Engineering Statistics Handbook Episode 6 Part 2 docx

In order to see more detail, we generate a full-size plot of the residuals versus the predictor variable, as shown above. This plot suggests that the assumption of homogeneous variances is now met. 4.6.2.4. Transformations to Improve Fit and Equalize Variances http://www.itl.nist.gov/div898/handbook/pmd/section6/pmd624.htm (6 of 6) [5/1/2006 10:22:40 AM] The fit output and plot from the replicate variances against the replicate means shows that the a linear fit provides a reasonable fit with an estimated slope of 1.69. Note that this data set has a small number of replicates, so you may get a slightly different estimate for the slope. For example, S-PLUS generated a slope estimate of 1.52. This is caused by the sorting of the predictor variable (i.e., where we have actual replicates in the data, different sorting algorithms may put some observations in different replicate groups). In practice, any value for the slope, which will be used as the exponent in the weight function, in the range 1.5 to 2.0 is probably reasonable and should produce comparable results for the weighted fit. We used an estimate of 1.5 for the exponent in the weighting function. Residual Plot for Weight Function 4.6.2.5. Weighting to Improve Fit http://www.itl.nist.gov/div898/handbook/pmd/section6/pmd625.htm (2 of 6) [5/1/2006 10:22:40 AM] The residual plot from the fit to determine an appropriate weighting function reveals no obvious problems. Numerical Output from Weighted Fit Dataplot generated the following output for the weighted fit of the model that relates the field measurements to the lab measurements (edited slightly for display). LEAST SQUARES MULTILINEAR FIT SAMPLE SIZE N = 107 NUMBER OF VARIABLES = 1 REPLICATION CASE REPLICATION STANDARD DEVIATION = 0.6112687111D+01 REPLICATION DEGREES OF FREEDOM = 29 NUMBER OF DISTINCT SUBSETS = 78 PARAMETER ESTIMATES (APPROX. ST. DEV.) T VALUE 1 A0 2.35234 (0.5431 ) 4.3 2 A1 LAB 0.806363 (0.2265E-01) 36. RESIDUAL STANDARD DEVIATION = 0.3645902574 RESIDUAL DEGREES OF FREEDOM = 105 REPLICATION STANDARD DEVIATION = 6.1126871109 4.6.2.5. Weighting to Improve Fit http://www.itl.nist.gov/div898/handbook/pmd/section6/pmd625.htm (3 of 6) [5/1/2006 10:22:40 AM] REPLICATION DEGREES OF FREEDOM = 29 This output shows a slope of 0.81 and an intercept term of 2.35. This is compared to a slope of 0.73 and an intercept of 4.99 in the original model. Plot of Predicted Values The plot of the predicted values with the data indicates a good fit. Diagnostic Plots of Weighted Residuals 4.6.2.5. Weighting to Improve Fit http://www.itl.nist.gov/div898/handbook/pmd/section6/pmd625.htm (4 of 6) [5/1/2006 10:22:40 AM] We need to verify that the weighting did not result in the other regression assumptions being violated. A 6-plot, after weighting the residuals, indicates that the regression assumptions are satisfied. Plot of Weighted Residuals vs Lab Defect Size 4.6.2.5. Weighting to Improve Fit http://www.itl.nist.gov/div898/handbook/pmd/section6/pmd625.htm (5 of 6) [5/1/2006 10:22:40 AM] In order to check the assumption of homogeneous variances for the errors in more detail, we generate a full sized plot of the weighted residuals versus the predictor variable. This plot suggests that the errors now have homogeneous variances. 