1 the odds are against auditing statistical sampling plans steven walfish statistical outsourcing...

1

The Odds Are Against AuditingStatistical Sampling Plans

Steven WalfishStatistical Outsourcing Services

Olney, MD301-325-3129

[email protected]

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.


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.


Common Sampling Strategies

Simple random sample.

Stratified sample.

Systematic sample.

Haphazard

Probability proportional to size


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-


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.


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.


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


Binomial Confidence Intervals


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


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.


Hypothesis Testing - Poisson

( )!

xeP x

x

P(x) = probability of exactly x occurrences.

is the mean number of occurrences.


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%)


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


Major Drawback

What is missing in ALL calculations for the Poisson?

No reference to sample size.

Assumes a large population (np>5)


Comparison


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


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.


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%


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.


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.


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


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.


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


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.


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


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.


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


Questions

Steven Walfish

[email protected]

301-325-3129 (Phone)

240-559-0989 (Fax)

1 the odds are against auditing statistical sampling plans steven walfish statistical outsourcing...

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