Chapter Seventeen
Correlation and Regression
172
Chapter Outline
1) Overview
2) ProductMoment Correlation
3) Partial Correlation
4) Nonmetric Correlation
5) Regression Analysis
6) Bivariate Regression
7) Statistics Associated with Bivariate Regression
Analysis
8) Conducting Bivariate Regression Analysis
i.
Scatter Diagram
ii.
Bivariate Regression Model
173
Chapter Outline
iii.
Estimation of Parameters
iv.
Standardized Regression Coefficient
v.
Significance Testing
vi.
Strength and Significance of Association
vii.
Prediction Accuracy
viii.
Assumptions
9)
Multiple Regression
10)
Statistics Associated with Multiple Regression
11)
Conducting Multiple Regression
i.
Partial Regression Coefficients
ii.
Strength of Association
iii.
Significance Testing
iv.
Examination of Residuals
174
Chapter Outline
12) Stepwise Regression
13) Multicollinearity
14) Relative Importance of Predictors
15) Cross Validation
16) Regression with Dummy Variables
17) Analysis of Variance and Covariance with
Regression
18) Internet and Computer Applications
19) Focus on Burke
20) Summary
21) Key Terms and Concepts
175
Product Moment Correlation
The
product moment correlation
,
r
,
summarizes the strength of association
between two metric (interval or ratio scaled)
variables, say
X
and
Y
.
It is an index used to determine whether a
linear or straightline relationship exists
between
X
and
Y
.
As it was originally proposed by Karl Pearson, it
is also known as the
Pearson correlation
coefficient
.
It is also referred to as
simple
correlation
,
bivariate correlation
, or merely the
correlation coefficient
.
176
From a sample of
n
observations,
X
and
Y
, the
product moment correlation,
r
, can be
calculated as:
r
=
(
X
i

X
)(
Y
i

Y
)
i
=1
n
(
X
i

X
)
2
i
=1
n
(
Y
i

Y
)
2
i
=1
n
Division of the numerator and denominator by (
n
1) gives
r
=
(
X
i

X
)(
Y
i

Y
)
n
1
i
=1
n
(
X
i

X
)
2
n
1
i
=1
n
(
Y
i

Y
)
2
n
1
i
=1
n
=
COV
xy
S
x
S
y
Product Moment Correlation
177
Product Moment Correlation
r
varies between 1.0 and +1.0.
The correlation coefficient between two
variables will be the same regardless of their
underlying units of measurement.
178
Explaining Attitude Toward the
City of Residence
Table 17.1
Respondent No Attitude Toward
the City
Duration of
Residence
Importance
Attached to
Weather
1
6
10
3
2
9
12
11
3
8
12
4
4
3
4
1
5
10
12
11
6
4
6
1
7
5
8
7
8
2
2
4
9
11
18
8
10
9
9
10
11
10
17
8
12
2
2
5
179
Product Moment Correlation
The correlation coefficient may be calculated as follows:
X
= (10 + 12 + 12 + 4 + 12 + 6 + 8 + 2 + 18 + 9 + 17 + 2)/12
= 9.333
Y
= (6 + 9 + 8 + 3 + 10 + 4 + 5 + 2 + 11 + 9 + 10 + 2)/12
= 6.583
(
X
i

X
)(
Y
i

Y
)
i
=1
n
= (10 9.33)(66.58) + (129.33)(96.58)
+ (129.33)(86.58) + (49.33)(36.58)
+ (129.33)(106.58) + (69.33)(46.58)
+ (89.33)(56.58) + (29.33) (26.58)
+ (189.33)(116.58) + (99.33)(96.58)
+ (179.33)(106.58) + (29.33)(26.58)
= 0.3886 + 6.4614 + 3.7914 + 19.0814
+ 9.1314 + 8.5914 + 2.1014 + 33.5714
+ 38.3214  0.7986 + 26.2314 + 33.5714
= 179.6668
1710
Product Moment Correlation
(
X
i

X
)
2
i
=1
n
= (109.33)
2
+ (129.33)
2
+ (129.33)
2
+ (49.33)
2
+ (129.33)
2
+ (69.33)
2
+ (89.33)
2
+ (29.33)
2
+ (189.33)
2
+ (99.33)
2
+ (179.33)
2
+ (29.33)
2
= 0.4489 + 7.1289 + 7.1289 + 28.4089
+ 7.1289+ 11.0889 + 1.7689 + 53.7289
+ 75.1689 + 0.1089 + 58.8289 + 53.7289
= 304.6668
(
Y
i

Y
)
2
i
=1
n
= (66.58)
2
+ (96.58)
2
+ (86.58)
2
+ (36.58)
2
+ (106.58)
2
+ (46.58)
2
+ (56.58)
2
+ (26.58)
2
+ (116.58)
2
+ (96.58)
2
+ (106.58)
2
+ (26.58)
2
= 0.3364 + 5.8564 + 2.0164 + 12.8164
+ 11.6964 + 6.6564 + 2.4964 + 20.9764
+ 19.5364 + 5.8564 + 11.6964 + 20.9764
= 120.9168
Thus,
r
=
179.6668
(304.6668) (120.9168)
=
0.9361
1711
Decomposition of the Total
Variation
r
2
=
Explained variation
Total variation
=
SS
x
SS
y
=
Total variation  Error variation
Total variation
=
SS
y

SS
error
SS
y
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 Fall '19