1. Chapter 12: Linear Regression 1

2. Introduction Regression analysis and Analysis of variance are the two most widely used statistical procedures. Regression analysis: –Description –Prediction –Estimation 2

3. 12.1 Simple Linear Regression 3 (12.1) (12.2)

4. 12.1 Simple Linear Regression 4

5. Table 12.1 Quality Improvement Data 5 MonthTime Devoted to Quality Impr. # of Non- conforming January5620 February5819 March5520 April6216 May6315 June6814 July6615 August6813 September7010 October6713 November729 December648

6. Figure 12.1 Scatter Plot 6

7. Figure 12.1a Scatter Plot 7

8. 12.1 Simple Linear Regression 8

9. The regression equation is Y = 55.9 - 0.641 X Predictor Coef SE Coef T P Constant 55.923 2.824 19.80 0.000 X -0.64067 0.04332 -14.79 0.000 S = 0.888854 R-Sq = 95.6% R-Sq(adj) = 95.2% Analysis of Variance Source DF SS MS F P Regression 1 172.77 172.77 218.67 0.000 Residual Error 10 7.90 0.79 Total 11 180.67 9

10. 12.1 Simple Linear Regression 10

11. 12.2 Worth of the Prediction Equation 11 ObsXYFitSE FitResidualSt Resid 156.020.00020.0460.464-0.046-0.06 258.019.00018.7650.3950.2350.30 355.020.00020.6870.500-0.687-0.93 462.016.00016.2020.286-0.202-0.24 563.015.00015.5610.270-0.561-0.66 668.014.00012.3580.2891.6421.95 766.015.00013.6390.2611.3611.60 868.013.00012.3580.2890.6420.76 970.010.00011.0770.338-1.077-1.31 1067.013.00012.9990.2720.0010.00 1172.09.0009.7950.400-0.795 1274.08.0008.5140.470-0.514-0.68

12. 12.2 Worth of the Prediction Equation 12 (12.4)

13. 12.3 Assumptions 13 (12.1)

14. 12.4 Checking Assumptions through Residual Plots 14

15. 12.4 Checking Assumptions through Residual Plots 15

16. 12.5 Confidence Intervals 16

17. 12.5 Hypothesis Test 17

18. 12.6 Prediction Interval for Y 18

19. 12.6 Prediction Interval for Y 19

20. 12.7 Regression Control Chart 20 (12.5) (12.6)

21. 12.8 Cause-Selecting Control Chart 21 The general idea is to try to distinguish between quality problems that occur at one stage in a process from problems that occur at a previous processing step. Let Y be the output from the second step and let X denote the output from the first step. The relationship between X and Y would be modeled.

22. 12.9 Linear, Nonlinear, and Nonparametric Profiles 22 Profile refers to the quality of a process or product being characterized by a (Linear, Nonlinear, or Nonparametric) relationship between a response variable and one or more explanatory variables. A possible way is to monitor each parameter in the model with a Shewhart chart. –The independent variables must be fixed –Control chart for R 2

23. 12.10 Inverse Regression 23 An important application of simple linear regression for quality improvement is in the area of calibration. Assume two measuring tools are available – One is quite accurate but expensive to use and the other is not as expensive but also not as accurate. If the measurements obtained from the two devices are highly correlated, then the measurement that would have been made using the expensive measuring device could be predicted fairly from the measurement using the less expensive device. Let Y = measurement from the less expensive device X = measurement from the accurate device

24. 12.10 Inverse Regression 24

25. 12.10 Inverse Regression Example 25 YX 2.32.4 2.52.6 2.42.5 2.82.9 2.93.0 2.62.7 2.42.5 2.22.3 2.12.22.7

26. 12.11 Multiple Linear Regression 26

27. 12.12 Issues in Multiple Regression 12.12.1 Variable Selection 27

28. 12.12.3 Multicollinear Data Problems occur when at least two of the regressors are related in some manner. Solutions: –Discard one or more variables causing the multicollinearity –Use ridge regression 28

29. 12.12.4 Residual Plots 29

30. 12.12.6 Transformations 30