Introduction to Business Statistics QM 220 QM 220 Chapter 15 Dr. - - PowerPoint PPT Presentation

Department of Quantitative Methods & Information Systems

Introduction to Business Statistics QM 220 QM 220 Chapter 15

• Dr. Mohammad Zainal

Chapter 15: Multiple linear regression 15.1 Multiple Regression Analysis The model used in the simple regression includes one independent variable, which is denoted by x, and one dependent variable which is denoted by y dependent variable, which is denoted by y. Usually a dependent variable is affected by more than one independent variable. independent variable. When we include two or more independent variables in a regression model, it is called a multiple regression model. g , p g Remember, whether it is a simple or a multiple regression model, it always includes one and only one dependent variable.

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Chapter 15: Multiple linear regression 15.1 Multiple Regression Analysis A multiple regression model with y as a dependent variable and x1, x2, x3, …, xk as independent variables is written as written as h A t th t t t B B B B

1 1 2 2

... (1) ε = + + + + +

k k

y A B x B x B x

where A represents the constant term, B1, B2, B3, …, Bk are the regression coefficients of independent variables x1, x2, x3 xk respectively and ε represents the random error x3, …, xk, respectively, and ε represents the random error term. This model contains k independent variables x1, x2, x3, …, s

• de co

s depe de v b es

1, 2, 3, …,

and xk.

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Chapter 15: Multiple linear regression 15.1 Multiple Regression Analysis This model contains k independent variables x1, x2, x3, …, and xk. A lti l i d l l b d h th A multiple regression models can only be used when the relationship between the dependent variable and each independent variable is linear. independent variable is linear. There can be no interaction between two or more of the independent variables. p In regression model (1), A represents the constant term, which gives the value of y when all independent variables assume zero values. The coefficients B1, B2, B3, …, and Bk are called the partial regression coefficients.

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Chapter 15: Multiple linear regression 15.1 Multiple Regression Analysis For example B is a partial regression coefficient of x For example, B1 is a partial regression coefficient of x1. It gives the change in y due to a one-unit change in x1 when all other independent variables included in the model when all other independent variables included in the model are held constant. In other words, if we change x1 by one unit but keep x2, x3, …, and xk unchanged, then the resulting change in y is measured by B1. In model (1) above A B B B and B are called the In model (1) above, A, B1, B2, B3, …, and Bk are called the true regression coefficients or population parameters. A positive value for a particular Bi in model (1) will p p

i

( ) indicate a positive relationship between y and the corresponding xi variable and vice versa.

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Chapter 15: Multiple linear regression 15.1 Multiple Regression Analysis For example B is a partial regression coefficient of x For example, B1 is a partial regression coefficient of x1. It gives the change in y due to a one-unit change in x1 when all other independent variables included in the model when all other independent variables included in the model are held constant. In other words, if we change x1 by one unit but keep x2, x3, …, and xk unchanged, then the resulting change in y is measured by B1. In model (1) above A B B B and B are called the In model (1) above, A, B1, B2, B3, …, and Bk are called the true regression coefficients or population parameters. A positive value for a particular Bi in model (1) will p p

i

( ) indicate a positive relationship between y and the corresponding xi variable and vice versa.

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Chapter 15: Multiple linear regression 15.1 Multiple Regression Analysis If model (1) is estimated using sample data which is If model (1) is estimated using sample data, which is usually the case, the estimated regression equation is written as In equation 2, a, b1, b2, b3, …, and bk are the sample

1 1 2 2

ˆ ... (2) = + + + +

k k

y a b x b x b x

statistics, which are the point estimators of the population parameters A, B1, B2, B3, …, and Bk, respectively. The degrees of freedom for the model is The degrees of freedom for the model is df = n – k – 1 The estimated equation 2 obtained by minimizing the sum The estimated equation 2 obtained by minimizing the sum

• f squared errors is called the least squares regression

equation.

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Chapter 15: Multiple linear regression 15.2 Assumptions of the Multiple Regression Model Assumption 1: The mean of the probability distribution of Assumption 1: The mean of the probability distribution of ε is zero Assumption 2: The errors associated with different sets of Assumption 2: The errors associated with different sets of values

• f

independent variables are independent. Furthermore, these errors are normally distributed and have a constant standard deviation which is denoted by σ have a constant standard deviation, which is denoted by σε. Assumption 3: The independent variables are not linearly

• related. However, they can have a nonlinear relationship.
• related. However, they can have a nonlinear relationship.

When independent variables are highly linearly correlated, it is referred to as multicollinearity. Assumption 4: There is no linear association between the random error term ε and each independent variable xi.

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Chapter 15: Multiple linear regression 15.5 Computer Solution of Multiple Regression

Example: A researcher wanted to find the effect of driving experience and the number

• f driving violations on auto insurance

Monthly Premium Driving Experience Number of Driving Violation 148 5 2

• premiums. A random sample of 12 drivers

insured with the same company and having similar auto insurance policies was selected

76 14 100 6 1 126 10 3 194 4 6

from a large city. The table lists the monthly auto insurance premiums (in dollars) paid by these drivers, their driving experiences (i ) d th b f d i i

194 4 6 110 8 2 114 11 3 86 16 1

(in years), and the numbers of driving violations committed by them during the past three years. Using MINITAB, find the regression eq ation of monthl premi ms

86 16 1 198 3 5 92 9 1 70 19

regression equation of monthly premiums paid by drivers on the driving experiences and the numbers of driving violations.

70 19 120 13 3 QM-220, M. Zainal 9