ctors affecting the fish community. It is also known 
at the establishment of a nuclear reactor on the banks 
f the river will progressively raise the temperature of 
1e water. The question is: “How will the rise in tem- 
erature of the water affect the populations of the fish 
ving in the river?” The expected rises in temperatures 
ith time could be entered into the specification equa- 
ms. The other factors could be assumed to be constant 
estimates of their probable values might be entered, 
id the predicted population levels of all species of fish 
the river estimated for time x. A weakness of the 
odel is that it can never predict a species becoming 
tinct although it will approach zero frequency as a 
nit. 
DISCUSSION 
Like any other statistical technique, factor analysis 
anipulates data in an attempt to reveal the underlying 
uses and their importance to the variables measured. 
wee important assumptions are made about the data 
en factor analysis is employed (Cattell nekee ics ab) 
lividual variables and factors are linearly interrelated, 
) two factors act additively in respect to any given 
riable, and (3) there are no interaction effects among 
: variables. 
No assumptions are made about the distributions of 
‘variables. Various tests for significance of factors do 
ke assumptions regarding the distributions of variables 
1, for that reason, have been avoided in this paper. It 
srobable that in any real, relatively large community 
organisms all three assumptions will be violated at 
» time or another. Because of the likelihood of some 
vilinear or higher polynomial relationships between 
tors and variables and because of the existence of non- 
litive factors, it is important to know how closely the 
ar model assumed by the factor analysis approximates 
situation where there are some nonlinear relation- 
9s between variables and factors. 
Cattell & Dickman (1962), using variables and 
‘ors between which the relationships were known, 
wed that if variables are not linearly related to the 
ors, the factor analysis approximates the determina- 
| of the variable by representing a product by a sum. 
‘Ta small range this is usually considered to be a good 
roximation. For example, if a species were deter- 
ed by two factors acting multiplicatively, 
Species = sF F a 
1 the factor-analysis model approximates it by 
Species — sF, + sF, 
Miter the analysis has been carried out and the num- 
and nature of the factors determined, the linear 
lel can be modified and the predictions improved by 
rimentally locating nonadditive factors and modify- 
the series of specification equations. The same can 
lone with nonlinear relationships between variables 
factors. Often the mathematical relationship of 
13 
a factor to a community of species, if not linear, will 
be roughly the same for all species (i.e. if the relation- 
ship is exponential, it will be exponential for all species) . 
Two other common situations that modify the rela- 
tionships between factors and variables are threshold 
levels and competition for a limited resource. Some- 
times a factor influencing a set of variables may op- 
erate only above or below a critical value. For exam- 
ple, dispersal in some animals occurs when the popula- 
tion of a species reaches a critical density. The sigmoid 
curve of population ecology assumes that reaction to in- 
creasing density is gradual: the closer the population 
approaches the carrying capacity of the environment, 
the slower the rate of growth. It is also possible that 
there may be a situation where the curve is completely 
exponential until the carrying capacity has been reached, 
or surpassed, and a point is reached where density-de- 
pendent factors act suddenly. In some predators, search 
images are formed on abundant species of prey and, 
when the population of a prey species reaches a critical 
level, a predator population may begin to attack it to 
the exclusion of other less common species. 
Competition between the members of a community 
may prove to be more of a problem, and depends on 
whether populations are controlled by density-dependent 
or density-independent factors. It is the author’s opin- 
ion that both types of factors are important in animal 
communities. One factor influencing a group of species 
in a community may be a common food resource, such 
as in a group of insects all feeding on the same species 
of plant. In the situation of two insect species feeding 
on one plant species, the feeding of species “a” re- 
duces the amount of factor ‘““X” (the plant) and there- 
fore indirectly influences species ‘tb,’ the other species 
feeding on the same species of plant. Factors of this 
type are referred to as “expendable” and, when they 
are shown to exist, the specification equations can be 
modified to take them into account. 
The computational steps in the factor analysis tech- 
nique presented in this paper are outlined in Fig. 14. 
The assumptions underlying each step of the procedure 
have been discussed in the Techniques section and will 
Standard Scores 
Data 
Invert -——Correlation Matrix 
Basic Diagonal Principal Axis 
Factor Analysis 
Rotation 
Factor Pattern 
Matrix 
Factor Scere 
Matra 
Specification Equations 
Fig. 14.—Sequence of steps in creating a factor analytical 
model of a community. 
