Audit sampling: Get started

Introduction

Welcome to the ‘Audit sampling’ vignette of the jfa package. Here you can find a detailed explanation of the functions in the package that facilitate the statistical audit sampling workflow. Specifically, these functions implement standard audit sampling techniques to calculate sample sizes, select items from a population, and evaluate misstatement in a data sample. The jfa package enables users to create a prior probability distribution to perform Bayesian audit sampling using these functions. For more detailed explanations of each function, read the other vignettes on the package website.

Cheat sheet

The cheat sheet below will help you get started with jfa’s’ intended audit sampling workflow. You can download a pdf version of the cheat sheet here.

cheatsheet

Functions and intended usage

Below you can find an explanation of the available functions in jfa sorted by their occurrence in the standard audit sampling workflow.

auditPrior()
planning()
selection()
evaluation()
report()

Create a prior distribution with `auditPrior()`

The auditPrior() function is used to specify a prior distribution for Bayesian audit sampling. The interface allows a complete customization of the prior distribution as well as a formal translation of pre-existing audit information into a prior distribution. The function returns an object of class jfaPrior which can be used with associated summary() and plot() methods. Objects with class jfaPrior can also be used as input for the prior argument in other functions. Moreover, jfaPrior objects have a corresponding predict() function to produce the predictions of the prior distribution on the data level.

Full function with default arguments:

auditPrior(method = c(
             "default", "strict", "param", "impartial", "hyp",
             "arm", "bram", "sample", "factor", "nonparam"
           ),
           likelihood = c(
             "poisson", "binomial", "hypergeometric",
             "normal", "uniform", "cauchy", "t", "chisq",
             "exponential"
           ),
           N.units = NULL,
           alpha = NULL,
           beta = NULL,
           materiality = NULL,
           expected = 0,
           ir = NULL,
           cr = NULL,
           ub = NULL,
           p.hmin = NULL,
           x = NULL,
           n = NULL,
           factor = NULL,
           samples = NULL,
           conf.level = 0.95)

Supported options for the method argument:

default: Indifferent / noninformative prior distribution.
param: Manually set the prior parameters.
strict: Improper prior distribution (matches classical results).
impartial: Impartial prior distribution (Derks et al., 2022).
hyp: Manually provide the prior probability for tolerable misstatement (Derks et al., 2021).
arm: Manually provide the inherent risk and internal control risk (Derks et al., 2021).
bram: Manually provide the upper bound of the prior distribution (Touw & Hoogduin, 2011).
sample: Manually provide an equivalent prior sample (Derks et al., 2021).
factor: Manually provide and weigh an equivalent prior sample (Derks et al., 2021).
nonparam: Nonparametric prior based on samples.

Supported options for the likelihood argument:

poisson: Poisson likelihood and conjugate gamma prior distribution (Stewart, 2013).
binomial: Binomial likelihood and conjugate beta prior distribution (Steele, 1992).
hypergeometric: Hypergeometric likelihood and conjugate beta-binomial prior distribution (Dyer & Pierce, 1991).
normal: Normal prior distribution.
uniform: Uniform prior distribution.
cauchy: Cauchy prior distribution.
t: Student-t prior distribution.
chisq: Chi-squared prior distribution.
exponential: Exponential prior distribution.

Example usage:

# Default beta(1, 1) prior distribution
x <- auditPrior(method = "default", likelihood = "binomial")

# Custom gamma(1, 10) prior distribution
x <- auditPrior(method = "param", likelihood = "poisson", alpha = 1, beta = 10)

# Beta prior distribution incorporating inherent risk (70%) and control risk (50%)
x <- auditPrior(method = "arm", likelihood = "binomial", materiality = 0.05, ir = 0.7, cr = 0.5)

summary(x) # Prints information about the prior distribution

## 
##  Prior Distribution Summary
## 
## Options:
##   Likelihood:                    binomial 
##   Specifics:                     ir = 0.7; cr = 0.5; dr = 0.1428571 
## 
## Results:
##   Functional form:               beta(α = 1, β = 21) 
##   Mode:                          0 
##   Mean:                          0.045455 
##   Median:                        0.032468 
##   Variance:                      0.0018865 
##   Skewness:                      1.7442 
##   Information entropy (nat):     -2.0921 
##   95 percent upper bound:        0.13295 
##   Precision:                     0.13295

predict(x, n = 20, cumulative = TRUE) # Predictions for a sample of n = 20

##      x<=0      x<=1      x<=2      x<=3      x<=4      x<=5      x<=6      x<=7 
## 0.5121951 0.7682927 0.8930582 0.9521576 0.9793114 0.9913797 0.9965519 0.9986816 
##      x<=8      x<=9     x<=10     x<=11     x<=12     x<=13     x<=14     x<=15 
## 0.9995206 0.9998352 0.9999468 0.9999841 0.9999956 0.9999989 0.9999998 1.0000000 
##     x<=16     x<=17     x<=18     x<=19     x<=20 
## 1.0000000 1.0000000 1.0000000 1.0000000 1.0000000

