**Unformatted text preview: **بسم الله الرحمن
الرحيم Correlation &
Regression
Dr. Moataza Mahmoud Abdel Wahab
Lecturer of Biostatistics
High Institute of Public Health
University of Alexandria Correlation
Finding the relationship between two
quantitative variables without being
able to infer causal relationships
Correlation is a statistical technique
used to determine the degree to
which two variables are related Scatter diagram
• 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 Example Scatter diagram of weight and systolic blood
pressure Scatter diagram of weight and systolic blood pressure Scatter plots
The pattern of data is indicative of the type of
relationship between your two variables: positive relationship negative relationship no relationship Positive relationship Negative relationship Reliability Age of Car No relation Correlation Coefficient
Statistic showing the degree of relation
between two variables Simple Correlation coefficient (r)
It is also called Pearson's correlation
or product moment correlation
coefficient. It measures the nature and strength
between two variables of
the quantitative type. The sign of r denotes the nature of
association
while the value of r denotes the
strength of association. If the sign is +ve this means the relation
is direct (an increase in one variable is
associated with an increase in the
other variable and a decrease in one
variable
is
associated
with
a
decrease in the other variable). While if the sign is -ve this means an
inverse or indirect relationship (which
means an increase in one variable is
associated with a decrease in the other). The value of r ranges between ( -1) and ( +1)
The value of r denotes the strength of the
association as illustrated
by the following diagram.
strong 1- intermediate -0.75 -0.25 weak weak 0 indirect
perfect
correlation intermediate 0.25 strong 0.75 1 Direct
no relation perfect
correlation If r = Zero this means no association or
correlation 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. How to compute the simple correlation
coefficient (r) r xy ( x)
x n 2 2 x y
n
2 ( y) 2
. y n :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.
serial
No Age
(years) Weight (Kg) 1 7 12 2 6 8 3 8 12 4 5 10 5 6 11 6 9 13 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: r xy ( x) 2
2
x n x y
n ( y)2
2
. y n Serial
.n Age
(years)
(x) Weight
(Kg)
(y) xy X2 Y2 1 7 12 84 49 144 2 6 8 48 36 64 3 8 12 96 64 144 4 5 10 50 25 100 5 6 11 66 36 121 6 9 13 117 81 169 Total =x∑
41 =y∑
66 xy=∑
461 =x2∑
291 =y2∑
742 41 66
461 6
r (41) 2 (66) 2 291 .742 6 6 r = 0.759
strong direct correlation EXAMPLE: Relationship between Anxiety and
Test Scores
Anxiety
)X( 10
8
2
1
5
6
X = 32∑ Test
score (Y) X2 Y2 XY 2
100
4
20
3
64
9
24
9
4
81
18
7
1
49
7
6
25
36
30
5
36
25
30
Y = 32∑ X2 = 230∑ Y2 = 204∑ XY=129∑ Calculating Correlation Coefficient
r (6)(129) (32)(32) 6(230) 32 6(204) 32 2 2 774 1024 .94
(356)(200) r = - 0.94 Indirect strong correlation Spearman Rank Correlation Coefficient
(rs)
It is a non-parametric measure of correlation.
This procedure makes use of the two sets of
ranks that may be assigned to the sample values
of x and Y.
Spearman Rank correlation coefficient could be
computed in the following cases:
Both variables are quantitative.
Both variables are qualitative ordinal.
One variable is quantitative and the other is
qualitative ordinal. :Procedure
1. 2.
3. 4. Rank the values of X from 1 to n where n
is the numbers of pairs of values of X and
Y in the sample.
Rank the values of Y from 1 to n.
Compute the value of di for each pair of
observation by subtracting the rank of Yi
from the rank of Xi
Square each di and compute ∑di2 which
is the sum of the squared values. 5. Apply the following formula
2 6 (di)
rs 1 n(n 2 1) The value of rs denotes the magnitude
and nature of association giving the same
interpretation as simple r. Example
In a study of the relationship between level education
and income the following data was obtained. Find the
relationship between them and comment. sample
numbers
A
B
C
D
E level education
(X) Income
(Y) Preparatory.
