Correlation

1
Correlation

• A bit about Pearsons r

2
Questions

• Why does the maximum value of r equal 1.0?
• What does it mean when a correlation is positive?
  Negative?
• What is the purpose of the Fisher r to z
  transformation?
• What is range restriction? Range enhancement?
  What do they do to r?

• Give an example in which data properly analyzed
  by ANOVA cannot be used to infer causality.
• Why do we care about the sampling distribution of
  the correlation coefficient?
• What is the effect of reliability on r?

3
Basic Ideas

• Nominal vs. continuous IV
• Degree (direction) closeness (magnitude) of
  linear relations
• Sign ( or -) for direction
• Absolute value for magnitude
• Pearson product-moment correlation coefficient

4
Illustrations
Positive, negative, zero
5
Simple Formulas
Use either N throughout or else use N-1
throughout (SD and denominator) result is the
same as long as you are consistent.
Pearsons r is the average cross product of z
scores. Product of (standardized) moments from
the means.
6
Graphic Representation

1. Conversion from raw to z.

2. Points quadrants. Positive negative
products.
3. Correlation is average of cross products.
Sign magnitude of r depend on where the points
fall.
4. Product at maximum (average 1) when points
on line where zXzY.
7
Descriptive Statistics N Minimum Maximum Mean
Std. Deviation Ht 10 60.00 78.00 69.0000 6.05530
Wt 10 110.00 200.00 155.0000 30.27650 Valid N
(listwise) 10
r 1.0
8
r1
Leave X, add error to Y.
r.99
9
r.99
Add more error.
r.91
10
With 2 variables, the correlation is the z-score
slope.
11
Review

• Why does the maximum value of r equal 1.0?
• What does it mean when a correlation is positive?
  Negative?

12
Sampling Distribution of r
Statistic is r, parameter is ? (rho). In
general, r is slightly biased.
The sampling variance is approximately
Sampling variance depends both on N and on ?.
13
(No Transcript)
14
Fishers r to z Transformation
r .10 .20 .30 .40 .50 .60 .70 .80 .90
z .10 .20 .31 .42 .55 .69 .87 1.10 1.47
Sampling distribution of z is normal as N
increases. Pulls out short tail to make better
(normal) distribution. Sampling variance of z
(1/(n-3)) does not depend on ?.
15
Hypothesis test
Result is compared to t with (N-2) df for
significance.
Say r.25, N100
plt .05
t(.05, 98) 1.984.
16
Hypothesis test 2
One sample z test where r is sample value and ?
is hypothesized population value.
Say N200, r .54, and ? is .30.
4.13
Compare to unit normal, e.g., 4.13 gt 1.96 so it
is significant. Our sample was not drawn from a
population in which rho is .30.
17
Hypothesis test 3
Testing equality of correlations from 2
INDEPENDENT samples.
Say N1150, r1.63, N2175, r270.
-1.18, n.s.
18
Hypothesis test 4
Testing equality of any number of independent
correlations.
Compare Q to chi-square with k-1 df.
Study r n z (n-3)z zbar (z-zbar)2 (n-3)(z-zbar)2
1 .2 200 .2 39.94 .41 .0441 8.69
2 .5 150 .55 80.75 .41 .0196 2.88
3 .6 75 .69 49.91 .41 .0784 5.64
sum 425 170.6 17.21Q
Chi-square at .05 with 2 df 5.99. Not all rho
are equal.
19
Hypothesis test 5 dependent r
Hotelling-Williams test
Say N101, r12.4, r13.6, r23.3
t(.05, 98) 1.98
See my notes.
20
Review

• What is the purpose of the Fisher r to z
  transformation?
• Test the hypothesis that
• Given that r1 .50, N1 103
• r2 .60, N2 128 and the samples are
  independent.
• Why do we care about the sampling distribution of
  the correlation coefficient?

21
Range Restriction/Enhancement
22
Reliability
Reliability sets the ceiling for validity.
Measurement error attenuates correlations.
If correlation between true scores is .7 and
reliability of X and Y are both .8, observed
correlation is 7.sqrt(.8.8) .7.8 .56.
Disattenuated correlation
If our observed correlation is .56 and the
reliabilities of both X and Y are .8, our
estimate of the correlation between true scores
is .56/.8 .70.
23
Review

• What is range restriction? Range enhancement?
  What do they do to r?
• What is the effect of reliability on r?

24
SAS Power Estimation
proc power onecorr distfisherz corr 0.35
nullcorr 0.2 sides
1 ntotal 100 power
. run
proc power onecorr corr 0.35
nullcorr 0 sides 2
ntotal . power .8 run
Computed N Total Alpha .05 Actual Power
.801 Ntotal 61
Computed Power Actual alpha .05 Power .486
25
Power for Correlations
Rho N required against Null rho 0
.10 782
.15 346
.20 193
.25 123
.30 84
.35 61
Sample sizes required for powerful conventional
significance tests for typical values of the
correlation coefficient in psychology. Power
.8, two tails, alpha is .05.