agenda correlation. correlation co-relation 2 variables tend to “go together” does knowing a...
TRANSCRIPT
Agenda• Correlation
CORRELATION
• Co-relation
• 2 variables tend to “go together”
• Does knowing a person’s score on one variable give you an idea of their score on other variable?
• Predict the degree of co-occurrence or association between 2 variables
Correlation• A measure of association between
– Two ordinal variables– An ordinal and an interval/ratio variables– Two variables
• Correlation analysis examines if one variable changes by a certain amount, how much the other variable would change in which direction
Example: Test Score Ten students
took a Chemistry class and a Biology class together
Compared final exam scores in two classes
J
I
H
G
F
E
D
C
B
A
9894
9390
9089
8388
8580
8378
8276
6972
6870
6562
BiologyChemistry
Example: Scatter Plot
60
65
70
75
80
85
90
95
100
60 65 70 75 80 85 90 95 100
Chemistry
Bio
logy
Students who get higher score in the Chemistry class also get higher score in the Biology class
Positive Correlation• When scores of two variables move
together in the same direction, we say that these variables are positively (or directly) correlated
• There is a positive correlation between between the chemistry final score and the biology final score
• When two variables are positively correlated, the scatter plot shows a trend line that runs from lower-left to upper-right
Example: Test Score The same
students also took the Art class
Compared final exam scores in Chemistry and Art
J
I
H
G
F
E
D
C
B
A
7194
7390
7489
7588
7780
7778
7876
7472
8070
9062
ArtChemistry
Example: Scatter Plot
60
65
70
75
80
85
90
95
60 65 70 75 80 85 90 95 100
Chemistry
Art
Students who get higher score in the Chemistry class get lower score in the Art Classc
Negative Correlation• When scores of two variables move in
opposite directions, we say that these variables are negatively (or inversely) correlated
• There is a negative correlation between between the chemistry final score and the art final score
• When two variables are negatively correlated, the scatter plot shows a trend line that runs from upper-left to lower-right
Example: Test Score The same ten
students also took an English class together
Compare the English final score with the Chemistry final score
J
I
H
G
F
E
D
C
B
A
7694
8990
6089
7588
9080
8078
7576
9072
6070
8562
EnglishChemistry
Example: Test Score
60
65
70
75
80
85
90
95
60 65 70 75 80 85 90 95 100
Chemistry
Engl
ish
Score in Chemistry and Score in English are not related
No Correlation• When the change in one variable does not
affect the change in another variable, we say these variables have no correlation
• There is no correlation between the chemistry final score and the English final score
• When two variables have no correlation, the scatter plot shows the dots scattered throughout the grids
SIGN
• 0: No relationship
• Positive: + • As one variable gets bigger, so does
the 2nd
• Negative: -• As one variable gets
bigger, the 2nd gets smaller
Should your height in inches have anything to do with….• How much you weigh?• What size shoe your wear?• How old you are?• What portion of your tuition you pay?• Your college GPA?• Your skill in basketball?• Your attractiveness to the opposite sex?• Your eye color?
TASK 1• Get in groups of 8-10
• Line up in order of height
• Name Height
• Line up in order of shoe size (men, add 2 sizes)
• Name Shoe size
• Line up in order of how much $ you spent the last time you bought something
• Name $ spent
• Where any of the 3 orders similar?
Do they go together?
• Draw a scatterplot• Each person has 2 scores, 1 on each variable
• Plot each person’s score
• Is there a pattern? E.g. as one variable gets bigger, does the other get bigger too? (Or smaller?)
• A strong relationship makes a diagonal line
Shoe size by height
Height in inches
80706050
Sh
oe
size
(w
om
en
's s
ize
s)16
14
12
10
8
6
4
Weight by height
Height in inches
80706050
We
igh
t in
po
un
ds
220
200
180
160
140
120
100
80
GPA by height
Height in inches
80706050
Gra
de
po
int
ave
rag
e4.5
4.0
3.5
3.0
2.5
2.0
Height in CM by Height in Inches
Height in inches
80706050
He
igh
t in
ce
ntim
ete
rs200
190
180
170
160
150
140
Draw a line through the dots
Height in Cm by Height in Inches
60.00 65.00 70.00 75.00
Height in inches
150.00
160.00
170.00
180.00
190.00
Hei
gh
t in
cen
tim
eter
s
Shoe size by Height
60.00 65.00 70.00 75.00
Height in inches
6.0
8.0
10.0
12.0
14.0S
ho
esiz
e (w
om
en's
siz
es)
GPA by Height
60.00 65.00 70.00 75.00
Height in inches
2.50
3.00
3.50
4.00
Gra
de
po
int
aver
age
Scatter of midterm scores by time to complete test
Fall 2002
Midterm Score
706050403020
# of Minutes to complete test
70
60
50
40
30
20
r = - .4p = .01
Fall 2002
Midterm score
Minutes
Correlation between midterm score and time to finish test
CorrelationsMidterm Score Min. to finish test
Midterm Score Pearson Correlation 1 -0.4Sig. (2-tailed) . 0.01N 40 40
Min. to complete testPearson Correlation -0.4 1Sig. (2-tailed) 0.01 .N 40 40
* Correlation is significant at the 0.05 level (2-tailed).
