Factor 2 
In the principal axis method the first factor is cal- 
culated to account for as much of the variance in the 
correlation matrix as possible. The method attempts to 
have this factor loaded as heavily as possible with all 
of the variables. It is possible that a factor such as 
temperature would influence all of the species strongly, 
and in this case the calculated factor as it comes from 
the analysis would accurately reflect the actual environ- 
mental factor. However, it is also possible that a factor 
may be important to only two or three species and rela- 
tively unimportant to the others in a community. In 
this second case the factors coming from the principal 
axis analysis would not fit the real situation and must 
be rotated to a position where they do. The above situ- 
ation is satisfied by rotation to what is known as simple 
structure. The factors coming from the principal axis 
analysis are orthogonal to each other, but very often, 
probably usually, the factors operating on the species 
are correlated with each other. By rotating to simple 
structure, the factors are allowed to be correlated 
with each other. Mathematically, rotation to simple 
structure attempts to correlate a factor with the smallest 
number of variables possible. In other words each fac- 
tor should affect only a few variables. 
In rotation, the original factor matrix (V,) 1s mul- 
tiplied by a transformation matrix (T) giving a new 
matrix referred to as the reference vector matrix (Vrs) 
Vr =—V,I 
The reference vector matrix does not give the new 
loadings of the factors on the variables for reasons dis- 
cussed by Cattell (1965). To calculate the new factor 
Fig. 3.—Loadings of the 10 species of 
Drosophila on factors 1 and 2. 
Factor 1 
loadings, a new matrix termed the factor-pattern matrix 
is calculated as 
Vip = Virsa 
where D is the diagonal matrix of the reciprocal square 
roots of the diagonal elements of the inverted matrix of 
the reference-vector correlations. The reference corre- 
lations are computed by multiplying the transformation 
matrix by its transpose 
Crs = 1’ T 
where Crs is the matrix of correlations between tht 
reference vectors, T the transformation matrix, and T" 
the transpose of the transformation matrix. 
Several mechanical programs are available for rota: 
tion to simple structure. The program Oblimax ( Pinz 
ka & Saunders 1954) was found to give the most reason 
able answers in this case and has been used in thi 
Taste 4. — New factor loadings (the factor-pattern matrix, 
after rotation to simple structure using the Oblimax program 0! 
the 10 species of Drosophila. 
Factor 1 Factor 2 Factor 3 
melanogaster = 57/9) 2648 —.0889 
pseudoobscura —.8942 0453 -.0339 
bandeirantorum —.9589 .0280 .0925 
“tripunctata 20” —.4056 —.6187 —.2004 
hydet =1262 0744 .7833 
immigrans —.2098 -.8148 -.0725 
viracochi —,0988 0462 8881 
mesophragmatica .8618 3365 2956 
brncici —.7255 .2903 —.1641 
gasici .2786 -.9021 0152 
