evaluation() is used to perform statistical inference
about the misstatement in a population after auditing a statistical sample.
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 jfaEvaluation that can be used with associated
summary() and plot() methods.
evaluation(
materiality = NULL,
method = c(
"poisson", "binomial", "hypergeometric",
"inflated.poisson", "hurdle.beta",
"stringer.poisson", "stringer.binomial", "stringer.hypergeometric",
"stringer.meikle", "stringer.lta", "stringer.pvz", "stringer",
"rohrbach", "moment", "coxsnell", "mpu", "pps",
"direct", "difference", "quotient", "regression"
),
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
)a numeric value between 0 and 1 specifying the
performance materiality (i.e., the maximum tolerable misstatement in the
population) as a fraction of the total number of units in the population.
Can be NULL. Not used for methods direct, difference,
quotient, and regression.
a character specifying the statistical method. Possible
options are poisson (default), binomial,
hypergeometric, stringer.poisson, stringer.binomial,
stringer.hypergeometric, stringer.meikle,
stringer.lta, stringer.pvz, rohrbach, moment,
mpu, pps, direct, difference, quotient,
or regression. See the details section for more information.
a character indicating the alternative hypothesis and
the type of confidence / credible interval returned by the function.
Possible options are less (default), two.sided, or
greater.
a numeric value between 0 and 1 specifying the confidence level (i.e., 1 - audit risk / detection risk).
a data frame containing a data sample.
a character specifying name of a numeric column in
data containing the book values of the items.
a character specifying name of a numeric column in
data containing the audit (i.e., true) values of the items.
a character specifying name of a factor column in
data indicating to which stratum the item belongs.
a character specifying name of an integer column in
data containing the number of times an item should be counted due
to (not) being selected (multiple times) for the sample. Items for which
this value is 0 will not be included in the evaluation.
a numeric value or vector of values equal to or larger
than 0 specifying the sum of (proportional) misstatements in the sample or,
if this is a vector, the sum of taints in each stratum. If this argument is
specified, the input for the data, values and
values.audit arguments is discarded and it is assumed that the data
come from summary statistics specified by x and n.
an integer or vector of integers larger than 0
specifying the sum of (proportional) misstatements in the sample or, if
this is a vector, the sum of taints in each stratum. If this argument is
specified, the input for the data, values and
values.audit arguments is discarded and it is assumed that the data
come from summary statistics specified by x and n.
a numeric value or vector of values than 0 specifying
the total number of units in the population or, if this is a vector, the
total number of units in each stratum of the population.
This argument is strictly required for the hypergeometric,
direct, difference, quotient, and regression
methods, but is also used in stratification to weigh the estimates of each
individual stratum to arrive at the population estimate. If NULL,
each stratum is assumed to be equally represented in the population.
an integer larger than 0 specifying the number of items
in the population. Only used for methods direct, difference,
quotient, and regression.
a character specifying the type of model to use when
analyzing stratified samples. Possible options are none (default)
for no pooling (i.e., no information is shared between strata),
complete for complete pooling (i.e., all information is shared
between strata) or partial for partial pooling (i.e., some
information is shared between strata). The latter two options fit a
Bayesian model to the data using a MCMC sampling procedure whose options
can be set globally using options("mc.iterations") (otherwise:
2000), options("mc.warmup") (otherwise: 1000),
options("mc.chains") (otherwise: 4) and options("mc.cores")
(otherwise: 1).
a logical specifying whether to use a prior
distribution, or an object of class jfaPrior or jfaPosterior.
If this argument is specified as FALSE (default), the function
performs classical evaluation. If this argument is specified as TRUE
or as a prior from auditPrior, this function performs Bayesian
evaluation using the specified prior.
