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August 2012
This month's newsletter is the first in a multi-part series on using
the ANOVA
are using are shown in the table below.
Operator Part Results
A
1 3.29 3.41 3.64
2 2.44 2.32 2.42
3 4.34 4.17 4.27
4 3.47 3.5 3.64
5 2.2 2.08 2.16
B
1 3.08 3.25 3.07
2 2.53 1.78 2.323 4.19 3.94 4.34
4 3.01 4.03 3.2
5 2.44 1.8 1.72
C
1 3.04 2.89 2.85
2 1.62 1.87 2.04
3 3.88 4.09 3.67
4 3.14 3.2 3.11
5 1.54 1.93 1.55
The ANOVA Table for Gage R&R
In most cases, you will use computer software to do the calculations. Since this is a
relatively simple Gage R&R, we will show how the calculations are done. This
helps understand the process better. The software usually displays the results in
an ANOVA table. The basic ANOVA table is shown in the table below for the
following where k = number of operators, r = number of replications, and n=
number of parts.
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The first column is the source of variability. Remember that a Gage R&R study is a
study of variation. There are five sources of variability in this ANOVA approach: the
operator, the part, the interaction between the operator and part, the equipment
and the total.
The second column is the degrees of freedom associated with the source of
variation. The third column is the sum of squares. The calculations with these two
columns were covered in the first part of this series.
The fourth column is the mean square associated with the source of variation. The
mean square is the estimate of the variance for that source of variability (not
necessarily by itself) based on the amount of data we have (the degrees of
freedom). So, the mean square is the sum of squares divided by the degrees of
freedom. We wi l l use the mean squ are informat ion to est imate the var iance of
each source of var iat ion this is th e key to analyzing the Gage R&R resul ts.
The fifth column is the F value. This is the statistic that is calculated to determine if
the source of variability is statistically significant. It is based on the ratio of two
variances (or mean squares in this case).
The ANOVA Table Results
The data was analyzed using the SPC for Excel software. The results for theANOVA table are shown below.
Source df SS MS F p Value
Operator 2 1.630 0.815 100.322 0.0000
Part 4 28.909 7.227 889.458 0.0000
Operator by Part 8 0.065 0.008 0.142 0.9964
Equipment 30 1.712 0.057
Total 44 32.317
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Note that there is an additional column in this output the p values. This is the
column we want to examine first. If the p value is less than 0.05, it means that the
source of variation has a significant impact on the results. As you can see in the
table, the operator by part source is not significant. Its p value is 0.9964. Many
software packages contain an option to remove the interaction if the p value isabove a certain value most often 0.25. In that case, the interaction is rolled into
the equipment variation. We will keep it in the calculations here though it has little
impact since its mean square is so small.
The next column we want to look at is the mean square column. This column is an
estimate of the variance due to the source of variation. So,
MSOperators = 0.815
MSParts = 7.227
MSOperators*Parts = 0.008
MSEquipment = 0.057You might be tempted to assume, for example, that the variance due to the
operators is 0.815. However, this would be wrong. There are other sources of
variation present in all put one of these variances. We must use the Expected
Mean Square to find out what other sources of variation are present. We will use 2
to denote a variance due to a single source.
Expected Mean Squares
As stated above, the mean square column contains a variance that is related to the
source of variation in the first column. To find the variance of each source ofvariation, we have to use the expected mean square (EMS). The expected mean
square represents the variance that the mean square column is estimating.
There are algorithms that allow you to generate the expected mean squares. This
is beyond the scope of this newsletter. So, we will just present the expected mean
squares.
Lets start at the bottom with the equipment variation. This is really the within
variation (also called error). It is the repeatability portion of the Gage R&R study.
The expected mean square for equipment is the repeatability variance. The
repeatability variance is the mean square of the equipment from the ANOVA table.
Now consider the interaction expected mean square which is given by:
Note that the EMS for the interaction tern contains the repeatability variance as
well as the variance of the interaction between the operators and parts. This is
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what is estimated by the mean square of the interaction. The parts expected mean
square is shown below.
Note that the EMS for parts contains the variances for repeatability, the interactionand parts. This is what is estimated by the mean square for parts. And last, the
expected mean square for the operators is given by:
The EMS for operators contains the variances for repeatability, the interaction and
operators. This is what the mean square for operators is estimating.
The Variances of the Components
We can solve the above equations for each individual 2. Repeatability is already
related directly to the mean square for equipment so we dont need to do anything
there. The other three can be solved as follows:
We can now do the calculations to estimate each of the variances.
Note that the value of the variance for the interaction between the operators and
parts is actually negative. If this happens, the variance is simply set to zero.
% Gage R&R
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The Measurement Systems Analysis manual published by AIAG (www.aiag.org)
provides the following definition: The measurement system variation for
repeatability and reproducibility (or GRR) is defined as the following:
GRR2=EV
2+ AV
2
where EV is the equipment variance and AV is the appraiser (or operator)variance. Thus:
The total variance is the sum of the components:
We can use the total variance to determine the % contribution of each source to
the total variance. This is done by dividing the variance for each source by the total
variance. For example, the % variation due to GRR is given by:
The results for all the sources of variation are shown in the table below.
Source Variance% ofTotalVariance
GRR 0.1109 12.14%
Equipment(Repeatability)
0.0571 6.25%
Operators(Reproducibility)
0.0538 5.89%
Interaction 0.0000 0.00%
Parts 0.8021 87.86%
Total 0.9130 100.00%
Based on this analysis, the measurement system is responsibility for 12.14% of the
total variance. This may or may not be acceptable depending on the process and
what your customer needs and wants. Note that this result is based on the total
variance. It is very important that the parts you use in the Gage R&R study
represent the range of values you will get from production.One of the major problems people have with Gage R&R studies is selecting
samples that do not truly reflect the range of production. If you have to do that, you
can begin to look at how the results compare to specifications. We will take a look
at that next month as we compare the ANOVA method to the Average and Range
method for analyzing a Gage R&R experiment. You could also use a variance
calculated directly from a month's worth of production in place of the total variance
in the analysis.
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Summary
In this newsletter, we continued our exploration of the using ANOVA to analyze a
Gage R&R experiment. We completed the ANOVA table, presented the expected
mean squares and how to use those to estimate the variances of the components,
and showed how to determine the %GRR as a percent of the total variance.In the next newsletter, we will compare the ANOVA method to the Average and
Range method for Gage R&R.
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