Distribution Plots

The first thing that your team wants to know is how the process behaved, in terms of Ys, over the six-week period reviewed by the previous crisis team. But you first run a Distribution analysis for all of the variables (except Batch Number). In your training, you learned that this is an important first step in any data analysis.

You review the resulting plots. Recall that, in the VOC analysis, you learned that MFI should fall between 192 and 198, and CI should be 80 or higher. The histogram for MFI shows that it varies far beyond the desired limits (Exhibit 9.13). The histogram for CI shows a significant percentage of values below 80.

Of particular interest are the Yield values that fall below 85 percent. Select these in the box plot to the right of the Yield histogram. To do this, click and drag a rectangle that includes these points as shown in Exhibit 9.13.

This action selects the corresponding rows in the data table—check to see that 14 rows have been selected. Consequently, the values corresponding to these 14 low-yielding batches are highlighted in the histograms for all of the variables. For example, you see that the 14 crisis Yield batches have very low SA values. To remove the selection of the 14 rows, select Rows > Clear Row States.

Looking at the histograms, one of the team members points out that the distribution of CI is not bell-shaped. You agree and explain that this is not unusual, given that CI has a natural upper bound of 100 percent. The team is now aware that they must keep this in mind when using certain statistical techniques, such as individual measurement control charts and capability analysis, both of which assume that measured values are normally distributed.

Control Charts

At this point, you are eager to see how the three Ys behave over time. To see this you construct control charts.

For the most part, MFI seems to be stable. On the Individuals chart for MFI, there are no points outside the control limits (results not shown). But what is troubling is that MFI averages 198.4 over this time period. The team's VOC analysis indicated that 198 is the upper specification limit for MFI!

The control chart for CI immediately shows the problem with applying a control chart based on normally distributed data to highly skewed data (Exhibit 9.14). The control limits do not reflect the skewed distribution. Nonetheless, there are indications of special causes. The plot also shows some very large and regular dips in CI. There are many excursions below the lower specification limit of 80. This leaves the team members puzzled, especially because the dips do not align with the one crisis period that is so evident in the Yield control chart (Exhibit 9.15).

You briefly consider the idea of running capability analyses for these three responses. However, the Individuals chart clearly shows that MFI is not capable, since the process average exceeds the upper specification limit. Each of CI and Yield is affected by special causes. The corresponding points could be removed prior to running capability analyses, but it is not all that easy to tell exactly which points result from special causes. The team decides that there is not much to gain by running formal capability analyses on these data.

Now you turn your attention to the crisis team's modeling efforts. Reading through the crisis team's documentation, you see that the team analyzed the data using multiple regression. The crisis team hoped to determine which Xs had a significant effect on the three key responses. The team identified M% and Viscosity as being significantly related to MFI, but it did not find any of the Xs to be related to CI.

This last result seems especially curious. You realize that there are several reasons that such an analysis might lead to no significant factors:

• One or more key Xs are missing.
• One or more higher-order terms involving the specified Xs are missing.
• Measurement variation in the Ys or Xs is too large and is masking the systematic patterns in the Ys caused by process variation in the Xs.

With this as background, you meet with members of the crisis team to discuss the reasoning that led to their final choice of Xs and Ys. They convince you that they did not overlook any critical Xs. After this meeting, you reanalyze their data, introducing higher-order terms. This uncovers some significant relationships, but they don't seem conclusive in terms of the process, especially as they relate to CI. At this point, you have a strong suspicion that measurement variation may be clouding results.

You meet with your team to determine how to proceed. Your team members agree that it is possible that measurement variation in the Ys or Xs could be large relative to the systematic variation caused by the Xs. They fully support your proposal to assess the magnitude of the measurement variation.

Properties of a Good Measurement System

You meet with the technicians and their manager to discuss the desired consistency of the measurement process and to enlist their support for an MSA. You discuss the fact that, for characteristics with two-sided specification limits, a guideline that is often used is that the measurement system range, measured as six standard deviations, should take up at most 10 percent of the tolerance range, which is the difference between the upper and the lower specification limits. Since the upper and lower specification limits for MFI are 198 and 192, respectively, the guideline would thus require the measurement system range not to exceed 10% × (198 – 192) = 0.6 MFI units.²

You also mention that there are guidelines as to how precise the measurement system should be relative to part-to-part, or process, variation: The range of variability of a highly capable measurement system should not exceed 10 percent of the part-to-part (or, in this case, the batch-to-batch) variability.

