ISO PWI 28596: two-stage sampling plans forauditing

Rainer Göb1, Jens Bischoff1

1Institute for Applied Mathematics and StatisticsUniversity of Würzburg, Germany

ENBIS-1611 – 15 September 2016, Sheffield

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Outline

1 Standards environment

2 Two-stage sampling plans for a proportion

3 ISO PWI 28596

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Outline

1 Standards environment

2 Two-stage sampling plans for a proportion

3 ISO PWI 28596

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ISO standards attributes samplingISO 2859-1 single sampling plans for series of lots, AQL index

ISO 2859-2 single sampling plans for isolated lots, LQL index

ISO 28591 sequential sampling plans for isolated lots

ISO 28592 double sampling plans for isolated lots

ISO 28593 accept-zero sampling plans for series of lots

ISO 28594 alternative to 2859-1, with AOQL instead of AQL

ISO 28597 ppm quality levels, isolated lots and series of lots

ISO 28598 increasing trust level reduces sample size

PWI 28596 two-stage sampling for auditing under prior information4 / 20

auditing standards

American Institute of CertifiedPublic Accountants (AICPA)

=⇒ Statements on AuditingStandards (SAS)

International Federation ofAccountants (IFAC)

=⇒ International Standardson Auditing (ISA)

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ISA 530 (2009) “Audit Sampling”

sampling condition“Audit sampling (sampling) – The application of audit procedures toless than 100 % of items within a population of audit relevance suchthat all sampling units have a chance of selection in order toprovide the auditor with a reasonable basis on which to drawconclusions about the entire population.”

ISA 530: judgmental sampling (purposive selection byprofessional judgment) violates the sampling condition

ISA 530: judgmental sampling is not audit sampling

statistical sampling mandatory

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auditing environment

sample sizes small to moderate: 50 ≤ n ≤ 150

simple sampling procedures (two stage possible, no sequential)prior information (past audits, previous steps in present audit)

control both risks: erroneous acceptance, erroneous rejection =⇒two-sided confidence intervals

enable estimation of quality parameter (confidence interval)

no existing ISO attributes sampling standard is adequate!

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auditing risk model

auditrisk

=inherent

risk× control

risk×

analyticalprocedures

risk×

test ofdetails

riskAR = IR × CR × AP × TD

0.05 = 1.00 × 0.20 × 0.8 × TD

TD =AR

IR× CR× AP=

0.051.00× 0.20× 0.80

= 0.3125

confidencelevel γ

= 1- TD = 1− 0.3125 = 0.6875

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Outline

1 Standards environment

2 Two-stage sampling plans for a proportion

3 ISO PWI 28596

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decision target and methodology

target parameterp = proportion nonconformingerroneous entries in accounts, malfunction of internal controlsystem, ...

methodologydecision based on two-sided confidence intervals for pprior information included in CI, see Göb & Lurz (2013)prior information modelled by beta distribution

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decision by sampling plan n1, (c1, r1), n2, c2

sample n1 items −→ find total of x1 nonconforming items↙ ↓ ↘

x1

]

c1

[

r1 x1

]

c1

[

r1 x1

]

c1

[

r1

accept second stage rejectysample n2 items −→ find total of x2 nonconforming items

↙ ↘

x1 + x2

]

c2 x1 + x2

]

c2

accept reject11 / 20

design of sampling plans

specify tolerance p0

specify prior information via beta distribution of p

specify confidence level γ

classical designs impose r = n2/n1, e. g. r = 1,2, Duncan (1965)

no ratio r = n2/n1 prescribed here, further restrictions instead,namely:

acceptance in first stage must be possible!

acceptance in second stage must be possible!

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Outline

1 Standards environment

2 Two-stage sampling plans for a proportion

3 ISO PWI 28596

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ISO PWI 28596 table of contents

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tolerances

tolerance p0 = tolerable proportion of nonconforming units inauditing classes (for example invoices)

ISO PWI 28596: p0 = 0.01, 0.02, ..., 0.05

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eliciting prior information

prior information elicited by specifying two quantiles of random p

quantile points linked to tolerance:z(level1) = p0 and z(level2) = 3p0

current quantile level pairs: (55%,80%), (60%,85%), (65%,90%).

example quantile points for p0 = 0.05:z(0.55) = 0.05 andz(0.80) = 0.15 proportion nonconforming

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sampling plan table

prior: 55% quantile = 0.05, 80% quantile = 0.15cell content:n1 (acceptance number first stage, rejection number first stage)n2 acceptance number second stage

z(0.55)=p0 p0=z(0.80)=3p0 0.05 0.04 0.03 0.02 0.01

γ

0.95 60 (0, 9)118 8

75 (0, 9)149 8

101 (0, 9)199 8

153 (0, 9)296 8

307 (0, 9)585 8

0.85 39 (0, 6)69 5

49 (0, 6)87 5

66 (0, 6)116 5

100 (0, 6)176 5

208 (0, 6)344 5

0.7 30 (0, 4)46 3

39 (0, 4)55 3

54 (0, 4)71 3

93 (0, 4)90 3

207 (0, 4)156 3

0.6 29 (0, 4)48 3

38 (0, 4)57 3

53 (0, 4)73 3

88 (0, 4)97 3

151 (0, 4)239 3

0.5 28 (0, 3)31 2

37 (0, 3)36 2

52 (0, 3)44 2

87 (0, 3)54 2

150 (0, 3)236 3

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sampling plan table

prior: 60% quantile = 0.05, 85% quantile = 0.15

z(0.60)=p0 p0=z(0.85)=3p0 0.05 0.04 0.03 0.02 0.01

γ

0.95 60 (0, 9)118 8

75 (0, 9)149 8

101 (0, 9)197 8

153 (0, 9)293 8

307 (0, 9)585 8

0.85 39 (0, 6)73 5

50 (0, 6)89 5

66 (0, 6)121 5

100 (0, 6)181 5

213 (0, 6)347 5

0.7 37 (0, 4)40 3

49 (0, 4)47 3

64 (0, 5)95 4

97 (0, 5)142 4

195 (0, 5)284 4

0.6 36 (0, 4)41 3

48 (0, 4)48 3

63 (0, 4)66 3

96 (0, 4)98 3

194 (0, 4)196 3

0.5 35 (0, 3)41 3

47 (0, 3)47 3

62 (0, 3)64 3

95 (0, 3)94 3

193 (0, 3)186 3

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Coverage (tolerance = 0.05)

prior 60% quantile = 0.05 85% quantile = 0.15γ 0.5 0.95

stage 1 35 (0,3) 60 (0,9)stage 2 41 3 118 8

actual coverage of combined CI

0.00 0.05 0.10 0.15 0.20 0.25 0.30

0.2

0.4

0.6

0.8

1.0

0.00 0.05 0.10 0.15 0.20 0.25 0.30

0.4

0.5

0.6

0.7

0.8

0.9

1.0

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Thank you for your attention

Rainer Göb, Jens BischoffInstitute for Applied Mathematics and StatisticsUniversity of Würzburg, Germany

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