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Comparison of Acceptance Criteria The acceptance rates of simulated samples with and without various problems were examined to compare different acceptance criteria E. Clarkson National Center for Advanced Materials Performance Presented to: Mil-HDBK-17 NCAMP December 2005, Santa Monica CA

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Qualifying Sample Variables Sample size –n = 18 & n = 30 Mean –Set at 100 –Also tested at the 5 th and 95 th percentiles to see the effect of an unusually high or low qualifying sample mean. Coefficient of variation (std dev / mean) –2%, 6%, and 10% –Since the mean is set to 100, the Coefficient of Variation in % and the standard deviation have the same value.

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Equivalence Sample Variables 50,000 samples generated for each set of conditions. Sample size –Tested at n = 2 to n = 10 Mean –Set at 100, at 95, and at 100 – 1σ. (σ = standard deviation) –Due to random generation of sample, the means will randomly vary around the preset value. Coefficient of variation (C.V.) –Set at 2%, 6%, and 10%, –Set at 1.5*C.V. of Qualifying sample. –Due to random generation of sample, the C.V. will randomly vary around the preset value.

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Acceptance Criteria Agate Criteria: –Detailed in DOT/FAA/AR-03/19 “Material Qualification and equivalency for Polymer Matrix Composite Material Systems: Updated Procedure”, Final Report Sept 2003 with alpha (α) set at 0.01. Cpk Criteria –A minimum Cpk of 1.33 is desired. The needed sample mean and minimum values are computed from the qualifying sample values based on that criteria. Criteria presented at Sept 1994 Mil-Hdbk-17 meeting. –A total of 8 different acceptance criteria used at that time by companies were tested to see how they performed compared to the recommended (AGATE) criteria.

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Acceptance Criteria Criteria presented at Sept 1994 Mil-Hdbk-17 meeting 1.Sample Average > B-Basis, Sample Minimum > A-basis 2.Sample Average > B-Basis, Sample Minimum > A-basis, Coef. Of Var < 7 3.Sample Minimum > B-Basis 4.Sample Average > Sample Minimum > B-Basis 5.Sample Average > Sample Minimum > 6.Sample Average > Sample Minimum > 7.Sample Average > 8.Sample Average > Sample Minimum > Where and s are the mean and standard deviation of the qualifying sample k is the one-sided tolerance factor based on the number of test values and n is the number of tests in the equivalence sample

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Agate Criteria –DOT/FAA/AR-03/19 specifies that the equivalency sample mean must be greater than: and the minimum value in the equivalency sample must be greater than: with: the qualifying sample mean the qualifying sample standard deviation n the size of the equivalency sample The k min and k mean values come from tables 20 and 21 in DOT/FAA/AR-03/19

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C pk Criteria –A minimum Cpk of 1.33 is desired. –The equivalency sample is compared to test values are computed as follows: 1.The equivalency sample mean must be greater than: 2.The minimum value of the equivalency sample must be greater than: is the qualifying sample mean is the qualifying sample standard deviation

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Comparison of Agate and C pk criteria Agate Criteria α set at 0.01 C pk Criteria C pk = 1.33 Mean Minimum

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Comparison of Agate and C pk criteria For n = 5 and producers risk (α) = 0.01 k mean = 1.1425 ≈ C pk = 0.852 k min.ind. = 3.0715 ≈ C pk = 1.024 By allowing for a single retest of samples that fail the Agate criteria, the producers risk (α) drops to 0.01 2 = 0.0001 For n = 5, k mean = 1.739 ≈ C pk = 1.296 k min.ind. = 4.2629 ≈ C pk = 1.421 Agate criteria with one retest ≈ C pk criteria with no retest.

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Type I and Type II Error Otherwise known as Producer’s risk and Consumer’s risk Claim that material is “good” Claim that material is “bad” Material is actually “good” No Error Type I Error (α) Producer’s Risk Material is actually “bad” Type II Error (β) Consumer’s Risk No Error

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Comparison of Criteria for producer’s risk Criteria #2, #3, #4, #5, #6, and #8 all reject far too many ‘good’ samples. The producers risk for those samples range from approx. 10% at the lowest to nearly 90% at the highest. To simplify the presentation, the remaining slides examine only criteria #1, #7, Cpk and Agate.

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Remaining Criteria CriteriaSample MeanSample Minimum # 1B-BasisA-Basis # 7 C pk Agate

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Effect of the size of the Equivalency Sample What effect does the size of the equivalency sample have on the probability of rejection of ‘good’ samples?

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Rejection rate of simulated “Good” samples Qualifying Sample Size 30

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What about the consumer’s risk? How well do these criteria identify ‘Bad’ Samples? To assess the impact of various types of problems, the mean and the coefficient of variation of the simulated samples were changed to determine acceptance rates for the different criteria.

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Samples were simulated to have the following problems: 1.The mean lowered from 100 to 95 2.The mean lowered by 1 std. dev. (σ) 3.The standard deviation (σ) was increased by 50% and the mean lowered by ½ σ. 4.Mean shifted randomly by 1 std dev. (σ) 5.Co-efficient of variation increased from 2 to 6 and to 10.

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What should happen when the mean changes? If mean shifts down, then the acceptance rate should go down If mean shifts randomly (both up and down) then the variance is increased and the acceptance rate should go down.

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Lowering the mean to 95

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Lowering the mean by one Std. Dev.

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Increasing the coefficient of variation and lowering the mean

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What should happen when the coefficient of variation increases for the equivalency sample? When the equivalence sample has a larger coefficient of variation (or standard deviation) than the qualifying sample, the rejection rates should go up.

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Rejection rate of simulated samples When C.V. is raised

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Rejection rate of simulated samples When mean shifts randomly

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What happens when the mean of the qualifying sample changes? The effect of shifting the mean of the qualifying sample up to the 95 th percentile or down to the 5 th percentile varies with the criteria being evaluated. Generally, as expected, a lower qualifying sample mean leads to a lower failure rate and vice versa For samples tested with the qualifying mean at the 95 th percentile (too high), the size of the equivalency sample impacted the failure rate.

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Rejection rate of simulated samples When Qualifying mean changes

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Effect of the size of the Equivalency Sample On almost all of the criteria tested, the size of the equivalency sample did have an effect. As sample size increases, so does the failure rate. This is due to the fact that all the other criteria involve checking the minimum value for the sample. As the sample size increases, the expected value for the sample minimum decreases. Unless the minimum value threshold is computed taking this into account (as the Agate criteria does), the probability of failure goes up with the sample size being evaluated. The one exception to this pattern was criteria #7, which only examined the mean, not the minimum value of the sample. In this case, as sample size increases, the sample mean converges to the true population mean and the failure rate falls.

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