Transcript Chapter 9 Correlation and Regression

STATISTICS
Chapter 5
Correlation/Regression
MVS 250: V. Katch
1
Overview
Paired Data
 is there a relationship
 if so, what is the equation
 use the equation for prediction
2
Correlation
3
Definition
Correlation
exists between two variables
when one of them is related to
the other in some way
4
Assumptions
1. The sample of paired data (x,y) is a
random sample.
2. The pairs of (x,y) data have a
bivariate normal distribution.
5
Definition
Scatterplot (or scatter diagram)
is a graph in which the paired
(x,y) sample data are plotted with
a horizontal x axis and a vertical
y axis. Each individual (x,y) pair
is plotted as a single point.
6
Scatter Diagram of Paired Data
7
Scatter Diagram of Paired Data
8
Positive Linear Correlation
y
y
y
(a) Positive
x
x
x
(b) Strong
positive
(c) Perfect
positive
Scatter Plots
9
Negative Linear Correlation
y
y
y
(d) Negative
x
x
x
(e) Strong
negative
(f) Perfect
negative
Scatter Plots
10
No Linear Correlation
y
y
x
(g) No Correlation
x
(h) Nonlinear Correlation
Scatter Plots
11
Definition
Linear Correlation Coefficient r
measures strength of the linear relationship
between paired x and y values in a sample
xy/n - (x/n)(y/n)
r=
(SDx) (SDy)
Where xy/n is the mean of the cross products;
(x/n) is the mean of the x variable; (y/n) is the
mean of the y variable; SDx is the standard
deviation of the x variable and SDy is the
standard deviation of the x variable
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Notation for the
Linear Correlation Coefficient
n
number of pairs of data presented

denotes the addition of the items indicated.
x/n denotes the mean of all x values.
y/n denotes the mean of all y values.
xy/n denotes the mean of the cross products [x
times y, summed; divided by n]
r
linear correlation coefficient for a sample

