Download Chapter 9 : Linear Correlation

Correlation
Correlational Research
Correlational research: describes the relationship between
two or more naturally occurring variables.
– Is age related to political conservativism?
– Are highly extraverted people less afraid of rejection
than less extraverted people?
– Is depression correlated with hypochondriasis?
– Is I.Q. related to reaction time?
measure two variables and determine whether
there is a relationship present
predictor <-> criterion
No causality because:
direction: there is no way to tell which is the cause or
the effect
third variable problem: some third variable that was not
measured could be responsible for the relationship.
?
Dr. Dimwit
?
speeding
ticket
red car
?
midnight
basketball
healthier
vitamins
?
larger feet
less
crime
reading
skills
3rd variable
problem
Estimate r
for each case:
No relationship
Positive linear
relationship
Curvilinear
Relationship:
(Linear corr.
not appropriate)
Negative linear
relationship
• Correlation coefficient (r)
+1.00 perfect positive correlation;
-1.00 perfect negative correlation;
0 lack of correlation
ABS|r| = magnitude of relationship
sign (r) direction of relationship
r2 = % variance of Y explained by X
Types of correlation coefficients
Pearson’s correlation coefficient: linear
relationship between two interval / ratio
variables.
Spearman’s rank-order correlation: linear
relationship between two variables measured
using ordinal (ranked) scores.
Point-biserial correlation: linear relationship
between the scores from one continuous
variable and one dichotomous (0 or 1) variable.
Conceptual Formula for
Pearson’s correlation:
Xi  X
zx 
sx
r
 zx z y
N
Positive r
[z’s from x and y same sign]
<-Neg zy | Pos zy->
<-Neg zy | Pos zy->
Negative r
[z’s from x and y different sign]
<-Neg zx | Pos zx->
<-Neg zx | Pos zx->
Example: Is education about other ethnicities correlated
with tolerant attitudes towards others?
Education Tolerance Zx
Zy
ZxZy
Score
Score
25
3
-1.05 -2.38 2.50
25
9
-1.05 -.62
.65
33
14
.24
.85
.20
35
11
.56
-.03 -.02
38
13
1.05 .56
.59
36
14
.72
.85
.61
31
12
-.08
.26
-.02
29
12
-.40
.26
-.10
22
9
-1.53 -.62
.95
41
14
1.53 .85 1.30
315
111
6.66
Xi  X
zx 
sx
Yi  Y
zy 
sy
s
 X
i
N 1
X  31.5 Y  11.1
Sx = 6.22
Sy = 3.41
X

