Lecture 5- 25 January 2013
1. Table 2.2
a. Multiplying the diagonals of a matrix should equal each other.
b. Spearman did this before factor analysis
2. Three graphs
a. The loading of g is more or less the correlation between the vectors and the axis
3. Table 2.2
a. Measures correlation about a thing but not causation. Once you find factors, you want to
figure out what causes them and then try to manipulate them to see if it works. In a sense,
this is what we are going to do for the rest of the course.
4. But correlation matrices don’t always show this tendency
a. Spearman’s position: all that’s important about how smart you are is a single factor-
cleverness. A single ability- intelligence. It begs the question, where and how do you draw
the line about what is intelligence.
b. If you want to draw the line a little closer, then you are switching over to Thurston- using
memory tests, puzzle tests, perceptual tests- all measure different abilities (not in terms of
Spearman’s s but in terms of important abilities that are in some sense independent of one
another). So you might have somebody smart at visual and not at verbal tasks. Its like a
phys ed class where one person better at high jump and another better at sprinting. He also
says they aren’t trivial.
c. We have another matrix here. If you look at the numbers, they are positive, but it doesn’t
all go smaller along the line. There are clusters of higher correlation and lower correlations.
This suggests the primary abilities Thurston talked about- the verbal and visuo-spatial one.
This demonstrates that there isn’t a single component.
d. A general positive manifold means that if you geometrically represented it on a cd and
put all those things on it put your finger in the middle and balanced it, you’d have the
positive stuff on the top because there are more positive stuff, even if there are a few
negative. In IQ testing, you almost never get even 0 correlation so it’s a very good positive
manifold. So all positive manifold is, is the centre of gravity (whether or not the
correlations are even. So if half are negative and half are positive then you would have no
manifold). If all were negative then you would have a negative manifold. If you had a
perfect positive manifold it would be one. This graph’s manifold is positive and agrees with
e. When you talk about the grouping factor, Spearman said your test performance is a
function of two things: your general intelligence (principle component) and then a bunch
test specific stuff (s). What you could say that these tests correlate highly because what is
specific to those tests, which isn’t important, is really high and similar. But the high
correlation isn’t a function of primary abilities but that the tests are all the same. It is
difficult to prove so ends up being like a kid’s argument.
5. Thurston’s illustration of how…
a. Famous illustration found in the textbook: test vectors on the surface of the sphere.
b. They are three primary mental abilities and two sets of 12 tests that you have chosen. In
this case, 8 are correlated with C and 8 with A. G is your average. So in the first diagram,
there are 8 tests for C and only 2 for A and B so the g would naturally tend towards C. its
like if you were using waist and height measure and you used 4 waist measures and only 2 height measures then you would naturally tend towards waist because the ratio is off. The
same goes for the second diagram. However, g is still in the same cognitive space. It just
moves around as you move the tests. Spearman says it’s a nice even manifold. You’ve got
good and bad measures of g. his argument is that if you take any sort of groups of tests, it’s
going to be even.
c. Gould argues that intelligence testing is useless. Once you put a principal component up
there and give it a name then people think of it as real- but it may or not be.
6. Thurstone’s primary mental abilities
a. He used the same technique- factor analysis to reveal primary mental abilities by not
restricting yourself to 2 dimensions, which by nature, accounts for more variance.
b. So he gave 56 tests to Chicago undergraduates and found 7 factors (mental abilities). G
loading was taken away. But he used a highly selective group. G variance would be very
low in this case. He later retracted the statement that there is no g
a. There is a positive manifold