# R R and confidence limits

Six Sigma – iSixSigma Forums Old Forums General R R and confidence limits

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• #33904 David Neagle
Participant

I have recently performed an R & R study on a new gauge. The gauge has been made in house, I hasten to add that we are not gauge makers, but it was felt that the cost could be cut by making in house. The gauge failed with a total R & R figure of some 62% Needless to say the management were not happy bunnies. However, as a great deal of time and money has been invested in the gauge they are keen to use it. Now, the gauge is not going to be used for any SPC, just like a go no go gauge, with the numerical results recorded as individual readings. The management have asked me to provide a statement stating the capability of the gauge, up to what dimension is it safe to accept the readings. I am not sure that they know what they want themselves, but I asume they are talking about confidence limits? Now the question is, how do I go about calculating confidence limits for the gauge?
David

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#92673 Andy Urquhart
Participant

David,
This is absolutely the right question to ask. What you’re intersted in is the operating curve or  ‘Confidence of a Correct Decision’ of the metrology. We obtained our curves in 1985 using some simulation written by Mike Wolfe of Motorola’s MOS 3, and published internally. This work saved Motorola millions of Dollars.
The reason was because all tolerances at that time were too tight, so that that the probability of a correct decision was only about 50%, and one might as well flip a coin. Once we widened the tolerance limits to provide a minimum of a 95% confidence at the tolerance limits, we eliminated most of the scrap and without any effect on the parametric or final test yield, and the rolled yield shot through the roof.
(The curve almost takes the shape of a ‘W’ with high confidence outside the tolerance limit, slowly reducing in confidence to the value at the tolerance limit, say 95% then rising inside the limits to hopefully a high value again in the middle of the tolerance window, then as before with the other limit.)
There is a fine chapter on this subject in ‘Design for Six Sigma’ by Creveling, on page 498, on the topic of the P/T ratio. The result is similar to the one I mentioned. Good luck!

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#92708

David,

This does not have to be that complicated.

What I believe the management is interested in is the sensitivity of the test.  The test sensitivity is usually expressed in the ratio of the minimum detectable difference divided by the test/retest standard deviation.

Sensitivity = d/s

And will be dependent on the number of repeat measures made on each unit, the alpha risk (probability that you detect a difference when the difference is just the random variation of the test), and beta risk (probability that you determine no difference due to the random variation of the test when one actually exists.  Since you have done a gage R&R, you have the information you need to do this.

Couple of assumptions that I am making of your study
1. This is not a destructive test so you can make repeat measures on the same unit
2.  There was little or no operator bias in the R&R variation (small contribution due to reproducibility)

The formula for Sensitivity is:

d/s = (ta/2,df +  tb,df)/sqrt(n)

where n is the number of repeat measures on each unit and t is the value from the students t distribution.  The degrees of freedom (df) will come from the number of degrees of freedom used to estimate the repeatability standard deviation.  The formula for the degrees of freedom is (# of parts)*(# of operators)*(n-1).  So if you did a study with 5 parts and 3 operators each doing 3 repeat measures on each part, degrees of freedom is 5*3*(3-1) = 30.

A table and charts can easily be set up in excel.  Create a column for alpha, a column for beta, a column for n and compute the sensitivity using the formula

=(TINV(A1/2,30)+TINV(B1,30))/SQRT(C1)  in this case alpha is column A, beta is in column B, n is in column c and I am using 30 as the degrees of freedom.  You can then make various sub-tables and graphs.  For example, you can graph the effect on the sensitivity of changing n at a given level of alpha and beta.  Lets say you had 30 df for repeatability and we hold alpha and beta at .05 and .10, you would get a table like below:

Alpha  Beta    n          sensitivity
0.05    0.1       1          4.057
0.05    0.1       2          2.869
0.05    0.1       3          2.342
0.05    0.1       4          2.028
0.05    0.1       5          1.814
0.05    0.1       6          1.656
0.05    0.1       7          1.533
0.05    0.1       8          1.434
0.05    0.1       9          1.352
0.05    0.1       10        1.283

You can then plot sensitivity vs. n.  You can also calculate the minimum detectable difference by multiplying the sensitivity with the repeatability standard deviation from your Gage R&R.  For example, lets say that your std. dev. for repeatability is 5 and you are doing 3 repeat measures on each unit, then the minimum detectable difference is 2.342*5 = 11.71 with an alpha risk of .05 and a beta risk of .10.

Hope this helps,

Statman

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#92725 David Neagle
Participant

Thanks for replying Statman. The results of the R & R (average and range method) give me the following. %EV = 39% % AV = 48% and % R&R = 62% Does this mean the % contribution is too great for this to work?
David

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#92754

Hi David,

First of all, with such a high level of %AV (it is greater than the EV) there is great opportunity to improve this gage by reducing the operator bias.  This should be your first priority.  Even though eliminating the operator bias you will still have a %R&R at approximately 44% and above the standard, this is about a 40% improvement.

What you can do with the test sensitivity that I presented in the last post is look at both the total R&R standard deviation and the EV standard deviation.  Since the sensitivity is a ratio of the minimum difference to the standard deviation, it does not depend on the standard deviation only the size of the study, number of repeat measures and the alpha and beta risk.  So once you have determined the sensitivity, you can then multiply it by the two estimates of the standard deviation and determine the minimum detectable difference.

So for example, if I take the study from my previous post (30 df for EV) and my EV is 0.39 and R&R is 0.62 (I am using the data from your study with a total study standard deviation of 1), I would get the table below.  This will show how much improvement in the minimum detectable difference (delta) from eliminating the operator bias.

Alpha  Beta    n          Sensitivity       Delta (totalRR)           Delta (EV)
0.05    0.1       1          4.057                          2.515                          1.582
0.05    0.1       2          2.869                          1.779                          1.119
0.05    0.1       3          2.342                          1.452                          0.913
0.05    0.1       4          2.028                          1.257                          0.791
0.05    0.1       5          1.814                          1.125                          0.707
0.05    0.1       6          1.656                          1.027                          0.646
0.05    0.1       7          1.533                          0.950                          0.598
0.05    0.1       8          1.434                          0.889                          0.559
0.05    0.1       9          1.352                          0.838                          0.527
0.05    0.1       10        1.283                          0.795                          0.500

Last point, you should use the ANOVA method for determining the variance components.  The range method will give unbiased estimates but you lose degrees of freedom in the estimate of EV with the range method so the formula for DF I had in my last post is not correct.  If you dont have software that will do the ANOVA, let me know.  This can quite easily be done in excel and I will send you a spreadsheet for it.

Statman

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#92775 Gabriel
Participant

Hi Statman
As an alternative, you can attach it here so it is avaliable to everybody.
Thanks!

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#92776

Gabriel,
Did you want a spreedsheet for Gage R&R variance components using ANOVA or a spreadsheet for determining test sensitivity for various alpha, beta, and repeat measurements?
Statman

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#92778 Gabriel
Participant

For the r&R using ANOVA

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#92793 David Neagle
Participant

Hi statman. Could you please send me the spreadsheet. I would be very grateful.
Many thanks
David

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#92850

Hi David,
Regards,
Statman

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#92861 David Neagle
Participant

Sorry Statman. Should have included the address. My email address is [email protected]
Once again, many thanks
David

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#107117

Hi Statman,
I am interested in that spreadsheet too. Would you kindly send to my email at [email protected]
Many thanks.
Cheers

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