For an example of this uncertainty, assume that the upper specification limit is six standard deviations above the mean. The original model with alpha = 1.0 would then have 912 ppm nonconforming. But with alpha = 2.1, the fitted model would predict only 8 ppm nonconforming. And with alpha = 0.5, the fitted model would predict 8,151 ppm nonconforming. These two estimates only differ by a factor of a thousand.
As a result, even though we may compute exact critical values based on our estimated parameters, the results will never have the expected precision. While we may want to filter out exactly 95% of the noise, we’ll only filter out approximately 95%. The uncertainty is hidden by the complexity of the computations, but it remains there all the same.
The second unexpected characteristic of the family of gamma distributions is seen in the top curve of Figure 3 (k = 3), which shows the areas within three standard deviations of the mean. Although this area does drop slightly at first, it stabilizes for the J-shaped gammas at about 97.6%. This means that a fixed-width, three-standard-deviation central interval for a gamma distribution will always contain at least 97.6% of that distribution.
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Figure 7 shows the values in each column of Figure 6 plotted against skewness. The bottom curve shows that the middle 92% of a gamma will shift toward the mean with increasing skewness. The 95% fixed-coverage intervals are remarkably stable until the increasing mass near the mean eventually begins to pull this curve down. The 97.5% fixed-coverage intervals initially grow until they plateau near three standard deviations. The spread of the top three curves shows that for the gamma models it’s primarily the outermost 2% that gets stretched into the extreme upper tail.
Figure 2: Some properties of gamma models
When you fit a model to your data you will be fitting the central portion, not the tails. And the uncertainty in the estimated parameter values will always result in huge differences in the infinitesimal areas under the extreme tails. This is what makes fitting a model to your data and then using that model to compute a parts per million nonconforming value completely bogus. You can do the computation, but the results have nothing to do with the underlying process. This is why, when people start talking about parts per million nonconforming, you can be sure that reality has left the building.
The purpose of analysis
While the infinitesimal areas under the extreme tails will move farther away from the mean with increasing skewness, the tail as a whole does not get heavier. Rather, it actually gets much lighter with increasing skewness. To move the outer few parts per thousand farther away from the mean, you must compensate by moving a much larger percentage of the area closer to the mean. This compensation is unavoidable and inevitable. To stretch the long tail, you have to pack an ever-increasing proportion into the center of the distribution. An illustration of this compensation is shown in Figure 4.
You can investigate this uncertainty with a simple simulation study. I used an exponential distribution (where alpha and beta were both equal to 1.00) to generate 5,000 data sets of size n = 100. For each data set, I estimated the value of alpha and beta as described above. The estimates for the shape parameter alpha ranged from 0.495 to 2.103.
When working with experimental data, where researchers are spending time and money trying to detect potential signals, it’s reasonable to very carefully model the routine variation. By modeling the routine variation, we can package some specified percentage of the probable noise to be ignored (usually 95%) and then look for values outside that package that look like potential signals. This approach is like using the table in Figure 6. Fit a model, choose a specific area to filter out, and then find the exact width of interval to use in packaging the noise.
So how do you filter out the noise when you think your data are modeled by a gamma distribution?
So, while the central 920 parts per thousand are shifting toward the mean, and while another 60 parts per thousand get slightly shifted outward and then stabilize, it’s primarily the outer 20 parts per thousand that bear the brunt of the stretching and elongation that goes with increasing skewness.
Fitting a gamma distribution to your data
So what’s changing as you select different gamma probability models? To answer this question, Figure 2 considers 19 different gamma models. For each model, we have the skewness and kurtosis, the areas within one, two, and three standard deviations on either side of the mean, and the z-score for the 99.9th percentile of the model.
Figure 1: Five standardized gamma distributions
Or, you could use the one-size-fits-all approach of three-sigma limits. This approach is guaranteed to filter out at least 97.6% of the probable noise regardless of which gamma model may fit your data. This is why a process behavior chart for individual values will work even when you think the data might be modeled by a skewed gamma distribution.
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Figure 7: Widths of fixed-coverage central intervals
The purpose of analysis is insight. To gain insight, we have to detect any potential signals within the data. To detect potential signals, we must filter out the probable noise. And filtering out the probable noise is the objective of statistical analysis.
Clear thinking and simplicity of analysis require concise, clear, and correct notions about probability models and how to use them. Here, we’ll examine the basic properties of the family of gamma and chi-square distributions that play major roles in the development of statistical techniques. An appreciation of how these probability models function will allow you to analyze your data with confidence that your results are reliable, solid, and correct.
The gamma family of distributions
With industrial data, a simpler approach is feasible. Here, we’re trying to do the same thing over and over, and the signals of interest are changes that are large enough to have an economic effect. In this case, we bundle up nearly all of the routine variation as probable noise and react only to potential signals that are clearly not part of the routine variation.
The z-scores for the last part per thousand show that the upper tails get elongated with increasing skewness. But there is a surprise contained in the other columns of Figure 2.
Figure 3: How the coverages vary with skewness for gamma distributions
Figure 4: The compensation for increasing skewness for gammas
Gamma distributions are widely used in all areas of statistics, and are found in most statistical software. Since software facilitates our use of the gamma models, the following formulas are given here only in the interest of clarity of notation. Gamma distributions depend upon two parameters, denoted here by alpha and beta. The probability density function for the gamma family has the form:
