Multiple Linear Regression

• Many applications of regression
analysis involve situations in which
there are more than one regressor
variables.
• A regression model that contains more
than one regressor variables is called a
multiple regression model.
Introduction
Multiple Linear Regression Model
Rizwan Manzoor (MSc Finance) rizman2004@yahoo.com Contact me on Email Address to
make Project Reports or using SPSS to evaluate results

 A study conducted to assess the satisfaction
levels of staff from an educational institution
with branches in a number of locations across
the country.
 Staff were asked to complete a short,
anonymous questionnaire (shown later in this
Appendix) containing questions about their
opinion of various aspects of the organization
and the treatment they have received as
employees.
Problem Statement
Rizwan Manzoor (MSc Finance)
rizman2004@yahoo.com

• There are many variables exists in the research
therefore factor analysis has been conducted to
make decision which variables are best fit for the
model by Extraction Method.
• The factor analysis extracted Q1,Q2,Q3 but when
Including the variables Q4,Q5and Q6 also gave the
right results
Factor Analysis of Variable to fit into the model

 The error term has a normal distribution
with a mean of 0.
 The variance of the error term is constant
across cases and independent of the
variables in the model.
 There is no multicolinearity
Multiple Linear Regression
Model
Assumptions:

• For example, suppose that the Satisfaction level of staff
is depend on the Q1,Q2,Q3,Q4,Q5 and Q6. A possible
multiple regression model could be
Model
Y=β0+ β1Q1+ β2Q2+ β3Q3 +β4Q4+β5Q5+β6Q6+E

where
Y - Total Staff Satisfaction Scale (Dependent)
Q1 - Is it clear what is expected of you at work? (Independent)
Q2 - At work have you been provided with all the equipment and materials
required for you do your work efficiently? (Independent)
Q3 - Does the organization keep you up to date with information concerning
development and changes? (Independent)
Q4 - Do you receive recognition from the organization for doing good Work ?(Independent)
Q5 - Does your manager or supervisor encourage your development at work?(Independent)
Q6 - Do you feel that your opinions seem to count to the organization? (Independent)
E - Error Term (Influence of other Variables/factors)
Introduction

Model
Example: Table 1.1
http://www.allenandunwin.com/spss/data_files.html

Example: Table 1.2
Education Institution
Staff Survey
conducted in across
Australia
Example. 536
Observations has
been taken to check
the reliability of the
Multiple Linear
Regression Model.
By taking Staff Total Satisfaction as dependent
variable and Q1,Q2,Q3,Q4,Q5 and Q6 as
Independent Variables

For running the Multiple Linear Regression
and testing the result in SPSS the variables
has been set according their dependency and
independency. By Using Enter Method

Interpreting Regression Output
•The Coefficient of determination or adjusted R2 can be interpreted because
of more than one independent variables.
•This is the percentage(%) of total variation in the dependent variable due to
the independent variable.
•We can see that Adjusted R square value 0.920 that our independent
variables explain 92.% of the variability of our dependent variable.
•Std. Error of the estimate shows that the other variables, that have not been
taken in the model, have influence 1.995%. The std Error is low due the
number of variables. The larger the size of variables the smaller the std.Error.Rizwan Manzoor (MSc Finance)

•The F-ratio in the ANOVA table tests whether the
overall regression model is a good fit for the data.
The table shows that the independent variables
statistically significantly predict the dependent
variable, F = 944.111 Sig is < .05 (i.e., the regression model is a
good fit of the data).
Analysis of variance (ANOVA)

Co-linearity diagnostics /Co-linearity issue
• Colinearity (Multicolinearity) is the
undesirable situation where the
correlations among the independent
variable are strong.
•When two X variables are highly
correlated, they both convey
essentially the same
information. When this happens, the
X variables are colinear and the
results show multicolinearity.
•To check the issue Colinearity
diagnostics has been selected. The
Variation Inflation Factor and
Significant values determine the multicolinearity issue. If VIF value is greater than >2 or
>3 then the Multicolinearity exists and if Significant value >0.05 shows which
independent variable highly insignificant. The higher the significant value the higher the
insignificant independent variable. Rizwan Manzoor (MSc Finance)

•The higher the beta value shows the more influential the independent
variable and sign (Positive,Negative) shows the nature of the relationship
between independent and dependent variable. The Q5a has higher
influence.
•By having a glance on the sig column we can interpret. The significance
level should be below the <0.05.
•We can see that the all independent variables are <0.05 means that they
are statistically significance.
•The VIF of the all independent variables are <3 which shows that the
multicolinearity issue does not exist whereas the VIF should be near to
2 or <3.
Interpretation of Coefficients

Model
Histogram showing the normality of the data

Model
The PP Plot showing the best fit of linearity of the data

 1. Increasing the sample size is a common first
step since when sample size is increased,
standard error decreases (all other things equal).
This partially offsets the problem that high
multicollinearity leads to high standard errors .
 2. Remove the most intercorrelated variable(s)
from analysis. This method is misguided if the
variables were there due to the theory of the
model, which they should have been.
 3. Take transformation of variables which is the
best fir for model.
Remedies of Multicolinearity

Remedy of Colinearity
1. Enter Method: It embraces all the variables that exist in the model and determines
whether the variables are significant or insignificant.
2. Stepwise Method: it embraces significant variables and excludes insignificant
variable.
3. Forward Method: It embraces the highly significant variables and order/rank
them but not to insignificant variables.
4. Backward Method: It will consider the highly insignificant variables and
order/rank them but not to significant variables.

 Using Enter method in Multiple Linear Regression, I
have selected a "best" model for predicting Satisfaction
level of staff of Educational Institution across Australia.
 With this model, I found two Q4 and Q5 in the model
were high performing in all the across the institution,
while no predictors are insignificant.
 The histogram showing the normality of data and PP
Plots also reflects the best fit of the model because all
the points are on the linear line
Model
Summary:

Multiple Linear Regression

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Multiple Linear Regression