Plan a sample with `planning()`

The planning() function is used to calculate a minimum sample size for audit samples. It allows specification of statistical requirements for the sample with respect to the performance materiality or the precision. The function returns an object of class jfaPlanning which can be used with associated summary() and plot() methods. To perform Bayesian planning, the input for the prior argument can be an object of class jfaPrior as returned by the auditPrior() function, or an object of class jfaPosterior as returned by the evaluation() function.

Full function with default arguments:

planning(materiality = NULL,
         min.precision = NULL,
         expected = 0,
         likelihood = c("poisson", "binomial", "hypergeometric"),
         conf.level = 0.95,
         N.units = NULL,
         by = 1,
         max = 5000,
         prior = FALSE)

Supported options for the likelihood argument:

poisson: Poisson likelihood (Stewart, 2012).
binomial: Binomial likelihood (Stewart, 2012).
hypergeometric: Hypergeometric likelihood (Stewart, 2012).

Example usage:

# Classical planning using the Poisson likelihood
x <- planning(materiality = 0.03, likelihood = "poisson")

# Bayesian planning using a default beta(1, 1) prior and binomial likelihood
x <- planning(materiality = 0.03, likelihood = "binomial", prior = TRUE)

# Bayesian planning using a custom beta(1, 10) prior and binomial likelihood
x <- planning(
  materiality = 0.03,
  prior = auditPrior(method = "param", likelihood = "binomial", alpha = 1, beta = 10)
)

summary(x) # Prints information about the planning

## 
##  Bayesian Audit Sample Planning Summary
## 
## Options:
##   Confidence level:              0.95 
##   Materiality:                   0.03 
##   Hypotheses:                    H₀: Θ > 0.03 vs. H₁: Θ < 0.03 
##   Expected:                      0 
##   Likelihood:                    binomial 
##   Prior distribution:            beta(α = 1, β = 10) 
## 
## Results:
##   Minimum sample size:           89 
##   Tolerable errors:              0 
##   Posterior distribution:        beta(α = 1, β = 99) 
##   Expected most likely error:    0 
##   Expected upper bound:          0.029807 
##   Expected precision:            0.029807 
##   Expected BF₁₀:                 54.479

Select sample items with `selection()`

The selection() function is used to perform statistical selection of audit samples. It offers flexible implementations of the most common audit sampling algorithms for attributes sampling and monetary unit sampling. The function returns an object of class jfaSelection which can be used with associated summary() and plot() methods. The input for the size argument can be an object of class jfaPlanning as returned by the planning() function.

Full function with default arguments:

selection(data,
          size,
          units = c("items", "values"),
          method = c("interval", "cell", "random", "sieve"),
          values = NULL,
          order = NULL,
          decreasing = FALSE,
          randomize = FALSE,
          replace = FALSE,
          start = 1)

Supported options for the units argument:

items: Sampling units are items (rows) (Leslie, Teitlebaum, & Anderson, 1979).
values: Sampling units are monetary units (Leslie, Teitlebaum, & Anderson, 1979).

Supported options for the method argument:

interval: Select a fixed unit from each interval.
cell: Select a random unit within each interval.
random: Select random units without an interval.
sieve: Select units using modified sieve sampling (Hoogduin, Hall, & Tsay, 2010).

Example usage:

# Selection using random record (attributes) sampling
x <- selection(data = BuildIt, size = 100, units = "items", method = "random")

# Selection using fixed interval monetary unit sampling (using column 'bookValue' in BuildIt)
x <- selection(
  data = BuildIt, size = 100, units = "values",
  method = "interval", values = "bookValue"
)

summary(x) # Prints information about the selection

## 
##  Audit Sample Selection Summary
## 
## Options:
##   Requested sample size:         100 
##   Sampling units:                monetary units 
##   Method:                        fixed interval sampling 
##   Starting point:                1 
## 
## Data:
##   Population size:               3500 
##   Population value:              1403221 
##   Selection interval:            14032 
## 
## Results:
##   Selected sampling units:       100 
##   Proportion of value:           0.037014 
##   Selected items:                100 
##   Proportion of size:            0.028571

Evaluate a sample with `evaluation()`

The evaluation() function takes a sample or summary statistics of the sample and performs evaluation according to the specified method and sampling objectives. The function returns an object of class jfaEvalution which can be used with associated summary() and plot() methods. To perform Bayesian evaluation, the input for the prior argument can be an object of class jfaPrior as returned by the auditPrior() function, or an object of class jfaPosterior as returned by the evaluation() function.

