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Unit 23: Regression Analysis



             The best estimate (an estimate having minimum sum of squares of errors) of Y, independently  Notes

            of X, is given by  Y =  Y .
                           C
            Remarks: If X and Y are independent variables, the two lines of regression are  Y =Y  and
                                                                               C
            X = X .
             C
            Very often, when we use X for the estimation of Y, we are interested in knowing how far the use
            of X enables us to explain the variations in Y values from Y  or, in other words, how much of the
            variations in Y, from  Y , are being explained by the regression equation Y  = a + bX ? To answer
                                                                      Ci      i
            this question, we write
                       Y - Y =Y - Y +Y - Y   (Subtracting and adding Y )
                        i
                                   Ci
                                        Ci
                                i
                       or Y - Y = Y - Y h+ Y - Yi
                                           d
                                c
                                                                   Ci
                         i        i   Ci     Ci
            Squaring both sides and taking sum over all the observations, we have
                    2            2           2
                          Y -
                                                           Y
                                      Y -
                                               2
               Y -
                                                   Y -
            å  ( i  Y ) = å  ( i  Y Ci  ) + å ( Ci  Y ) + å ( i  Y Ci  )( Ci  - Y )  ....(1)
            Consider the product term
                                     Y -
                Y -
                       Y -
            2å  ( i  Y Ci  )( Ci  Y ) =  2å é ë { i  Y -  ( b X -  X )} ( { b X -  X )} ù û
                                               i
                                                         i
                                                    2
                                   Y -
                             =  b 2 å  ( i  Y )( X -  X ) 2-  b å ( X -  X ) 2
                                                         i
                                           i
                                           2             2
                                 2             2
                                                   X -
                             =  b 2 å ( X -  X  ) -  b 2 å  ( i  X  ) =  0
                                      i
            Thus, equation (1) becomes
                                2           2           2
                                                  Y -
                        å  ( i  Y ) = å  ( i  Y Ci ) + å ( Ci  ) Y        .... (2)
                          Y -
                                     Y -
            From the above figure, we note that Y - Y  is the deviation of the estimated value from   .
                                           Ci                                         Y
            This deviation has occurred because X and Y are related by the regression equation Y  = a + bX ,
                                                                               Ci      i
            so that the estimate of Y is Y  when X = X . Similar type of deviations would occur for other
                                    Ci         i
                                                         2
                                                å  Y - Y )  gives the strength of the relationship,
            values of X. Thus, the magnitude of the term  ( Ci
            Y  = a + bX , between X and Y or, equivalently, the variations in Y that are explained by the
             Ci       i
            regression  equation.
                                              Figure  23.5











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