Educability-and-Group-Differences-1973-by-Arthur-Robert-Jensen

84 Educability and Group Differences
tx : Gx (Gain since t0)
t2 : CGl +G2 = S2 (Consolidated gain from time 1 to
time 2 plus unconsolidated gain at
time 2 = status at time 2.)
t$1 CGt + CG2 + G3 = 1S3
*4 :C G x+ CG2 + CG3 + G4 = 5 4
: C (Gj + G2 + G3 + G4 + . .. + G„_1) + G„ = S„
For some measures, like height, one can never observe in the
measurements themselves the gain G but only the consolidated
gain CG, so that one always finds S l < S2< S3, etc. This is not
always the case for other characteristics such as the growth of body
weight during development or the growth of intelligence or of
scholastic achievement.
An actual simplex can be created simply by assigning some
numerical values to C and G. Simulated individuals, for example,
can each be assigned a C value selected from randomly distributed
numbers from 0-10 to 1*00, and at each point in time G will be
some value from 0 to 9 also taken from a table of normal random
numbers. (To produce a growth curve which does not increase
linearly but logarithmically, i.e., at a negatively accelerated rate
characteristic of most growth curves, one can simply use the
natural logarithm of S at each point in time. This will produce a
quite typical looking growth curve, but the form of the growth
function is not an essential aspect of the simplex. In the absence of
an absolute scale, as is true of most psychological measurements,
the form of the average growth curve, aside from being an increasing
monotonic function of time, is quite arbitrary. The growth of
vocabulary, a good index of intellectual development, can be
measured on an absolute scale [number of words] and appears to
be sigmoid. Over the period of schooling, from about age 5 to
18 years, however, the growth curve of vocabulary is logarithmic.)
The S values at times tu t2, t3, etc. for 100 or more such simulated
individuals when intercorrelated yield a correlation matrix with
the simplex pattern. More complicated models can also produce
a simplex; but this is the simplest model that will do it. The
resulting simulated correlation matrix is virtually indistinguishable
from those obtained from actual longitudinal intelligence and
achievement test data.
Can we make a reasonable psychological interpretation of this