Primary. 25 University.
secondary 8 10
10
15 F secondary
illiterate G University. 60 50 Answer:
(X) (Y) Rank
X Rank
Y di di2 A Preparatory 25 5 3 2 4 B Primary. 10 6 5.5 0.5 0.25 C University. 8 1.5 7 -5.5 30.25 D secondary 10 3.5 5.5 -2 4 E secondary 15 3.5 4 -0.5 0.25 F illiterate 50 7 2 5 25 G university. 60 1.5 1 0.5 0.25 ∑ di2=64 6 64
rs 1 0.1
7(48)
Comment:
There is an indirect weak correlation
between level of education and income. exercise Regression Analyses
Regression: technique concerned with predicting
some variables by knowing others
The process of predicting variable Y using
variable X 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 Correlation and Regression Correlation describes the strength of a linear
relationship between two variables
Linear means “straight line”
Regression tells us how to draw the straight line
described by the correlation 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 line Regression minimizes residuals 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: ˆ
y a bX
ˆ
y y b(x x) b1 b xy x2 x y n
( x) 2 n Regression Equation Regression equation
describes the regression
line mathematically Intercept Slope Linear Equations
Linear Equations Y
ˆ
y =baX bX
Y
+ a
b = S lo p e Change
in Y C h a n g e in X
a = Y -in te r c e p t X Hours studying and
grades Regressing grades on hours Linear Regression Final gra de in co urse = 59 .95 + 3.17 * st udy
R-Square = 0 .8 8 90.00 80.00 7 0.00 2.00 4.00 6.00 8.00 10.00 Number of hours spent st udying 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) …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 Exercise A sample of 6 persons was selected the
value of their age ( x variable) and their
weight is demonstrated in the following
table. Find the regression equation and
what is the predicted weight when age is
8.5 years. .Serial no Age (x) Weight (y) 1
2
3
4
5
6 7
6
8
5
6
9 12
8
12
10
11
13 Answer
.Serial no Age (x) Weight (y) xy X2 Y2 49 144
36
64
64 144
25 100
36 121
81 169 1
2
3
4
5
6 7
6
8
5
6
9 12
8
12
10
11
13 84
48
96
50
66
117 Total 41 66 461 291 742 41
x 6.83
6 66
y 11
6 41 66
461 6
b
0.92
2
(41)
291 6
Regression equation ˆ
y (x) 11 0.9(x 6.83) ˆ
y (x) 4.675 0.92x
ˆ
y (8.5) 4.675 0.92 * 8.5 12.50Kg ˆ
y (7.5) 4.675 0.92 * 7.5 11 .58Kg we create a regression line by plotting two
estimated values for y against their X component,
then extending the line right and left. Exercise 2 The following are the
age (in years) and
systolic blood
pressure of 20
apparently healthy
adults. Age B.P Age B.P
(x)
(y)
(x)
(y)
20
43
63
26
53
31
58
46
58
70 120
128
141
126
134
128
136
132
140
144 46
53
60
20
63
43
26
19
31
23 128
136
146
124
143
130
124
121
126
123 Find the correlation between age
and blood pressure using simple
and Spearman's correlation
coefficients, and comment.
Find the regression equation?
What is the predicted blood
pressure for a man aging 25 years? Serial
1 x
20 y
120 xy
2400 x2
400 2 43 128 5504 1849 3 63 141 8883 3969 4 26 126 3276 676 5 53 134 7102 2809 6 31 128 3968 961 7 58 136 7888 3364 8 46 132 6072 2116 9 58 140 8120 3364 10 70 144 10080 4900 Serial
11
12
13
14
15
16
17
18
19
20 x
46
53
60
20
63
43
26
19
31
23 y
128
136
146
124
143
130
124
121
126
123 xy
5888
7208
8760
2480
9009
5590
3224
2299
3906
2829 x2
2116
2809
3600
400
3969
1849
676
361
961
529 11448 4167 b1 xy x y n
( x) 2
x2 n ˆ
y = 852 2630
114486 20
0.4547
2
852
41678 20 =112.13 + 0.4547 x for age 25
B.P = 112.13 + 0.4547 * 25=123.49 = 123.5 mm hg Multiple Regression
Multiple regression analysis is a
straightforward extension of simple
regression analysis which allows more
than one independent variable. ...

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- Fall '15
- Correlation, Regression Analysis, Pearson product-moment correlation coefficient, Spearman's rank correlation coefficient