Transcript No Slide Title

USES OF CORRELATION
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•
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Test reliability
Test validity
Predict future scores (regression)
Test hypotheses about relationships between
variables
Doing Correlational Research
• Any of the descriptive methods can be used
(observation, survey, archival, physical traces).
• Must have pairs of scores
Doing Correlational Research
• The pairs of scores should be independent.
• Both variables should be normally
distributed.
• If there is a relationship between variables, it
should be linear.
Correlation and Cause
• A correlation by itself does not show that one
variable causes the other.
• A correlation is consistent with a causal
relationship.
The Third Variable Problem
• A correlation between X and Y could be
caused by a third variable influencing both X
and Y.
• example: The use birth control is correlated
with the number of electrical appliances in the
household.
The Directionality Problem
• A correlation between X and Y could be a
result of X causing Y or Y causing X
• example: Amount of TV watching and the
level of aggression are correlated.
The Cross-Lagged Panel
• Design used to determine direction of cause
• Measure both variables at two different
points in time
• Cause cannot work backwards in time
Time 1
variable X
Time 2
variable X
variable Y
variable Y
The Correlation Coefficient
• Strength of relationship
– 0 means no relationship at all
– -1 or +1 means perfectly related
• Direction of relationship
– positive: variable X increases as variable Y
increases
– negative: variable X decreases as variable Y
increases
The Scatterplot
high
oo
oo
oo ooo o
oo o
oo o
oo o
o o
Y
low
positive r
high
X
high
Y
o
o oo
o o
o oo oo
oo
o
oo
o
o o
o o
o
o oo
low
negative r
high
X
high
o
o
o o
o
o
o
zero r
o
o
o
oo o o o
o o
o o o
o
o
o
o o o
o
Y
low
high
X
Non-linear
relationship
high
Y
o
o oo
o o
o oo
o
o
o
o
o
o
low
o
o o
o o
o
o
o
o
o
o o
high
X
Correlation Coefficients
X data
Y data
Coefficient
interval/ratio
ordinal
dichotomous
dichotomous
interval/ratio
ordinal
interval/ratio
dichotomous
Pearson r
Spearman rho
Point Biserial
Phi
dichotomous: having only two values
More on Dichotomous Variables
• With dichotomous variables, whether r is
negative or positive depends on how the
numbers were assigned
More on Dichotomous Variables
• If the correlation between gender and GPA is
positive, it could mean that
– females have higher GPAs, if males were 1’s and
females were 2’s
– males have higher GPAs, if females were 1’s and
males were 2’s
Pearson r formula
z x zy
r=
N
z x = z - score on x
z y = z - score on y
N = # of individuals
Computation of Pearson r
Example: Compute the correlation
between scores on Exam1 and Exam2.
Student Exam1 Exam2
1
97
86
2
82
95
3
74
79
4
89
95
5
93
90
STEP 1:Convert the x scores to z-scores.
Exam1
97
82
74
89
93
m=87
x-m
10
-5
-13
2
6
(x-m)2
100
25
169
4
36
zx
+1.22
-.61
-1.59
+.24
+.73
=334
sx = 334/5 = 8.17
STEP 2:Convert the y scores to z-scores.
Exam2
86
95
79
95
90
m=89
y-m
-3
6
-10
6
1
(y-m)2
9
36
100
36
1
zy
-.50
+1.00
-1.66
+1.00
+.17
=182
sy = 182/5 = 6.03
STEP 3: Multiply the z-scores.
zx
+1.22
-.61
-1.59
+.24
+.73
zy
-.50
+1.00
-1.66
+1.00
+.17
z xz y
-.61
-.61
+2.64
+.24
+.12
STEP 4: Add up the zxzy products.
z xz y
-.61
-.61
+2.64
+.24
+.12
 = 1.78
STEP 5: Divide zxzy by N.
1.78
r=
 .36
5
Coefficient of Determination
• Measures proportion of explained variance in
Y based on X.
• Square r to get r2.
Example: r = .36
r2 = .13
We can explain 13% of the differences in
Exam 2 scores by knowing Exam 1 scores.
What Could a Low r Mean?
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Lack of a relationship.
Unreliable measurement.
Non-linear relationship.
Restricted range : full range of scores not
measured on both variables