The Mismeasure of Man by Stephen Jay Gould

THE REAL ERROR OF CYRIL BURT

We can grasp the three-dimensional case, both mentally and

pictorially. But what about 20 dimensions, or 100? If we measured

100 parts of a growing body, our correlation matrix would contain

10,000 items. To plot this information, we would have to work in

a 100-dimensional space, with 100 mutually perpendicular axes

representing the original measures. Although these 100 axes pres-

ent no mathematical problem (they form, in technical terms, a

hyperspace), we cannot plot them in our three-dimensional Euclid-

ian world.

These 100 measures of a growing body probably do not repre-

sent 100 different biological phenomena. Just as most of the infor-

mation in our three-dimensional'example could be resolved into a

single dimension (the long axis of the football), so might our 100

measures be simplified into fewer dimensions. We will lose some

information in the process to be sure—as we did when we collapsed

the long and skinny football, still a three-dimensional structure,

into the single line representing its long axis. But we may be willing

to accept this loss in exchange for simplification and for the possi-

bility of interpreting the dimensions that we do retain in biological

terms.

Factor analysis and its goals

With this example, we come to the heart of what/actor analysis

attempts to do. Factor analysis is a mathematical technique for

reducing a complex system of correlations into fewer dimensions.

It works, literally, by factoring a matrix, usually a matrix of corre-

lation coefficients. (Remember the high-school algebra exercise

called "factoring," where you simplified horrendous expressions by

removing common multipliers of all terms?) Geometrically, the

process of factoring amounts to placing axes through a football of

points. In the 100-dimensional case, we are not likely to recover

enough information on a single line down the hyperfootball's long

axis—a line called the first principal component. We will need addi-

tional axes. By convention, we represent the second dimension by

a line perpendicular to the first principal component. This second

axis, or second principal component, is defined as the line that resolves

more of the remaining variation than any other line that could be

drawn perpendicular to the first principal component. If, for

example, the hyperfootball were squashed flat like a flounder, the