1 the odds are against auditing statistical sampling plans steven walfish statistical outsourcing...
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1
The Odds Are Against AuditingStatistical Sampling Plans
Steven WalfishStatistical Outsourcing Services
Olney, MD301-325-3129
April 21, 2010 ASQ Section 511 2
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 3
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 4
Common Sampling Strategies
Simple random sample.
Stratified sample.
Systematic sample.
Haphazard
Probability proportional to size
April 21, 2010 ASQ Section 511 5
Types of Risk
Decision
Reality
Accept Reject
Accept Correct Decision Type II Error () Consumer Risk
Reject Type I Error () Producer Risk
Correct Decision
Power (1-
April 21, 2010 ASQ Section 511 6
Normal Distribution
0.4
0.3
0.2
0.1
0.0
X
Densi
ty
-2
0.0228
2
0.0228
0
Distribution PlotNormal, Mean=0, StDev=1
Typical bell-shaped curve.
Z-scores determine how many standard deviations a value is from the mean.
April 21, 2010 ASQ Section 511 7
Continuous Data Sample Size
2 2
2
Z Z Sn
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 8
Binomial Distribution
Binomial Distribution
where:• n is the sample size
• x is the number of positives
• p is the probability
• is the probability of the observing x in a sample of n.
x)(nx p-1 p x
n
April 21, 2010 ASQ Section 511 9
Binomial Confidence Intervals
Binomial Distribution
Solve the equation for p given , x and n.
x=0, n=11 and =0.05 (95% confidence).• p=0.28 (table shows 0.30ucl)
x=2, n=27 and =0.01 (99% confidence).• p=0.298 (table shows 0.30ucl)
x)(nx p-1 p x
n
April 21, 2010 ASQ Section 511 10
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 11
Hypothesis Testing - Poisson
( )!
xeP x
x
P(x) = probability of exactly x occurrences.
is the mean number of occurrences.
April 21, 2010 ASQ Section 511 12
Example of Poisson If the average number () 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 13
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)
2 20.975;2 0.025;2( 1)
2 2
1 1
2 2
1.965 1.9651
2 2
x x
x x
April 21, 2010 ASQ Section 511 14
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 15
Comparison
April 21, 2010 ASQ Section 511 16
1N 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.
1N
April 21, 2010 ASQ Section 511 17
1N
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 18
Compare the PlansANSI/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 19
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 20
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 21
Sample Size Comparison
Sqrt(N+1) versus Z1.4
1
10
100
1000
1 10 100 1000 10000 100000 1000000
Lot Size
Sa
mp
le S
ize
Sqrt (N+1)
Z1.4
April 21, 2010 ASQ Section 511 22
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 23
ExampleLot Size Sample Size Ac=0 Ac=1
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
April 21, 2010 ASQ Section 511 24
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 25
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 26
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 27
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.
Conclusion
April 21, 2010 ASQ Section 511 28
Questions
Steven Walfish
301-325-3129 (Phone)
240-559-0989 (Fax)