Linear correlation

1
Chapter 9

• Linear correlation

2
Relationships between variables

• So far we have been testing hypotheses about
  differences.
• There are other things we can do with statistics.
• We can investigate the relationships between
  variables.

3
The value of relationshipsKnowledge compression

• If A depends on B and B depends on C and C
  depends on D and so on
• Then we can compress the amount of information we
  need to have about the world.
• For example, in the above, knowing about D tells
  us what we need to know about C, B, and A.
• Our mental capacity is finite, so it pays to be
  efficient.

4
The value of relationshipsStructure

• Imagine a world where all variables varied
  independently from each other.
• This would be a very chaotic and meaningless
  world.
• The relationships between variables gives
  structure and meaning to life.
• So, relationships are good, but what sort of
  relationships should we look for?
• And, how should be look for them?

5
Challenge 1

• Given Xi1, 2, 3, 4, 5, Yi10, 8, 6, 4,
  2
• Arrange these numbers into pairs, attempting to
  maximize the sum of the products of the pairs.
• For example
• 1 8
• 2 10
• 3 4
• 4 6
• 5 2
• 18 210 34 46 52 74

6
Challenge 2

• Given 200 yards of fencing, find the rectangular
  cattle pen dimensions with the maximal area
• For example
• 75yd 25yd 1875

7
What have we learned?

• By matching ______ coefficients, we maximize a
  product.
• Implicit in this pairing is a relationship
• When Xi is large Yi is large and when Xi is small
  Yi is small.
• Many pairs of variables have such a relationship.
• The more you drink in an hour the more
  intoxicated you are.
• The more gas you have in your tank, the farther
  you can drive.
• The colder it is, the thicker the ice on a lake.

8
Applying what weve learned

• We now have an interesting relationship and a way
  to measure it.
• The relationship As Xi increases Yi increases.
  As Xi decreases Yi decreases.
• The measure
• Divide by N because we dont care how many scores
  we have.
• If the Xi and Yi are properly matched, our
  measure will be high.
• If the pairing is random, it will be low.

9
Our measure of correlation is not perfect

• Consider an extremely simple example.
• Xi is perfectly correlated with Xi.
• Example Let Ci be the highs in a 6 day forecast.
• Ci -13, -15, -21, -17, 14, -8
• correlation (-13-13) (-15-15) (-21-21)
  (-17-17) (-14-14) (-8-8)
• Now, what if we substitute the Fahrenheit
  equivalents for the second variable?
• F9/5 C 32
• correlation (-138) (-155) (-21-5)
  (-172) (-147) (-88)
• Problem our correlation will change based purely
  on a change of unit.

10
Our measure of correlation is not perfect

• What happened?
• F9/5 C 32
• Multiplying by 9/5 scales all scores.
• Adding 32 translates the mean.
• We dont want things to change if we scale by
  some factor or add a constant.
• What did we do last time?

11
Solution

• Standardize by mapping our raw scores into z
  values.
• This neutralizes any scaling or translation of
  the mean.

12
Properties of r

• A set of z scores have an expected mean of 0.
• Thus, a z score is as likely to be positive as
  negative.
• Therefore, if the zXi and zYi are paired
  randomly, what will be the correlation r?
• So, no correlation implies r ___________________
  _

13
Properties of r

• We know that zi is perfectly correlated with
  itself.
• What correlation r does this give?
• This is the variance of z.
• What is the variance of z (the standard normal
  distribution)?
• Therefore, perfect correlation means r
  ___________

14
Graphing correlationwith SPSS

• Using grades.sav
• Graphs-gtScatter/Dot
• Simple Scatter
• Define
• Select total for X.
• Select percent for Y
• Guess if the correlation r will be large or
  small.
• OK

15
Computing correlationwith SPSS

• Now compute r.
• Analyze -gt Correlate -gt Bivariate
• Choose total and percent.
• Check Pearson.
• OK
• Next compute the graph and the correlation for
• Quiz1 and final
• Id and GPA
• Predict the degree of correlation before testing
  with SPSS.

16
Affine functions and relationships

• A straight line plot represents an affine
  function.
• Y aX b
• Often mistakenly called linear.
• Y aX
• Non-linear
• Sounds very sophisticated
• Could be almost anything

17
Correlation and non-affine relationships

• Some variables are related by non-affine
  relationships.
• This doesnt mean they are any less related.
• Correlation, as weve defined it, will not detect
  this relationship.

18
An efficient formula for computation of
correlation

• It eliminates the need for repeated subtraction
  of the means.
• It is partially derived on page 253.
• A bias is introduced in the case of estimating ?
  with s.

19
Given some r?0, can can we assume that there is
some correlation?

• Non-zero r might have arisen by chance.
• As usual, look to the null hypothesis
• ?????0, where ? is the correlation for the
  entire population.
• It turns out that the distribution of r, assuming
  ??0, is normal, or at least a t distribution.
• So we can do a t test to see if the r?0 we got
  arose by chance.

20
All we need is the standard error for r

• The gods tell us that the standard error for r
  is
• So, by substitution
• The gods also tell us that df for r is N-2.

21
New assumptions

• Distributions are normal bivariate.
• This can be assumed if the the sample is large
  (gt30) and the bivariate distribution is
  approximately normal.

22
Exercises

• Page 251 1-4, 9
• Page 262 1, 2, 5, 7 (Use SPSS.), 8