In many biological cases the variance of the response variable rises with the mean.
That is particularly true of counts of animals that tend to fit a negative binomial dis-
tribution. A transformation of the counts to logarithms after adding 0.1 (to knock
out the zeros) will usually stabilize the variances. If the animals are solitary rather
than gregarious their counts are more likely to fit a Poisson distribution, which is
characterized by the mean and the variance being identical. Transformation to
square roots after adding 3/8 to each will homogenize the variances. The variances
of almost all body measurements regress on their means and the data need log trans-
formation. Use an arcsine transformation if data come as percentages.Before any data are collected, let alone analyzed, we must decide precisely what ques-
tions are to be asked of them. Take the example of comparing counts of kangaroos
on successive days. We were asking whether the act of flying over the study area on
one day influenced the counts obtained the next day. The influence could con-
ceivably be negative (disturbance forced the kangaroos into cover in front of the
surveying aircraft) or positive (the kangaroos became progressively habituated to
aircraft noise).
Now take another question: do viewing conditions differ between days of survey?
That question would be answered by counts obtained by days sampled at random.
We would want those days to be spaced rather than consecutive as they were in answer-
ing the question about the effect of day order. Otherwise the answers to the two ques-
tions would be confounded and we would not know which was being answered by
a significant day effect. In the question concerning an order effect of days, the factor
DAYis said to be fixed. No arbitrary selection of any three days will do. The days have
to follow each other without gaps between them.
Whether a factor is declared fixed or random determines both the question being
asked and the denominator of the Fratio that answers it. Table 16.2 shows the appro-
priate choice of denominators. For two of the three-factor models there is no explicit
test for the significance of some of the factors. There are various messy approxima-
tions available (see Zar 1996, appendix) but it is far better to rephrase the question
to one logically answerable from consideration of the data.
Let us generalize the difference between fixed and random factors. A fixed factor
is one whose levels cover exhaustively the range of interest. MONTHStherefore usually
constitute a fixed factor because they index seasons. YEARSmay be fixed or random
depending on context. REGIONSmay be fixed if the levels of that factor are the only
ones of interest. If they are simply a random sample of regions, and any other selec-
tion of regions would serve as well, the factor REGIONSis random.
Note that questions change according to whether the factor is fixed or random.
Suppose that the response variable is growth rate of a species of pine and we wish
to test for a difference among three soil types (factor SOILS) covering the entire range
of soil types of interest. The factor SOILSis thus fixed. If the question concerns the
best region in which to plant a plantation of that species, and there are four and only
four regions that are possible candidates, the choice of regions is fixed and the appro-
priate denominator of the Ftesting a difference of growth rate among the soil types
of those regions is the residual mean square. However, if we ask the more general
question of whether grow rate differs among soil types across regions in general, any
random selection of a set of regions will suffice and the denominator of the Ftest-
ing SOILSis the mean square of the interaction between SOILSand REGIONS(Table 16.2).EXPERIMENTAL MANAGEMENT 285
16.6.5Are the
factors fixed or
random?WECC16 18/08/2005 14:47 Page 285