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Discussion of Simulation Results 127
9.4 DISCUSSION OF SIMULATION RESULTS
The seven methods included in this simulation can generally be classified into two
categories. The “assumed normal”, probability plotting, distribution-free tolerance
interval, weighted variance, and Wright’s index methods are classified as nontrans-
formation methods. Among the transformation methods are Clements, Box--Cox and
Johnson. There are two performance yardsticks in this investigation, namely accuracy
and precision given the sample size. For accuracy we look at the difference between
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the average simulated C pu values, C pu , and the target C pu value. For precision we look
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at the variation or spread in the average simulated C pu values. The smaller the vari-
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ation or spread in the average simulated C pu values, the better is the performance of
the method.
From Tables 9.3--9.6 we observe that the performance of the transformation methods
is consistently better than that of the non-transformation methods for all the different
underlying nonnormal distributions. There are only two exceptions to this statement.
Firstly, Clements’ method does not perform as well as the other two transformation
methods. In the case of the Weibull distribution (see Tables 9.5 and 9.6), the perfor-
mance of Clements’ method is inferior to that of the probability plotting method.
Secondly, we observe from the box plots in Figures 9.9--9.11 that the probability plot-
ting method is the only non-transformation method that manages to outperform the
transformation methods, but only in the case of the Weibull distribution with σ = 1.0
and η = 2.0. For the lognormal distribution the transformation methods consistently
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yield C pu values that are closer to the target C pu , while all the other methods over-
estimate it. The practical implication is that, though appealing in terms of simplicity
of calculation, the other nontransformation methods are found to be inadequate in
capturing the capability of the process except when the underlying distribution is
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close to normal. Moreover, the variations in the simulated C pu values are consistently
large than those of the transformation methods. The use of the other nontransforma-
tion methods is clearly inadequate for distributions that depart severely from normal.
This means that if one does not recognize the need to work with transformed normal
data, one may have a false sense of process capability where the potential process
fallout will be greater than expected. Indeed, many practitioners have the experience
of getting an acceptable PCI (≥1.33) but suffering from a relatively high process fallout.
Thus far, transformation methods seem to be adequate for handling nonnormal
data. The performance of the transformation methods is fairly consistent in terms
of accuracy and precision. However, it is noted that the transformation methods are
¯
quite sensitive to sample size, as the differences between C pu values estimated at
n = 50 and 150 are 33%, 11%, and 26% for the Clements, Box--Cox, and Johnson
transformation methods, respectively. These differences are higher compared with the
less than 10% difference in the nontransformation methods. Since, as noted earlier, the
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transformation methods exhibit a smaller variability in C pu , such a large difference
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betweenC pu s,computedfromsmallandlargesamplesizes,cannotbesolelyattributed
to natural statistical variation. This suggests that transformation methods are not
appropriate for small sample sizes. Some possible reasons are as follows.
1. Transformation methods usually involve a change from a scale--shape distribution
to a location--scale distribution. This may induce an unwanted process shift that
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distorts C pu , especially for small sample sizes.