Regression

Saurav Das
MURP 180405
Lecture 5

1. Linear regression analysis
2. Multi linear regression analysis
3. Standard error of the regression
4. Coefficient of determination
5. Application of regression analysis
Content

 Regression analysis is a powerful statistical method that allows you to
examine the relationship between two or more variables of interest.
 While there are many types of regression analysis, at their core they all
examine the influence of one or more independent variables on a
dependent variable.
 Regression analysis is widely used for prediction and forecasting such as
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Definition

Simple: In case of simple relationship only two variables are considered.
Multiple: In the case of multiple relationship, more than two variables are involved. On
this while one variable is a dependent variable the remaining variables are
independent
Linear: The linear relationships are based on straight-line trend, the equation of which
has no-power higher than one.
Non- Linear: In the case of non-linear relationship curved trend lines are derived.
Total: In the case of total relationships all the important variables are considered.
Partial: In the case of partial relationship one or more variables are considered, but not
all.

Regression
Linear
Simple Linear Multiple Linear
Simple Multiple
Non-
Linear
Total Partial

What is “Linear”?
Remember this:
Y= mX + B?
B
m
What’s Slope?
A slope of 2 means that every 1-unit
change in X yields a 2-unit change in Y.
If you know something about x, this knowledge helps you predict something about Y
(conditional probabilities)
Prediction

Linear regression equation
Y = a + bX
Where,
b = slope
a = intercept

Regression
Graphic Algebraic
Y = a + bX

X axis
Yaxis
. .
.
. .
.
.. .
1. Points are plotted on a graph paper representing
various parts of values
2. These points give a picture of a scatter diagram
3. A regression line may be drawn in between these
points
4. Line should be drawn faithfully as the line of best fit
leaving equal number of points on both sides

1. Obtain a random sample of n data pairs (X, Y), where X is the explanatory variable
and Y is the response variable.
2. Using the data pairs, compute ∑X, ∑Y, ∑X2, ∑Y2, and ∑XY. Then compute the sample
means X and Y.
3. With n = sample size, ∑X, ∑Y, ∑X2, ∑Y2, ∑XY, X and Y, you are ready to compute the
slope b and intercept a using the computation formulas
Slope: b =
𝒏∑XY− (∑X)(∑Y)
n∑X2 – (∑X)2 Intercept: a = Y— bX
4. The equation of the least-squares line computed from your sample data is, Y = a + bX

Find the regression equations from the following data:
Solution:
X =
∑X
n
=
𝟑𝟎
5
= 6
Y =
∑ 𝐘
n
=
𝟒𝟎
5
= 8
b =
𝒏∑XY− (∑X)(∑Y)
n∑X2 –(∑X)2 = -0.65
a = Y— bX = 11.9
Y = a + bX = 11.9 + 0.65X

Simple regression considers the relation between a single
explanatory variable and response variable
Multiple regression simultaneously considers the influence of
multiple explanatory variables on a response variable Y
A multiple regression equation expresses a linear relationship
between a response variable y and two or more predictor
variables (x1, x2, . . ., xk). The general form of a multiple
regression equation obtained from sample data is
y = b0 + b1 x1 + b2x2 + ………………. + bkxk

The standard error of the estimate is a measure of the accuracy of predictions made
with a regression line.
The square root of the average squared error of prediction is used as a measure of the
accuracy of prediction. This measure is called the standard error of the estimate and is
designated as σest.
where N is the number of pairs of (X,Y) points

The slope and intercept of the regression line
are 3.2716 and 7.1526 respectively.
Y' = 3.2716X + 7.1526

The coefficient of determination (denoted by R2) is a key output of regression analysis.
It is interpreted as the proportion of the variance in the dependent variable that is
predictable from the independent variable.
The coefficient of determination is the square of the correlation (r) between predicted
y scores and actual y scores; thus, it ranges from 0 to 1.
With linear regression, the coefficient of determination is also equal to the square of
the correlation between x and y scores.

1. An R2 of 0 means that the dependent variable cannot be predicted from the
independent variable.
2. An R2 of 1 means the dependent variable can be predicted without error from the
independent variable.
3. An R2 between 0 and 1 indicates the extent to which the dependent variable is
predictable. An R2 of 0.10 means that 10 percent of the variance in Y is predictable
from X; an R2 of 0.20 means that 20 percent is predictable; and so on.

Correlation coefficient, r =
𝒏∑𝑿𝒀− (∑𝑿)(∑𝒀)
𝒏∑𝑿 𝟐 – (∑𝑿) 𝟐 𝒏∑𝒀 𝟐 – (∑𝒀) 𝟐
= - 0.92
Coefficient of determination = r2 = (-0.92)2 = 0.85
Here,
n = 5
∑𝑿 = 30
∑𝒀 = 40
∑𝑿𝒀 = 214
∑𝑿 𝟐
= 220
(∑𝐗) 𝟐 = 900
∑𝒀 𝟐 = 340
(∑𝐘) 𝟐 = 1600
Interpretation: 0.85 means that 85 percent of the variance in Y is predictable from X

1. Regression analysis helps in establishing a functional relationship between two or more
variables.
2. Since most of the problems of economic analysis are based on cause and effect
relationships, the regression analysis is a highly valuable tool in economic and business
research.
3. Regression analysis predicts the values of dependent variables from the values of
independent variables.
4. We can calculate coefficient of correlation ( r) and coefficient of determination ( r2) with the
help of regression coefficients.
5. In statistical analysis of demand curves, supply curves, production function, cost function,
consumption function etc., regression analysis is widely used.
Application of regression analysis

Regression

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