Correlation - Coe College

Bivariate
Correlation
Lesson 11
Measuring Relationships
Correlation
 degree relationship b/n 2 variables
 linear predictive relationship
 Covariance
 If X changes, does Y change also?
 e.g., height (X) and weight (Y) ~

Covariance

Variance
 How much do scores (Xi) vary from mean?
2
 (standard deviation)


s 
2
2
(
X

X
)
 i
N 1
(X


i
 X )( X i  X )
N 1
Covariance
 How much do scores (Xi, Yi) from their
means

(X

cov(x, y) 
i
 X )(Yi  Y )
N 1
Covariance: Problem
How to interpret size
 Different scales of measurement
 Standardization
 like in z scores
 Divide by standard deviation
 Gets rid of units
 Correlation coefficient (r)


cov(X , Y )  ( X i  X )(Yi  Y )
r

s X sY
( N  1) s X sY
Pearson Correlation Coefficient
Both variables quantitative (interval/ratio)
 Values of r
 between -1 and +1
 0 = no relationship
 Parameter = ρ
(rho)
 Types of correlations
 Positive: change in same direction



X then Y; or X then Y
Negative: change in opposite direction

X then Y; or X then Y ~
Correlation & Graphs
Scatter Diagrams
 Also called scatter plots
 1 variable: Y axis; other X axis
 plot point at intersection of values
 look for trends
 e.g., height vs shoe size ~

Scatter Diagrams
84
78
Height
72
66
60
6
7
8
9
Shoe size
10
11
12
Slope & value of r
Determines sign
 positive or
Height
negative
 From lower left to
upper right
 positive ~

84
78
72
66
60
6
7
8
9
Shoe size
10
11
12
Slope & value of r

From upper left to
lower right
Weight
 negative ~
300
250
200
150
100
3
6
9
12
Chin ups
15
18
21
Width & value of r
Magnitude of r
 draw imaginary ellipse around most
points
 Narrow: r near -1 or +1
 strong relationship between variables
 straight line: perfect relationship (1 or -1)
 Wide: r near 0
 weak relationship between variables ~

Width & value of r
Weak relationship
Strong negative relationship
r near 0
r near -1
Weight
300
300
250
250
Weight
200
200
150
150
100
100
3
6
9
12
Chin ups
15
18 21
3
6
9
12
Chin ups
15
18 21
Strength of Correlation
R2
 Coefficient of Determination
 Proportion of variance in X
explained by relationship with Y
 Example: IQ and gray matter volume
 r = .25
(statisically significant)
2
 R = .0625
 Approximately 6% of differences in
IQ explained by relationship to gray
matter volume ~

Guidelines for interpreting
strength of correlation
Table 5.2 Interpreting a correlation coefficient
Size of Correlation (r)
General coefficient interpretation
.8 to 1.0
Very strong relationship
.6 to .8
Strong relationship
.4 to .6
Moderate relationship
.2 to .4
Weak relationship
.0 to .2
Weak to no relationship
*The same guidelines apply for negative values of r
*from Statistics for People Who (Think They) Hate Statistics: Excel 2007 Edition
By Neil J. Salkind
Factors that affect size of r

Nonlinear relationships
 Pearson’s r does not
detect more complex
relationships
 r near 0 ~
Peeps (Y)
Stress (X)
Factors that affect size of r

Range restriction
 eliminate values
from 1 or both Height
variable
 r
is reduced
 e.g. eliminate
people under 72
inches ~
84
78
72
66
60
6
7
8
9
Shoe size
10
11
12
Hypothesis Test for r



H 0: ρ = 0
rho = parameter
H 1: ρ ≠ 0
ρCV

df = n – 2

Table: Critical values of ρ

PASW output gives sig.
Example: n = 30; df=28; nondirectional

ρCV = + .361

decision: r = .285 ? r = -.38 ? ~
Using Pearson r
Reliability
 Inter-rater reliability
 Validity of a measure
 ACT scores and college success?
 Also GPA, dean’s list, graduation rate,
dropout rate
 Effect size
 Alternative to Cohen’s d ~

Evaluating Effect Size

Pearson’s r


Cohen’s d

r = ± .1

Small:

r = ± .3

Medium: d = 0.5
r = ±.5 ~

Large:
d = 0.2
d = 0.8
Note: Why no zero before decimal for r ?
Correlation and Causation
Causation requires correlation, but...
 Correlation does not imply causation!
 The 3d variable problem
 Some unkown variable affects both
 e.g. # of household appliances
negatively correlated with family size
 Direction of causality
 Like psychology  get good grades
 Or vice versa ~

Point-biserial Correlation
One variable dichotomous
 Only two values
 e.g., Sex: male & female
 PASW/SPSS
 Same as for Pearson’s r ~

Correlation: NonParametric
Spearman’s rs
 Ordinal
 Non-normal interval/ratio
 Kendall’s Tau
 Large # tied ranks
 Or small data sets
 Maybe better choice than Spearman’s ~

Correlation: SPSS
Data entry
 1 column per variable
 Menus
 Analyze  Correlate  Bivariate
 Dialog box
 Select variables
 Choose correlation type
 1- or 2-tailed test of significance ~

Reporting Correlation Coefficients

Guidelines
1.
2.
3.
4.
5.
No zero before decimal point
Round to 2 decimal places
significance: 1- or 2-tailed test
Use correct symbol for correlation type
Report significance level

There was a significant relationship between the number of
commercials watch and the amount of candy purchased, r =
+.87, p (one-tailed) < .05.

Creativity was negatively correlated with how well people did
in the World’s Biggest Liar Contest, rS = -.37, p (two-tailed) =
.001.
Correlation: Example
Correlations
WorkHours
WorkHours
Pearson Correlation
ExCurrHours
1
Sig. (2-tailed)
N
ExCurrHours
Pearson Correlation
Sig. (2-tailed)
N
-.313
.081
32
32
-.313
1
.081
32
32
Correlation: Example

Analysis using the Pearson’s r correlation
indicated that the there was moderately strong
negative relationship between the number of
work hours and the number of hours spent on
extracurricular activities, but the relationship was
not statistically significant, r = -.31, p (two-tailed)
= .08. The R2 = .097, indicating that the
relationship accounts for approximately 9.7% of
the variance in the number of hours spent in each
activity.
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