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. 2 QM-220, M. Zainal
Chapter 15: Multiple linear regression 15.1 Multiple Regression Analysis � A multiple regression model with y as a dependent variable and x 1 , x 2 , x 3 , …, x k as independent variables is written as written as = + + + + + ε y A B x B x B x ... (1) k k 1 1 2 2 where A represents the constant term, B 1 , B 2 , B 3 , …, B k are h A t th t t t B B B B the regression coefficients of independent variables x 1, x 2 , x 3 x 3 , …, x k , respectively, and ε represents the random error x k respectively and ε represents the random error term. � This model contains k independent variables x 1 , x 2 , x 3 , …, s ode co s depe de v b es 1 , 2 , 3 , …, and x k . 3 QM-220, M. Zainal
Chapter 15: Multiple linear regression 15.1 Multiple Regression Analysis � This model contains k independent variables x 1 , x 2 , x 3 , …, and x k . � A multiple regression models can only be used when the � A lti l i d l l b d h th 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 B 1 , B 2 , B 3 , …, and B k are called the partial regression coefficients. 4 QM-220, M. Zainal
Chapter 15: Multiple linear regression 15.1 Multiple Regression Analysis � For example B is a partial regression coefficient of x � For example, B 1 is a partial regression coefficient of x 1 . � It gives the change in y due to a one-unit change in x 1 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 x 1 by one unit but keep x 2 , x 3 , …, and x k unchanged, then the resulting change in y is measured by B 1 . � In model (1) above A B � In model (1) above, A , B 1 , B 2 , B 3 , …, and B k are called the B B and B are called the true regression coefficients or population parameters . � A positive value for a particular B i in model (1) will p p ( ) i indicate a positive relationship between y and the corresponding x i variable and vice versa. 5 QM-220, M. Zainal
Chapter 15: Multiple linear regression 15.1 Multiple Regression Analysis � For example B is a partial regression coefficient of x � For example, B 1 is a partial regression coefficient of x 1 . � It gives the change in y due to a one-unit change in x 1 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 x 1 by one unit but keep x 2 , x 3 , …, and x k unchanged, then the resulting change in y is measured by B 1 . � In model (1) above A B � In model (1) above, A , B 1 , B 2 , B 3 , …, and B k are called the B B and B are called the true regression coefficients or population parameters . � A positive value for a particular B i in model (1) will p p ( ) i indicate a positive relationship between y and the corresponding x i variable and vice versa. 6 QM-220, M. Zainal
Chapter 15: Multiple linear regression 15.1 Multiple Regression Analysis � If model (1) is estimated using sample data � If model (1) is estimated using sample data, which is which is usually the case, the estimated regression equation is written as = + + + + ˆ y a b x b x b x ... (2) k k 1 1 2 2 � In equation 2, a , b 1 , b 2 , b 3 , …, and b k are the sample statistics, which are the point estimators of the population parameters A , B 1 , B 2 , B 3 , …, and B k , 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 of squared errors is called the least squares regression equation. 7 QM-220, M. Zainal
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 of 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 σ ε . � 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 x i . 8 QM-220, M. Zainal
Chapter 15: Multiple linear regression 15.5 Computer Solution of Multiple Regression Example: A researcher wanted to find the Monthly Driving Number of Driving Premium Experience Violation effect of driving experience and the number 148 5 2 of driving violations on auto insurance 76 14 0 premiums. A random sample of 12 drivers insured with the same company and having 100 6 1 126 10 3 similar auto insurance policies was selected 194 194 4 4 6 6 from a large city. The table lists the monthly 110 8 2 auto insurance premiums (in dollars) paid 114 11 3 by these drivers, their driving experiences (in years), and the numbers of driving (i ) d th b f d i i 86 86 16 16 1 1 198 3 5 violations committed by them during the 92 9 1 past three years. Using MINITAB, find the 70 70 19 19 0 0 regression eq ation of monthl regression equation of monthly premiums premi ms 120 13 3 paid by drivers on the driving experiences and the numbers of driving violations. 9 QM-220, M. Zainal
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