Correlation - Coe College
Download
Report
Transcript 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.