This article by Lou Johnson, technical training specialist at Minitab
Inc., explains why a standard Gage R&R cannot adequately assess the
capability of many measurement systems. The article also demonstrates that when
a standard study is not enough, an Expanded Gage R&R is an ideal tool to
comprehensively characterize your measurement system.
If you can’t trust your measurement system, you can’t trust the data it produces. That’s why Measurement Systems Analysis (MSA) is a key component of establishing, improving and maintaining quality systems. Whether you’re engaged in a Six Sigma project or an ISO-9000 certification, an MSA helps you identify
problems with your measurement system and determine if you can trust your data.
The most common type of MSA is the Gage repeatability and reproducibility (R&R) study. Most Gage R&R studies assess the effects of two factors on variation in your measurement system—typically Operator and Part.
However, the effects of Operator and Part frequently are not enough to provide a complete understanding of the measurement system. Adding a third variable (typically “Gage”) to the standard study is often required.
When three or more factors are included in the analysis, we call the study an Expanded Gage R&R. In the following situations, a third factor is crucial to understanding the system:
The four main differences between a standard and an expanded study are:
Since developing the Expanded Gage R&R tool for Minitab, Minitab has helped dozens of companies implement Expanded Gage studies to correctly assess their measurement system and improve quality.
In using Expanded Gage R&R to evaluate systems for a wide range of measurement types—from surface roughness at Corning, Inc., to coating thickness at AzkoNobel—we have learned that simply running a separate standard gage R&R at each of the levels of the extra variable is rarely an efficient
design for answering the questions of interest.
To help more quality practitioners reap the benefits of this powerful tool, let’s take a step-by-step look at how to design, analyze and interpret the results of an Expanded Gage R&R Study. We will use a system for measuring film thickness from the microelectronics industry for illustration.
Photoresist coating is used in the microelectronics industry to etch integrated circuits for microprocessors, RAM, etc., onto silicon wafers.1 We need to assess the measurement system for the thickness of this photoresist coating. The thickness affects how coated silicon wafers
perform in microelectronics, so obtaining accurate measurements is critical.
The data collection plan is outlined below:
In a standard Gage R&R plan2, we would select 10 wafers at random to represent process performance. If a standard study was followed for each of the three gages, the total sample size would be:
That is an unacceptably large sample size. By decreasing the number of parts (wafers) from 10 to 5, the total study can be completed in 90 measurements.
Changing the sampling plan is commonly required to reduce the size of the Expanded Gage R&R study to a manageable level. This is an important difference between a standard and an expanded study. Later, we will demonstrate that reducing the number of parts from 10 to 5 did not compromise the quality of
As can be seen in the worksheet for this study’s 90-row dataset, each operator measures each wafer on each of the three gages, twice. Each row has a column that identifies the Operator, Gage, Wafer and Thickness reading. Even though missing data is not allowed in a Standard Gage R&R, an expanded study
accommodates missing data, as seen in Row 10 below.
To carry out the analysis in Minitab, choose Stat > Quality Tools
> Gage Study > Gage R&R Study (Expanded). Complete the dialog box as shown below. The analysis treats Operator, Part, and Gage as random factors because each of these factor levels (e.g., each operator) was randomly sampled from a larger population. (If our measurement system had only two gages
and our main goal was to compare them to each other, then our analysis should consider Gage as a fixed factor3, and we would identify it as a fixed factor in the dialog box.)
Next we select the terms we wish to evaluate by clicking the “Terms…” button and adding all main effects (Wafer, Operator, and Gage) as well as all second-order terms—Wafer*Operator, Wafer*Gage, and Operator*Gage. By including “Gage” in the study, not only do we determine the variability due to the gage
main effect, but also its interaction with the other two variables, Operator and Part. Finally, we select the graphs we would like to evaluate by clicking the “Graphs…” button and completing the dialog box as shown.
Minitab provides a great deal of numeric and graphical output. Let’s evaluate the two most important data tables first. The ANOVA table shows which sources of variation were statistically significant. Factors with p-values less than .05 in the ANOVA table below are statistically significant.
The ANOVA output indicates that gage-to-gage variation, the Wafer*Operator interaction, and the Wafer*Gage interaction are statistically significant. The high p-values for Operator and the Operator*Gage interaction indicate that these two sources of variation are not statistically significant, and therefore will
not be of concern when trying to reduce the variability of the measurement system. (Wafer-to-Wafer variability also is statistically significant, but since we are focusing on the measurement system, part-to-part variation is not a key concern in this study.)