2
N
6.66

10
 .67
14.00
Tolerance Score
r
 zx z y
16.00
12.00
10.00
8.00
6.00
4.00
2.00
0.00
0.00
10.00
20.00
30.00
40.00
Education Score
50.00
Could this be (1) due to chance, such as random error, or
(2) very UNLIKELY to occur due to chance (< 5%)?
Inferential statistics are needed.
Testing Pearson’s r for significance
Ho: ρ = 0
Ha: ρ ≠ 0
x<->y association does not exist
x<->y association exists (non-directional)
Using the t distribution:
t 
N  2r
1 r 2
Using the table of critical values
df = N – 2 (N is the number of pairs of scores)
= 10 – 2 = 8
Using a t-table
ha: an association exists between education & tolerance (two-tailed)
alpha = .05
df = N – 2
10-2 = 8
df
p=
0.05
1
12.71
2
3
4
5
6
7
8
9
10
11
12
13
14
4.30
3.18
2.78
2.57
2.45
2.36
2.31
2.26
2.23
2.20
2.18
2.16
2.14
p=
0.01
63.6
6
9.92
5.84
4.60
4.03
3.71
3.50
3.36
3.25
3.17
3.11
3.05
3.01
2.98
If t > 2.31,
reject ho,
left with ha.
If t <= 2.31,
retain ho
Hypotheses
Directional hypothesis – ha states whether the
correlation is expected to be positive or negative
(one-tailed test appropriate).
Nondirectional hypothesis – ha states that there
is an association, but does not specify the
direction (two-tailed test appropriate).
df = 60
alpha = .05
t = -2.0
t = +2.0
t = -1.67
Our example
t
tr = 3.96
tcrit
df = 8 =
N  2r
1 r
2
t
10  2  .67
1  .67
df = 10 – 2 = 8
2.31
APA Style: r(N) = value obtained, p = .##
r(10) = .67, p = .002
2
Hypothesis Testing
Rejecting the null hypothesis –concluding that the null
hypothesis is wrong.
Leaving us with the alternative hypothesis (ha) that there
is an association between predictor and criterion
Failing to reject the null hypothesis –concluding the null
hypothesis (no association) is a likely possibility.
We do not “accept” the null hypothesis (h0) , because
the null hypothesis can never be proven.
Errors
• Type I error – a researcher rejects the null
hypothesis when it is true (a false positive)
– Alpha –probability of Type I error
(most commonly p = .05).
• Type II error – a researcher fails to reject the
null hypothesis when it is false (a false
negative)
– Beta – the probability of Type II error
(most commonly beta = .20).
Statistical Decisions and
Outcomes
Reality (unknown)
Statistical Decision
Reject null
Fail to reject
hypothesis
(we retain) null
Correct:
Type II Error ():
Null hypothesis
false
Correlation exists Incorrectly conclude
No correlation
Null hypothesis Type I Error ():
Incorrectly conclude
true
there is a correlation
Correct:
Correlation does
not exist
Power
• Power is the probability that a study will
detect effects that are really present
(correctly reject the null hypothesis).
• Power = 1-beta. Typically set at .80, or
80% chance of observing an effect when
present.
• Power analysis is used to decide how
many participants are needed to detect a
significant effect, since increasing
participants increases power.
Power Table:
required n (rows) and r (columns)
n
.10
.20
.30
.40
.50
.60
.70
.80
.90
15
.06
.11
.19
.32
.50
.70
.88
.98
>.995
30
.08
.16
.37
.61
.83
.95
>.995
50
.11
.29
.57
.83
.97
>.995
100
.17
.52
.86
.99
>.995
200
.29
.81
.99
>.995
1000
.89
>.995
Power has a direct impact on likelihood of success and is often required for
Masters and Dissertation proposals and fellowship and grant applications.
Know your power, use your power!
Effect Size
Effect size: how strongly variables are related to
eachother.
Coefficient of determination (r2): the proportion of
variability in the DV that is due to the IV
(Range: .00 to 1.00). One indicator of effect size.
r2 =.672 = .45
45% of variance in the criterion (tolerance) is
explained by the predictor (education)
Limitations
• Pearson’s r only measures the degree of linear
correlation
• Problems in generalizing from sample
correlations
– Restricted or truncated ranges (results in smaller
value)
– Bivariate outliers
RESTRICTION OF RANGE
Full Range. r = .60
Restricted range, r = .20
Restriction of range
often decreases r
Marital Satisfaction
Marital Satisfaction Over Time
Wife
Husband
1
2
3
4
5
6
7
8
Years of Marriage
9
10
Marital Satisfaction Over Time
m
en
ire
R
et
y
pt
Em
Years of Marriage
t
t
N
es
lt
A
du
ng
en
t
Yo
u
A
do
le
sc
ho
ol
Sc
ol
ch
o
Pr
es
fa
nt
In
N
o
C
hi
ld
Marital Satisfaction
Previous slide data showed
restriction of range!
Outliers
• An outlier is a score that is so deviant from
the data that one can question whether it
belongs in the data set.
• > + / - 3 SD from the mean.
• On-line outliers fall in the same pattern as the
rest of the data artificially inflating r.
• Off-line outliers fall outside of the pattern of
the rest of the data artificially deflating r.
IMPACT OF OUTLIERS ON CORRELATION
On-line outlier
…
..
Off-line outlier
Assumptions of the significance test
Independent random sampling
Normal distribution (and bivariate normal
distribution)
Interval or ratio scale variables
SPSS
Pearson’s r:
Analyze → correlate → bivariate correlations
Select variable you wish to correlate and
place them in box
Make sure Pearson’s is checked
Choose one/two tailed
OK
SPSS
Scatter Plot:
GraphLegacy DialoguesScatter/Dot
Select Simple Scatter
Select variables for X and Y axis
Ok
Note: Select 3D Scatter to look at bivariate
normal assumption for r.
END
Null Hypothesis Testing and
Inferential Statistics
Partitioning of Variance
Systematic variance: the portion of the participant’s score (e.g.,
behavior) that is related to variables within the study
Error variance: the portion of the participant’s score that is
unaccounted for by variables within the study
total variance = systematic variance + error variance
t-test, F-test, r, β based on:
systematic variance
error variance