4.6.2.5. Weighting to Improve Fit http://www.itl.nist.gov/div898/handbook/pmd/section6/pmd625.htm (6 of 6) [5/1/2006 10:22:40 AM] Conclusion Although the original fit was not bad, it violated the assumption of homogeneous variances for the error term. Both the fit of the transformed data and the weighted fit successfully address this problem without violating the other regression assumptions. 4.6.2.6. Compare the Fits http://www.itl.nist.gov/div898/handbook/pmd/section6/pmd626.htm (2 of 2) [5/1/2006 10:22:41 AM] 3. Fit and validate initial model. 1. Linear fit of field versus lab. Plot predicted values with the data. 2. Generate a 6-plot for model validation. 3. Plot the residuals against the predictor variable. 1. The linear fit was carried out. Although the initial fit looks good, the plot indicates that the residuals do not have homogeneous variances. 2. The 6-plot does not indicate any other problems with the model, beyond the evidence of non-constant error variance. 3. The detailed residual plot shows the inhomogeneity of the error variation more clearly. 4. Improve the fit with transformations. 1. Plot several common transformations of the response variable (field) versus the predictor variable (lab). 2. Plot ln(field) versus several common transformations of the predictor variable (lab). 3. Box-Cox linearity plot. 4. Linear fit of ln(field) versus ln(lab). Plot predicted values with the data. 5. Generate a 6-plot for model validation. 6. Plot the residuals against the predictor variable. 1. The plots indicate that a ln transformation of the dependent variable (field) stabilizes the variation. 2. The plots indicate that a ln transformation of the predictor variable (lab) linearizes the model. 3. The Box-Cox linearity plot indicates an optimum transform value of -0.1, although a ln transformation should work well. 4. The plot of the predicted values with the data indicates that the errors should now have homogeneous variances. 5. The 6-plot shows that the model assumptions are satisfied. 6. The detailed residual plot shows more clearly that the assumption of homogeneous variances is now satisfied. 4.6.2.7. Work This Example Yourself http://www.itl.nist.gov/div898/handbook/pmd/section6/pmd627.htm (2 of 3) [5/1/2006 10:22:41 AM] 5. Improve the fit using weighting. 1. Fit function to determine appropriate weight function. Determine value for the exponent in the power model. 2. Examine residuals from weight fit to check adequacy of weight function. 3. Weighted linear fit of field versus lab. Plot predicted values with the data. 4. Generate a 6-plot after weighting the residuals for model validation. 5. Plot the weighted residuals against the predictor variable. 