This document outlines multiple linear regression. It begins with an introduction to regression and its objectives. It then presents the mathematical model of multiple linear regression as Y = a + b1X1 + b2X2 +...+ bkXk, where Y is the dependent variable, a is the intercept, and the bs are coefficients of the independent variables. An example is provided to illustrate the technique. Key assumptions and pros and cons are discussed, and references are presented.

1. MULTIPLE LINEAR REGRESSION
Presented By:
A.K.M.Ashek Farabi
Dept. of CSE

2. • Introduction
• Learning Objectives
• Mathematical Model
• Example
• Method
• Linear Regression Assumptions
• Pros and Cons
• Conclusion
• References
Outline
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3. Introduction
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Regression can be defined as a method that estimates the value of one variable when that of
other variable is known.
It was used for the first time by Sir Francis Galton in his research paper “Regression towards
mediocrity in hereditary stature”.
Types:
• Simple Linear Regression
• Multiple Linear Regression

4. Multiple linear regression
• A multiple linear regression model shows the relationship between the dependent variable
and multiple (two or more) independent variable.
• In MLR, the shape is not really a line. If there are three variables, the shape is a plane, and if
there are four or more variables, it is impossible to visualize or graph.
• However, by convention, we still refer to the regression equation as a regression 'line'.
Continued..
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5. Objectives of MLR:
• Articulate assumptions for multiple linear regression.
• Explain the primary components of multiple linear regression.
• Identify and define the variables included in the regression equation.
• Construct a multiple regression equation.
• Calculate a predicted value of a dependent variable using a multiple regression equation.
Learning Objectives
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6. Multiple linear regression for k independent variables x1 , x2 ….., xk is given by
• Y = a + b1X1 + b2X2 + b3X3 + ∙ ∙ ∙ bkXk
a= Intercept
b1 , b2 , b3 ……bk = Coefficients of partial regression
x1 , x2 ….., xk = Independent Variables
Mathematical Model
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7. Continued..
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b1=
For the equation we solve the following:
SS(X2) SP(X1Y)-SP(X1X2) SP(X2Y)
SS(X1) SS(X2) – [SP(X1X2) ]2
b2=
SS(X1) SP(X2Y)-SP(X1X2) SP(X1Y)
SS(X1) SS(X2) – [SP(X1X2) ]2

8. Example
• We are interested in knowing if going to restaurants frequently (five or
more times/week) can lead to higher cholesterol.
• We also know that age, gender, and race/ethnicity can affect cholesterol.
• How can we tell if going out to restaurants frequently, this factor alone,
will affect cholesterol levels?
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9. Continued..
• Do age, gender, ethnicity, and going out to eat frequently all affect
cholesterol levels?
• Dependent variable: cholesterol level
• Independent variables: age (years), gender (male/female),
race/ethnicity (Black, White, Asian, or Hispanic), frequency of going
out to eat (5+ times/week vs less than 5 times/week)
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10. Linear Regression Assumptions
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Linear regression is a parametric method and requires that certain assumptions be
met to be valid.
• The sample must be representative of the population
• The dependent variable must be of ratio/interval scale and normally distributed overall
and normally distributed for each value of the independent variables
• For every value of X, the distribution of Y scores must have approximately equal
variability (homoscedasticity)
• The relationship between X and Y must be linear
• The independent variables are not very strongly inter-correlated (no multicollinearity)

11. Pros and Cons
Pros:
• Multiple linear regression enables an investigation of the relationship between Y and
several independent variables simultaneously.
Cons:
• Increased complexity makes it more difficult to
• ascertain which model is best.
• interpret the models.
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12. Conclusion
• Multivariate linear regression can be used when the outcome of interest is of interval or
ratio scale and normally distributed
• Using the regression model, we can estimate the strength and direction of the
association from the adjusted partial regressión of Independent Variables
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13. References
[1] http://www.stat.yale.edu/Courses/1997-98/101/linmult.htm
[2] http://reliawiki.org/index.php/Multiple_Linear_Regression_Analysis
[3] https://www.researchgate.net/publication/242418171
[4] Numerical methods for engineers 6th edition (Page: 474-477)
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Thank you very much for
your time.