Lecture 4- 23 January 2013
1. Correlation, geometry, and the basis for factor analysis
2. Scatter diagram
a.Negative correlation. Hypothesis is that those who drink fewer beers get better
b.Almost every correlation will be a positive one
c. With a normal distribution, there are a lot of scores in the middle and fewer scores
are the ends.
d.If you know the correlation between things and one value, you can predict future
or other values. Things that are correlated are predictive of one another. This does
not imply causality.
3. High, low and zero correlations
a.Examples of what correlations look like
4. Correlation between three variables
a.Paleontologists use factor analysis and correlations. They measure bone lengths for
different species and see how they correlate and perform factor analysis. If you
find a new animal and they fit into factors of a known animal then there’s a good
chance they are related
b.Three-dimensional graph. If there is fourth dimension, it can exist but you cant
physically represent it. All the diagrams are there for approximations.
5. Scatter plot of height to weight Afghani boys
a.Highly correlated. But it still looks like it gets wider as the graph moves up. This
may indicate some kind of social condition
b. (Rest of the slide was basically inaudible)
6. What the previous chart tells us.
a. They tend to vary more in weight as they get taller
b. All causal things are dependent but not all dependent (in the sense of correlation)
things are causal
7. BMI Mystery
a.A person is a three-dimensional object, which means it has to be measured in
b.Area of the circle depends on the radius/diameter.
c. Volume is related to the cube of the diameter. So for a person, their volume or
BMI is dependent on the cube of their height.
d.Unlike a human, if you increase one of the dimensions of the circle, all will
increase because they are proportional. But that isn’t how humans work.
e. An example: as you make a shot put bigger, it gets heavier. Weight and mass are
correlated. A men’s shot put is 7.62kg and a female’s is 4kg. The ratio of men’s to
female’s shot put diameter is only 1.24, which isn’t that much. So even though the
difference in weight looks like a lot, in fact, there isn’t much difference.
f. BMI is an empirical measure. It is supposed to be fairly constant
8. Scatter plot height to weight Afghani boys
a.The two red lines signify equivalent weights in terms of heights (ratio of them) and
they both fall on the line (upward trend) so it is a pretty good line. So his theory about social influences is wrong because they are following the normal trajectory
in terms of BMI
b. Height and weight in terms of obesity is a bad measure. You should measure
weight in terms of BMI instead. Health professionals use it as an indirect measure
of your body fat. But it doesn’t always work
9. Numbers are sometimes an indirect measure and not always right
a. Picture shows him destroying two other members. His BMI is 33, which is
supposedly obese. But that is not the case for him
10. Definition of cosine
a.This is the geometrical interpretation of the correlation coefficient (factor analysis
b. Sine is opposite over hypotenuse. When the angle decreases, the hypotenuse and
the adjacent get to be almost the same size so the relationship will approach 1.
c. If you assume AB and BC to be vectors and you change them around then you are
changing the cosine. So the cosine, it turns out, is related to the correlation of 2
11. Cosine varies 1 to -1 over 360 degree
a.As you move along the cosine of y, which is a sine function where at an angle of 0
it is 1, at 90 its 0 and so on and so forth. This 90 and 0 is important
a.As you move angles around our cosine varies and you can see as it follows the
13. Correlations can be considered as vectors…
a.When you do a graph you have to use some sort of axis. The picture is just an axis
and two vectors from that axis and are of different lengths.
b. For the vectors, one has a smaller angle from the axis than the other. So if you
move along vector AB you will end up higher on the axis than if you move along
c. Things with higher correlations tell you more about the other variable. As the
angle gets smaller and the cosine gets bigger, if you move further along the vector,
you will move further along the axis. If you draw vertical lines, you would still end
up with the same correlation in terms of points B and C on that line.
d.All three angles have cosines and correlations with one another. Both vectors are
correlated with the axis (one more so than the other), the axis is correlated with
them and they are correlated with each other. Through geometry we can define
them in relation to one another.
e. This is an axis in a geometric space. If you have an axis that is at 90 degrees with
another axis, you have a cosine of 0. As you move along one axis, you aren’t
moving on the other at all. Perpendicular axes are independent. This is way you
use them in geometry. You don’t want a point on one axis to be correlated to the
other axis. So axes are arbitrary.
14. Here is a group of vectors that…
a.(You will see this diagram a number of times).
b.This is geometric space with 2 axes. There are groups of vectors that are correlated
with one axis and with each other. They are correlated more with I than J because the more you along a vector, the movement doesn’t change along J as much as it
would change on I.
c. Some people believe that your axes are important, that they mean something but
we getting ahead.
d.Axis I could be Spearman’s g- there is one general intelligence factor. So knowing
any one of the scores will tell you about g. His second factor ‘s’ is any one of the
following: digit span, reaction time, ravens, verbal comprehension etc. They are
going to have 2 sources of v