Designing the MSA

Given these guidelines, you suggest that an MSA for MFI might be useful, and the technicians and their manager agree. You learn from the technicians that the test is destructive. MFI is reported in units of grams per ten minutes. The protocol calls for the test to run over a half-hour period, with three measurements taken on each sample at prescribed times, although due to other constraints in the laboratory the technicians may not always be available precisely at these set times. Each of these three measurements is normalized to a ten-minute interval, and the three normalized values are averaged. From preparation to finish, a test usually takes about 45 minutes to run.

Using this information, you design the structure for the MSA. Since the test is destructive, true repeatability of a measurement is not possible. However, you reason, and the technicians agree, that a well-mixed sample from a batch can be divided into smaller samples that can be considered identical. The three technicians who perform the test all want to be included in the study, and they also want to have all four instruments included. You suggest that the MSA should be conducted using samples from three randomly chosen batches of polymer and that the technicians make two repeated measurements for each batch and instrument combination.

This leads to 72 tests: 3 batches × 3 technicians × 4 instruments × 2 measurements. Since each test is destructive, a sample from a given batch of polymer will have to be divided into 24 aliquots for testing. For planning purposes, it is assumed that the MSA design will permit three tests to be run per hour, on average, using three of the four instruments. This leads to a rough estimate of 24 hours for the total MSA. With other work intervening, the technicians conclude that they can finish the MSA comfortably in four or five workdays.

Designing the MSA Experiment

You construct the designed MSA experiment using JMP's design of experiments (DOE) capabilities.

The design table, partially shown in Exhibit 9.19, appears. Most likely, your table will be different because the run order is randomized.

Note that the Full Factorial Design dialog remains open. This is useful in case changes need to be made to the design that has been generated. In fact, to obtain the precise run order shown in Exhibit 9.19, go back to the Full Factorial Design window. There, select Set Random Seed from the red triangle menu, enter 123, and click OK. Then select Make Table.

When JMP creates the data table, it automatically includes two data table scripts, Model and DOE Dialog. The Model script specifies a model that you can fit, while the DOE Dialog script allows you to recreate the dialog that generated the data table.

Since the purpose of the design is to collect data for a MSA, you decide that the Model script is not necessary and so you delete it. You retain the DOE Dialog script in case you want to set up another study like this one.

As mentioned earlier, the runs are randomized. You stress the importance of running the trials in this order. The technicians conduct the experiment over the course of the next week and enter their results into the data table.

Conducting the Analysis

Your team regroups to analyze the data. The table that contains the design and results is called MSA_MFI_Initial.jmp.

Interpreting the Results—The Average and Range Charts

The report displays an Average Chart and a Range Chart (Exhibit 9.21). The Average Chart plots the average of the repeated MFI measurements and displays control limits calculated from the variation within subgroups. To get a better understanding, select Show Data from the Average Chart's red triangle menu to display the individual data points. The control limits are those for an X-bar R chart using the two-measurement subgroups. This means that the limits are for repeatability variation: the variability when the same Batch is measured by the same Operator using the same Instrument.

In a good measurement process, the measurement variability is small relative to the part—in this case, Batch—variation. The measurement system allows you to distinguish different parts. For a good measurement process, almost all averages should fall outside the repeatability control limits. But this is not the case here, indicating that the repeatability variation is large compared to the Batch variation.

The Range Chart shows that the repeatability variation is stable. Although this is a positive result, the level at which the process is functioning is not acceptable. The Average Range indicates that the mean difference between two measurements of the same batch by the same operator using the same instrument is 1.1036. The UCL indicates that this could be as large as 3.6050.

Given the fact that the MFI measurement system variability should not exceed 0.6 units, this signals a big problem. A sample that falls within the specification limits can easily give a measured MFI value that falls outside the specification limits. For example, when Bob used Instrument D to measure Batch 3, one of his measurements, 196.42 (see row 9 of the data table), indicated that the batch fell within the specification limits (192 to 198). But another measurement, 199.44 (row 62), indicated that it was not within the specification limits. The variability in measurements made on the same batch by the same operator with the same instrument is too large to permit accurate assessment of whether product is acceptable.

Interpreting the Results—Parallelism Plots

For additional graphical insight, select Parallelism Plots from the Measurement Systems Analysis red triangle menu (Exhibit 9.22). The Parallelism plots show the Mean MFI values across Batch for both Operator and Instrument. These plots help you see Operator differences across batches, Instrument differences across batches, and possible interactions between Operator and Batch or Instrument and Batch. Note the following:

• The Operator plot suggests that Janet's measurements have higher MFI than Bob's. The means of their measurements for the same batches differ by at least 0.5 units.
• It appears that there might be an interaction between Instrument and Batch. By an interaction between Instrument and Batch, denoted Instrument*Batch, we mean that instruments behave differently across batches. Here, Instrument C seems to give higher readings for Batch 2 and Batch 3 than would be expected based on the other instruments.