Full function with default arguments:

evaluation(materiality = NULL, 
           method = c(
             "poisson", "binomial", "hypergeometric",
             "stringer", "stringer.meikle", "stringer.lta", "stringer.pvz",
             "rohrbach", "moment", "coxsnell",
             "direct", "difference", "quotient", "regression", "mpu"
           ),
           alternative = c("less", "two.sided", "greater"),
           conf.level = 0.95,
           data = NULL,
           values = NULL,
           values.audit = NULL,
           strata = NULL,
           times = NULL,
           x = NULL,
           n = NULL,
           N.units = NULL,
           N.items = NULL,
           pooling = c("none", "complete", "partial"), 
           prior = FALSE)

Supported options for the method argument:

poisson: Poisson likelihood (Stewart, 2012).
binomial: Binomial likelihood (Stewart, 2012).
hypergeometric: Hypergeometric likelihood (Stewart, 2012).
stringer: Stringer bound (Bickel, 1992).
stringer.meikle: Stringer bound with Meikle’s correction (Meikle, 1972).
stringer.lta: Stringer bound with LTA correction (Leslie, Teitlebaum, & Anderson, 1979).
stringer.pvz: Modified Stringer bound (Pap & van Zuijlen, 1996).
rohrbach: Rohrbach’s augmented variance estimator (Rohrbach, 1993).
moment: Modified moment bound (Dworin & Grimlund, 1984).
coxsnell: Cox and Snell bound (Cox & Snell, 1979).
mpu: Mean-per-unit estimator (Touw & Hoogduin, 2011).
direct: Direct estimator (Touw & Hoogduin, 2011).
difference: Difference estimator (Touw & Hoogduin, 2011).
quotient: Quotient (ratio) estimator (Touw & Hoogduin, 2011).
regression: Regression estimator (Touw & Hoogduin, 2011).

Example usage:

# Classical evaluation using the Poisson likelihood (and summary statistics)
x <- evaluation(materiality = 0.03, x = 1, n = 100, method = "poisson")

# Bayesian evaluation using a default minimal information prior (and summary statistics)
x <- evaluation(materiality = 0.03, x = 1, n = 100, method = "poisson", prior = TRUE)

# Bayesian evaluation using a custom beta(1, 10) prior (and summary statistics)
x <- evaluation(
  materiality = 0.03, x = 1, n = 100,
  prior = auditPrior(method = "param", likelihood = "binomial", alpha = 1, beta = 10)
)

summary(x) # Prints information about the evaluation

## 
##  Bayesian Audit Sample Evaluation Summary
## 
## Options:
##   Confidence level:               0.95 
##   Materiality:                    0.03 
##   Hypotheses:                     H₀: Θ > 0.03 vs. H₁: Θ < 0.03 
##   Method:                         binomial 
##   Prior distribution:             beta(α = 1, β = 10) 
## 
## Data:
##   Sample size:                    100 
##   Number of errors:               1 
##   Sum of taints:                  1 
## 
## Results:
##   Posterior distribution:         beta(α = 2, β = 109) 
##   Most likely error:              0.0091743 
##   95 percent credible interval:   [0, 0.042399] 
##   Precision:                      0.033225 
##   BF₁₀:                            15.385

Create a report with `report()`

The report() function takes an object of class jfaEvaluation as returned by the evaluation() function and automatically creates a html or pdf report containing the analysis results and their interpretation.

Full function with default arguments:

report(object,
       file = "report.html",
       format = c("html_document", "pdf_document"))

Example usage:

# Generate an automatic report
report(object = x, file = 'myReport.html')

For an example report, see the following link.

Benchmarks

To validate the statistical results, jfa’s automated unit tests regularly verify the main output from the package against the following benchmarks:

Audit Sampling: Audit Guide (Appendix A and Appendix C)
AuditSampler
EZ-quant
MUS (R package version 0.1.6)
samplingbook (R package version 1.2.4)
SMASH21 + SMASH21-Bayes
SRA
Touw, P., and Hoogduin, L. (2011). Statistiek voor audit en controlling. Boom uitgevers, Amsterdam.

Statistical tables

Below you can find several informative tables that contain statistical sample sizes, upper limits, one-sided p values, and Bayes factors. These tables are created using the planning() and evaluation() functions provided in the package.

