Sample Size based on Process Capability Indices (Pp/Ppk)?

B

Barbara B

#1
One of my customers told me that one of his customers has shown him a paper which provides sample sizes depending on process capability indices (Pp/Ppk). With this method appropriate sample sizes for process validation could be picked out of a table (without further knowledge about mean, variation, tolerance limits, etc.)

A colleague said that there are "rumors" an interesting paper has been published in which this method is described in detail. (Unfortunately he doesn't know who the author is or where this paper was published and I couldn't find anything in the www).

AFAIK the sample size for Pp/Ppk could be determined based on the accuracy of process capability indices (width of confindence intervals for Pp/Ppk with respect to max. type I error alpha / confidence level 1-alpha), but I haven't seen a method for sample sizes in process validation which takes both error types into account (maximal tolerable risks for type I & type II error / alpha & beta).

Hopefully someone in the Cove could provide further informations about this "sample size based on Pp/Ppk"-method!

Regards,

Barbara :bigwave:
 
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B

Barbara B

#3
Thanks for your quick reply. The formulas within the SPC manual are based solely on the confidence level (1-alpha / 1-type I error, usually 95%) and do not take into account the power (1-beta / 1-type II error).

To calculate a sample size for a process capability index with respect to alpha AND beta a statistical test is needed, e. g. something like:
H0: Cpk >= 1.33 vs. H1: Cpk < 1.33
Based on the test static and the corresponding OC-curve the necessary sample size could be determined to get enough information for a statistical safe decision that the process capability is large enough.

But I haven't seen something like this test or another approach which deals with both error risks to determine an appropriate sample size based on Pp/Ppk. Hopefully someone in the Cove has seen it :)

Regards,

Barbara
 

Bev D

Heretical Statistician
Staff member
Super Moderator
#4
Barbara - this one sounds like an urban myth. no direct knowledge of something is usually a sign that we are talking about the Yeti.

I supppose one could come up with a 'safe' table for this btu the sampel sizes woudl have to be relatively large AND - I would think - based on categorical data?

Perhaps one of our other statisticians can shed some light on this one?
 
D

Darius

#6
As you said, as far as I know, must be alpha error and confidence intervals, I just add here a Function to Obtain sample size based on "Historical" Cpk, the "Sample's Cpk" and the Confidence percentage, a piece of cake.

It is based on Size matters: How good is your Cpk, really?

Code:
Function SampleSize_CpkNotLessThan(Cpk_Hist As Single, Cpk_Limit As Single, Conf100 As Byte) As Long
Dim N As Long
  N = 1: OLDDIF = 10000000000#
  Do
    N = N + 1
    DIF = ((Cpk_Hist - Cpk_Limit) / Evaluate("NormSInv(" & Conf100 / 100 & ")")) ^ 2 - (1 / (9 * N) + Cpk_Hist ^ 2 / (2 * N - 2))
    If Abs(DIF) < OLDDIF Then
      OLDDIF = Abs(DIF)
      NN = N
    End If
  Loop Until DIF >= 0 Or N = 10000
  If N = 10000 Then NN = N
  SampleSize_CpkNotLessThan = NN
End Function
Do you have the curve for Beta error (wich sample size?)?
 
B

Barbara B

#7
Thanks for the formula with historic and actual Cpk.

Do you have the curve for Beta error (wich sample size?)?
The posting about OC-curves etc. was just one idea how a sample size for a Cpk could be determined, but I haven't seen anything like this. Imho it would be a great challenge to develop some kind of formula for beta, because the distribution of Cp and Cpk is a little bit complicated, but sometimes humans do find incredible formulas ;)

Regards,

Barbara
 

bobdoering

Stop X-bar/R Madness!!
Trusted Information Resource
#8
Here is an interesting point. You can plug in all sorts of data into all sorts of equations, but there are some empirical statistical concepts that you need to consider. First what is the point of the question? Is the point to determine what your ongoing sample size should be if you know your Cpk (and...assuming your Cpk is even valid, which should always be called into question)?

If so, then the next empirical statistical point is that the lower the probability of defect (higher Cpk) the larger your sample size has to be. It reaches 100% very quickly. Why?The fewer rejects, the harder they will be to find - pure and simple. Think needle in a haystack. Would you assume the fewer needles you have, the less hay you would need to dig through to find them? Well, the same issue applies to sampling. So, seeking out any statistical method that would support lesser size sample when your capability is greater does not end up making any sense if you think of the point of the exercise.
 
B

Barbara B

#9
The relation "the smaller the ppm-rate, the higher the sample size" is a good point, but imho it has to be considered only for attribute data and not for numeric data. For attributes there have to be results out of the tolerance.

For numeric data all value can be within the specification limits and the risk for out of specification results (expected ppm) could nevertheless be calculated. This will only work properly if several assumptions are met (normality, randomess, etc.), but in case all assumptions are met, one could (probably) set up some kind of test for Pp/Ppk which could provide a required sample size to control alpha and beta risks.
 

bobdoering

Stop X-bar/R Madness!!
Trusted Information Resource
#10
For numeric data all value can be within the specification limits and the risk for out of specification results (expected ppm) could nevertheless be calculated. This will only work properly if several assumptions are met (normality, randomness, etc.), but in case all assumptions are met, one could (probably) set up some kind of test for Pp/Ppk which could provide a required sample size to control alpha and beta risks.
Yes, and there is a risk that the defect exists - which gets smaller, and the risk of detecting the defect if it exists, which gets larger. The larger the risk of detecting the defect, the "harder you have to look for it" - which means the better you are, the more you have to look - or the sooner you will reach 100% inspection.

OC curves are Probability of Acceptance, not Probability of Detection - which is what the customer really cares about. But, if you are into weighing your "risk of scrapping good product versus risk of getting caught" (alpha versus beta) here is a pretty nice link.
 
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