1. The fit to determine an appropriate weight function indicates that a an exponent between 1.5 and 2.0 should be reasonable. 2. The residuals from this fit indicate no major problems. 3. The weighted fit was carried out. The plot of the predicted values with the data indicates that the fit of the model is improved. 4. The 6-plot shows that the model assumptions are satisfied. 5. The detailed residual plot shows the constant variability of the weighted residuals. 6. Compare the fits. 1. Plot predicted values from each of the three models with the data. 1. The transformed and weighted fits generate lower predicted values for low values of defect size and larger predicted values for high values of defect size. 4.6.2.7. Work This Example Yourself http://www.itl.nist.gov/div898/handbook/pmd/section6/pmd627.htm (3 of 3) [5/1/2006 10:22:41 AM] [...]... http://www.itl.nist.gov/div898 /handbook/ pmd/section6/pmd631.htm (1 of 6) [5/1 /20 06 10 :22 :41 AM] 4 .6. 3.1 Background and Data 21 .4000 29 .1750 22 . 125 0 17.5 125 14 .25 00 9.4500 9.1500 7.9 125 8.4750 6. 1 125 80.0000 79.0000 63 .8000 57 .20 00 53 .20 00 42. 5000 26 . 8000 20 .4000 26 . 8500 21 .0000 16. 4 62 5 12. 525 0 10.5375 8.5875 7. 125 0 6. 1 125 5. 9 62 5 74.1000 67 .3000 60 .8000 55.5000 50.3000 41.0000 29 .4000 20 .4000 29 . 3 62 5 21 .1500 16. 7 62 5 13 .20 00 10.8750 8.1750... http://www.itl.nist.gov/div898 /handbook/ pmd/section6/pmd631.htm (2 of 6) [5/1 /20 06 10 :22 :41 AM] 4 .6. 3.1 Background and Data 32. 5000 12. 4100 13. 120 0 15. 560 0 5 .63 00 78.0000 59.9000 33 .20 00 13.8400 12. 7500 14 . 62 00 3.9400 76. 8000 61 .0000 32. 9000 13.8700 11.8100 13.3100 5.4400 78.0000 63 .5000 33.8000 12. 560 0 5 .63 00 12. 7500 13. 120 0 5.4400 76. 8000 60 .0000 47.8000 32. 0000 22 .20 00 22 .5700 18. 820 0 13.9500 11 .25 00 9.0000 6. 6700 75.8000 62 .0000 48.8000 35 .20 00 20 .0000... [5/1 /20 06 10 :22 :41 AM] 4 .6. 3.1 Background and Data 8.7400 80.7000 61 .3000 47.5000 29 .0000 24 .0000 17.7000 24 . 560 0 18 .67 00 16. 24 00 8.7400 7.8700 8.5100 66 .7000 59 .20 00 40.8000 30.7000 25 .7000 16. 3000 25 .9900 16. 9500 13.3500 8 . 62 00 7 .20 00 6. 6400 13 .69 00 81.0000 64 .5000 35.5000 13.3100 4.8700 12. 9400 5. 060 0 15.1900 14 . 62 00 15 .64 00 25 .5000 25 .9500 81.7000 61 .60 00 29 .8000 29 .8100 17.1700 10.3900 28 .4000 28 .69 00... 5. 9 62 5 5 . 62 50 81.5000 62 .4000 2. 2500 1.7500 2. 2500 2. 7500 3 .25 00 3.7500 4 .25 00 4.7500 5 .25 00 5.7500 0.5000 0 . 62 50 0.7500 0.8750 1.0000 1 .25 00 1.7500 2. 2500 1.7500 2. 2500 2. 7500 3 .25 00 3.7500 4 .25 00 4.7500 5 .25 00 5.7500 0.5000 0 . 62 50 0.7500 0.8750 1.0000 1 .25 00 1.7500 2. 2500 1.7500 2. 2500 2. 7500 3 .25 00 3.7500 4 .25 00 4.7500 5 .25 00 5.7500 0.5000 0.7500 http://www.itl.nist.gov/div898 /handbook/ pmd/section6/pmd631.htm... http://www.itl.nist.gov/div898 /handbook/ pmd/section6/pmd631.htm (3 of 6) [5/1 /20 06 10 :22 :41 AM] 4 .6. 3.1 Background and Data 10. 420 0 7.3100 7. 420 0 70.5000 59.5000 48.5000 35.8000 21 .0000 21 .67 00 21 .0000 15 .64 00 8.1700 8.5500 10. 