Interpreting the Results—Variance Components

For a more complete picture of the sources of measurement variation, select EMP Gauge RR Results from the Measurement Systems Analysis red triangle menu. This report (Exhibit 9.23) shows the standard deviations and variance components associated with the components of the measurement process.

A variance component is an estimate of a variance. The variance components listed in EMP Gauge R&R Results give estimates of the variance in MFI values due to:

• Repeatability: Repeated measurements of the same part by the same operator with the same instrument.
• Reproducibility: Repeated measurements of the same part by different operators using different instruments.
• Product Variation: Differences in the parts used in the MSA; here parts are represented by batches.
• Interaction Variation: Differences due to the interaction components. Here the interaction components are Operator*Instrument, Operator*Batch, and Instrument*Batch.

The bar graph in the EMP Gauge R&R Results panel shows that the Gauge R&R variance component (1.81) is very large compared to the Product Variation (Part-to-Part) variance component (0.09). This suggests that the measurement system has difficulty distinguishing batches. A control chart monitoring this process is mainly monitoring measurement variation. The bar graph also indicates that the repeatability and reproducibility variances, where Instrument variation is included in the latter, are large and very close in magnitude, again pointing out that both must be addressed.

The % of Total column shows the following:

• Both Reproducibility and Repeatability variation stand out as the large contributors, accounting for 39.9 percent and 46.7 percent of the variation, respectively.
• The combined effect of Reproducibility and Repeatability variation is given by the Gauge R&R component, and is 86.5 percent of the observed variation when measuring the three batches.
• Interaction Variation accounts for 9.3 percent of the total variation.
• The Batch variation (Product Variation) accounts for only 4.2 percent of the total variation.

Note the following:

• The variance components for Repeatability and Reproducibility sum to the variance component for Gauge R&R, which is an estimate of the measurement process variance.
• The Total Variation variance component is the sum of the Gauge R&R, Product Variation, and Interaction Variation variance components.
• The individual interaction components are given in the Variance Components report, which you can select from the red triangle menu.
• The Std Dev values are the square roots of the variance components.
• The % of Total is calculated from the Variance Component column.
• In an MSA, we are typically not directly interested in part-to-part variation. In fact, we often intentionally choose parts that represent a range of variability.

Recall that the measurement system range, measured as six standard deviations, should take up at most 10 percent of the tolerance range. Since the upper and lower specification limits for MFI are 198 and 192, respectively, the guideline requires that the measurement system range not exceed 10% × (198 – 192) = 0.6 MFI units.

The Std Dev column in the report indicates that the measurement system standard deviation is 1.344. Six times 1.344 is 8.064 MFI units! The measurement variation vastly exceeds the desired precision relative to the specification limits.

Also, the range of variability of a highly capable measurement system should not exceed 10 percent of the Product Variation. Here the Batch standard deviation is 0.295 and the Gauge R&R standard deviation is 1.344.

Interpreting the Results—Intraclass Correlation

Select EMP Results from the Measurement Systems Analysis red triangle menu. This gives you the report in Exhibit 9.24. The intraclass correlation (with bias and interactions) is defined as the variance component of Product Variation divided by the variance component of Total Variation. In other words, it is the part variance divided by the sum of the part variance and measurement error variance. When the measurement error is zero, the intraclass correlation is one. When the measurement error variation is large compared to the part variation, the intraclass correlation is small.

The EMP Results report indicates that the Intraclass Correlation is 0.0418, assuming that you include the reproducibility (bias) and interaction variation inherent in the measurement process. In other words, actual Batch variation (0.087) is only about 4 percent of the variation that is measured (2.088).

The EMP Results report goes on to classify this measurement system as a Fourth Class monitor.³ This means that even a large shift in the true Batch measurements is unlikely to be detected quickly by a process control chart. For example, the probability of detecting a large three-sigma shift in the true Batch values in the next ten subgroups, using the single test of a point beyond the control limits, is very low (between 0.03 and 0.40, see the Monitor Classification Legend at the bottom of Exhibit 9.24).

