lecture_chp12_9 - Lecture and Examples Topic 9: Testing...

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Lecture and Examples Topic 9: Testing Portions of a Model Suppose we use both Model I and Model II to fit the same data and want to know which model is better in fitting the data. Model I: ε + β + + β + β + β = g g x x x y 2 2 1 1 0 Model II: ε + β + + β + β + + β + β + β = + + k k g g g g x x x x x y 1 1 2 2 1 1 0 Asking whether Model II contributes more information for the prediction of y than Model I is equivalent to asking whether 0 2 1 = β = = β = β + + k g g ; that is, to test the following hypothesis: Hypothesis: . zero not is one least At : : a 2 1 0 i k g g H H β β = = β = β + +
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The test statistic for the above hypothesis is: ( 29 ( 29 ( 29 ) 1 ( SSE SSE SSE II II I c + - - - = k n g k F where (1) SSE I is the Sum of Squared Errors for Model I; (2) SSE II is the Sum of Squared Errors for Model II; (3) k - g is the number of parameters in the null hypothesis; (4) n is total sample size. The rejection region for the above test is: α - - - ), 1 ( ), ( c k n g k F F . Since Model I has fewer parameters, we call Model I the reduced model. The model that has more parameters is called the complete model.
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Example 12.17: A firm likes to forecast its annual sales in each of its sales regions. The firm has decided to base its forecasts on regional population size and its yearly regional advertising expenditures. The data are given in Table 12.15. Table 12.15 Data for Example 12.17 Population Advertising Square of Square of Sales of Region Expenditure Population Expenditure Interaction 65 200 8.0 40000 64.00 1600 80 210 10.0 44100 100.00 2100 85 205 9.0 42025 81.00 1845 100 300 8.5 90000 72.25 2550 108 320 12.0 102400 144.00 3840 114 290 10.0 84100 100.00 2900 40 90 6.0 8100 36.00 540 45 85 8.0 7225 64.00 680 150 450 9.0 202500 81.00 4050 42 87 9.0 7569 81.00 783 220 480 13.0 230400 169.00 6240 200 500 15.0 250000 225.00 7500
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SAS Printout for the First Order Model: Model: EQ1 Dependent Variable: Y Sales Analysis of Variance Sum of Mean Source DF Squares Square F Value Prob>F Model 2 36704.49428 18352.24714 78.862 0.0001 Error 9 2094.42239 232.71360 C Total 11 38798.91667 Root MSE 15.25495 R-square 0.9460 Dep Mean 104.08333 Adj R-sq 0.9340 C.V. 14.65648 Parameter Estimates Parameter Standard T for H0: Variable DF Estimate Error Parameter=0
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lecture_chp12_9 - Lecture and Examples Topic 9: Testing...

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