An object of class jfaEvaluation containing:
a numeric value between 0 and 1 giving the confidence level.
a numeric value between 0 and 1 giving the most likely misstatement in the population as a fraction.
a numeric value between 0 and 1 giving the upper bound for the misstatement in the population.
a numeric value between 0 and 1 giving the lower bound for the misstatement in the population.
a numeric value between 0 and 1 giving the difference
between the most likely misstatement and the bound relative to
alternative.
for classical tests, a numeric value giving the p-value.
an integer larger than, or equal to, 0 giving the number of misstatements in the sample.
a value larger than, or equal to, 0, giving the sum of proportional misstatements in the sample.
an integer larger than 0 giving the sample size.
if materiality is specified, a numeric value
between 0 and 1 giving the performance materiality as a fraction.
a character indicating the alternative hypothesis.
a character the method used.
if N.units is specified, in integer larger than 0
indicating the number of units in the population
if N.items is specified, in integer larger than 0
indicating the number of items in the population.
if method = 'hypergeometric', an integer indicating the
assumed total errors in the population.
an object of class jfaPrior that contains the prior
distribution.
an object of class jfaPosterior that contains the
posterior distribution.
a data frame containing the relevant columns from the
data.
a data frame containing the relevant statistical results for the strata.
a character giving the name of the data.
This section lists the available options for the method
argument.
poisson: Evaluates the sample with the Poisson
distribution. If combined with prior = TRUE, performs Bayesian
evaluation using a gamma prior.
binomial: Evaluates the sample with the binomial
distribution. If combined with prior = TRUE, performs Bayesian
evaluation using a beta prior.
hypergeometric: Evaluates the sample with the
hypergeometric distribution. If combined with prior = TRUE,
performs Bayesian evaluation using a beta-binomial prior.
inflated.poisson: Inflated Poisson model incorporating
the explicit probability of misstatement being zero. If
prior = TRUE, performs Bayesian evaluation using a beta
prior.
hurdle.beta: Hurdle beta model incorporating the
explicit probability of a taint being zero, one, or in between. If
prior = TRUE, this setup performs Bayesian evaluation using a
beta prior.
stringer.poisson: Evaluates the sample with the Stringer
bound using the Poisson distribution.
stringer.binomial: Evaluates the sample with the Stringer
bound using the binomial distribution (Stringer, 1963).
stringer.hypergeometric: Evaluates the sample with the Stringer
bound using the hypergeometric distribution.
stringer.meikle: Evaluates the sample using the
Stringer bound with Meikle's correction for understatements
(Meikle, 1972).
stringer.lta: Evaluates the sample using the
Stringer bound with LTA correction for understatements (Leslie,
Teitlebaum, and Anderson, 1979).
stringer.pvz: Evaluates the sample using the
Stringer bound with Pap and van Zuijlen's correction for understatements
(Pap and van Zuijlen, 1996).
rohrbach: Evaluates the sample using Rohrbach's
augmented variance bound (Rohrbach, 1993).
moment: Evaluates the sample using the
modified moment bound (Dworin and Grimlund, 1984).
coxsnell: Evaluates the sample using the Cox and
Snell bound (Cox and Snell, 1979).
mpu: Evaluates the sample with the
mean-per-unit estimator using the Normal distribution.
pps: Evaluates the sample with the
proportional-to-size estimator using the Student-t distribution.
direct: Evaluates the sample using the direct
estimator (Touw and Hoogduin, 2011).
difference: Evaluates the sample using the
difference estimator (Touw and Hoogduin, 2011).
quotient: Evaluates the sample using the
quotient estimator (Touw and Hoogduin, 2011).
regression: Evaluates the sample using the
regression estimator (Touw and Hoogduin, 2011).
Cox, D. and Snell, E. (1979). On sampling and the estimation of rare errors. Biometrika, 66(1), 125-132. doi:10.1093/biomet/66.1.125 .