Designing the MSA

A single dedicated oven is used for the test. Three instruments are available within the laboratory to perform the weight measurements, and there are six different technicians who do the testing. An MSA involving all six technicians would be too time-consuming, so three of the technicians are randomly chosen to participate. You design a study using samples from three randomly chosen batches of polymer. These will be measured on each of the three instruments by each of the three technicians. Again, two repetitions of each measurement will be taken.

Your design results in a total of 54 tests, with each technician performing 18 tests. At an hour per test, this will take each technician about 18 hours in total; however, the technicians can complete other work while the samples are in the oven. The laboratory manager agrees that this is an acceptable amount of time. The constraint on the duration of the study is the oven, which can be used for only one sample at a time. So, a two-week period is designated for the study, with the intent that Xf samples will be worked in between other ash tests performed as part of the laboratory's regular work. To come as close to true repeated measurements as possible, a single sample is taken from each of the three batches of polymer, and the resulting sample is divided into the 18 required aliquots.

You construct the design in a fashion similar to the way you constructed the design for MFI. The Xf study is completely randomized. Run the script DOE Dialog in the data table MSA_Xf_Initial.jmp to review how you designed the study.

Conducting the Analysis

The study is conducted and the results are recorded in the data table MSA_Xf_Initial.jmp.

The Average Chart, shown in Exhibit 9.25, indicates that Batches are not distinguished by the measurement process. Individual Xf measurements range from about 12 to 18. To see this, select Show Data from the Average Chart's red triangle menu. The Range Chart shows that repeated measurements on the same batch can differ by up to 2.96 units, almost half of the range of the observed data.

The EMP Gauge R&R Results are shown in Exhibit 9.26. Repeatability variation is on a par with Product Variation, and Reproducibility variation is more than double the Product Variation.

Your team members observe that, given the variability in readings, it is impossible to differentiate the batches. For example, a team member points out that measurements made by one of the technicians, Eduardo, using Instrument B do not distinguish among the three batches. Repeatability variation seems large as well. Consider Eduardo's two measurements of Batch 2 using Instrument C—click on the point in the Average Chart and view the selected points in the data table. (See Exhibit 9.27.) The two measurements differ by about 2.5 units. Moreover, it appears that measurements made with Instrument A are systematically lower than those made with the other two instruments.

Since no tolerance range for Xf has ever been determined, whether Xf is being measured with enough precision is determined by whether measurements can distinguish different batches, in this case the three batches that were used for the study. (In the production setting, two or three measurements are typically taken, which increases precision. But your intent in this MSA is to estimate the precision of a single measurement.)

The EMP Gauge R&R Results indicate that the Gauge R&R variance component (1.27) is much larger than the Product Variation variance component (0.38). As was the case for MFI, this is indicative of a measurement system in trouble. The Reproducibility and Repeatability variance components are both large. The Gauge R&R standard deviation (1.1278) is almost twice the Product Variation standard deviation (0.614).

Select EMP Results from the Measurement Systems Analysis red triangle menu (Exhibit 9.28). The EMP Results report indicates that the Intraclass Correlation (with bias and interactions) is 0.2187. This means that actual Batch variation accounts for only about 22 percent of the variation that is measured.

The EMP Results report also indicates that this measurement system is a Third Class monitor. The probability of detecting a three-sigma shift in the true Batch values in the next ten subgroups, using the single test of a point beyond the control limits, is only between 0.40 and 0.88.

Note that the variance components given in the EMP Gauge R&R Results report can also be obtained using Analyze > Quality and Process > Variability / Attribute Gauge Chart or using Fit Model. In the MSA_Xf_Initial.jmp data table, the scripts Variability Chart and Var Comps Model (see the Random Effects tab) illustrate how this is done. Although variances are never negative, sometimes their estimates are negative. Note that, using the script Var Comps Model, the variance components are estimated using the bounded method so that negative components are set to zero.

Follow-Up for MFI

The follow-up MSA for MFI has the same structure as the initial study. The results are given in the table MSA_MFI_Final.jmp. The Average and Range Charts are shown in Exhibit 9.29 and the EMP Gauge R&R Results are given in Exhibit 9.30. The script is EMP Analysis.

The team members and technicians look at the Average Chart and are delighted! Measurements clearly distinguish the three parts. All measurements fall beyond the control limits and, for a given batch, they are very close. The Range Chart indicates that an upper control limit on the range of measurements is 0.4083. There is very little repeatability or reproducibility variation evident.

