10 
variables is heavily loaded on any of the three factors, 
although minimum temperature has a moderately heavy 
loading on factor 2, suggesting that factor 2 may in some 
way be associated with minimum temperature. Rainfall 
is lightly loaded on all three factors, and it is thus un- 
likely that it has any association with the three factors. 
The factor matrix was then rotated to simple struc- 
ture using the Oblimax method, and the resulting refer- 
ence-vector structure matrix is given in Table 9. The 
calculated factor-pattern matrix is given in Table 10. 
Predictive Equations 
The next stage is the formulation of what are known 
as specification equations. These equations specify the 
weights to be given to each factor in accounting for the 
score (observed measurement of some kind) of each 
variable. The specification equation can be written in a 
general form as 
Vii =suF i + 8joF oi + -----SFaa + 8iF ji + SieFet 
as given by Cattell (1965). If there are k observations, 
the score on a variable on one of these observations is 
equal to the sums of the scores of the factors (Fj.) in- 
fluencing the variable as modified by the significance or 
weight of each factor to the variable (the sya) aelinese 
factors include a series of common factors, any specific 
factors there may be, and an error factor. The specifi- 
cation equations will be the basic predictive equations. 
In the example analyzed in this paper there are 10 species 
measured at 28 observations, giving a total of 280 specifi- 
cation equations. To formulate the set of equations for 
all species in the community, it is necessary to calculate 
first the factor-score matrix (F,)and secondly the fac- 
tor-pattern matrix (Vp) which gives the necessary values 
of the 5;s. 
The factor-score matrix is computed by multiplying 
the reference-vector structure matrix by the basic diagon- 
al of the original correlation matrix. In computation 
this step was done by inverting the correlation matrix, 
multiplying that by the matrix of standard scores for the 
variables standardized by rows, and multiplying the re- 
sulting matrix by the reference-vector structure matrix 
(Vrs) or 
Fp = Vrs 8 
where Fp is the factor score matrix, Vrs the reference- 
vector structure matrix, and 8 the basic diagonal of the 
correlation matrix. The resulting factor-score matrix for 
the 28 observations is given in Table 11. The factor 
scores are the standard scores for the factors calculated 
for a particular rotation. If the factors have been ro- 
tated to where they correspond to real parts of the en- 
vironment, the factor-score matrix gives estimated stand- 
ard scores for the environmental factors. If the rota- 
tion is not the correct one, the numbers are only numbers 
that will reproduce the scores on the variables. It is, of 
course, impossible to use them predictively if they are 
not real. 
Having calculated the factor-score matrix and the 
factor-pattern matrix, it is now possible to estimate the 
value of a variable on any observation. As an example, 
the standard score of Drosophila pseudoobscura at ob- 
Tape 11.— Calculated factor score matrix for the 28 ol 
servations from the Oblimax rotation to simple structure. 
Observation Factor 1 Factor 2 Factor 3 
1 4840 1.0297 —.5129 
2 5836 .9663 —-1.0332 
3 1.7794 ye —.6943 
4 .7821 —.1245 =] tea 
5 —.4793 —.0247 8224 
6 —.1264 3034 2.3076 
7 —,3645 8503 1.6247 
8 1.1017 4221 7145 
9 1.2421 .0689 —.0601 
10 3.2278 —1.3368 —1.9288 
11 1.1410 —1.0098 -.6190 
12 1.1349 3455 5760 
13} 8159 —.0459 —.4046 
14 .9906 = S290 —.8530 
15 —1.4451 .9605 1.8509 
16 —1.9348 1.2871 1.7579 
17 —1.2463 6319 7126 
18 —2.1194 1e3 56) 2.2356 
19 —,3433 = 26/2 .0467 
20 —1.2586 4976 1.0491 
21 —.4868 —3.9399 .4878 
22 4718 —1.9417 —1.0491 
23 —.6639 —.4351 —1.1375 
24 —.8399 1.2284 — 1013 
25 0412 0449 —2 Tiel 7 
26 —.5968 —.8071 —.5184 
27 —.8115 —.2905 —.5392 
28 —1.0795 .0832 —.8220 
servation 4 (December, 1961) equals the sums of t 
factor scores as weighted by the factor loadings for tk 
period plus specific factor scores, plus an error ter 
In other words 
Drosophila pseudoobscura (.9097) (.7821) 
(.1218) (—.1245) + (.0126) (—7125) + specific f 
tors,,, + error factors,,) 
Drosophila pseudoobscura;,, = .6873 + specific fi 
tors;,, -—+ error factors,,). All scores are in stande 
form. 
Theoretically if the scores for the common factc 
the specific factors, and the error factors were known, ! 
predicted scores would exactly fit the actual scores 
the variables (species population levels) . However, in t 
case nothing is known of the specific factors and | 
error factors, and the predictions are based only on | 
variance attributable to common factors. Where 0 
mon factors account for a large percentage of the vé 
ance of a species, the predictions should be fairly ac 
rate. In a species population influenced to a large 
tent by specific factors and error factors, the predicti 
will not be as good. To a certain extent, the reliabi 
of the estimates can be judged from the size of the s 
cies population’s communality, species with large ¢ 
munalities being more predictable than those with sn 
communalties. This procedure, in essence, pretends t 
specific and error factors do not exist. 
Graphs of the predicted and observed abundances 
standard scores) of the 10 species are given in Fig. + 
It is clear that for many of the species, particularly 
common ones, predicted and actual values agree 4 
well, although there are still some deviations. De 
