A.2.2 Results

Figure A-1 illustrates how the coverage of the 95% UCL varies for the Studentís-t and Chebyshev UCL equations for a range of sampling designs applied to lognormal distributions with a range of variability. The table below the graph gives the coverage statistics as well as the average RPD (based on the full distribution of UCLs calculated).

CV n r Student's t 95UCL Chebyshev 95UCL
Coverage mean RPD Coverage mean RPD
2.0 15 2 90.2% 156% 86.4% 108%
2.0 15 3 88.2% 66% 93.5% 99%
2.0 15 5 86.9% 40% 97.1% 82%
2.0 15 6 86.9% 32% 97.9% 72%
2.0 30 2 91.3% 139% 88.5% 96%
2.0 30 3 88.9% 61% 94.3% 91%
2.0 30 5 87.7% 39% 97.1% 79%
2.0 30 7 87.2% 31% 98.2% 70%
 
4.0 15 2 85.3% 237% 82.8% 163%
4.0 15 3 81.3% 102% 90.5% 152%
4.0 15 5 79.8% 63% 95.7% 129%
4.0 15 7 80.1% 51% 97.2% 115%
4.0 30 2 88.9% 129% 83.8% 187%
4.0 30 3 84.4% 119% 90.6% 80%
4.0 30 5 83.5% 100% 94.9% 49%
4.0 30 7 83.3% 90% 97.1% 40%

 


Figure A-1. Examples of simulation results using lognormal probability distributions with CV equal to 2 and 4, increments of 15 and 30, replicates ranging 2–7, and two 95% UCL calculation methods (Cheby = Chebyshev; t-UCL = Student’s-t).

These examples are useful for illustrating the following general patterns that emerge from the simulation experiments with lognormal distributions:

  1. The Chebyshev UCL generally yields higher coverage than the Student’s-t UCL, with the exception of scenarios in which two replicates (r = 2) are selected. The upper critical value of the Student’s-t distribution (i.e., t-value) varies with the degrees of freedom (df = r – 1), as noted below. For r = 2, the t-value is 6.3, which introduces an additional factor of 2 or more to the calculation of the UCL compared to sampling designs with three or more replicates.
  2. The coverage of the Chebyshev UCL generally increases with increasing sample sizes (increments and replicates) but with diminishing returns. The table below lists examples of combination of replicates and increments that can be expected to yield approximately 95% coverage.
    Replicates df = r – 1 t -value for alpha = 0.05
    2 1 6.3
    3 2 2.9
    4 3 2.4
    5 4 2.1
    6 5 2.0
    7 6 1.9

    The coverage of the Student’s-t UCL does generally not achieve 95% and does not increase with increasing samples sizes (increments and replicates) within a practical range.

  3. The RPD between the 95% UCL and the population mean is generally greater for the Chebyshev than the Student’s-t, particularly for trials in which the 95% UCL actually exceeds the population mean. Therefore, the tradeoff with the Chebyshev UCL is that it achieves more reliable coverage but also higher UCLs. See Section 4 for a side-by-side comparison of the range of 95% UCL RPDs for different scenarios for both the Chebyshev and Student’s-t UCL.

    CV Increments Replicates Coverage CV Increments Replicates Coverage
    1 15 3 96% 4 30 4 94%
    30 3 97% 50 4 95%
    2 15 3 93% 100 3 93%
    15 4 95% 100 4 96%
    30 3 94% 7 30 5 95%
    30 4 96% 100 5 95%
    3 15 5 95%  
    30 4 95%  
    50 4 96%  
    100 3 95%  
  4. The simulations with lognormal distributions yield unbiased estimates of the mean.