The Gauge R&R Std Dev, given in the EMP Gauge R&R Results report, is 0.105 (Exhibit 9.30). This means that the measurement system only takes up about 6(0.105) = 0.63 units, which is almost exactly 10 percent of the tolerance range (recall that the specification limits are 192 and 198). The measurement system is now sufficiently reliable in classifying batches as good or bad relative to the specification limits on MFI.

Also, note that the Intraclass Correlation (with bias and interactions) is 0.9954 (Exhibit 9.30). This value indicates that there is very little measurement variation relative to batch variation. The EMP Results report also indicates that the current system is a First Class monitor. This means that a control chart for MFI will signal within ten subgroups with probability of at least 0.99, based on a single point beyond the control limits, if there is shift in the mean that exceeds three standard errors.

Shift Detection Profiler

You can investigate the sensitivity of a control chart for MFI in more detail by selecting the Shift Detection Profiler from the topmost red triangle menu in the report. The Shift Detection Profiler, which estimates the probability of detecting shifts in the process mean, appears as shown in Exhibit 9.31 (MSA_MFI_Final.jmp).

The control limit calculations for the chart include the sources of measurement variation. For the part variation, the control limit calculations use the In Control Part Std. Dev. This is initially set to the part standard deviation as estimated from parts used in the MSA. But because parts for MSAs are often not selected at random from process output, you can set the In Control Part Std Dev to an appropriate value using a red triangle option.

The Profiler shows six cells:

• Number of Subgroups: The number of subgroups over which the probability of a warning is computed. This is set to 10 by default.
• Part Mean Shift: The shift in the part mean that you want the control chart to detect. The initial value is set to one standard deviation of the part variation estimated by the MSA analysis. (See Product Variation in the EMP Gauge R&R Results report, Exhibit 9.32.)
• Part Std Dev: The value of the part standard deviation for new points. The initial value is set to one standard deviation of the part variation estimated by the MSA. (See Exhibit 9.32.) You can set the Part Std Dev value to reflect changes in the process.
• Bias Factors Std Dev: The standard deviation of factors related to reproducibility, including operator and instrument variability. (See Exhibit 9.32.)
• Test-Retest Std Dev: The standard deviation of the test-retest, or repeatability, variation in the model. The initial value is the standard deviation of the Repeatability component estimated by the MSA. (See Exhibit 9.32.)
• Subgroup Size: The sample size for each subgroup. This is set to 1 by default.

The initial settings of the Shift Detection Profiler are shown in Exhibit 9.31. These settings indicate that, given the current process, the probability of detecting a one standard deviation shift in the mean in the next 10 subgroups, using an individual measurements control chart (Subgroup Size = 1), is about 0.205 (Probability of Warning = 0.204815).

What if you were to monitor the process with an Xbar and S chart using a Subgroup Size of 5? Slide the vertical bar in the rightmost cell to 5. Then the probability of seeing a warning in the next 10 subgroups is 0.917. (See Exhibit 9.33.)

You can also explore other scenarios, such as the consequences of reducing part or measurement variation. What if you were able to reduce the In-Control Part Std Dev to 1.2? Select the option to Change In-Control Part Std Dev from the red triangle menu for the Profiler. Also, click above Part Std Dev in the third cell and set that standard deviation to 1.2. Change the Subgroup Size to 4. (See Exhibit 9.34.) You learn that the probability of detecting the 1.5374 mean shift in the next 10 subgroups, using an X-bar chart based on subgroups of size 4, is about 0.981.

The Shift Detection Profiler is a versatile tool, enabling you to interactively explore various scenarios relating to further changes or improvements in how MFI is measured, how the process changes, and how the chart itself is constructed. See Help > Books > Quality and Process Methods for more details.

Follow-Up for Xf

As for Xf, the follow-up MSA is conducted with the three technicians who were not part of the original study. The results are given in MSA_Xf_Final.jmp. The script is EMP Analysis.

The Average and Range Charts show that the measurement system now clearly distinguishes among the parts. The EMP Gauge R&R Results are shown in Exhibit 9.35. The bar graph shows that most of the variation is due to the product.

Once again, the team members and technicians are pleased. Compared to the variability among the three batches, there is very little repeatability or reproducibility variation.

The Gauge R&R Std Dev is 0.034, compared to the Product Variation Std Dev of 0.292. The EMP Results report indicates that the Intraclass Correlation (with bias and interactions) is 0.982, indicating that 98.2 percent of the observed variation is due to part (results not shown).

To ensure that both the MFI and Xf measurement systems continue to operate at their improved levels, measurement control systems are introduced in the form of periodic checks, monthly calibration, semiannual training, and annual MSAs.

