Biology Reference
In-Depth Information
the total model. Differences in the variance explained by the various combinations of inde-
pendent variables sort out (i.e. decompose) the variance into the portions explained by
each term of the model. Or perhaps we should say that they do so under ideal experimen-
tal conditions, including an ideal (i.e. balanced) design. Significance testing of each contri-
bution is done using F-tests, which are ratios between the variances explained by different
terms. As discussed earlier, variance can be characterized using variance
covariance
matrices (based on SSCP), or squared Procrustes distances, and there are both analytic and
numerical approaches to estimating significances of the observed F-ratios, which will be
discussed later. The important feature here is that all of this depends on being able to
decompose variance.
Balanced and Unbalanced Design
In a balanced design, each possible combination of levels of the various factors has the
same sample size. If we lay the factors out along the rows and columns of a matrix, each
“cell” in that matrix is a unique combination of levels of all factors. In a balanced design,
the number of individuals within all cells is equal. For example, say that we have two fac-
tors, handedness and sex. The levels of handedness are right-handed and left-handed, and
the levels of sex are male and female. In a balanced design, we would have equal numbers
of right- and left-handed individuals and equal numbers of males and females; and fur-
thermore, equal numbers of right-handed females, left-handed females, right-handed
males and left-handed males). But we can see that numbers are not equal in all cells of
Table 9.1
. We have equal numbers of right- and left-handed females but we have an
excess of right-handed males. This kind of design is called “unbalanced”.
Unbalanced designs present serious problems for statistical analyses because the
experimental design, not biology, induces a correlation between factors. In this case,
handedness is correlated with sex. If the sex of a particular specimen is female, then the
chances are equal that the individual is right- or left-handed but that is not the case for
males. For males, the chance of being right handed is 70%. That is a problem for the fol-
lowing reason. Suppose that being right-handed causes a detectable change in shape of
the hand (or other part of the limb or even the brain). If we compare the variance
explained by sex, without considering handedness, we would see that on average, most
men had the shape associated with right-handedness (because 70% of the males
are right-handed). In contrast, the average female would not have that shape because
half do and half do not. If we were to consider handedness alone, we would see the
impact of being right-handed, but it would be difficult to tell if that effect was due to
TABLE 9.1
Number of Individuals Sampled by Sex and Handedness
Sex
Handedness
Right
Left
Female
50
50
Male
70
30