linear correlation coefficient for a
population
13
Rounding the
Linear Correlation Coefficient r
 Round to three decimal places
Use calculator or computer if possible
14
Properties of the
Linear Correlation Coefficient r
1. -1  r  1
2. Value of r does not change if all values of
either variable are converted to a different
scale.
3. The r is not affected by the choice of x and y.
Interchange x and y and the value of r will
not
change.
4. r measures strength of a linear relationship.
15
Interpreting the Linear
Correlation Coefficient
If the absolute value of r exceeds the
value in Sig. Table, conclude that there is
a significant linear correlation.
Otherwise, there is not sufficient
evidence to support the conclusion of
significant linear correlation.
Remember to use n-2
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Common Errors Involving Correlation
1. Causation: It is wrong to conclude that
correlation implies causality.
2. Averages: Averages suppress individual
variation and may inflate the correlation
coefficient.
3. Linearity: There may be some relationship
between x and y even when there is no
significant linear correlation.
17
Common Errors Involving Correlation
250
Distance
(feet)
200
150
100
50
0
0
1
2
3
4
5
6
7
8
Time (seconds)
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Correlation is Not Causation
A
B
C
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Correlation Calculations
Rank Order Correlation - Rho
Pearson’s - r
20
Rank Order Correlation
Hits
1
Rank
10
HR
3
Rank
8
D
2
D2
4
2
3
4
5
9
8
7
6
4
5
1
7
7
6
10
4
2
2
-3
2
4
4
9
4
6
7
8
5
4
3
6
2
10
5
9
1
0
-5
2
0
25
4
9
10
2
1
9
8
2
3
0
2
0
4
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Rank Order Correlation, cont
Rho = 1- [6
2
(∑D )
Hits
Rank
HR
Rank
D
D2
1
10
3
8
2
4
2
9
4
7
2
4
3
8
5
6
2
4
4
7
1
10
-3
9
5
6
7
4
2
4
6
5
6
5
0
0
7
4
2
9
-5
25
8
3
10
1
2
4
9
2
9
2
0
0
10
1
8
3
2
4
N=10
/N
2
(N -1)]
Rho = 1- [6(58)/10(102-1)]
Rho = 1- [348 / 10 (100 -1)]
Rho = 1- [348 / 990]
Rho = 1- 0.352
Rho = 0.648
(∑D2 = 58)
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Pearson’s r
Hits
HR
xy
1
3
3
2
4
8
3
5
15
4
1
4
5
7
35
6
6
36
7
2
14
8
10
80
9
9
81
10
8
80
x/n x/n xy/n
=5.5 = 5.5 =32.86
xy/n - (x/n)(y/n)
r=
(SDx) (SDy)
r = 32.86 - (5.5) (5.5)/(3.03) (3.03)
r = 35.86 - 30.25 / 9.09
r = 5.61 / 9.09
r = 0.6172
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Pearson’s r
Excel Demonstration
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Is there a significant linear correlation?
Data from the Garbage Project
x Plastic (lb)
y Household
0.27 1.41
2
3
2.19
2.83
2.19
1.81
0.85
3.05
3
6
4
2
1
5
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Is there a significant linear correlation?
Data from the Garbage Project
x Plastic (lb)
y Household
0.27 1.41
2
2.19
2.83
2.19
1.81
0.85
3.05
3
6
4
2
1
5
3
P las tic
0 .2 7
1 .4 1
2 .1 9
2 .8 3
2 .1 9
1 .8 1
0 .8 5
3 .0 5
H ous ehold
2
3
3
6
4
2
1
5
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Is there a significant linear correlation?
Data from the Garbage Project
x Plastic (lb)
y Household
0.27 1.41
2
3
2.19
2.83
2.19
1.81
0.85
3.05
3
6
4
2
1
5
Household size
Plastic Garbage v Household size
7
6
5
4
3
2
1
0
r = 0.842
R2 2= 0.7096
R = 0.71
0
0.5
1
1.5
2
2.5
3
3.5
Plastic (lbs)
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Is there a significant linear correlation?
n=8
 = 0.05
=0
:  0
H 0:
H1
Test statistic is r = 0.842
Critical values are r = - 0.707 and 0.707
(Table R with n = 8 and  = 0.05)
TABLE R Critical Values of the Pearson Correlation Coefficient r
n
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
25
30
35
40
45
50
60
70
80
90
100
 = .05
.950
.878
.811
.754
.707
.666
.632
.602
.576
.553
.532
.514
.497
.482
.468
.456
.444
.396
.361
.335
.312
.294
.279
.254
.236
.220
.207
.196
 = .01
.999
.959
.917
.875
.834
.798
.765
.735
.708
.684
.661
.641
.623
.606
.590
.575
.561
.505
.463
.430
.402
.378
.361
.330
.305
.286
.269
.256
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Is there a significant linear correlation?
0.842 > 0.707, That is the test statistic does fall within the
critical region.
Therefore, we REJECT H0:  = 0 (no correlation) and conclude
there is a significant linear correlation between the weights of
discarded plastic and household size.
Reject
 =0
-1
r = - 0.707
Fail to reject
=0
0
Reject
 =0
r = 0.707
1
Sample data:
r = 0.842
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Method 1: Test Statistic is t
(follows format of earlier chapters)
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Formal Hypothesis Test
 To determine whether there is a
significant linear correlation
between two variables
 Two methods
 Both methods let H0:  =
(no significant linear correlation)
H1:  
(significant linear correlation)
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Method 2: Test Statistic is r
(uses fewer calculations)
Test statistic: r
Critical values: Refer to Table R
(no degrees of freedom)
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Method 2: Test Statistic is r
(uses fewer calculations)
Test statistic: r
Critical values: Refer to Table A-6
(no degrees of freedom)
Reject
 =0
-1
r = - 0.811
Fail to reject
=0
0
Reject
 =0
r = 0.811
1
Sample data:
r = 0.828
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Method 1: Test Statistic is t
(follows format of earlier chapters)
Test statistic:
t=
r
1-r2
n-2
Critical values:
use Table T with
degrees of freedom = n - 2
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Start
Testing for a
Linear Correlation
Let H0:  = 0
H1:   0
Select a
significance
level 
Calculate r using
Formula 9-1
METHOD 1
METHOD 2
The test statistic is
t=
The test statistic is
r
r
Critical values of t are from
Table A-6
1-r2
n -2
Critical values of t are from Table A-3
with n -2 degrees of freedom
If the absolute value of the
test statistic exceeds the
critical values, reject H0:  = 0
Otherwise fail to reject H0
If H0 is rejected conclude that there
is a significant linear correlation.
If you fail to reject H0, then there is
not sufficient evidence to conclude
that there is linear correlation.
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Why does the critical value of r
increase as sample size decreases?
A correlation by chance is more likely.
36
Coefficient of Determination
(Effect Size)
r2
The part of variance of one variable that can be
explained by the variance of a related variable.
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Justification for r Formula
r=
 (x -x) (y -y)
(n -1) Sx Sy
(x, y)
centroid of sample points
x=3
y
x - x = 7- 3 = 4
(7, 23)
•
24
20
y - y = 23 - 11 = 12
Quadrant 1
Quadrant 2
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•
12
8
•
Quadrant 3
••
4
y = 11
(x, y)
Quadrant 4
x
0
0
1
2
3
4
5
6
7
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