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Two variable linear regressionmodel
Dr. Aziz JavedDepatt: of Business Administration
Gomal University Dera Ismail Khan KPK
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Chapter 02 topics are.
The nature of regression analysis and correlationSimple, rank and partial correlation co0efficient
Types and sources of data
The concepts of linearity of the regression model
The significance of the error termThe model of Ordinary Least Squares (OLS)
The assumptions of the linear regression model
Least square estimators and their properties
Gauss Markov Theorem
The co-efficient of determination
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Correlation
Finding the relationship between two
quantitative variables without being able to
infer causal relationships
Correlationis a statistical technique used to
determine the degree to which two
variables are related
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Rectangular coordinate
Two quantitative variables
One variable is called independent (X) and
the second is called dependent (Y)
Points are not joined
No frequency table
Scatter diagram
Y
* *
*
X
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Wt.
(kg)
67 69 85 83 74 81 97 92 114 85
SBP
mHg)
120 125 140 160 130 180 150 140 200 130
Example
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Scatter diagram of weight and systolic blood
pressure
80
100
120
140
160
180
200
220
60 70 80 90 100 110 120wt (kg)
SBP(mmHg)Wt.
(kg)
67 69 85 83 74 81 97 92 114 85
SBP
mHg)
120 125 140 160 130 180 150 140 200 130
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80
100
120
140
160
180
200
220
60 70 80 90 100 110 120Wt (kg)
SBP(mmHg)
Scatter diagram of weight and systolic blood pressure
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Scatter plots
The pattern of data is indicative of the type of
relationship between your two variables:
positive relationship
negative relationship
no relationship
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Positive relationship
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0
2
4
6
8
10
12
14
16
18
0 10 20 30 40 50 60 70 80 90
Age in Weeks
Heightin
CM
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Negative relationship
Reliability
Age of Car
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No relation
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Correlation Coefficient
Statistic showing the degree of relation
between two variables
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Simple Correlation coefficient (r)
It is also called Pearson's correlation
or product moment correlation
coefficient.
It measures the natureand strength
between two variables of
the quantitativetype.
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The signof rdenotes the nature of
association
while the valueof rdenotes the
strength of association.
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If the sign is +ve this means the relationis direct (an increase in one variable isassociated with an increase in theother variable and a decrease in onevariable is associated with a
decrease in the other variable).
While if the sign is -ve this means an
inverse or indirect relationship (whichmeans an increase in one variable isassociated with a decrease in the other).
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The value of r ranges between ( -1) and ( +1)
The value of r denotes the strength of theassociation as illustrated
by the following diagram.
-110
-0.25-0.75 0.750.25
strong strongintermediate intermediateweak weak
no relation
perfect
correlation
perfect
correlation
Directindirect
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If r= Zero this means no association orcorrelation between the two variables.
If 0 < r< 0.25= weak correlation.
If 0.25 r< 0.75= intermediate correlation.
If 0.75 r< 1= strong correlation.
If r = l= perfect correlation.
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n
y)(y.
n
x)(x
n
yxxy
r2
2
2
2
How to compute the simple correlation
coefficient (r)
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Example:
A sample of 6 children was selected, data about their
age in years and weight in kilograms was recorded as
shown in the following table . It is required to find the
correlation between age and weight.
Weight
(Kg)
Age
(years)
serial
No
1271
862
1283
1054
1165
1396
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These 2 variables are of the quantitative type, one
variable (Age) is called the independent and
denoted as (X) variable and the other (weight)is called the dependent and denoted as (Y)
variables to find the relation between age and
weight compute the simple correlation coefficient
using the following formula:
n
y)(y.
n
x)(x
n
yxxy
r2
22
2
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Y2X2xy
Weight
(Kg)
(y)
Age
(years)
(x)
Serial
n.
14449841271
643648862
14464961283
10025501054
12136661165
169811171396
y2=
742
x2=
291
xy=
461
y=
66
x=
41
Total
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r = 0.759
strong direct correlation
6
(66)742.
6
(41)291
6
6641461
r22
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EXAMPLE: Relationship between Anxiety and
Test Scores
Anxiety
(X)
Test
score (Y)X2 Y2 XY
10 2 100 4 20
8 3 64 9 242 9 4 81 18
1 7 1 49 7
5 6 25 36 306 5 36 25 30
X = 32 Y = 32 X2= 230 Y2= 204 XY=129
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Calculating Correlation Coefficient
94.
)200)(356(
1024774
32)204(632)230(6
)32)(32()129)(6(22
r
r = - 0.94
Indirect strong correlation
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Spearman Rank Correlat ion Coeff ic ient
(rs)
It is a non-parametric measure of correlation.
This procedure makes use of the two sets ofranks that may be assigned to the sample
values of x and Y.Spearman Rank correlation coefficient could becomputed in the following cases:
Both variables are quantitative.
Both variables are qualitative ordinal.
One variable is quantitative and the other isqualitative ordinal.
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Procedure:
1. Rank the values of X from 1 to n where nis the numbers of pairs of values of X and
Y in the sample.
2.
