Regression and corelation (Biostatistics)

‫الرحيم‬ ‫الرحمن‬ ‫هللا‬ ‫بسم‬

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

• 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

Wt.
(kg)
67 69 85 83 74 81 97 92 114 85
SBP
mmHg)
120 125 140 160 130 180 150 140 200 130
Example

Scatter diagram of weight and systolic blood
pressure
80
100
120
140
160
180
200
220
60 70 80 90 100 110 120
wt (kg)
SBP(mmHg)
Wt.
(kg)
67 69 85 83 74 81 97 92 114 85
SBP
(mmHg)
120 125 140 160 130 180 150 140 200 130

80
100
120
140
160
180
200
220
60 70 80 90 100 110 120
Wt (kg)
SBP(mmHg)
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

0
2
4
6
8
10
12
14
16
18
0 10 20 30 40 50 60 70 80 90
Age in Weeks
HeightinCM

Negative relationship
Reliability
Age of Car

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.
-1 10-0.25-0.75 0.750.25
strong strongintermediate intermediateweak weak
no relation
perfect
correlation
perfect
correlation
Directindirect

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.





















  
  
n
y)(
y.
n
x)(
x
n
yx
xy
r
2
2
2
2
How to compute the simple correlation
coefficient (r)

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

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
yx
xy
r
2
2
2
2

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

r = 0.759
strong direct correlation
















6
(66)
742.
6
(41)
291
6
6641
461
r
22

EXAMPLE: Relationship between Anxiety and
Test Scores
Anxiety
(X)
Test
score (Y)
X2 Y2 XY
10 2 100 4 20
8 3 64 9 24
2 9 4 81 18
1 7 1 49 7
5 6 25 36 30
6 5 36 25 30
∑X = 32 ∑Y = 32 ∑X2 = 230 ∑Y2 = 204 ∑XY=129

Calculating Correlation Coefficient
  
94.
)200)(356(
1024774
32)204(632)230(6
)32)(32()129)(6(
22





r
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. 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.
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.

5. Apply the following formula
1)n(n
(di)6
1r 2
2
s



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.
Income
(Y)
level education
(X)
sample
numbers
25Preparatory.A
10Primary.B
8University.C
10secondaryD
15secondaryE
50illiterateF
60University.G

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

Comment:
There is an indirect weak correlation
between level of education and income.
1.0
)48(7
646
1 

sr

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
80
100
120
140
160
180
200
220
60 70 80 90 100 110 120
Wt (kg)
SBP(mmHg)

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
yx
xy
b 2
2
1
)xb(xyyˆ  b
bXayˆ 

Regression Equation
 Regression equation
describes the
regression line
mathematically
 Intercept
 Slope
80
100
120
140
160
180
200
220
60 70 80 90 100 110 120
Wt (kg)
SBP(mmHg)

Linear Equations
Y
Y = bX + a
a = Y-intercept
X
Change
in Y
Change in X
b = Slope
bXayˆ 

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
Finalgradeincourse












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)

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
Predicted final grade in class = 59.95 + 3.17*(hours of study)

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.

Weight (y)Age (x)Serial no.
12
8
12
10
11
13
7
6
8
5
6
9
1
2
3
4
5
6

Answer
Y2X2xyWeight (y)Age (x)Serial no.
144
64
144
100
121
169
49
36
64
25
36
81
84
48
96
50
66
117
12
8
12
10
11
13
7
6
8
5
6
9
1
2
3
4
5
6
7422914616641Total

6.83
6
41
x  11
6
66
y
92.0
6
)41(
291
6
6641
461
2




b
Regression equation
6.83)0.9(x11yˆ (x) 

0.92x4.675yˆ (x) 
12.50Kg8.5*0.924.675yˆ (8.5) 
Kg58.117.5*0.924.675yˆ (7.5) 

11.4
11.6
11.8
12
12.2
12.4
12.6
7 7.5 8 8.5 9
Age (in years)
Weight(inKg)
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.
B.P
(y)
Age
(x)
B.P
(y)
Age
(x)
128
136
146
124
143
130
124
121
126
123
46
53
60
20
63
43
26
19
31
23
120
128
141
126
134
128
136
132
140
144
20
43
63
26
53
31
58
46
58
70

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?

x2xyyxSerial
4002400120201
18495504128432
39698883141633
6763276126264
28097102134535
9613968128316
33647888136587
21166072132468
33648120140589
4900100801447010

x2xyyxSerial
211658881284611
280972081365312
360087601466013
40024801242014
396990091436315
184955901304316
67632241242617
36122991211918
96139061263119
52928291232320
416781144862630852Total




 



n
x)(
x
n
yx
xy
b 2
2
1 4547.0
20
852
41678
20
2630852
114486
2




=
=112.13 + 0.4547 x
for age 25
B.P = 112.13 + 0.4547 * 25=123.49 = 123.5 mm hg
yˆ

Multiple Regression
Multiple regression analysis is a
straightforward extension of simple
regression analysis which allows more
than one independent variable.

Regression and corelation (Biostatistics)

Regression and corelation (Biostatistics)

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