Regression Analysis 124839

Regression Analysis
•Regression is the determination of a statistical

relationship btn Two or More variables.
•Regression is useful in:/ or Functions of

Regression:
Studying the dependence of dependent
variable on one or more independent
variable
Predicting or forecasting the value of
dependent variable for a given value of
independent variable
Types of Regression Analysis
1. Simple and
2. Multiple Regression Analysis
• Simple Regression involves analysis of TWO variable only:

One of which is Dependent and the other Independent
• Multiple Regression Analysis involves One dependent

variable and Two or More independent variable
• Besides, regression analysis may involve linear model or

nonlinear model
Thus, from what we’ve just said,
•Simple Regression is one of the techniques for

Bivariate data analysis whereas
•Multiple Regression is a multivariate data

analysis technique
** Data need to be at Interval-Ratio Level

Simple Regression Analysis
• Studying the Relationship btn One Independent variable
and One Dependent variable
• As it was the case with correlation, our focus here is to

answer the questions:
Is there a relationship btn the variables?
If yes, how strong is the relationship? and Relationship
What is the direction of relationship?
• But in Regression we are also able to Predict …..
*Remember - the functions of Regression analysis include:

Studying the Relationship btn dependent variable and
independent variable(s) and Predicting the value of dependent
variable for a given value of independent variable
• The relationship btn independent variable and
dependent variable (simple regression analysis)
together with Prediction can be portrayed using
a Scattergram and
a model/equation called Simple Regression Model

• Construction and Uses of Scattergram (Scatter Diagram) in relation
to Regression Analysis
Example: Suppose a researcher wonders if size of household is

related to the size of land cultivated by the household, and the
researcher happens to collect the following data
Table 1
Household A B C D E F G H I J K L
Household Size 9 3 5 5 6 6 1 2 9 8 3 1
Farm Size (Acres) 5 4 5 3 5 4 2 3 6 4 2 1
(a)Construct a scattergram depicting the data

(b)State the relationship between household size and farm size
(c)Predict the size of farm that will be cultivated by a household with
4 members
Household Size 9 3 5 5 6 6 1 2 9 8 3 1
Farm Size (Acres) 5 4 5 3 5 4 2 3 6 4 2 1
Farm Size 6
Regression Line
0 1 2 3 4 5 6 7 8 9
Household Size (# of individuals)
NB.
•Regression line is inserted in the Scattergram through

eyeball approximation such that a straight line touches
every dot or come close to every dot as possible
•Functions of the regression line include enhancing the

clarity of the pattern and predicting the value of dependent
variable (Y) when the value of independent variable (X) is
known.
Prediction
• As stated earlier, prediction is one of the functions of
regression analysis and the scattergram can be used to
accomplish it.
• Suppose that, basing on the relationship btn household size

and farm size, we want to predict size of land that will be
cultivated by a household with 4 members
• Please note that in the data collected we do not have a

household with 4 members
• To predict the size of land that will be cultivated by a

household with 4 members, we proceed as follows:
1. Locate 4 on X axis (independent Variable)
2. Draw a straight line from 4 to the reg line & from there to Y
3. The predicted Y score is found where the line crosses Y axis = 3
6
Farm Size (Acres) Regression Line
1
0 1 2 3 4 5 6 7 8 9
•It is worth noting that prediction using
scattergram is crude as one individual may
insert regression line slightly different from the
other
•As such, slightly different predictions can be

made
•Nonetheless, this is the best starting point

•To overcome this weakness, the simple regression equation can be
used. The equation provides a single straight regression line which best
fits the pattern of the data.
•The Simple Regression Equation is given as

Y = a + bX
Where: Y = Score on the dependent variable
a = Y intercept (the point at which regression line
crosses Y axis)
b = Slope of regression line (the amount of change
produced in Y (dependent variable) by a unit
change in X (independent variable)
X = Score on the independent variable
To use the formula to predict the value of Y, one need to find
the Y intercept (a) and the slope of regression line (b). These are
given as:
Suppose we want to predict the farm size for the household

with 4 individuals as in the case above (Data on Table 1 which
was this one)
Household Size 9 3 5 5 6 6 1 2 9 8 3 1
Farm Size (Acres) 5 4 5 3 5 4 2 3 6 4 2 1
*Let us start by finding b because its value is needed in calculating a

Insert the data into:
We may tabulate as follows:

b= 39.32/91.68
(X) (Y)
b= 0.4 (The slope)
9 5 4.2 1.3 5.46 17.64
3 4 -1.8 0.3 -0.54 3.24
5 5 0.2 1.3 0.26 0.04
5 3 0.2 -0.7 -0.14 0.04 a = 3.7 – (0.4 x 4.8)
6 5 1.2 1.3 1.56 1.44 a = 3.7 – 1.92
6 4 1.2 0.3 0.36 1.44 a = 1.78 (Y intercept)
1 2 -3.8 -1.7 6.46 14.44 Back to our Simple Regression
Equation:
2 3 -2.8 -0.7 1.96 7.84
Y = a + bX
9 6 4.2 2.3 9.66 17.64
Y = 1.78 + (0.4 x 4)
8 4 3.2 0.3 0.96 10.24
3 2 -1.8 -1.7 3.06 3.24
Y = 3.38
1 1 -3.8 -2.7 10.26 14.44 *Comment on this value of Y in
relation to the predictions we
∑X=58 ∑Y=44 ∑=39.32 ∑=91.68 made using Scattergram above.
X =4.8 Y=3.7
Please recall that, we have Multiple Regression, just as we have
Multiple Correlation
Multiple Regression is computed using the formula
•This formula represent the relationships between One

dependent variable (Y) and Two independent variable
•When independent variables are more than Two the equation

is extended to incorporates all the variable, where numbers 1
and 2 will increase to 3, 4, 5 etc

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Regression Analysis

•Regression is the determination of a statistical

•Regression is useful in:/ or Functions of

• Simple Regression involves analysis of TWO variable only:

• Multiple Regression Analysis involves One dependent

• Besides, regression analysis may involve linear model or

•Simple Regression is one of the techniques for

•Multiple Regression is a multivariate data

** Data need to be at Interval-Ratio Level

• As it was the case with correlation, our focus here is to

*Remember - the functions of Regression analysis include:

a model/equation called Simple Regression Model

Example: Suppose a researcher wonders if size of household is

(a)Construct a scattergram depicting the data

•Regression line is inserted in the Scattergram through

•Functions of the regression line include enhancing the

• Suppose that, basing on the relationship btn household size

• Please note that in the data collected we do not have a

• To predict the size of land that will be cultivated by a

•As such, slightly different predictions can be

•Nonetheless, this is the best starting point

•The Simple Regression Equation is given as

Suppose we want to predict the farm size for the household

*Let us start by finding b because its value is needed in calculating a

We may tabulate as follows:

Multiple Regression is computed using the formula

•This formula represent the relationships between One

•When independent variables are more than Two the equation

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