Rank the values of Y from 1 to n.3. Compute the value of di for each pair of
observation by subtracting the rank of Yi
from the rank of Xi
4. Square each di and compute di2 which
is the sum of the squared values.
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5. Apply the following formula
1)n(n
(di)61r2
2
s
The value of rsdenotes the magnitude
and nature of association giving the same
interpretation as simple r.
E l
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Example
In a study of the relationship between leveleducation and income the following data was
obtained. Find the relationship between themand comment.
Income
(Y)
level education
(X)
sample
numbers
25Preparatory.A
10Primary.B
8University.C
10secondaryD15secondaryE
50illiterateF
60University.G
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Answer:
di2diRank
Y
Rank
X(Y)(X)423525PreparatoryA
0.250.55.5610Primary.B
30.25-5.571.58University.C
4-25.53.510secondaryD
0.25-0.543.515secondaryE
2552750illiterateF
0.250.511.560university.G
di2=64
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Comment:
There is an indirect weak correlation
between level of education and income.
1.0)48(7
646
1
sr
i
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exercise
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Regression Analyses
Regression: technique concerned with predictingsome variables by knowing others
The process of predicting variable Y usingvariable X
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Regression
Uses a variable (x) to predict some outcome
variable (y)
Tells you how values in y change as a function
of changes in values of x
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Correlation and Regression
Correlation describes the strength of a linear
relationship between two variables
Linearmeans straight line
Regressiontells us how to draw the straight line
described by the correlation
Regression
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Regression
Calculates the best-fit line for a certain set of data
The regression line makes the sum of the squares of
the residuals smaller than for any other lineRegression minimizes residuals
80
100
120
140
160
180
200
220
60 70 80 90 100 110 120Wt (kg)
SBP(mmHg)
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By using the least squares method (a procedure
that minimizes the vertical deviations of plotted
points surrounding a straight line) we are
able to construct a best fitting straight line to the
scatter diagram points and then formulate a
regression equation in the form of:
n
x)(x
n
yxxy
b2
2
1)xb(xyy b
bXay
R i E ti
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Regression Equation
Regression equation
describes the
regression line
mathematically
Intercept Slope
80
100
120
140
160
180
200
220
60 70 80 90 100 110 120Wt (kg)
SBP(mmHg)
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Linear Equations
Y
Y = bX + a
a = Y-intercept
X
Changein Y
Change in X
b = Slope
bXay
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Hours studying and grades
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Regressing grades on hours
Linear Regression
2.00 4.00 6.00 8.00 10.00
Number of hours spent studying
70.00
80.00
90.00
Finalgr
ade
in
course
Final grade in course = 59.95 + 3.17 * study
R-Square = 0.88
Predicted final grade in class =
59.95 + 3.17*(number of hours you study per week)
Predicted final grade in class = 59.95 + 3.17*(hours of study)
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Predict the final grade of
Someone who studies for 12 hours
Final grade = 59.95 + (3.17*12)
Final grade = 97.99
Someone who studies for 1 hour:
Final grade = 59.95 + (3.17*1)
Final grade = 63.12
g ( y)
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Weight (y)Age (x)Serial no.
12
8
1210
11
13
7
6
85
6
9
1
2
34
5
6
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Answer
Y2X2xyWeight (y)Age (x)Serial no.
144
64
144100
121
169
49
36
6425
36
81
84
48
9650
66
117
12
8
1210
11
13
7
6
85
6
9
1
2
34
5
6
7422914616641Total
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6.836
41x 11
6
66y
92.0
6
)41(
291
6
6641461
2
b
Regression equation
6.83)0.9(x11y (x)
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0.92x4.675y (x)
12.50Kg8.5*0.924.675y(8.5)
Kg58.117.5*0.924.675y(7.5)
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11.4
11.6
11.8
12
12.2
12.4
12.6
7 7.5 8 8.5 9
Age (in years)
Weight(i
nKg)
we create a regression line by plotting two
estimated values for y against their X component,
then extending the line right and left.
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Exercise 2
The following are theage (in years) andsystolic bloodpressure of 20
apparently healthyadults.
B.P
(y)
Age
(x)
B.P
(y)
Age
(x)
128136
146
124
143
130
124
121126
123
4653
60
20
63
43
26
1931
23
120128
141
126
134
128
136
132140
144
2043
63
26
53
31
58
4658
70
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Find the correlation between ageand blood pressure using simple
and Spearman's correlationcoefficients, and comment.
Find the regression equation?
What is the predicted bloodpressure for a man aging 25 years?
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x2xyyxSerial
4002400120201
1849550412843239698883141633
6763276126264
28097102134535
9613968128316
33647888136587
21166072132468
336481201405894900100801447010
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x2xyyxSerial
211658881284611
280972081365312360087601466013
40024801242014
396990091436315184955901304316
67632241242617
3612299121191896139061263119
52928291232320
416781144862630852Total
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n
x)(x
n
yxxy
b2
21
4547.0
2085241678
20
2630852114486
2
=
=112.13 + 0.4547 x
for age 25
B.P = 112.13 + 0.4547 * 25=123.49 = 123.5 mm hg
y
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Multiple Regression
Multiple regression analysis is a
straightforward extension of simple
regression analysis which allows more
than one independent variable.
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