H15 approach
A more statistically sound approach is to include the OOS result directly using a robust procedure based on the median and
deviations from it, because medians are less influenced by outlying values.
The method briefly described here is based on Huber's H15 method (6). This method is more difficult to calculate than the
standard confidence interval as it relies on an iterative procedure. However, it is easily accomplished using an Excel spreadsheet
without the necessity for macros. The mathematical details are given in Ellison, Barwick, and Duguid Farrant (7) and are not
covered here.
This robust statistical procedure has a major advantage in that the calculation includes the outlier as part of the data set
and provides values for the robust mean and robust standard deviation. This calculation procedure meets the FDA requirement
outlined earlier because active consideration is given to the OOS result. In addition, it calculates the 99% confidence interval
rather than the 95% that provides greater assurance of compliance with specification.
Briefly, the algorithm used is as follows:
1. Calculate the initial estimates of the median and robust standard deviation
2. Evaluate all the data points using
This step replaces points outside the range with values at the calculated 99% confidence limits but leaves other values unchanged.
3. Update the estimate for the standard deviation based on the current estimate with a correction factor, β, for the normal
distribution.
4. Repeat steps 2 and 3 for the updated data until the values for the robust mean and robust standard deviation no longer
change, for example, by 0.1% (i.e., the calculation converges).
5. Calculate the final 99% confidence interval from 2.96 Ŝi/β where i is the cycle at which it converges.
Figure 3: Illustration of the H15 approach for calculating the robust mean and robust standard deviation. LSL is lower specification
limit.
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It is easier to understand the process graphically (Figure 3) rather than the maths, and this approach is illustrated using the same data set as the original example. The initial OOS
result is shown as a red circle and the retest data are shown as green circles (initial data line). The LSL is also marked
on the plot. The first iteration moves both the initial OOS result and one retest result to the calculated robust 99% confidence
interval shown as blue dots with a red outline. Note that five of the six retest results are unchanged throughout all the
iterative calculations. As the iterations proceed, the robust mean and robust standard deviation change until the values converge.
In our example, it took 28 iterations to obtain convergence. The biggest change during the iterations is to the robust standard
deviation. The final 99% confidence interval is well within the specification and the robust mean value for the reportable
result (based on all seven data points) is 96.64.
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