The%20Odds%20Are%20Against%20Auditing%20Statistical%20Sampling%20Plans

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The Odds Are Against AuditingStatistical
Sampling Plans

• Steven WalfishStatistical Outsourcing
  ServicesOlney, MD301-325-3129steven_at_statistical
  outsourcingservices.com

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Topics of Discussion

• The Paradox
• Different types of sampling plans.
• Types of Risk
• Statistical Distribution
• Normal
• Binomial
• Poisson
• When to Audit.

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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.

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Common Sampling Strategies

• Simple random sample.
• Stratified sample.
• Systematic sample.
• Haphazard
• Probability proportional to size

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Types of Risk
Decision Reality Reality Reality
Decision Accept Reject
Decision Accept Correct Decision Type II Error (b) Consumer Risk
Decision Reject Type I Error (a) Producer Risk Correct Decision Power (1-b)
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Normal Distribution

• Typical bell-shaped curve.
• Z-scores determine how many standard deviations a
  value is from the mean.

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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.

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Binomial Distribution

• 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.

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Binomial Confidence Intervals

• Binomial Distribution
• Solve the equation for p given a, x and n.
• x0, n11 and a0.05 (95 confidence).
• p0.28 (table shows 0.30ucl)
• x2, n27 and a0.01 (99 confidence).
• p0.298 (table shows 0.30ucl)

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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.

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Hypothesis Testing - Poisson

• P(x) probability of exactly x occurrences.
• l is the mean number of occurrences.

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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
• (x0 x1 x2 x3 x4)
• 0.0041 0.0225 0.0618 0.1133 0.1558
  0.358 (35.8)

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Poisson Confidence Interval

• The central confidence interval approach can be
  approximated in two ways
• 95 CI for x6 would be (2.2,13.1)

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Major Drawback

• What is missing in ALL calculations for the
  Poisson?
• No reference to sample size.
• Assumes a large population (npgt5)

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Comparison
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• 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.

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• 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.

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Compare the Plans

• ANSI/ASQ Z1.4
• Lot Size N1000
• Sample size n32
• Acceptance Ac0
• Rejection Re1
• AQL0.160
• LQ 6.94

• Square root N plus one
• Lot Size N1000
• Sample size n33
• Acceptance Ac0
• Rejection Re1
• AQL0.153
• LQ 6.63

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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.

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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.

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Sample Size Comparison
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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 Ac0.

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Example
Lot Size Sample Size Ac0 Ac0 Ac1 Ac1
AQL LQ AQL LQ
4 3 1.69 54 13.50 80
10 4 1.27 44 9.78 68
25 6 0.85 32 6.30 51
50 8 0.64 25 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 11
10000 101 0.05 2.3 0.35 3.8
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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.

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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

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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.

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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.

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Questions

• Steven Walfish
• steven_at_statisticaloutsourcingservices.com
• 301-325-3129 (Phone)
• 240-559-0989 (Fax)