auditPrior() is used to create 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 that can be used in the planning() and
evaluation() functions via their prior argument. Objects with
class jfaPrior can be further inspected via associated
summary() and plot() methods. They can also be used to compute
a convoluted prior using the + (for addition) and * (for
weighing) operators.
auditPrior(
method = c(
"default", "param", "strict", "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
)a character specifying the method by which the prior
distribution is constructed. Possible options are default,
strict, impartial, param, arm, bram,
hyp, sample, and factor. See the details section for
more information.
a character specifying the likelihood for updating the
prior distribution. Possible options are poisson (default) for a
conjugate gamma prior distribution, binomial for a conjugate beta
prior distribution, or hypergeometric for a conjugate beta-binomial
prior distribution. See the details section for more information.
a numeric value larger than 0 specifying the total number
of units in the population. Required for the hypergeometric
likelihood.
a numeric value specifying the \(\alpha\) parameter of
the prior distribution. Required for method param.
a numeric value specifying the \(\beta\) parameter of
the prior distribution. Required for method param.
a numeric value between 0 and 1 specifying the
performance materiality (i.e., the maximum tolerable misstatement in the
population) as a fraction. Required for methods impartial,
arm, and hyp.
a numeric value between 0 and 1 specifying the expected
(tolerable) misstatements in the sample relative to the total sample size.
Required for methods impartial, arm, bram, and
hyp.
a numeric value between 0 and 1 specifying the inherent
risk (i.e., the probability of material misstatement occurring due to
inherent factors) in the audit risk model. Required for method arm.
a numeric value between 0 and 1 specifying the internal
control risk (i.e., the probability of material misstatement occurring due
to internal control systems) in the audit risk model. Required for method
arm.
a numeric value between 0 and 1 specifying the
conf.level-% upper bound for the prior distribution as a
fraction. Required for method bram.
a numeric value between 0 and 1 specifying the prior
probability of the hypothesis of tolerable misstatement (H1: \(\theta <\)
materiality). Required for method hyp.
a numeric value larger than, or equal to, 0 specifying the
sum of proportional misstatements (taints) in a prior sample. Required for
methods sample and factor.
a numeric value larger than 0 specifying the number of
units in a prior sample. Required for methods sample and
factor.
a numeric value between 0 and 1 specifying the weight of
a prior sample specified via x and n. Required for method
factor.
a numeric vector containing samples of the prior
distribution. Required for method nonparam.
a numeric value between 0 and 1 specifying the confidence level (1 - audit risk).
An object of class jfaPrior containing:
a string describing the functional form of the prior distribution.
a list containing a description of the prior distribution, including the parameters of the prior distribution and the implicit sample on which the prior distribution is based.
a list containing statistics of the prior distribution, including the mean, mode, median, and upper bound of the prior distribution.
a list containing specifics of the prior distribution that
vary depending on the method.
if materiality is specified, a list containing
information about the hypotheses, including prior probabilities and odds
for the hypothesis of tolerable misstatement (H1) and the hypothesis of
intolerable misstatement (H0).
a character indicating the method by which the prior distribution is constructed.
a character indicating the likelihood of the data.
if materiality is specified, a numeric value
between 0 and 1 giving the materiality used to construct the prior
distribution.
a numeric value larger than, or equal to, 0 giving the input for the number of expected misstatements.
a numeric value between 0 and 1 giving the confidence level.
if N.units is specified, the number of units in the
population.
To perform Bayesian audit sampling you must assign a prior
distribution to the parameter in the model, i.e., the population
misstatement \(\theta\). The prior distribution can incorporate
pre-existing audit information about \(\theta\) into the analysis, which
consequently allows for a more efficient or more accurate estimates. The
default priors used by jfa are indifferent towards the possible
values of \(\theta\), while still being proper. Note that the default
prior distributions are a conservative choice of prior since they, in most
cases, assume all possible misstatement to be equally likely before seeing
a data sample. It is recommended to construct an informed prior
distribution based on pre-existing audit information when possible.
