Chapter 15
                              Chapter 15
                        Multiple Regression
                        Multiple Regression
        Multiple Regression Model
          Multiple Regression Model
        Least Squares Method
          Least Squares Method
        Multiple Coefficient of 
          Multiple Coefficient of 
          Determination
        Determination
          Model Assumptions
          Model Assumptions
        Testing for Significance
          Testing for Significance
        Using the Estimated Regression 
          Using the Estimated Regression 
          Equation
          Equation
             for Estimation and Prediction
           for Estimation and Prediction
          Qualitative Independent 
          Qualitative Independent 
          Variables
        Variables
          Residual Analysis
          Residual Analysis
        Logistic 
          Logistic 
          Regression
          Regression
                                                                           
                                                                           
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                   Multiple Regression Model
                   Multiple Regression Model
        Multiple Regression Model
         Multiple Regression Model
                The equation that describes how the 
                The equation that describes how the 
           dependent variable y is related to the independent 
           dependent variable y is related to the independent 
           variables x , x , . . . x  and an error term is:
           variables x , x , . . . x  and an error term is:
                       1  2       p
                       1  2       p
                y =   +  x  +  x +. . . +  x  + 
                                                        
                y =   +  x  +  x +. . . +  x  + 
                       0     1 1     2 2            p p
                       0     1 1     2 2             p p
           where:
           where:
            ,  ,  , . . . ,   are the parameters, and
                          
             0,  1,  2, . . . ,  p are the parameters, and
              0  1   2          p
             is a random variable called the error term
             is a random variable called the error term
                                                                           
                                                                           
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               Multiple Regression Equation
               Multiple Regression Equation
       Multiple Regression Equation
        Multiple Regression Equation
               The equation that describes how the 
               The equation that describes how the 
          mean value of y is related to x , x , . . . x  
          mean value of y is related to x , x , . . . x  
                                             1  2       p
                                             1  2       p
          is:
          is:
               E(y) =   +  x +  x + . . . +  x
                                               
               E(y) =   +  x +  x + . . . +  x
                        0    1 1     2 2           p p
                        0    1 1     2 2           p p
                                                                     
                                                                     
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        Estimated Multiple Regression Equation
        Estimated Multiple Regression Equation
    
       Estimated Multiple Regression Equation
       Estimated Multiple Regression Equation
                    ^
                    ^
                y = b  + b x + b x + . . . + b x
                y = b  + b x + b x + . . . + b x
                       0    1 1     2 2            p p
                       0    1 1     2 2            p p
         A simple random sample is used to compute 
         A simple random sample is used to compute 
         sample statistics b , b , b , . . . , b  that are 
         sample statistics b , b , b , . . . , b  that are 
                              0   1   2         p
                               0  1   2         p
         used as the point estimators of the parameters 
         used as the point estimators of the parameters 
          ,  ,  , . . . ,  .
                       
          0,  1,  2, . . . ,  p.
          0   1  2         p
                                                                     
                                                                      
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                        Estimation Process
                        Estimation Process
           Multiple Regression Model
           Multiple Regression Model
                                                      Sample Data:
  E(y) =   +  x +  x +. . .+  x  +               Sample Data:
                                            
   E(y) =   +  x +  x +. . .+  x  + 
             0     1 1     2 2           p p
             0     1 1     2 2           p p         x   x   . . .  x    y
                                                     x   x   . . .  x    y
                                                      1   2       p
         Multiple Regression Equation                 1   2       p
         Multiple Regression Equation
                                                    .     .          .     .
     E(y) =   +  x +  x +. . .+  x              .     .          .     .
                                      
     E(y) =     +  x +  x +. . .+  x  
              0     1 1     2 2           p p
              0     1 1     2 2           p p       .     .          .     .
                                                    .     .          .     .
           Unknown parameters are                             
           Unknown parameters are                             
                ,  ,  , . . . , 
                               
                 0,  1,  2, . . . ,  p
                 0   1   2         p
                                           Estimated Multiple
                                           Estimated Multiple
                                          Regression Equation
         b , b , b , . . . , b            Regression Equation
         b , b , b , . . . , b
          0   1   2          p
          0   1    2         p          ˆ
                                        ˆ
                                       yb bx bx ...bx
                                       yb bx bx ...bx
                                            0    1 1   2 2        p p
       provide estimates of                 0    1 1   2 2        p p
        provide estimates of
          ,  ,  , . . . ,            Sample statistics are
                                     Sample statistics are
          0,  1,  2, . . . ,  p
          0   1   2          p
                                            b , b , b , . . . , b
                                            b , b , b , . . . , b
                                             0   1   2          p 
                                             0   1   2          p 
                                                                           
                                                                           
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