It is also important to evaluate the ANOVA table for the number of degrees of freedom (an indicator of the number of repeat measurements) available to estimate the repeatability of the gage. Here we see 57 degrees of freedom, well above the 30 to 45 recommended by simulation studies.4 Therefore,
the reduced number of Parts in the study has not hindered our ability to estimate the contribution of the gage repeatability to the overall variation of the measurement system.
Next we’ll examine the Gage Evaluation table. The Automotive Industry Action Group2 has set guidelines for %Study Variation and Number of Distinct Categories at a maximum of 30% and a minimum of 5 categories, respectively. Here we see that both measures of capability indicate that this measurement system
just narrowly achieves both of these guidelines.
The Gage Evaluation table also shows the relative importance of each of the sources of variation. The variation due to Gage and Wafer*Gage are the two strongest contributors to the overall variation, each accounting for about 16% of Study Variation. We can see the contribution of Gage
to the variation in the main effects plot below. The average reading by gage varies from 111 to 123 microns.
However, this is not the full story, because the Wafer*Gage interaction was also a strong contributor to the measurement system variation, as shown in the figure below.
The general agreement seen in the three gages on parts 3 and 5 indicates that there is not a consistent bias between the three gages. However, Gage 2 has a strong positive bias for wafers 1 and 4. Even though the measurement system is acceptable, determining why the gage exhibited bias when measuring wafers 1 and
4—and fixing this problem—will reduce overall variation in the measurement system.
Finally, we return to the question of the effect of reducing the number of parts from 10 to 5. Our capability estimators % Study Variation and Number of Distinct Categories are a function of the part-to-part variability which can be calculated from the parts in the study or from historical data. With only 5
parts, one would expected more reliable results from using the historical standard deviation. The ratio of the measurement system variation to the process variation calculated from historical data is called the % Process shown in the Gage Evaluation table. The general specification on % Process (less than 30%) is
the same as that for % Study Variation. When reducing the number of parts below 10, entering a historical standard deviation and focusing on % Process instead of % Study Variation is strongly recommended. In this way, the size of the study can be reduced without concern that the quality of the results have been
compromised. In this case, we see that the % Process and % Study Var are nearly equal. Therefore, our conclusions remain the same.
The Expanded Gage R&R study has provided a comprehensive assessment of the measurement system for the photoresist thickness measurement. With the Number of Distinct Categories = 5, the system meets the minimum acceptance criteria for a measurement used to study the process.
Since Gage and the Wafer*Gage interaction were the strongest contributors to the measurement variation, determining the cause of the differences between Gages, particularly for certain parts, will reduce overall measurement system variation. The within
gage repeatability was also a reasonably large source of variation. Identifying ways to make the gage more repeatable will also reduce variation in the system.
As we have seen, a standard Gage R&R cannot adequately assess the capability of many measurement systems. When a standard study is not enough, an Expanded Gage R&R is an ideal tool to comprehensively characterize your measurement system.
Lou Johnson brings 24 years of process engineering and Six Sigma experience to his role as a Minitab Trainer. He has trained engineers, project leaders and technicians to use data effectively when analyzing processes. He has also consulted and trained with dozens of companies, from Arrow to Xerox. Lou is a
senior member of the American Society for Quality and enjoys sharing techniques for improving Quality with ASQ members. He has published articles in Industry Week, Quality Progress and ASQ Six Sigma Forum. He is also a frequent presenter at ASQ conferences including the World Conference,
Lean Six Sigma Conference, Fall Technical Conference and Service Quality Conference. Lou earned B.S. degrees in Chemistry, Education, and Chemical Engineering from the University of Illinois. After ten years of manufacturing experience, he returned to school to complete his M.S. in Statistics from Penn
State. He also has ASQ Black Belt and Oriel Master Black Belt certifications.
1. Johnson, L., and S. P. Bailey (2012), “Implementing an Expanded Gage R&R Study.” ASQ World Conference on Quality and Improvement, Anaheim, Ca.
2. AIAG Measurement Systems Analysis, Reference Manual, 3rd
ed. (2003). Automotive Industry Action Group, Southfield, Mich.
3. Dolezal, K. K., R. K. Burdick, and N. J. Birch (1998). “Analysis of a Two-Factor R&R Study with Fixed Operators.” Journal of Quality Technology, Vol 30, p163.
4. Zuo, Y., (2009) “Effect of Sample Size on Variance Component Estimates in Gage R & R Studies.” Minitab Technical White Paper.
Get our free monthly e-newsletter for the latest Minitab news, tutorials, case studies, statistics tips and other helpful information.
See a sample issue.
A Statistical Analysis of Boston’s 2015 Record Snowfall
Using Nonlinear Regression in Minitab to Model Diffusion in a High Intensity Discharge Lamp
Teaching DoE with Paper Helicopters and Minitab