Non-Normal Distribution Data - Tolerance Intervals and Minitab

[jameswalsh]

Apologies if there is already a thread which discusses Minitab and tolerance intervals, I couldn’t find it.
I have two questions on using Minitab to calculate tolerance intervals.

1.) When calculating tolerance intervals using Minitab and the data is found to be non normal, you can use the nonparametric test result; however the associated confidence level is usually lower than the required 95% (we typically looks for 95% confidence). The confidence level can be increased if additional samples are taken from the sample pool.

Is it correct to go back and take additional samples if the desired confidence level is not achieved when using the nonparametric test result? When I say take additional samples, let me clarify what I mean. When carrying out a process validation we might process 500 parts, we would then usually take 30 samples, at random, form this pool of 500 parts. We test the 30 samples and calculate tolerance intervals, using Minitab, on the results.

I personally don’t see why we couldn’t then go back to the pool of 500 parts and take, say an additional 30 samples to add to our Minitab data, if we wanted to increase the confidence level. Does anyone have any thoughts on this?

2.) When the data is found to be non normal, I usually just use the nonparametric value that Minitab calculates for you. Someone suggested we try and find a more suitable distribution that best fits the data. For example a Weibull or lognormal distribution. I have done this for capability analysis, but I don’t see how you can use a different distribution to calculate tolerance intervals using Minitab. Any thoughts?

Thanks,

 

[Miner]

Minitab's Help menu for this analysis states that a sample size of 90 or more is necessary when the nonparametric analysis is used.

"For the Nonparametric Method, you must use a relatively large sample size, approximately 90 or more depending upon the minimum percentage of the population to cover. Larger minimum percentages require larger sample sizes. If your sample size is not large enough, the Nonparametric Method will still produce a tolerance interval but the achieved confidence level will be lower than desired."

Regarding the fitting of a distribution, you can certainly do this, but be forewarned that the tolerance limits for a fitted distribution will generally be tighter than the tolerance limits for a nonparametric distribution. Also, distribution fitting can be tricky for the inexperienced. For example, you should never just take the distribution with the highest p-value. You may select a distribution that must always be > 0 when in fact your data can go negative.

 

[jameswalsh]

Thanks for the feedback Miner.
What if you don’t know in advance of testing that the data is going to non normal? Our protocols usually call for 30 samples for variable data, do you see any problem with going back to the original pool of test samples and testing an additional 30 or more samples to establish normality or at least increase the achieved confidence level?

On using a distribution (for example weibull) that best fits the data, I don’t understand how you can use that for tolerance interval analysis using Minitab? I can see how a transformation (such as Johnson’s) would work because the transformed data appears in a new column and then you can carry out the tolerance interval analysis on the transformed data. But Minitab’s ID distribution (goodness of fit test) just seems to produce graphs with an AD and a P value. I know how to use these distributions for capability analysis, if the data is non normal, but the tolerance intervals analysis function in Minitab seems to be more basic and doesn’t have any options to use different distributions.
Appreciate any advice.

 

[Miner]

See attached files.

I created a Weibull distribution for a 100-piece sample. I performed the Tolerance Analysis for a baseline reference then did a probability plot using the Weibull distribution. I also confirmed that the Distribution Identification properly identified Weibull as the best fit. I then added percentile lines to the probability plot at 2.5 and 97.5.

Notice that these 95% Tolerance limits are tighter than the Nonparametric limits.

Regarding increasing your sample size from the original pool of parts, you can do this, but remember that you still need > 90 parts to use the Nonparametric analysis. Also, you will probably have lost any time ordering of your data and will probably not be able to assess the data for stability.

 

[jameswalsh]

Thanks again Miner for the help,
I understand how you created the histogram and the probability plot of the data. I also understand the tolerance interval plot. Why did you add percentile lines at 2.5 and 97.5?


Regards,

James

 

[Miner]

The Tolerance Limits are set to capture 95% of the product variation. That is 2.5% per tail, hence the percentile lines at 2.5% and 97.5%.

 

[jameswalsh]

Thanks Miner,
Just a couple of more questions, do I still add percentile lines at 2.5% and 97.5% if the data is one sided, for example there is only a lower specification (lower bound)?

What if I want to carry out the tolerance interval analysis with 99% reliability and 95% confidence? What percentile lines do I need to add to the Weibull distribution graph then?

 

[Miner]

If you only have a lower bound, Minitab applies the 5th percentile only.

Regarding your question about 95% and 99% confidence, we have to make a distinction between the confidence and the % of population covered.

The % of population covered dictates the location of the percentile line. The % Confidence dictates the size of the confidence interval/limit. You are interested in the intersection of the percentile line with the confidence limit.