Gage R. & R. vs. ANOVA - MSC Conf

[Pages:26]Gage R. & R. vs. ANOVA

Dilip A. Shah E = mc3 Solutions 197 Great Oaks Trail # 130 Wadsworth, Ohio 44281-8215 Tel: 330-328-4400 Fax: 330-336-3974 E-mail: emc3solu@

ABSTRACT Quality and metrology technicians are faced with choosing different analytical techniques when determining measurement uncertainty budgets. Gage Repeatability & Reproducibility is a popular technique used by automotive industry suppliers while Analysis of Variance (ANOVA) is thought to be a statistician's analysis tool. This paper discusses the pros and cons of both techniques, using sample data. Metrology technicians will find the presentation of data useful in determining analytical methods for their own requirements.

Page 1 of 9

INTRODUCTION

Quality, calibration and test technicians face many challenges in analyzing data when determining uncertainty budgets. The use of computers and software applications makes it easier to perform data calculations. But the user still has to ensure that correct data is entered, and that a correct judgment is made based on the results provided by the software. Computer spreadsheet models can be easily developed to perform mundane and repetitive data calculations.

Gage Repeatability and Reproducibility studies are performed on a routine basis by automotive suppliers. The "GM Long Form" method is now referred to as the "standard method"1 for performing the Gage R & R studies. Once performed on a paper form using a calculator, this is now performed either using a spreadsheet template or using a software application.

Analysis of Variance (ANOVA) is a statistical tool. Before the availability of personal computers and spreadsheets, the calculations for ANOVA were cumbersome. Mainframe software applications such as SAS and Minitab made those tasks easier.

For those with access to mainframe use, analyzing the ANOVA data required statistician assistance. Now, PC versions of Minitab and similar software exist. Computer spreadsheets usually have add-in ANOVA analysis applications. Simple data analysis can be performed observing a set of statistical rules.

This paper uses sample data to compare use of both Gage R. & R. and ANOVA techniques. Both GRR3 and ANOVA methods can be found in many textbooks and references and is not discussed in this paper. The resultant data between the two methods is compared.

UNCERTAINTY ANALYSIS The U. S. Guide to Expression of Uncertainty of Measurements in Measurements2 defines Type A uncertainty as that obtained by statistical means.2 One of the simplest ways of determining Type A uncertainty is to calculate standard deviation of a set of data from repeated measurements.

When two or more technicians make a measurement, the variability between technicians need to be identified/quantified. If two or more measurement systems are used, then the variability between the instruments also needs to be identified /quantified.

Either ANOVA or the GRR method can be set up to quantify Type A uncertainty components for uncertainty budgets.

Page 2 of 9

DATA PREPARATION

As with any analysis, a set of rules needs to be observed when collecting data.

Ensure that the instrument is calibrated. Ensure that the process/method is validated. Ensure that the process/method is in statistical control. Use trained technicians/Metrologists to perform the study. Ensure repeated measurements are made in a random manner. Use more than one appraiser/technician.

More detailed rules on GRR methods are found in the references noted.1, 3 and 4

DATA

The data collected among three technicians is displayed in Table 1. Each technician measured the ten artifacts for a total of three trials. The tolerance for the artifact is defined as 0.0005 units.

Technician Alan

Nancy

James

1 1.00007 1.00003 1.00006 1.00004 1.00003 1.00001 1.00009 1.00005 1.00006

2 1.00004 1.00010 1.00003 1.00003 1.00005 1.00004 1.00009 1.00008 1.00006

3 1.00005 1.00001 1.00006 1.00002 1.00005 1.00000 1.00008 1.00010 1.00001

4 1.00003 1.00007 1.00003 1.00004 1.00004 1.00005 1.00003 1.00007 1.00002

5 1.00007 1.00007 1.00007 1.00000 1.00003 1.00001 1.00001 1.00002 1.00007

6 1.00003 1.00010 1.00005 1.00003 1.00003 1.00002 1.00007 1.00002 1.00005

7 1.00006 1.00004 1.00002 1.00003 1.00002 1.00000 1.00005 1.00009 1.00005

8 1.00009 1.00009 1.00004 1.00001 1.00005 1.00005 1.00003 1.00002 1.00002

9 1.00003 1.00009 1.00003 1.00002 1.00004 1.00004 1.00001 1.00003 1.00001

10 1.00008 1.00005 1.00008 1.00001 1.00001 1.00001 1.00006 1.00005 1.00005

Table 1

GAGE REPEATABILITY AND REPRODUCIBILITY (GRR) METHOD

The AIAG publication, Measurement Systems Analysis (3rd Edition) 3 defines Gage Repeatability and Reproducibility as "An estimate of the combined variation of repeatability and reproducibility for a measurement system. The GRR variance is equal to the sum of within-system and between-system variances."

Several methods to calculate GGR are documented1, 3, and 4. The method described in AIAG publication, Measurement Systems Analysis 3rd Edition3 was used for this paper.

