The Odds Are Against Auditing Statistical Sampling Plans Steven Walfish Statistical Outsourcing Services Olney, MD 301-325-3129 steven@statisticaloutsourcingservices.com

Topics of Discussion The Paradox Different types of sampling plans. Types of Risk Statistical Distribution Normal Binomial Poisson When to Audit. April 21, 2010 ASQ Section 511

The Paradox During an audit you increase the sample size if you have a finding… But, no findings might be because your sample size is too small to find errors. April 21, 2010 ASQ Section 511

Common Sampling Strategies Simple random sample. Stratified sample. Systematic sample. Haphazard Probability proportional to size April 21, 2010 ASQ Section 511

Types of Risk Decision Reality Accept Reject Correct Decision Type II Error (b) Consumer Risk Type I Error (a) Producer Risk Power (1-b) April 21, 2010 ASQ Section 511

Normal Distribution Typical bell-shaped curve. Z-scores determine how many standard deviations a value is from the mean. April 21, 2010 ASQ Section 511

Continuous Data Sample Size As the effect size decreases, the sample size increases. As variability increases, sample size increases. Sample size is proportional to risks taken. April 21, 2010 ASQ Section 511

Binomial Distribution where: n is the sample size x is the number of positives p is the probability a is the probability of the observing x in a sample of n. April 21, 2010 ASQ Section 511

Binomial Confidence Intervals Binomial Distribution Solve the equation for p given a, x and n. x=0, n=11 and a=0.05 (95% confidence). p=0.28 (table shows 0.30ucl) x=2, n=27 and a=0.01 (99% confidence). p=0.298 (table shows 0.30ucl) April 21, 2010 ASQ Section 511

Poisson Distribution Describes the number of times an event occurs in a finite observation space. For example, a Poisson distribution can describe the number audit findings. The Poisson distribution is defined by one parameter: lambda. This parameter equals the mean and variance. As lambda increases, the Poisson distribution approaches a normal distribution. April 21, 2010 ASQ Section 511

Hypothesis Testing - Poisson P(x) = probability of exactly x occurrences. l is the mean number of occurrences. April 21, 2010 ASQ Section 511

Example of Poisson If the average number (l) of audit findings is 5.5. What is the probability of a sample with exactly 0 findings? 0.0041 (0.41%) What is the probability of having 4 or less findings in a sample (x=0 + x=1 + x=2 + x=3 + x=4) 0.0041 + 0.0225 + 0.0618 + 0.1133 + 0.1558 = 0.358 (35.8%) April 21, 2010 ASQ Section 511

Poisson Confidence Interval The central confidence interval approach can be approximated in two ways: 95% CI for x=6 would be (2.2,13.1) April 21, 2010 ASQ Section 511

Major Drawback What is missing in ALL calculations for the Poisson? No reference to sample size. Assumes a large population (np>5) April 21, 2010 ASQ Section 511

Comparison April 21, 2010 ASQ Section 511

was an unpublished report by the AOAC in 1927. It was intended to be a quick rule of thumb for inspection of foods. Since it was unpublished, there was not a description of the statistical basis of it. April 21, 2010 ASQ Section 511

There is no known statistical justification for the use of the square root of n plus one’ sampling plan. “Despite the fact that there is no statistical basis for a ‘square root of n plus one’ sampling plan, most firms utilize this approach for incoming raw materials.” Henson, E., A Pocket Guide to CGMP Sampling, IVT. April 21, 2010 ASQ Section 511

Compare the Plans ANSI/ASQ Z1.4 Lot Size N=1000 Sample size n=32 Acceptance Ac=0 Rejection Re=1 AQL=0.160% LQ = 6.94% Square root N plus one Lot Size N=1000 Sample size n=33 Acceptance Ac=0 Rejection Re=1 AQL=0.153% LQ = 6.63% April 21, 2010 ASQ Section 511

Is it a Real Sampling Plan? Yes, it meets the Z1.4 definition of a sampling plan. It is statistically valid in that it defines the lot size, N, the sample size, n, the accept number, Ac, and the reject number, Re. The Operational Characteristic, OC, curve can be calculated for any square root N plus one plan. April 21, 2010 ASQ Section 511

Sample Size Comparison It is very common to use Z1.4 General Level I as the plan for audits. The sample sizes for square root N plus one are very close to the sample sizes for Z1.4 GL I. Square root N plus one can be used any where that Z1.4 GL I is or could be used. April 21, 2010 ASQ Section 511

Sample Size Comparison April 21, 2010 ASQ Section 511

Is it a Good Plan? Like Z1.4 GL I it can be used for audits. Any plan is justified by AQL and LQ It is easy to use and calculate. Works best with an Ac=0. April 21, 2010 ASQ Section 511

Example Lot Size Sample Size Ac=0 Ac=1 AQL LQ 4 3 1.69 54 13.50 80 10 1.27 44 9.78 68 25 6 0.85 32 6.30 51 50 8 0.64 4.60 41 100 11 0.46 19 3.30 31 250 17 0.30 13 2.10 21 500 23 0.22 9.5 1.57 16 1000 33 0.16 6.7 1.09 10000 101 0.05 2.3 0.35 3.8 April 21, 2010 ASQ Section 511

Using Statistics How do you determine when you have too many findings? How do you determine the correct sample size for an audit? Would a confidence interval approach work? As long as the observed number is lower than the upper confidence interval, the system is in control. April 21, 2010 ASQ Section 511

Deciding to Audit Need to use risk or statistical probability to determine when to audit: Critical components Low rank High Volume suppliers No third party data available April 21, 2010 ASQ Section 511

Results of an Audit The results of an audit can help to establish acceptance controls. Better audit results would have less risk, and require smaller sample sizes for incoming inspection. Can use AQL or LTPD type of acceptance plans based on audit results. April 21, 2010 ASQ Section 511

Conclusion Using the correct sampling strategy helps to assure coverage during an audit. Using confidence intervals to determine if a system is in control. More compliant systems require larger sample sizes. April 21, 2010 ASQ Section 511

Questions Steven Walfish steven@statisticaloutsourcingservices.com 301-325-3129 (Phone) 240-559-0989 (Fax) April 21, 2010 ASQ Section 511