analysis. The factor-pattern matrix after rotation to 
simple structure is shown in Table 4. A comparison of 
Tables 3 and 4 shows few significant changes because 
of rotation to simple structure, using the Oblimax pro- 
gram (the signs have been changed in factor DAN 
The new factors produced by rotation to simple 
structure are not necessarily orthogonal and may be cor- 
related (oblique). The correlation matrix of these three 
factors is given in Table 5. 
TaBLE 5.— Correlations among the factors after rotation 
to simple structure using the Oblimax program. 
7 
The calculated communalities are listed in Table 6. 
Other techniques of communality estimation were tried: 
(1) replacing the diagonal entry of a variable by the 
square of the multiple R of each variable with all other 
variables, and (2) replacing the diagonal entry of a 
row by the square root of the average r” across the row. 
The estimated communalities using these two methods 
are also given in Table 6. The Varimax-rotation pro- 
TABLE 6.— Estimated communalities of the 10 species of 
Drosophila using the following methods: 1) (r*¥ ix) (Si—r*ix)/ 
(Sx—r*ix), 2) square of multiple R, 3) square root of average 
r’, 4) iterative. 
Factor 1 Factor 2 Factor 3 
Factor 1 1.0000 = Mes —.1987 
Factor 2 = eis 1.0000 .0469 
Factor 3 —.1987 .0469 1.0000 
Communalities 
In the principal components analysis 1’s are entered 
in the diagonal of the correlation matrix because the 
correlation of a variable with itself is 1. In factoring the 
matrix this presumes that all of the variance of a spe- 
cies can be accounted for by factors common to other 
species. However, a species is normally affected not 
only by common factors, but by factors specific to it, 
and also error factors. 
The variance of a species (o?,) is equal to the vari- 
ance explicable by common factors (os) plus the vari- 
mce of the species due to specific factors (O75) 8 plus 
wn error term (0.5), 
oO"; at Oy; + One + Ou 
The term o?,, .is usually referred to as a variable’s 
ommunality. 
To remove the variance of a species due to specific 
actors and error terms, communalities for each species 
nust be calculated and substituted for the diagonal ele- 
nents of the correlation matrix. Unfortunately there 
re many different techniques used to estimate com- 
nunalities and none of them is “the best.” Also, the 
ubject of communalities is a controversial one. 
In a practical sense, with large initial matrices the 
fect of not calculating communalities on the estimates 
f the factors is minimal and becomes less and less im- 
ortant for larger and larger matrices. The calculated 
ommunalities are important, however, in estimating the 
liability of the predictive equations presented later. 
Communalities in the factor analysis carried out in 
1e following pages were calculated by replacing the 
iagonal entry for each row by 
(r* ix) (Sir*i) / (Si-t* ix) 
here 
ry, = maximum absolute Yij 
Si = absolute rj; 
Sx = absolute Tj 
1 2 3 4 
melanogaster .2919 8454 4075 .3780 
pseudoobscura .8338 .9825 5499 .7999 
bandeirantorum .8638 .9097 .5708 .8846 
“tripunctata 20” .7150 .7567 .4940 .7077 
hydet 2395 HIRE 3484 9921 
immigrans .8314 1779 .4669 .7801 
viracochi .6267 .9399 3.67 7641 
mesophragmatica .9345 SP 5795 9524 
brncici 4282 .6633 4592 .6079 
gasici .4036 9522 4113 8021 
gram (Kaiser 1958) also gives iterative solutions for 
the communalities. The calculated communalities using 
this iterative technique are also given in Table 6. 
Factor Identification 
The purpose of the analysis is to arrive, mathematic- 
ally, at a set of factors corresponding to the real factors 
in the environment that cause changes in populations 
of the species in the community. This problem has been 
partially discussed under rotation. There it was shown 
that factors calculated by the principal axis analysis do 
not necessarily correspond to any real factors. To make 
these factors useful, the factor vectors must be rotated 
in hyperspace to a position where they do correspond 
to real parts of the environment. The problem of identi- 
fication can be broken into two stages: (1) rotation of 
computed factors to where they correlate heavily with 
real factors of the environment, and (2) the identifi- 
cation of the environmental factors. I will discuss the 
second stage first. 
A set of factors has been calculated that explain part 
of the variation in the population of a species. How- 
ever, to be useful these factors must correspond to real 
parts of the environment that can be identified. Basic- 
ally we want to know that factor 1 is so highly cor- 
related with rainfall that rainfall, for practical purposes, 
can be taken as factor 1. Often a person knows a priori, 
or suspects, that species “a” is heavily influenced by 
some factor such as maximum temperature. Therefore 
if this species has a heavy loading on one of the factors 
derived from the factor analysis, it is a good indication 
that this factor is either maximum temperature or is, in 
some way, Closely correlated with maximum tempera- 
ture. It is also possible, if measurements of maximum 
