Phase I Analysis of Linear Profiles With Calibration Applications

1. Phase I Analysis of Linear Profiles With Calibration Applications 指導教授：童超塵 作者：Mahmoud A. Mahmoud and William H. Woodall 主講人：廖莉芳

2. Outline • Introduction • Phase I Approaches • An Alternative Approach • Performance Comparisons • Calibration Example • Conclusions

3. Introduction • For phase I applications, one compares the competing control chart methods in terms of the probability of deciding whether or not the process is stable. • We propose a method based on using indicator variables in a multiple regression model. • We study a method for phase I analysis based on the indicator variables technique combined with a control chart for detecting changes in variation about the regression line.

4. Introduction • We assume that k random samples from a historical data set. • For each sample, we assume that the model relating X to Y is：

5. Phase I Approaches • Stover and Brill(1998) proposed a method for a phase I linear profile calibration process： • The method is T2 chart： However, a more appropriate UCL is：

6. Phase I Approaches • Kang and Albin(2000) proposed two methods for monitoring of linear profiles： • The first method is T2 chart：

7. Phase I Approaches • The second method is an R-chat to be used in conjunction with EWMA chart. • Kim et al.(2003) recommend replacing the three EWMA charts by three Shewhart charts for monitoring the Y intercept, slope, and process standard deviation.

8. Phase I Approaches • For monitoring the B0： • For monitoring the B1： • For monitoring the σ2： (first)

9. An Alternative Approach • We apply the global F test in conjunction with a univariate control chart to check for the stability of the variation about the regression line. • Suppose we have k samples and we need to test the identity of the regression lines.

10. An Alternative Approach • Test the hypothesis：H0：β01=β02=…=β11=β12=…=β1k’=0H1： H0 is not true

11. An Alternative Approach • If an out-of-control is obtained from the F test, we code the X values such that the average coded value is 0 and applying 3-sigma control charts for the Y intercept and slope separately. • For the Y intercept： • For the slope： • Test for normality • Test for linearity：lack-of-fit F test

12. Performance Comparisons • Method A：T2 chart by Stover and Brill.Method B：T2 chart by Kang and Albin. Method C：3 Shewhart by Kim et al.Method D：F-test in conjunction with control limit in (15) • For k=20 and underlying in-control model with Yij=Xi+εij, i=1,2,…,n X=0, 0.2, 0.4, 0.6, 0.8,…, 1.8n=10, x-bar=0.9, Sxx=3.3 • For k=20, we considered fixed shifts in m out of k individual model parameters for m=2, 5, and10.

13. 0.4 0.2 0.18 0.02

17. Calibration Example • The purposewas to study the stability of the calibration curve in the photometric determination of Fe3+ with sulfosalicylic acid. • We have some diagnostics to check for the normality assumption and linearity of X and Y for each of the samples. • These suggested the reasonableness of the normality assumption of error terms.

18. Calibration Example • Use the lack-of-fit test for linearity. median max min

19. Calibration Example • Using a principal components analysis to identity the primary patterns of variation among the 22 calibration curves.

22. Calibration Example

23. Calibration Example • For method D, the control chart for monitoring the process variance is exactly the same as that of Figure 10. • LCL=0.0911 and UCL=3.66316 • F=75.8019, p-value=0 => reject H0 • We code X values and then applied separated 3-sigma control charts for intercept and slope. • For intercept, LCL=202.99 and UCL=205.4 • Only sample 19 is in control. • For slope, LCL=2.03 and UCL=2.064 • All samples are in control.

24. Conclusions • Method D is much more effective than the other methods in detecting shifts affecting much of phase I data. • For shifts for the slope and Y intercept affecting only a few samples of the phase I data, the method B and C both gave better results. • For shifts in the process standard deviation, method C is much more interpretable than method B. • We recommend using either method D or method C.