The spreadsheet GRR template calculation is shown in Table 2. Whenever a spreadsheet template

is developed, it is important to validate it. For the spreadsheet template used for this paper, sample data from a Measurement Systems Analysis 3rd Edition3 was used to validate the calculations.

Page 3 of 9

Appraiser/ Trial# Alan 1

2

3

Average

Range Nancy 1

2

3

Average

Range James 1

2

3

Average

Range Part Average

1

1.00007 1.00003 1.00006

1.000 3.9E-05

1.00004 1.00003 1.00001

1.000 2.7E-05

1.00009 1.00005 1.00006

1.000 4.2E-05

1.000051

2

1.00004 1.00010 1.00003

1.000 6.7E-05

1.00003 1.00005 1.00004

1.000 1.5E-05

1.00009 1.00008 1.00006

1.000 2.4E-05

1.000058

3

1.00005 1.00001 1.00006

1.000 4.3E-05

1.00002 1.00005 1.00000

1.000 4.8E-05

1.00008 1.00010 1.00001

1.000 8.5E-05

1.000042

4

1.00003 1.00007 1.00003

1.000 4.4E-05

1.00004 1.00004 1.00005

1.000 8.4E-06

1.00003 1.00007 1.00002

1.000 5.5E-05

1.000044

Part

5

6

1.00007 1.00007 1.00007

1.000 3.5E-06

1.00003 1.00010 1.00005

1.000 6.9E-05

1.00000 1.00003 1.00001

1.000 2.8E-05

1.00003 1.00003 1.00002

1.000 1.8E-05

1.00001 1.00002 1.00007

1.000 6.5E-05

1.00007 1.00002 1.00005

1.000 5.1E-05

1.000039 1.000046

7

1.00006 1.00004 1.00002

1.000 4.2E-05

1.00003 1.00002 1.00000

1.000 3E-05

1.00005 1.00009 1.00005

1.000 4.5E-05

1.000040

8

1.00009 1.00009 1.00004

1.000 4.6E-05

1.00001 1.00005 1.00005

1.000 3.6E-05

1.00003 1.00002 1.00002

1.000 8.7E-06

1.000043

9

1.00003 1.00009 1.00003

1.000 6.4E-05

1.00002 1.00004 1.00004

1.000 2E-05

1.00001 1.00003 1.00001

1.000 2.2E-05

1.000034

10

1.00008 1.00005 1.00008

1.000 2.9E-05

1.00001 1.00001 1.00001

1.000 6.8E-06

1.00006 1.00005 1.00005

1.000 1E-05

1.000045 Rp R-Dbar XBarDiff UCLR EV AV GRR PV TV

Average

1.000055 1.000066 1.000048 1.000056 0.000045

1.000025 1.000036 1.000023 1.000028 0.000024

1.000052 1.000054 1.000040 1.000048 0.000041

3.000133 0.0000242 0.0000364

0.0000285 9.40E-05 2.15E-05 1.44E-05 2.59E-05 7.62E-06 2.70E-05

Table 2

Page 4 of 9

ANALYSIS OF VARIANCE (ANOVA) METHOD

The AIAG publication, Measurement Systems Analysis (3rd Edition)3 defines ANALYSIS OF VARIANCE (ANOVA) as "A statistical method (ANOVA) often used in designed experiments (DOE), to analyze variable data from multiple groups in order to compare means and analyze sources of variation."

The collected data is analyzed using the Microsoft Excel? spreadsheet Analysis (ANOVA Twoway with replication) add-in and is shown in Tables 3 and 4.