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. doi:10.1111/ijau.12240
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. doi:10.21105/joss.02733
Derks, K., de Swart, J., Wagenmakers, E.-J., & Wetzels, R. (2024). The Bayesian approach to audit evidence: Quantifying statistical evidence using the Bayes factor. Auditing: A Journal of Practice & Theory. doi:10.2308/AJPT-2021-086
Derks, K., de Swart, J., Wagenmakers, E.-J., & Wetzels, R. (2022). An impartial Bayesian hypothesis test for audit sampling. PsyArXiv. doi:10.31234/osf.io/8nf3e
Derks, K., de Swart, J., Wagenmakers, E.-J., & Wetzels, R. (2022). Bayesian generalized linear modeling for audit sampling: How to incorporate audit information into the statistical model. PsyArXiv. doi:10.31234/osf.io/byj2a
Dworin, L. D. and Grimlund, R. A. (1984). Dollar-unit sampling for accounts receivable and inventory. The Accounting Review, 59(2), 218-241. https://www.jstor.org/stable/247296
Leslie, D. A., Teitlebaum, A. D., & Anderson, R. J. (1979). Dollar-unit Sampling: A Practical Guide for Auditors. Copp Clark Pitman; Belmont, CA. ISBN: 9780773042780.
Meikle, G. R. (1972). Statistical Sampling in an Audit Context. Canadian Institute of Chartered Accountants.
Pap, G., and van Zuijlen, M. C. (1996). On the asymptotic behavior of the Stringer bound. Statistica Neerlandica, 50(3), 367-389. doi:10.1111/j.1467-9574.1996.tb01503.x .
Rohrbach, K. J. (1993). Variance augmentation to achieve nominal coverage probability in sampling from audit populations. Auditing, 12(2), 79.
Stringer, K. W. (1963). Practical aspects of statistical sampling in auditing. In Proceedings of the Business and Economic Statistics Section (pp. 405-411). American Statistical Association.
Touw, P., and Hoogduin, L. (2011). Statistiek voor Audit en Controlling. Boom uitgevers Amsterdam.
# Using summary statistics
evaluation(materiality = 0.05, x = 0, n = 100) # Non-stratified
#>
#> Classical Audit Sample Evaluation
#>
#> data: 0 and 100
#> number of errors = 0, number of samples = 100, taint = 0, p-value =
#> 0.0067379
#> alternative hypothesis: true misstatement rate is less than 0.05
#> 95 percent confidence interval:
#> 0.00000000 0.02995732
#> most likely estimate:
#> 0
#> results obtained via method 'poisson'
evaluation(materiality = 0.05, x = c(2, 1, 0), n = c(50, 70, 40)) # Stratified
#>
#> Classical Audit Sample Evaluation
#>
#> data: 3 and 160
#> number of errors = 3, number of samples = 160, taint = 3, p-value = NA
#> alternative hypothesis: true misstatement rate is less than 0.05
#> 95 percent confidence interval:
#> 0.00000000 0.06702395
#> most likely estimate:
#> 0.018095
#> results obtained via method 'poisson' + 'no-pooling'
# Using data
data("BuildIt")
BuildIt$inSample <- c(rep(1, 100), rep(0, 3400))
levs <- c("low", "medium", "high")
BuildIt$stratum <- factor(c(levs[3], levs[2], rep(levs, times = 1166)))
sample <- subset(BuildIt, BuildIt$inSample == 1)
# Non-stratified evaluation
evaluation(
materiality = 0.05, data = sample,
values = "bookValue", values.audit = "auditValue"
)
#>
#> Classical Audit Sample Evaluation
#>
#> data: sample
#> number of errors = 5, number of samples = 100, taint = 3, p-value =
#> 0.26502
#> alternative hypothesis: true misstatement rate is less than 0.05
#> 95 percent confidence interval:
#> 0.00000000 0.07753631
#> most likely estimate:
#> 0.03
#> results obtained via method 'poisson'
# Stratified evaluation
evaluation(
materiality = 0.05, data = sample, values = "bookValue",
values.audit = "auditValue", strata = "stratum"
)
#>
#> Classical Audit Sample Evaluation
#>
#> data: sample
#> number of errors = 5, number of samples = 100, taint = 3, p-value = NA
#> alternative hypothesis: true misstatement rate is less than 0.05
#> 95 percent confidence interval:
#> 0.0000000 0.1049822
#> most likely estimate:
#> 0.029946
#> results obtained via method 'poisson' + 'no-pooling'