This section elaborates on the available input options for the
method argument.
default: This method produces a gamma(1, 1),
beta(1, 1), beta-binomial(N, 1, 1), normal(0.5, 1000)
, cauchy(0, 1000), student-t(1), or chi-squared(1)
prior distribution. These prior distributions are mostly indifferent about
the possible values of the misstatement.
param: This method produces a custom
gamma(alpha, beta), beta(alpha, beta),
beta-binomial(N, alpha, beta) prior distribution,
normal(alpha, beta), cauchy(alpha, beta),
student-t(alpha), or chi-squared(alpha). The alpha and
beta parameters must be set using alpha and beta.
strict: This method produces an improper gamma(1, 0),
beta(1, 0), or beta-binomial(N, 1, 0) prior distribution.
These prior distributions match sample sizes and upper limits from
classical methods and can be used to emulate classical results.
impartial: This method produces an impartial prior
distribution. These prior distributions assume that tolerable misstatement
(\(\theta <\) materiality) and intolerable misstatement (\(\theta >\)
materiality) are equally likely.
hyp: This method translates an assessment of the prior
probability for tolerable misstatement (\(\theta <\) materiality) to a
prior distribution.
arm: This method translates an assessment of inherent
risk and internal control risk to a prior distribution.
bram: This method translates an assessment of the
expected most likely error and x-% upper bound to a prior
distribution.
sample: This method translates the outcome of an earlier
sample to a prior distribution.
factor: This method translates and weighs the outcome of an
earlier sample to a prior distribution.
nonparam: This method takes a vector of samples from the prior
distribution (via samples) and constructs a bounded density
(between 0 and 1) on the basis of these samples to act as the prior.
This section elaborates on the available input options for the
likelihood argument and the corresponding conjugate prior
distributions used by jfa.
poisson: The Poisson distribution is an approximation of
the binomial distribution. The Poisson distribution is defined as:
$$f(\theta, n) = \frac{\lambda^\theta e^{-\lambda}}{\theta!}$$. The
conjugate gamma(\(\alpha, \beta\)) prior has probability density
function:
$$p(\theta; \alpha, \beta) = \frac{\beta^\alpha \theta^{\alpha - 1}
e^{-\beta \theta}}{\Gamma(\alpha)}$$.
binomial: The binomial distribution is an approximation
of the hypergeometric distribution. The binomial distribution is defined as:
$$f(\theta, n, x) = {n \choose x} \theta^x (1 - \theta)^{n - x}$$. The
conjugate beta(\(\alpha, \beta\)) prior has probability density
function: $$p(\theta; \alpha, \beta) = \frac{1}{B(\alpha, \beta)}
\theta^{\alpha - 1} (1 - \theta)^{\beta - 1}$$.
hypergeometric: The hypergeometric distribution is defined as:
$$f(x, n, K, N) = \frac{{K \choose x} {N - K \choose n - x}}
{{N \choose n}}$$. The conjugate beta-binomial(\(\alpha, \beta\))
prior (Dyer and Pierce, 1993) has probability mass function:
$$f(x, n, \alpha, \beta) = {n \choose x}
\frac{B(x + \alpha, n - x + \beta)}{B(\alpha, \beta)}$$.
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. (2022). An impartial Bayesian hypothesis test for audit sampling. PsyArXiv. doi:10.31234/osf.io/8nf3e
# Default beta prior
auditPrior(likelihood = "binomial")
#>
#> Prior Distribution for Audit Sampling
#>
#> functional form: beta(α = 1, β = 1)
#> parameters obtained via method 'default'
# Impartial prior
auditPrior(method = "impartial", materiality = 0.05)
#>
#> Prior Distribution for Audit Sampling
#>
#> functional form: gamma(α = 1, β = 13.863)
#> parameters obtained via method 'impartial'
# Non-conjugate prior
auditPrior(method = "param", likelihood = "normal", alpha = 0, beta = 0.1)
#>
#> Prior Distribution for Audit Sampling
#>
#> functional form: normal(μ = 0, σ = 0.1)T[0,1]
#> parameters obtained via method 'param'