Fall 2002
Correlation Co-efficient
• A number that indicates how strongly and in which direction 2 variables are correlated with each other
• A correlation co-efficient varies –1 to +1
• Indicated as r
• r = +1: Perfect positive correlation– If one variable increases by x%, another
variable also increases by x%
• r = - 1: Perfect negative correlation
• r = 0: No correlation
Correlation Co-efficient
+1-1 0Negative Positive
Stronger StrongerWeaker
Perfect PerfectNone
Significance Test• Correlation co-efficient also comes with
significance test (p-value)
• p=.05: .05 probability of no correlation in the population = 5% risk of TYPE I Error = 95% confidence level
• If p<.05, reject H0 and support Ha at 95% confidence level
Chemistry & BiologyChemistry Biology
A 62 65
B 70 68
C 72 69
D 76 82
E 78 83
F 80 85
G 88 83
H 89 90
I 90 93
J 94 98
r = .939 p = .000
Significant, Positive and
Strong Correlation
Chemistry & ArtChemistry Art
A 62 90
B 70 80
C 72 74
D 76 78
E 78 77
F 80 77
G 88 75
H 89 74
I 90 73
J 94 71
r = - .839 p = .002
Significant, Negative
and Strong Correlation
Chemistry & EnglishChemistry English
A 62 85
B 70 60
C 72 90
D 76 75
E 78 80
F 80 90
G 88 75
H 89 60
I 90 89
J 94 76
r = -.133 p = .714
Non-Significant
Correlation
SIZE / STRENGTH• Ranges from –1 to + 1• 0 or close to 0 indicates NO relationship• +/- .2 - . 39 weak • +/- .4 - .6 moderate• +/- .>6 - .8 strong• +/- .>8 - .9 very strong• +/- 1.00 perfectNegative relationships are NOT weaker!
USE THIS FOR YOUR ANALYSIS!
Types of Correlation r
• Use Spearman rho’s correlation if one or both of your variables are ordinal
• Use Pearson’s r correlation if both of your variables are interval or ratio
• You can interpret both kinds of correlation in the same way
STATISTICAL SIGNIFICANCE
• How different from 0 must r be for there to be some kind of relationship?
• Depends on size of sample, other factors
• IF statistically significant, safe to conclude there is a relationship
• If p < .05– May also be indicated by * next to the correlation
Limitations of r (correlation)
• Correlation does NOT equal cause
C
A B
• Linear relations only
A B
B A “Third variable”
?
?
“Problem of direction”
TASK 2 with your neighbor:
• Write down 3 examples.
1. 2 variables expect to have no association.
2. 2 variables expect to have positive association.
3. 2 variables expect to have negative association.
• Write down 2 examples.
1. 2 variables expect a weak to medium association.
2. 2 variables expect a strong or very strong association.
TASK: Graph job aptitude vs. performance Graph self-esteem vs. depression
100520John
91420Jake
84621Milly
83822Homer
67725Leslie
55827June
09829Bob
181028Jane
210930Joe
DepressionSelf-EsteemPerformance
Job AptitudeName
Job Aptitude by Performance
Supervisor's performance rating
11109876543
Job
ap
titu
de
te
st s
core
32
30
28
26
24
22
20
18
r = .84, p < .005
Self-Esteem by Depression
Depressive symptoms
121086420-2
Se
lf-e
ste
em
12
10
8
6
4
2
0
-2
r = -.92, p < .001
Example from class data
Height in inches, weight, age, GPA, # semesters completed, height in cm,
who pays tuition (high score = self), where sit in class (high score = towards front)
• For which pairs do you expect no relation? Positive? Negative? Strongest? Weakest?
1.00 .032-.021 .725 * .544 *1.00 *-.012Hgt cm
1.00-.018 .058 .144 .032 .234*Sems
1.00-.050-.115-.021 .075GPA
1.00 .724 * .725 * .016Shoe
1.00 .544 * .108Wt.
1.00-.012Ht.
1.00Age
Hgt cm
Sems
compGPAShoeWt.Ht. Age