The Use of PPK to Determine If Confidence and Reliability Statement is met.

Thank you for the quick responses to date. In trying to give a bit more information please see below. These are directly from the procedure governing how we do our validation. A pre-requisite for this is that the process is stable (I-MR Charts) and Normal.


*Sampling plan may be established as per the table and indications provided below. Other sampling plans fulfilling the requirements of confidence and reliability pre-defined and justified are also acceptable.
*The Ppk established from pre-PQ data (e.g. OQ, DOE / Characterization Report etc.) is used to establish the Sample Size and the Acceptance Criteria (Ppk and Pp).
*All High Residual Total Risk Sampling Plans within this table provide 95% Confidence the Reliability of the Process is at least 99% (LTPD0.05 ≤ 1%).
All Low Residual Total Risk Sampling Plans within this table provide 95% Confidence the Reliability of the Process is at least 95% (LTPD0.05 ≤ 5%).

You will find detailed descriptions in Wayne Taylors book. I have not read it, but I attended a training in which the trainer discussed these topics and this was the reference he provided.

The calculations are pretty straight forward, if you are comfortable with estimating the uncertainty of the Cpk value. However, if you plan to do the calculations yourself, do NOT use the "exact formula" for the uncertainty of the Cpk stated on NIST. The formula is wrong. Either use the formula from the original Zhang paper, or use the "common approximation".

I used the normal assumption and calculated the following values:
[1] "nSample = 15 => Ppk >= 1.18 and 95% (single-sided) CI = [1.18, 1.57]"
[2] "nSample = 20 => Ppk >= 1.1 and 95% (single-sided) CI = [1.1, 1.42]"
[3] "nSample = 30 => Ppk >= 1.03 and 95% (single-sided) CI = [1.03, 1.27]"
[4] "nSample = 40 => Ppk >= 0.99 and 95% (single-sided) CI = [0.99, 1.19]"
[5] "nSample = 50 => Ppk >= 0.96 and 95% (single-sided) CI = [0.96, 1.14]"
[6] "nSample = 60 => Ppk >= 0.94 and 95% (single-sided) CI = [0.94, 1.1]"
[7] "nSample = 80 => Ppk >= 0.92 and 95% (single-sided) CI = [0.92, 1.05]"
[8] "nSample = 100 => Ppk >= 0.9 and 95% (single-sided) CI = [0.9, 1.02]"
I used the "common approximation" to obtain the standard deviations. Although these numbers do not match your stated result, they are pretty close.

Thank you Ill have a look at your recommendations and see if I can figure it out.

Thanks.

I just realised that I did not describe the math behind my calculations. Here it goes ...

To calculate the Cpk value, we use the concept of tolerance intervals. For a single-sided specification we use the non-central t-distribution,

As you can see, we need three inputs to calculate the k_1:
(a) the coverage p = 1 - RQL,
(b) the confidence level 1-alpha, and
(c) the sample size n.
Hence, we just pick a sequence of sample sizes and redo the calculation until we are satisfied. E.g. for n=20, p=0.99, alpha=0.05 we get k_1=3.295157.

Next, we user the relationship Cpk = k_1/3. This yields the required (minimal) Cpk-value. In the above example we get Cpk=1.098386. This number is always rounded upwards. I round to the second digit, which yields Ppk >= 1.10.
Note, that translating the RQL=1% value directly into a Cpk-value would yield Cpk=0.7754493. By using the tolerance interval method we "add the required 95% confidence" into the result. Hence, the Cpk-value 1.10 is an upper limit of the corresponding tolerance interval.

Finally, I use the "common approximation" of the uncertainty of the Cpk value (see NIST link above), and calculate the single-sided confidence interval. This yields 1.416078 as an upper limit. This corresponds to the value you stated in your pre-PQ column.

Please note, that k_1 is for a single sided specification. The calculation is similar for two-sided specifications, but we are not allowed to use the non-central t-distribution.

Thank you so much for above.

Semoi said:

Next, we user the relationship Cpk = k_1/3.

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Could you expand on where this comes from? What's the statistical basis for this relationship?

I absolutely hate stating design verification/validation acceptance criteria as cpk/ppk criteria. Just state the required assurance level such as 95/95. It makes no sense to describe single sample criteria as cpk/ppk. While one can derive the CpK from the Z-statistic calculation so its 'statistically justifiable' (though statisticians may quibble with it) it really confuses things because people don't inherently know the assurance level demonstrated when quoting cpk/ppk.

Based on a prior comment it seems maybe this practice is being done thinking this is what Dr. Taylor is recommending. Is this just another misapplication of his work? Just stick to calculating the assurance level the sample demonstrates and leave the CpK/PpK estimates for the operations team when predicting their yields.

d_addams said:

While one can derive the CpK from the Z-statistic calculation so its 'statistically justifiable'

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Being able to do the math does not make it statistically justifiable. This is the hammer and nail argument. Not saying YOU are trying to justify it, just taking exception to considering it "statistically justifiable". As far as I can tell, it is not.

d_addams said:

I absolutely hate stating design verification/validation acceptance criteria as cpk/ppk criteria. Just state the required assurance level such as 95/95.

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Totally agree. Any statistical jaberwocky you might be able to contrive is much harder and nebulous than just doing a tolerance interval. I don't get the need for reinventing a really well established and easy to perform wheel. Six Sigma strikes again I suppose.

I don't have a degree in stats so I'm not well equipped to explain to PhD development partners waiving Dr. Taylor's book in my face why this is isn't appropriate beyond 'this is a dumb idea'. Of course that doesn't go over well.

Their justification has been 'well my python script can easily output cpk'. My response of 'you mean your validated python script, right?' is just met with blank stares.

Just because you can doesn’t mean you should.

The manipulation of mathematical formulas is no substitute for thinking.

By the way, most real statisticians made fun of Cpk when it was first introduced in the eighties. It simply wasn’t serious theoretical statistics. There are many articles from that time rebutting the use of Cpk for anything.