Model Building Chap 5 p251

Transcription

1 Model Building Chap 5 p251 Models with one qualitative variable, 5.7 p277 Example 4 Colours : Blue, Green, Lemon Yellow and white Row Blue Green Lemon Insects trapped Descriptive Statistics: Insects Variable Colour N N* Mean Insects B G L W The regression equation is Insects trapped = Blue Green Lemon Predictor Coef StDev T P Constant Blue Green Lemon S = R-Sq = 82.1% R-Sq(adj) = 79.4% 1

2 Analysis of Variance Source DF SS MS F P Regression Residual Error Estimate the betas using the means (Descriptive statistics) State whether the following statements are true or false. a) The value of the F-statistic for testing any differences among the colours is b) We have evidence at p < 0.01 that the means for green and white are different. c) We have evidence at p < 0.01 that means for blue and white are different. d) A 95% confidence interval for the difference between means for lemon yellow and white is (23.3, 39.7) e) We may say that 82.1% of the variation in the number of insects trapped has been accounted for by the above model. 2

3 Models with two qualitative variables, 5.8 p282 Data Display Row C1 perform F2 F3 B2 F2B2 F3B2 F B 1 F1B F1 B1 2 F1B F1 B1 3 F1B F1 B1 4 F1B F1 B2 5 F2B F2 B1 6 F2B F2 B1 7 F2B F2 B2 8 F2B F2 B2 9 F3B F3 B1 10 F3B F3 B1 11 F3B F3 B2 12 F3B F3 B2 Example 5.10 p286 Main effects model The regression equation is perform = F F B2 Predictor Coef StDev T P Constant F F B S = R-Sq = 36.2% R-Sq(adj) = 12.3% Analysis of Variance Source DF SS MS F P Regression Residual Error Total Source DF Seq SS F F B

4 Interaction model Descriptive Statistics: perform Variable C1 N N* Mean StDev perform F1B F1B * F2B F2B F3B F3B The regression equation is perform = F F B F2B F3B2 Predictor Coef StDev T P Constant F F B F2B F3B S = R-Sq = 97.1% R-Sq(adj) = 94.8% Analysis of Variance Source DF SS MS F P Regression Residual Error Total Source DF Seq SS F F B F2B F3B

5 Interaction Plot - Data Means for perform B 80 B1 B2 70 Mean F1 F2 F3 F Estimate the regression equation using the means (descriptive statistics) Test whether there is an interaction between brand and fuel type. 5

6 Variable Screening methods, Chap 6 p321 Stepwise regression p323 A hospital Surgical unit was interested in predicting the survival times of patients undergoing a particular type of liver operation. A random sample of patients was available for analysis. From each patient record, the following info was extracted from the preoperation evaluation: X1 = blood clotting score X2 = prognostic index X3 = enzyme function test score X4 = liver function test score X5 = age in years X6 = indicator variable for gender (0 = M, 1 = F) X7 and X8 = indicator variables for history of alcohol use (categorical: none, moderate, severe) X7 = indicator of moderate X8 = indicator of severe Data Display Row X1 X2 X3 X4 X5 X6 X7 X8 Y lny

7 The regression equation is Y = X X X X X X X X8 Predictor Coef StDev T P Constant X X X X X X X X S = R-Sq = 78.2% R-Sq(adj) = 74.3% Analysis of Variance Source DF SS MS F P Regression Residual Error Total Residuals Versus the Fitted Values (response is Y) 800 Residual Fitted Value 7

8 Normal Probability Plot of the Residuals (response is Y) 2 Normal Score Residual Histogram of the Residuals (response is Y) Frequency Residual

9 The regression equation is lny = X X X X X X X X8 Predictor Coef StDev T P Constant X X X X X X X X S = R-Sq = 84.6% R-Sq(adj) = 81.9% Analysis of Variance Source DF SS MS F P Regression Residual Error Total Residuals Versus the Fitted Values (response is lny) Residual Fitted Value 7.5 9

10 Normal Probability Plot of the Residuals (response is lny) 2 Normal Score Residual Histogram of the Residuals (response is lny) Frequency Residual 10

11 Stepwise Regression F-to-Enter: 4.00 F-to-Remove: 4.00 Response is lny on 8 predictors, with N = 54 Step Constant X T-Value X T-Value X T-Value X T-Value 3.86 S R-Sq

12 Minitab commands for stepwise regression 12

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14 All possible Regressions Selection Procedure (6.3) p327 R-sq Criterion: 2 SSR SSE R = = 1 SST SST Response is lny Adj. X X X X X X X X Vars R-Sq R-Sq C-p s X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X 14

15 Best Subsets Regression Response is lny Adj. X X X X X X X X Vars R-Sq R-Sq C-p s X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X R-sq vars 15

16 Ex: Response is crimes p b o h d p t p s e o o 1 p g g v u t 8 o r r e n p - p a e r e Adj. o 3 6 d e t m Vars R-Sq R-Sq C-p s p 4 5 s s y p X X X X X X X X X X X X X X X X X X X X X X X X X X X X R-sq Vars

17 Other Criteria R-sq (Adj) 2) 2 MSE R = 1 Adj SST /( n 1) 3) C p criterion p328 Cp SSE p = + 2( p+ 1) n MSE k C p criterion selects as the best model, the subset model with 1) a small value of C p 2) value of C p near p + 1 (p is the number of predictors) 17

18 MINITAB commands 18

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