ANOVA: Two-Factor With Replication

SUMMARY

Alan

Count Sum

Average Variance

1

3 3.00017 1.00006

4E-10

2

3 3.00017 1.00006 1.4E-09

3

3 3.00012 1.00004 5.1E-10

4

3 3.00014 1.00005 5.9E-10

5

3 3.00022 1.00007

3E-12

6

3 3.00018 1.00006 1.2E-09

7

3 3.00011 1.00004 4.5E-10

8

3 3.00022 1.00007 7.1E-10

9

3 3.00015 1.00005 1.3E-09

10

3 3.00021 1.00007 2.3E-10

Total

30 30.0017 1.00006 6.3E-10

Nancy

Count Sum

Average Variance

3 3.00008 1.00003 1.9E-10

3 3.00012 1.00004 6.7E-11

3 3.00007 1.00002 5.8E-10

3 3.00013 1.00004 1.8E-11

3 3.00005 1.00002 2.1E-10

3 3.00008 1.00003 8.9E-11

3 3.00005 1.00002 2.4E-10

3 3.00011 1.00004 4.4E-10

3 3.00011 1.00004 1.1E-10

3 3.00003 1.00001 1.3E-11

30 30.0008 1.00003 2.5E-10

James

Count Sum

Average Variance

3 3.0002 1.00007 5.2E-10

3 3.00024 1.00008 1.6E-10

3 3.00019 1.00006

2E-09

3 3.00012 1.00004 8.6E-10

3 3.00009 1.00003 1.2E-09

3 3.00015 1.00005 6.5E-10

3 3.00019 1.00006 6.3E-10

3 3.00006 1.00002 1.9E-11

3 3.00005 1.00002 1.5E-10

3 3.00016 1.00005 3.1E-11

30 30.0015 1.00005 8.5E-10

Total

Count Sum

Average Variance

9 9.00046 1.00005 5.9E-10

9 9.00052 1.00006 6.9E-10

9 9.00038 1.00004 1.1E-09

9 9.00039 1.00004 3.7E-10

9 9.00036 1.00004

1E-09

9 9.00041 1.00005 7.2E-10

9 9.00036 1.00004 7.3E-10

9 9.00039 1.00004 8.6E-10

9 9.00031 1.00003 6.1E-10

9 9.00041 1.00005

8E-10

Table 3

Page 5 of 9

Source of Variation Sample Columns

Interaction Within

SS 1.30E-08 3.53E-09 1.67E-08 3.00E-08

ANOVA

df

MS

2 6.50E-09

9 3.93E-10

18 9.27E-10

60 5.01E-10

F 12.98 0.78 1.85

P-value 2.1E-05 6.3E-01 3.9E-02

F crit 3.2E+00 2.0E+00 1.8E+00

Total

6.32E-08 89 7.11E-10

Table 4

In Table 4:

? The Source of Variation column is due to each cause contributor. ? The SS or Sum of Squares column is the squared sum of deviation around the mean of

each source. ? The df column is the degrees of freedom associated with each source. ? The mean square (MS) value is also defined as the sample variance and is calculated by

dividing the Sum of Squares by degrees of freedom (SS/df). ? The F column refers to the ratio of the MS source divided by the error (Within) and is used

to test null hypothesis against the F-table value (F crit). For this ANOVA, it was set at 95% confidence interval. The hypothesis tests were not considered for this paper.

Table 5 further defines the Expected Mean Square (EMS) as the linear combination of variance components of each Mean Square (MS).

Source of Variation

df

Sample (Technician) k-1

Columns (Parts) n-1

Interaction (Tech. x Part) (n-1)(k-1)

Within (Gage Error) nk(r-1)

EMS 2 + r2 + nr2 2 + r2 + kr2

2 + r2 2

EMS(for this example)

2 + 32 + 302 2 + 32 + 92

2 + 32 2

Table 5

Page 6 of 9

Table 6 breaks down the variance components and associated standard deviations.

Variance

Std. Dev.

Appraiser-Technician (AV) 2

1.86E-10 1.36E-05

Part 2

-5.94E-11 0.00E+00

Interaction(Tech. x Part) 2

1.42E-10

1.19E-05

Equipment(EV) 2

5.01E-10 2.24E-05

Table 6

Please note that when calculating individual component variances, negative variance components are possible and they are set to zero as shown in the Part component.

The ANOVA result is compared with the GRR. In some cases (depending on the method chosen), one may have to multiply the standard deviation from the ANOVA by 5.15 or conversely, divide the GRR values by 5.15 to compare the data.

The comparison is shown in Table 7.

Source of Variation Appraiser-Technician (AV)

Part Interaction(Tech. x Part)

Equipment(EV) Total

ANOVA Std Dev. 1.36E-05 0.00E+00

1.19E-05 2.24E-05 2.77E-05

GRR 1.44E-05 7.62E-06

0.00 2.15E-05 2.70E-05

Table 7

Equipment Variation (EV) is generally associated with Repeatability. Appraiser Variation (AV) is generally associated with Reproducibility.

Page 7 of 9

SUMMARY

It can be concluded that:

? Equipment Variation (EV) and Appraiser Variation (AV) can be calculated by both methods. The difference in the two methods can be attributed to the fact that the GRR method uses estimated method (range method).

? The ANOVA method is more accurate because it does not use estimates. ? One cannot obtain interaction (Technician x Part) from the GRR method. The ANOVA

method will provide the interaction. ? The part variation was zero using the ANOVA method while the GRR method yielded part

variation. It is possible that this comes from the interaction which the GRR cannot differentiate. ? Both methods provide a close value for the Total Variation (TV). The total is calculated using the Root-Sum-Square (RSS) method. ? If using the data in the uncertainty budgets, ensure that the standard deviation is used instead of the EV or AV value. Some GRR packages calculate EV and AV values with 99% confidence interval. To obtain standard deviation from those GRR AV and EV values, the data is divided by 5.15. The number in this paper reflects standard deviations.

Guidelines for GRR acceptability4

GRR is divided by the part tolerance and expressed as a percentage. The following general guidelines are used to determine gage acceptability.

? %GRR is 10% and 30% Gage is considered unacceptable and may need

improvement/replacement.

If the Repeatability is large compared to Reproducibility:

? Gage/Instrument needs maintenance, repair or calibration. ? Gage may need redesign. ? Gage location (environment) needs evaluation. ? Method needs evaluation. ? Excessive within part variation.

If the Reproducibility is large compared to Repeatability:

? Operator training is required. ? Gage Resolution is too large. ? More refinement on fixture required.

Page 8 of 9

 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download