Sample sizes

Upper limits

One-sided p values

Bayes factors

References

Bickel, P. J. (1992). Inference and auditing: The Stringer bound. International Statistical Review, 60(2), 197–209. - View online
Cox, D. R., & Snell, E. J. (1979). On sampling and the estimation of rare errors. Biometrika, 66(1), 125-132. - View online
Derks, K. (2022). jfa: Statistical methods for auditing. R package version 0.6.5. - View online
Derks, K., de Swart, J., van Batenburg, P., Wagenmakers, E.-J., & Wetzels, R. (2021). Priors in a Bayesian audit: How integration of existing information into the prior distribution can improve audit transparency and efficiency. International Journal of Auditing, 25(3), 621-636. - View online
Derks, K., de Swart, J., Wagenmakers, E.-J., & Wetzels, R. (2021). The Bayesian Approach to Audit Evidence: Quantifying Statistical Evidence using the Bayes Factor. PsyArXiv - View online
Derks, K., de Swart. J., Wagenmakers, E.-J., Wille, J., & Wetzels, R. (2021). JASP for Audit: Bayesian Tools for the Auditing Practice. Journal of Open Source Software, 6(68), 2733. - View online
Derks, K., de Swart, J., Wagenmakers, E.-J., & Wetzels, R. (2022). An Impartial Bayesian Hypothesis Test for Audit Sampling. PsyArXiv - View online
Derks, K., de Swart, J., Wagenmakers, E.-J., & Wetzels, R. (2022). Bayesian Generalized Linear Modeling: How to Incorporate Audit Information into the Statistical Model. PsyArXiv - View online
Dworin, L. D. and Grimlund, R. A. (1984). Dollar-unit sampling for accounts receivable and inventory. The Accounting Review, 59(2), 218–241. - View online
Dyer, D., & Pierce, R. L. (1993). On the choice of the prior distribution in hypergeometric sampling. Communications in Statistics - Theory and Methods, 22(8), 2125-2146. - View online
Hoogduin, L. A., Hall, T. W., & Tsay, J. J. (2010). Modified sieve sampling: A method for single-and multi-stage probability-proportional-to-size sampling. Auditing: A Journal of Practice & Theory, 29(1), 125-148. - View online
International Auditing and Assurance Standards Board (IAASB). (2018). Handbook of international quality control, auditing review, other assurance, and related services pronouncements (Vol. I). New York: International Federation of Accountants.
Meikle, G. R. (1972). Statistical Sampling in an Audit Context. Canadian Institute of Chartered Accountants.
Leslie, D. A., Teitlebaum, A. D., & Anderson, R. J. (1979). Dollar-unit Sampling: A Practical Guide for Auditors. London: Pitman.
Pap, G., & van Zuijlen, M. C. (1996). On the asymptotic behaviour of the Stringer bound. Statistica Neerlandica, 50(3), 367-389. - View online
Rietveld, C. (1978). De zeefmethode als selectiemethode voor statistische steekproeven in de controlepraktijk (1). Compact: Computer en Accountant, 15, 2–11.
Rohrbach, K. J. (1993). Variance augmentation to achieve nominal coverage probability in sampling from audit populations. Auditing: A Journal of Practice & Theory, 12(2), 79-97.
Steele, A. (1992). Audit Risk and Audit Evidence: The Bayesian Approach to Statistical Auditing. San Diego: Academic Press.
Stewart, T. R. (2012). Technical Notes on the AICPA Audit Guide Audit Sampling. American Institute of Certified Public Accountants, New York. - View online
Stewart, T. R. (2013). A Bayesian Audit Assurance Model with Application to the Component Materiality problem in Group Audits. VU University, Amsterdam. - View online
Talens, E. (2005). Statistical Auditing and the AOQL-method. University of Groningen, Groningen. - View online
Touw, P., and Hoogduin, L. (2011). Statistiek voor Audit en Controlling. Boom uitgevers, Amsterdam.
Weiler, H. (1965). The use of incomplete beta functions for prior distributions in binomial sampling. Technometrics, 7(3), 335-347. - View online

Koen Derks

2023-03-21

Introduction

Cheat sheet

Functions and intended usage

Create a prior distribution with `auditPrior()`

Plan a sample with `planning()`

Select sample items with `selection()`

Evaluate a sample with `evaluation()`

Create a report with `report()`

Benchmarks

Statistical tables

References

Audit sampling: Get started

Koen Derks

2023-03-21

Introduction

Cheat sheet

Functions and intended usage

Create a prior distribution with auditPrior()

Plan a sample with planning()

Select sample items with selection()

Evaluate a sample with evaluation()

Create a report with report()

Benchmarks

Statistical tables

References

Create a prior distribution with `auditPrior()`

Plan a sample with `planning()`

Select sample items with `selection()`

Evaluate a sample with `evaluation()`

Create a report with `report()`