Hello,I have been using Box-Cox transformation in Minitab for normalizing data being used for calculating CPK. However, I noticed that some times Box-Cox transformation doesn’t achieve normality, therefore I would like to know if it’s valid to estimate CPK on this particular case using box-cox transformation or If I should use some other technique.If anyone has any information about what Minitab is doing behind stages would be great,Best regards,
The real issue, before determining a Cpk calculation, is to make the determination that the process that is producing the data is stable. Once you have that you have your short-term estimate of variability to feed into the calculation. Stability around the estimate of variability should be your primary concern.
In regards to Box-Cox, the appropriate transform can be chosen from anywhere within the 95% CI. If you let Minitab choose the optimal lamda it might not really be the right one for the data. Take a look at the total range that you can use for a transform and see if there is one that makes more practical sense. For instance, if 0 is within the 95% CI one would typically use that as justification to use the log transform (assuming time series data).
Eric is right about the stability. That is why the AIAG SPC reference manual recommends creating control charts before caclulating & assessing the capability indices. MINITAB offers that capability via their capability sixpack – control charts, normal probability plot, and metrics.
Eric is also right about your ability to use any transformation within the 95% confidence interval presented by the Box-Cox transformation technique. The advice is to use a value that is easy to explain, for example, a lambda value of 2 represents a “square” transformation - that is, x’ = x^2. (this means the transformed value, x’, equals x^2, which means the square of x).
Lambda = 1 means x’ = x [no transform]Lambda = 0.5 means x’ = SQRT(x) [square root transform]Lambda = 0 means x’ = ln(x) [a natural log transform]Lambda = -0.5 means x’ = 1/SQRT(x) [inverse square transform]Lambda = -1 means x’ = 1/x [inverse transform]Lambda = -2 means x’ = 1/(x^2) [inverse square transform] you get the idea . . .
Now, to your question of what to do if the Box-Cox transformation doesn’t provide normality . . .
Well, technically you have two choices:
1. Use MINITAB’s Weibull Capability Sixpack to see if the Weibull distribution can provide a decent fit to your data. If yes, life is good.
2. Use “nonparametric” emperical percentiles to calculate capability indices. Instead of using 3*sigma or 6*sigma in the denominators, use the difference between the median and the 99.87th percentile (the equivalent of 3*sigma) or the difference between the 99.87th percentile and the 0.13rd percentile (the equivalent of 6*sigma). The problem here is that it will takea VERY large sample size to be able to calculate these percentiles accurately. Maybe 10,000 or more – to be honest I really don’t know the sample size needed – maybe someone out there does.
MINITAB, as shipped, doesn’t calculate nonparametric percentiles, but you can download a macro that does from their website (go to Support, then the Macro Catalog).
Beyond these three techniques (Box-Cox, Weibull, Empirical), some say that a mild deviation from normality shouldn’t affect the capability indices too much. How much is some and too much, I don’t know.
Can anyone give a reference to an article that clears this up?
The article “Six Sigma Special Topics: Z-Shifts, Statistics & Non-Standard Data Analysis” from GE’s R&D center will help you in this topic. Get the article here http://www.crd.ge.com/cooltechnologies/pdf/2001crd120.pdf
try reading this article from Thomas Pyzdek.
Has anyone use the Johnson Curves ?
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