R: Audit Sampling: Prior Distributions

auditPrior {jfa}

R Documentation

Audit Sampling: Prior Distributions

Description

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.

Usage

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
)

Arguments

`method`	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.
`likelihood`	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.
`N.units`	a numeric value larger than 0 specifying the total number of units in the population. Required for the `hypergeometric` likelihood.
`alpha`	a numeric value specifying the `\alpha` parameter of the prior distribution. Required for method `param`.
`beta`	a numeric value specifying the `\beta` parameter of the prior distribution. Required for method `param`.
`materiality`	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`.
`expected`	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`.
`ir`	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`.
`cr`	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`.
`ub`	a numeric value between 0 and 1 specifying the `conf.level`-% upper bound for the prior distribution as a fraction. Required for method `bram`.
`p.hmin`	a numeric value between 0 and 1 specifying the prior probability of the hypothesis of tolerable misstatement (H1: `\theta <` materiality). Required for method `hyp`.
`x`	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`.
`n`	a numeric value larger than 0 specifying the number of units in a prior sample. Required for methods `sample` and `factor`.
`factor`	a numeric value between 0 and 1 specifying the weight of a prior sample specified via `x` and `n`. Required for method `factor`.
`samples`	a numeric vector containing samples of the prior distribution. Required for method `nonparam`.
`conf.level`	a numeric value between 0 and 1 specifying the confidence level (1 - audit risk).

Details

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)}

.

Value

An object of class jfaPrior containing:

`prior`	a string describing the functional form of the prior distribution.
`description`	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.
`statistics`	a list containing statistics of the prior distribution, including the mean, mode, median, and upper bound of the prior distribution.
`specifics`	a list containing specifics of the prior distribution that vary depending on the `method`.
`hypotheses`	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).
`method`	a character indicating the method by which the prior distribution is constructed.
`likelihood`	a character indicating the likelihood of the data.
`materiality`	if `materiality` is specified, a numeric value between 0 and 1 giving the materiality used to construct the prior distribution.
`expected`	a numeric value larger than, or equal to, 0 giving the input for the number of expected misstatements.
`conf.level`	a numeric value between 0 and 1 giving the confidence level.
`N.units`	if `N.units` is specified, the number of units in the population.

Author(s)

Koen Derks, k.derks@nyenrode.nl

References

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

Examples

# Default beta prior
auditPrior(likelihood = "binomial")

# Impartial prior
auditPrior(method = "impartial", materiality = 0.05)

# Non-conjugate prior
auditPrior(method = "param", likelihood = "normal", alpha = 0, beta = 0.1)

[Package jfa version 0.7.1 Index]