Module 2_ Regression Models..pptx

Module 2: Regression
Models.
CSC601.2: Apply various regression models on given data
set and perform prediction.

CONTENTS
Introduction to simple Linear Regression:
● The Regression Equation
● Fitted value and Residuals
● Least Square
Introduction to Multiple Linear Regression:
● Assessing the Model
● Cross-Validation
● Model Selection and Stepwise Regression
● Prediction Using Regression
Logistic Regression:
● Logistic Response function and logic
● Logistic Regression and GLM
● Generalized Linear model
● Predicted values from Logistic Regression
● Interpreting the coefficients and odds ratios
Linear and Logistic Regression:
● Similarities and Differences
● Assessing the models.

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

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

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

⮚ 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 1
0
-0.25
-0.75 0.75
0.25
strong strong
intermediate intermediate
weak weak
no relation
perfect
correlation
perfect
correlation
Direct
indirect

• 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)

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:

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

r = 0.759
strong direct correlation

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

Calculating Correlation Coefficient
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

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

5. Apply the following formula
• 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
level education
(X)
Income
(Y)
A Preparatory. 25
B Primary. 10
C University. 8
D secondary 10
E secondary 15

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

Comment:
There is an indirect weak correlation
between level of education and income.

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:
b

Regression Equation
⮚ Regression equation
describes the
regression line
mathematically
■ Intercept
■ Slope

Regressing grades on hours
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)
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.

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
1
2
3
4
5
6
7
6
8
5
6
9
12
8
12
10
11
13
84
48
96
50
66
117
49
36
64
25
36
81
144
64
144
100
121
169
Total 41 66 461 291 742

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
(x)
B.P (y) Age
(x)
B.P (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 x y xy x2
1 20 120 2400 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

Serial x y xy x2
11 46 128 5888 2116
12 53 136 7208 2809
13 60 146 8760 3600
14 20 124 2480 400
15 63 143 9009 3969

=
=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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