between

reds and grays, 957 as against 1490, is more than a quirk of sampling, that gray

kangaroos really were more numerous than reds on the Cunnamulla block at the

time of survey.

In like manner we test for a day effect. The trend in day totals – 625, 825, and

997 kangaroos – suggests that the animals are becoming habituated to aircraft noise

and hence progressively more visible, day-by-day. The Ftables show however that,

with degrees of freedom 2 and 42, a one-tail probability of 5% or better would require

F=3.23. Ours reached only 1.91, equivalent to a probability of 16%, and so we are

not tempted to replace our null hypothesis (no day effect) with the alternative expla-

nation suggested by a superficial look at the data.

The Fratio for interaction was less than 1, indicating that the mean square for

interaction is less than the residual mean square. It cannot, therefore, be significant

and we do not even bother to look up the probability associated with it.

In the last section we tested an interaction, and found it non-significant, without really

exploring what question we were answering. A non-significant interaction implies

that the effect of one factor on the response variable is independent of any effect that

may be exerted on it by the other factor, that the two factors are each operating alone.

The effect of the two factors acting together is exactly the addition of the effects of

the two factors each acting in the absence of the other.

If an analysis produces a significant interaction, the relationship should be

examined by graphing the response variable against the levels of the first factor.

Figure 16.5 shows the kind of trends most commonly encountered. A significant inter-

action is telling you that no statement can be made as to the effect on the response

variable of a particular level of the first factor unless we know the prevailing level

of the second factor. The graph will make that clear.

It is entirely possible for an ANOVAto reveal no main effect of the first factor, no

main effect of the second, but a massive interaction between them. A graphing of the

response variable will reveal a crossing over pattern as in the last graph of Fig. 16.5.

The main assumption underlying ANOVAis that the variance of the response variable

is constant across treatments. The means may differ (and that is in fact what we are

testing to discover) but the variances remain the same. A violation of this assump-

tion can seriously bias the test. Consequently, we need to test for heterogeneity

of variance and, if we find it, either transform the data to render the variances

homogeneous or use an alternative method such as analysis of deviance that does

not employ the assumption of homogeneity.

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16.6.3What is an

interaction?

16.6.4Heterogeneity

of variance

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