Distribution Plots

Your first step is to run a Distribution analysis for all of the variables except Batch Number.

The first five histograms are shown in Exhibit 9.37 (script is Distribution). You note the following:

• MFI appears to have a mound-shaped distribution, except for some values of 206 and higher.
• CI is, as expected, left-skewed.
• Yield is also left-skewed, and may exhibit some outliers in the form of low values.

You select those points that reflect low Yield values, specifically, those five points that fall below 60 percent, by clicking and drawing a rectangle that includes these points using the arrow tool inside the box plot area. The highlighting in the other histograms in Exhibit 9.38 indicates that these five very low Yield rows correspond to four very high MFI values, one very low CI value, and generally low SA values. This is good news, since it is consistent with knowledge that the crisis team obtained. It also suggests that crisis yields are related to Ys, such as MFI and CI, and perhaps influenced by Xs, such as SA. Interestingly, though, four of these rows have CI values that exceed the lower specification limit of 80.

A striking aspect of the histograms for MFI and CI is the relationship of measurements to the specification limits. Recall that MFI has lower and upper specification limits of 192 and 198, respectively, and that CI has a lower specification limit of 80.

From the Quantiles panel for MFI, notice that all 110 observations exceed the lower specification limit of 192 and that about 50 percent of these exceed the upper specification of 198. From the Quantiles panel for CI, notice that about 25 percent of CI values fall below the lower specification of 80. To see how often both MFI and CI meet the specification limits, you use the Data Filter.

The completed Data Filter dialog is shown in Exhibit 9.39. The Data Filter panel indicates that only 34 rows match the conditions you have specified. This means that only 34 out of 110 batches conform to the specifications for both responses.

All rows that meet both specification limits are highlighted in all of the plots obtained using the Distribution platform (Exhibit 9.40). As you review the plots, you notice that the batches that meet the joint MFI and CI specifications tend to result in Yield values of 85 percent and higher. They also tend to be in the upper part of the SA distribution and the lower part of the M% and Ambient Temp distributions. This suggests that there might be relationships between the Xs and these two Ys.

Before you close the Data Filter dialog, you click the Clear button at the top of the dialog to remove the Select row states imposed by the Data Filter.

Control Charts

With this as background, the team proceeds to see how the three Ys behave over time. As before, you construct individual measurement charts for these three responses using the Control Chart Builder. But, this time, you start by constructing an Individuals chart for MFI and then use the Column Switcher to construct charts for CI and Yield.

These control charts are quite informative (see Exhibit 9.42). You learn that the average yield is roughly 87 percent, and that there are five batches below the lower control limit. Four of these are consecutive batches. You see that the grouping of four batches with low yields greatly exceed the upper control limit for MFI.

From the control chart for CI you learn that the fifth yield outlier, corresponding to row 14, falls below the lower control limit. However, you are reminded that the control limits for CI are suspect because of the extreme non-normality of its distribution.

You decide that it might be a good idea to assign markers to the five batches that are Yield outliers in order to identify them in subsequent analyses.

Exhibit 9.42 shows the three control charts with the markers you constructed for the five crisis batches.

Relationships between the Ys

MFI and Yield

To explore the relationship between MFI and Yield, you use Graph Builder.

The points and a default smoother are plotted (Exhibit 9.43). If you wish, you can deselect the smoother by clicking the second icon from the left at the top of the plot before you click Done (or, right-click on the graph and select Smoother > Remove).

The plot suggests that there is a strong nonlinear relationship between these two Ys. Four of the five outliers clearly suggest that high MFI values are associated with crisis level yields. You recognize the outlier with a Yield of about 40 as row 14, the point with the low CI value identified on the control chart for CI. (If you hover over this point, its row number, MFI, and Yield will appear.)

CI and Yield

You follow the instructions above, replacing MFI with CI, to construct a plot of Yield versus CI (see Exhibit 9.44—the script is Graph Builder—Yield and CI).

The plot shows a general tendency for Yield to increase as CI increases. Four of the five outliers have high values of CI. The outlier with the low value of CI is row 14.

MFI and CI

Now you investigate the relationship between MFI and CI. You construct the plot shown in Exhibit 9.45, replacing Yield with MFI (the script is Graph Builder—MFI and CI).

MFI and CI seem to have a weak positive relationship. There may be process factors that affect both of these Ys jointly. In the next section, we look at relationships between the process factors and these responses.

