ARTICLES
# Nested Gauge R&R (Multiple Equipment, Multiple Appraisers, One Part

### Tabular Output

### Dart Board

### Appraiser Reproducibility Chart

### Probability Plot

There are many applications where an organization has multiple sets of measuring devices (equipment) opening the possibility of variation between the different measurement devices, which can introduce an additional source of error to measurements.

A nested gauge R&R is performed when appraisers cannot take measurements using all the measuring devices available for the study. For example, a measuring device may be kept in one laboratory, another in another and so on. Each laboratory has its own set of testers. Schematically, operators are nested in Equipment (Measuring device) as shown

Equipment 1 | Equipment 2 | Equipment 3 | ....Equipment n | |

Appraiser | A B C | D E F | G H I | XYZ |

For this layout a Nested Analysis of Variance is appropriate. Currently there are several restrictions

- Each appraiser must perform at least 2 tests.
- The number of appraisers per equipment must be greater than one.
- The number of appraisers per equipment must be the same number
- No appraiser will test more than one equipment
- If different models are used, then do so only if the repeatability is known to be the same.

If one location has 3 appraisers, another 5 and another 2 then select two randomly appraisers for each

If a location has more than 1 equipment, test on a randomly selected measuring device. To test all devices in the location, perform a separate analysis, such as using the Gauge R&R Multiple Equipment, Multiple Devices, 1 Part module.

For this type of study each appraiser must test the same part at least twice with each measuring device.

An example of input is shown below.

The column headings are arbitrary codes for the equipment and appraisers.

The BIS.Net APP provides the following output.

The Analysis of Variation table is included for completeness for statisticians. For non-statisticians the last column conveniently lists whether an effect is significant or not. The Appraiser effect is commonly called Reproducibility and Equipment Error is called repeatability. The Equipment effect is the variation caused by differences in the measuring devices.

For this application, Reproducibility is the reproducibility of Appraisers, nested in Equipment. Alternatively, it is the effect taking into account differences in equipment.

If a result is significant then there is statistical evidence that the differences are not due to chance alone. In this instance there is significant variation between equipment and appraisers, not explained by chance alone. The fact that there is equipment variation, implies that the bias differs for some measuring devices. A BIAS study should then be performed to establish why there is a difference.

The approximate confidence intervals are the intervals within which the reproducibility and repeatability, and variation between equipment, measured by standard deviation are likely to fall at the chosen level of significance. If the default of .05 has been used, then the confidence coefficient is equal to 100-.05 *100=95

The measurement system performance table shows the how much the percentage of the various components of variation, (all measured by Sd) takes up relative to the total study variation. The last column uses variance instead of standard deviation. A zero value is used if there is no statistically significant interaction effect.

Gauge R&R is the total variation due to appraisers and equipment error, but not differences equipment. This follows a common approach used in industry. However, since equipment variation is part of the measuring system, argument exists for including this component. If this is desired use the values in the Total row for Gauge R&R.

It is also possible to use process variation instead of study variation as a basis of computing percentages. This is the standard deviation of measurements of parts produced by a production process.

The dart board is a visual tool which enables the analyst to see at a glance how reproducible the measurements are and how good or bad the repeatability is and the effect of the different equipment.

The circles are placed at 1 standard deviation (green), 2 standard deviations (yellow), 3 standard deviations (orange) and beyond (red) around zero. The standard deviation is the total standard deviation of all measurements, NOT the process variation. It is designed to place perspective on the components of variation relative to the study variation, NOT process variation.

Each of the small circles correspond to pure equipment measurement error after removing the effect of the Equipment differences and Appraisers.

Using a sophisticated algorithm, the circles have been randomly placed around the centre just as if they were thrown darts. This is an effective way for visualizing repeatability. Each coloured point corresponds to a different appraiser.

The larger black coloured clusters reflect reproducibility, considering differences caused by the equipment.

The example indicates that repeatability is very small, compared to the total study variation, as seen by the tight cluster of small circles. The reproducibility is however large, as shown by the black circles. he dart board is hence a very powerful visualization tool.

The appraiser reproducibility chart is used to identify appraisers that differ significantly from expectation, considering differences in equipment. All appraisers should fall inside the two red limits. Those that fall outside the limits may need to be retrained.

The above instance shows that there is a reproducibility problem as three appraisers fall outside the red limits. This is confirmed by the Analysis of Variance.

Each shaded region corresponds to a different measuring device.

The probability plot is used to establish normality of the measurement error (residuals). BIS.Net MSA uses the Anderson Darling Statistic and will advise if the if there is evidence of non-normality. The ANOVA method does assume normality. Fortunately, measurement error tends to follow a normal distribution.

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