jfa: Bayesian and Classical Audit Sampling

jfa is an R package for statistical audit sampling. The package provides functions for planning, performing, evaluating, and reporting an audit sample compliant with the International Standards on Auditing. Specifically, these functions implement standard audit sampling techniques for calculating sample sizes, selecting items from a population, and evaluating misstatement from a data sample or from summary statistics. Additionally, the jfa package allows the user to create a prior probability distribution to perform Bayesian audit sampling using these functions.

The package and its intended workflow are also implemented with a graphical user interface in the Audit module of JASP, a free and open-source statistical software program.

Overview

For complete documentation of jfa, visit the package website or download the package manual.

1. Installation
2. Cheat sheet
3. Intended workflow
4. Benchmarks
5. Statistical tables
6. References
7. Contributing

1. Installation

The most recently released version of jfa can be downloaded from CRAN by running the following command in R:

install.packages('jfa')

Alternatively, you can download the development version from GitHub using:

devtools::install_github('koenderks/jfa')

After installation, the jfa package can be loaded with:

library(jfa)

2. Cheat sheet

The cheat sheet below can help you get started with the jfa package and its intended workflow. You can download a pdf version of the cheat sheet here.

3. Intended workflow

Below you can find an explanation of the available functions in jfa sorted by their occurrence in the standard audit sampling workflow. For detailed examples of how to use these functions, visit the Get started section on the package website.

• 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"
           ), likelihood = c("poisson", "binomial", "hypergeometric"),
           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, conf.level = 0.95)

Supported options for the method argument:

• default: Indifferent / noninformative prior distribution.
• strict: Improper prior distribution (matches classical results).
• impartial: Impartial prior distribution (Derks et al., 2022).
• param: Manually set the prior parameters.
• 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).

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).

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

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

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

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, min.precision = 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, times = NULL,
           x = NULL, n = NULL, N.units = NULL, N.items = NULL,
           r.delta = 2.7, m.type = "accounts", cs.a = 1, cs.b = 3, cs.mu = 0.5,
           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

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.

4. 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
• Touw, P., and Hoogduin, L. (2011). Statistiek voor audit en controlling. Boom uitgevers, Amsterdam.

5. 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

• Sample sizes based on the Poisson distribution
• Sample sizes based on the binomial distribution
• Sample sizes based on the hypergeometric distribution

Upper limits

• Upper limits based on the Poisson distribution
• Upper limits based on the binomial distribution
• Upper limits based on the hypergeometric distribution

One-sided p values

• One-sided p values based on the Poisson distribution
• One-sided p values based on the binomial distribution
• One-sided p values based on the hypergeometric distribution

Bayes factors

• Impartial Bayes factors based on the gamma distribution
• Impartial Bayes factors based on the beta distribution
• Impartial Bayes factors based on the beta-binomial distribution

6. 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: Bayesian and classical audit sampling. R package version 0.6.3. - 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
• 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

7. Contributing

jfa is an open-source project that aims to be useful for the audit community. Your help in benchmarking and extending jfa is therefore greatly appreciated. Contributing to jfa does not have to take much time or knowledge, and there is extensive information available about it on the Wiki of this repository.

If you are willing to contribute to the improvement of the package by adding a benchmark, please check out the Wiki page on how to contribute a benchmark to jfa. If you are willing to contribute to the improvement of the package by adding a new statistical method, please check the Wiki page on how to contribute a new method to jfa.