Relationships between Ys and Xs

As a more efficient way to view bivariate relationships, you decide to create a scatterplot matrix of all responses (Ys) with all factors (Xs), shown in Exhibit 9.46.

The matrix shows a scatterplot for each Y and X combination, including scatterplots that involve the two nominal variables, Quarry and Shift. For the nominal values involved in these scatterplots, the points are jittered randomly within the appropriate level. Density ellipses assume a joint bivariate normal distribution, so they are not shown for the plots involving Quarry and Shift because these factors are nominal.

The five outliers appear prominently in the MFI and Yield scatterplots. In fact, they affect the scaling of the plots, making it difficult to see other relationships. To better see the other points, select the five points in one of the Yield plots, then right-click in an empty part of the plot and choose Row Hide and Exclude from the menu that appears. (Alternatively, run the script Hide and Exclude Outliers.) Notice the following:

• Excluding the points causes the ellipses to be automatically recalculated.
• Hiding the points ensures that they are not shown in any of the plots.

Check the Rows panel in the data table to verify that the five points are excluded and hidden. Rescale the vertical axes to remove whitespace and accommodate all density ellipses. The updated scatterplot matrix is shown in Exhibit 9.47 (script is Scatterplot Matrix 2).

Viewing down the columns from left to right, you see evidence of moderate-to-strong relationships between:

• MFI and SA
• MFI and M%
• MFI and CI (and Yield) and Xf
• MFI and Ambient Temp

The relationship between CI and Xf appears to be highly nonlinear. To see this relationship more clearly, you again use the Graph Builder (see Exhibit 9.48—the script is Graph Builder—CI and Xf).

Everyone on your team notes the highly nonlinear relationship. An engineer on your team observes that the relationship is not quadratic. She speculates that it might be cubic. You also observe that both low and high values of Xf are associated with CI values that fail to meet the lower specification limit of 80.

The engineer also states that the relationship between MFI and SA might also be nonlinear, based on the underlying science. In a similar fashion, you construct a plot for these two variables using Graph Builder (script is Graph Builder—MFI and SA). The plot is shown in Exhibit 9.49.

The relationship does appear to be nonlinear. The bivariate behavior suggests that the underlying relationship might be quadratic or even cubic.

One team member starts speculating about setting a specification range on Xf, maybe requiring Xf to fall between 14.0 and 15.5. You point out that in fact the team must find operating ranges for all of the Xs. You caution that setting these operating ranges one variable at a time is not a good approach. The operating ranges have to simultaneously satisfy specification limits on two Ys, both MFI and CI. A statistical model relating the Xs to the Ys would reveal appropriate operating ranges and target settings for the Xs. This brings the team to the Model Relationships step of the Visual Six Sigma Data Analysis Process.

Excluding and Hiding the Crisis Rows

The five crisis observations have already been excluded and hidden in connection with your scatterplots for the three responses. You can check the Rows panel of the data table to make sure that five points are Excluded and Hidden. However, if you have cleared row states since then, reselect the outliers as shown in Exhibit 9.50, right-click in the plot, and select Row Hide and Exclude (the script is Hide and Exclude Outliers).

Saving the Specification Limits as Column Properties

In the interests of expediency for subsequent analyses, you decide to store the specification limits for MFI and CI in the data table. You do this by entering the information in the Spec Limits column property for each column.

In the Columns panel of the data table, you see that asterisks have appeared to the right of CI and MFI. Click on one of the asterisks to see that the Spec Limits property is listed (Exhibit 9.52). If you click on Spec Limits, the Column Info window opens and displays the Spec Limit property panel.

Stepwise Variable Selection

The Stepwise report appears (Exhibit 9.53), showing two main outline nodes, one entitled Stepwise Fit for MFI and the other entitled Stepwise Fit for CI (the Current Estimates panels have been minimized). The Stepwise Regression Control panel in each of these reports gives you control over how stepwise selection is performed. You decide to accept the JMP default settings, which specify a Minimum BIC stopping rule, the Forward direction, and the Combine rule that combines effects in determining their significance.

The Bayesian Information Criterion, or BIC, is a measure of model fit based on the likelihood function. It includes a penalty for the number of parameters in the model. A lower value of BIC indicates a better model. For details on the BIC and other selections, see www.jmp.com/support/help/Fitting_Linear_Models.shtml.

Model Specification windows for MFI and CI are shown in Exhibit 9.54.

The Hot Xs for MFI are SA, M%, and Xf. The only Hot X for CI is Xf. You observe that SA and its quadratic and cubic effects appear in the model for MFI and that Xf and its quadratic and cubic effects appear in the model for CI.

