Standards

Hello Members,

I am working out a process to identify data which is out of trend. I have proposed to use the I-MR chart for this purpose. However, the challenge I face is using this tool to handle non-normal data. I normally transform non normal data using the Johnson's Transformation. The resultant data, which is normal, is not same as the original data and therefore co-relating the UCL and LCL with the original data becomes difficult. Can someone please advise me on how this situation can be handled?


Siddharth Sanghvi   

Hi Siddharth,


Let me offer a couple of observations. If I am in error, someone please correct me.


First, I believe the Individual and Moving Range approach would not be effective. This is because the three sigma control limits would not be appropriate for a variable with a significantly non-normal distribution. One could substitute 99% control limits calculated using Chebyshev's inequality, but this would be relatively insensitive to shifts in mean.


On the other hand, one should not rule out X-bar (averages) charts for non-normal data, as they can be relatively robust against some forms of non-normality, especially skewness, kurtosis, or censoring.


To see this in action, using your statistical software, conduct this experiment:


- draw 1000 samples of size 10 from a uniform distribution


- take the average of each sample


- construct a histogram of the averages


- construct a normal probability plot with the averages


You should see that the distribution of the averages strongly resembles a normal distribution.


You can do the same for an exponential distribution and see that, while the departure from normality is more pronounced, it isn't terrible. There will be more frequent  out of control indications, but mean shifts should be perceptible.


If, of course, the underlying distribution is something like a Weibull distribution, this approach may break down.


On the other hand, there is nothing wrong with control charting data that has been transformed to make it normal. Yes, the control limits for the transformed data cannot be compared to the specifications for the untransformed data. But that is also true for any X-bar chart. Averages data cannot be compared to individual observations. Control limits and specification limits should not be compared.

Harry Rowe is spot on in his advice.   I would also point you to an analysis in Dr Don Wheeler's paper The Empirical Rule.  Make sure your non-normal distribution is not just an out of control process appearing to be non-normal.  If the underlying distribution is truly non-normal as Harry suggested, Average charts follow the Central Limit Theorem and will detect shifts in the process regardless of the shape of the underlying distribution.  

Thank you Gentlemen for your response. Truly helpful. I thought of using the I-MR chart since the data I am dealing with is generated individually and therefore is difficult to set in groups. For data that is generated in groups, like for example weights, I use either the Xbar-R or Xbar-S chart and this works out well.


The problem I am facing is in trying to co-relate the control limits obtained after transformation against original values generated from the process. This is important for me since these values (ie the UCL and LCL) are to be used as a guidance for identifying an Out of Trend value during future production runs.


I understand the control limits cannot be compared with specification limits but I do check if the control limits are too close to the specification limits. If this is the case, I classify the process as in poor control since this increases the chances of the process producing a product which crosses the specification.    


Regards,

Siddharth 

Siddharth,

Remember when you transform the data to chart it, the chart limits follow the same transformation.  If you want to compare them to specification, you must reverse the transformation on the limit to see where it lies compared to the specification.  Ideally, there should be at least 1 sigma between the specification and the chart limit which provides a little wiggle room for shifts.  That translates to a Cpk of 1.33.  If it is a two sided specification that rule would apply to the chart limit closest to the specification.  If you calculate for both sides and one is greater than one sigma and the other is less, you need to center your process.  Anything greater than 1 sigma between the limits and your specification is just more assurance that your process is producing good product.  If that distance is less than 1 sigma on both sides of the process, then you need to take measures to reduce variability.   Remember a process in control ( no points outside the chart limits) ONLY means that it is behaving naturally, with no assignable causes of variation.  That is why you can't just tighten the control limits on a process and make it behave better. 

I hope that clarifies it for you. 


John Finley  

Thank you Mr. Finley. Truly helpful. 


Regards,

Siddharth 

My inclination would be to fit the non-normal distribution in question, which Minitab and StatGraphics will do automatically. (This assumes that you know what the actual distribution is, e.g. from prior experience or documentation of similar applications).


Then set the upper control limit at the 0.99865 quantile (same as for a  3-sigma chart for a normal distribution), the lower control limit at the 0.00135 quantile, and the center line at the median. The false alarm risks will be the same as for a traditional control chart. Minitab, StatGraphics, and presumably other software packages will also calculate accurate process performance indices for non-normal distributions based on the estimated nonconforming fraction.


I would also recommend always performing tests for goodness of fit to whatever distribution is selected. These include histograms, the chi square test for goodness of fit, and the probability plot with the Anderson-Darling test for goodness of fit. (Kolmogorov-Smirnov is not as good.)


Note also that the specification limit may be one-sided, especially if there is a physical limit. If for example the quality characteristic involves trace elements or impurities, the measurement cannot be less than zero. If the lower control limit of a traditional chart (one that assumes a bell curve) is outside this physical limit, it is a sign that the distribution may be non-normal.


--Bill