120 0 78.0000 66 .0000 62 .0000 58.0000 47.7000 37.8000 20 .20 00 21 .0700 13.8700 9 .67 00 7. 760 0 5.4400 4.8700 4.0100 3.7500 24 .1900 25 . 760 0 18.0700 11.8100 12. 0700 16. 120 0 70.8000 54.7000 48.0000 39.8000 29 .8000 23 .7000 29 . 62 00 23 .8100... 92. 9000 0.5000 78.7000 0 . 62 50 64 .20 00 0.7500 64 .9000 0.8750 57.1000 1.0000 43.3000 1 .25 00 31.1000 1.7500 23 .60 00 2. 2500 31.0500 1.7500 23 .7750 2. 2500 17.7375 2. 7500 13.8000 3 .25 00 11.5875 3.7500 9.4 125 4 .25 00 7. 725 0 4.7500 7.3500 5 .25 00 8. 025 0 5.7500 90 .60 00 0.5000 76. 9000 0 . 62 50 71 .60 00 0.7500 63 .60 00 0.8750 54.0000 1.0000 39 .20 00 1 .25 00 29 .3000 1.7500 http://www.itl.nist.gov/div898 /handbook/ pmd/section6/pmd631.htm... 12. 0700 4.0000 5.0000 6. 0000 0.5000 0.7500 1.0000 1.5000 2. 0000 2. 0000 2. 5000 3.0000 4.0000 5.0000 6. 0000 0.5000 0 . 62 50 0.7500 0.8750 1.0000 1 .25 00 2. 2500 2. 2500 2. 7500 3 .25 00 3.7500 4 .25 00 4.7500 5 .25 00 5.7500 3.0000 3.0000 3.0000 3.0000 3.0000 3.0000 0.5000 0.7500 1.0000 1.5000 2. 0000 2. 5000 2. 0000 2. 5000 3.0000 4.0000 5.0000 http://www.itl.nist.gov/div898 /handbook/ pmd/section6/pmd631.htm (4 of 6) ... 6. 0000 0.5000 0.7500 1.0000 1.5000 2. 0000 2. 5000 2. 0000 2. 5000 3.0000 4.0000 5.0000 6. 0000 0.5000 0.7500 1.0000 1.5000 2. 0000 2. 5000 2. 0000 2. 5000 3.0000 4.0000 5.0000 6. 0000 3.0000 0.5000 0.7500 1.5000 3.0000 6. 0000 3.0000 6. 0000 3.0000 3.0000 3.0000 1.7500 1.7500 0.5000 0.7500 1.7500 1.7500 2. 7500 3.7500 1.7500 1.7500 http://www.itl.nist.gov/div898 /handbook/ pmd/section6/pmd631.htm (5 of 6) [5/1 /20 06. .. 1.7500 1.7500 http://www.itl.nist.gov/div898 /handbook/ pmd/section6/pmd631.htm (5 of 6) [5/1 /20 06 10 :22 :41 AM] 4 .6. 3.1 Background and Data 81.3000 60 .9000 16. 6500 10.0500 28 .9000 28 .9500 0.5000 0.7500 2. 7500 3.7500 1.7500 1.7500 http://www.itl.nist.gov/div898 /handbook/ pmd/section6/pmd631.htm (6 of 6) [5/1 /20 06 10 :22 :41 AM] ... 20 .0000 20 . 320 0 19.3100 12. 7500 1.5000 3.0000 3.0000 3.0000 6. 0000 0.5000 0.7500 1.5000 3.0000 3.0000 3.0000 6. 0000 0.5000 0.7500 1.5000 3.0000 3.0000 3.0000 6. 0000 0.5000 0.7500 1.5000 3.0000 6. 0000 3.0000 3.0000 6. 0000 0.5000 0.7500 1.0000 1.5000 2. 0000 2. 0000 2. 5000 3.0000 4.0000 5.0000 6. 0000 0.5000 0.7500 1.0000 1.5000 2. 0000 2. 0000 2. 5000 3.0000 http://www.itl.nist.gov/div898 /handbook/ pmd/section6/pmd631.htm . 5 .25 00 6. 1 125 5.7500 80.0000 0.5000 79.0000 0 . 62 50 63 .8000 0.7500 57 .20 00 0.8750 53 .20 00 1.0000 42. 5000 1 .25 00 26 . 8000 1.7500 20 .4000 2. 2500 26 . 8500 1.7500 21 .0000 2. 2500 16. 4 62 5 2. 7500 . 1.7500 20 .4000 2. 2500 29 . 3 62 5 1.7500 21 .1500 2. 2500 16. 7 62 5 2. 7500 13 .20 00 3 .25 00 10.8750 3.7500 8.1750 4 .25 00 7.3500 4.7500 5. 9 62 5 5 .25 00 5 . 62 50 5.7500 81.5000 0.5000 62 .4000 0.7500 4 .6. 3.1 Data http://www.itl.nist.gov/div898 /handbook/ pmd/section6/pmd631.htm (1 of 6) [5/1 /20 06 10 :22 :41 AM] 21 .4000 2. 2500 29 .1750 1.7500 22 . 125 0 2. 2500 17.5 125 2. 7500 14 .25 00 3 .25 00 9.4500 3.7500 9.1500 4 .25 00 7.9 125 4.7500