Checking and Revising Stepwise Models

Next, you run each model to check the fit, starting with the model for MFI. In the Model Specification window for MFI, you click Run. Exhibit 9.55 shows the Effect Summary report and the Actual by Predicted Plot.

In the Actual by Predicted plot, you see a point that might be considered an outlier. Given its location, though, it doesn't appear to have high influence on the model fit. You don't see any other issues with the Actual by Predicted Plot, so you proceed to look at the effect tests.

The Stepwise Stopping Rule, Minimum BIC, can result in a model containing terms that are not active or significant. At this point, you reduce the model by removing terms that are not significant.

In the Effect Summary report, you notice that the interaction effect M%*Xf is not significant. Click on M%*Xf in the Source list to select it, as shown in Exhibit 9.56. Then click Remove. This removes the effect from the model and updates all the reports in the window. Now all remaining effects are significant.

You want to save the model you have constructed to the data table as documentation for your analysis. To do this, from the Response MFI report's red triangle menu, select Script > Relaunch Analysis. Then save the model script to the data table. (The script for the saved model is Model for MFI.)

Next, run the Stepwise model for CI. Exhibit 9.57 shows the Effect Summary report and the Actual by Predicted Plot.

Based on the Actual by Predicted plot, the model seems to fit adequately. Because there is only one variable, namely Xf, in the model, JMP provides a Regression Plot. This shows how the model fits CI as a function of Xf and gives you an additional view of how the model fits. You see that the nonlinearity seems to be well modeled.

The Effect Summary report shows that all three effects are significant. The script for this model is saved to the data table as Model for CI.

All Possible Models

This seemed easy. You realize that your initial model has a large number of terms (70, to be exact, not counting the intercept). With that many terms, there are 2^70 possible models (with intercept). This is a huge number, on the order of 10^21. Stepwise takes a selected path through the space of possible models that is based on p-values. This can be limiting.

You're tempted to do a quick check of your newfound models against other models that could be considered, using the All Possible Models option. You know that you will have to limit the search somehow because the total number of all possible models is daunting.

For MFI, the model with the smallest BIC, in row 1, is precisely the five-term model that you derived from the Stepwise model, once you removed the insignificant M%*Xf terms. For CI, the model with the smallest BIC, in row 1, is precisely the model selected by Stepwise.

You can view the model information in the data tables produced by All Possible Models graphically by constructing a Graph Builder plot of BIC versus Number. The tables created by the scripts All Possible Models for MFI and All Possible Models for CI contain Graph Builder scripts to construct these plots. Exhibit 9.58 shows the plot for MFI, with your final model selected.

Your All Possible Models analysis confirms that the terms that you included in your models for MFI and CI are appropriate.

Saving the Prediction Formulas

Your plan is to obtain a prediction equation for each response and then to use the profiler to conduct multiple optimization.

A column called Pred Formula MFI is added to the data table. To view the formula, right-click in the column header area and select Formula. This opens a formula editor box that shows the prediction equation.

Alternatively, in the Columns panel, the Pred Formula MFI column is now listed. To the right of the column name, you see a plus sign. Click on the plus sign to open the formula editor window displaying the formula.

Save the prediction formula for CI in the same fashion. The saved script is Pred Formula for CI.

Determining Optimal Process Settings

You realize that there are many combinations of settings of the factors that optimize both MFI and CI. When you select Maximize Desirability, the profiler provides one such set of settings.

You have seen that setting SA to 64.5 results in essentially the same predicted values of MFI and CI as did the settings provided when you maximized the desirability function. You decide to use 64.5 as the setting for SA. But you wonder if, for this setting of SA, there might be better settings for M% and Xf.

Exhibit 9.66 shows that the optimal setting for M% is still 0 and that the optimal setting for Xf has changed only slightly, to 15.12.

Controlling SA and Xf

Your sensitivity analysis makes it clear that both SA and Xf must be controlled tightly around the optimal settings that you have derived. Your team members get started on developing an understanding of why these two factors vary and on proposing procedures that will ensure tighter control over them. They propose a new control mechanism to monitor SA (amps for slurry tank stirrer) and to keep it on target at the optimal value of 64.5 and within narrow tolerances. They also develop procedures to ensure that filler and water will be added at regular intervals to ensure that Xf (percent of filler in the polymer) is kept at a constant level close to 15.12 percent.

The team tests these new controls and collects data from which you can estimate the variability in the SA and Xf settings in practice.