In an audit sampling test the auditor generally assigns performance materiality, \(\theta_{max}\), to the population which expresses the maximum tolerable misstatement (as a fraction or a monetary amount). The auditor then inspects a sample of the population to make a decision between the following two hypotheses:
\[H_1:\theta<\theta_{max}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, H_0:\theta\geq\theta_{max}\].
The evaluation() function allows you to make a statement
about the credibility of these two hypotheses after inspecting a sample.
Note that this requires that you specify the materiality
argument in the function.
Classical hypothesis testing uses the p value to make a decision about whether to reject the hypothesis \(H_0\) or not. As an example, consider that an auditor wants to verify whether the population contains less than 5 percent misstatement, implying the hypotheses \(H_1:\theta<0.05\) and \(H_0:\theta\geq0.05\). They have taken a sample of 100 items, of which 1 contained an error. They set the significance level for the p value to 0.05, implying that a p value < 0.05 will be enough to reject the hypothesis \(H_0\).
result_classical <- evaluation(materiality = 0.05, x = 1, n = 100)
summary(result_classical)##
## Classical Audit Sample Evaluation Summary
##
## Options:
## Confidence level: 0.95
## Materiality: 0.05
## Materiality: 0.05
## Hypotheses: H₀: Θ >= 0.05 vs. H₁: Θ < 0.05
## Method: poisson
##
## Data:
## Sample size: 100
## Number of errors: 1
## Sum of taints: 1
##
## Results:
## Most likely error: 0.01
## 95 percent confidence interval: [0, 0.047439]
## Precision: 0.037439
## p-value: 0.040428As we can see, the p value is lower than 0.05 implying that the hypothesis \(H_0\) is rejected.
Bayesian hypothesis testing uses the Bayes factor, \(BF_{10}\) or \(BF_{01}\), to make a statement about the
evidence provided by the sample in support for one of the two hypotheses
\(H_1\) or \(H_0\). The subscript The Bayes factor
denotes which hypothesis it favors. By default, the
evaluation() function returns the value for \(BF_{10}\).
As an example of how to interpret the Bayes factor, the value of
\(BF_{10} = 10\) (provided by the
evaluation() function) can be interpreted as: the data
are 10 times more likely to have occurred under the hypothesis \(H_1:\theta<\theta_{max}\) than under the
hypothesis \(H_0:\theta\geq\theta_{max}\). \(BF_{10} > 1\) indicates evidence for
\(H_1\), while \(BF_{10} < 1\) indicates evidence for
\(H_0\).
| \(BF_{10}\) | Strength of evidence |
|---|---|
| \(< 0.01\) | Extreme evidence for \(H_0\) |
| \(0.01 - 0.033\) | Very strong evidence for \(H_0\) |
| \(0.033 - 0.10\) | Strong evidence for \(H_0\) |
| \(0.10 - 0.33\) | Moderate evidence for \(H_0\) |
| \(0.33 - 1\) | Anecdotal evidence for \(H_0\) |
| \(1\) | No evidence for \(H_1\) or \(H_0\) |
| \(1 - 3\) | Anecdotal evidence for \(H_1\) |
| \(3 - 10\) | Moderate evidence for \(H_1\) |
| \(10 - 30\) | Strong evidence for \(H_1\) |
| \(30 - 100\) | Very strong evidence for \(H_1\) |
| \(> 100\) | Extreme evidence for \(H_1\) |
Again, consider the same example of an auditor who wants to verify whether the population contains less than 5 percent misstatement, implying the hypotheses \(H_1:\theta<0.05\) and \(H_0:\theta\geq0.05\). They have taken a sample of 100 items, of which 1 contained an error. The prior distribution is assumed to be a default beta(1,1) prior.
The output below shows that \(BF_{10}=515\), implying that there is extreme evidence for \(H_1\), the hypothesis that the population contains misstatements lower than 5 percent of the population.
prior <- auditPrior(materiality = 0.05, method = "default", likelihood = "binomial")
result_bayesian <- evaluation(materiality = 0.05, x = 1, n = 100, prior = prior)## Warning in evaluation(materiality = 0.05, x = 1, n = 100, prior = prior): using
## 'method = binomial' from 'prior'summary(result_bayesian)##
## Bayesian Audit Sample Evaluation Summary
##
## Options:
## Confidence level: 0.95
## Materiality: 0.05
## Materiality: 0.05
## Hypotheses: H₀: Θ > 0.05 vs. H₁: Θ < 0.05
## Method: binomial
## Prior distribution: beta(α = 1, β = 1)
##
## Data:
## Sample size: 100
## Number of errors: 1
## Sum of taints: 1
##
## Results:
## Posterior distribution: beta(α = 2, β = 100)
## Most likely error: 0.01
## 95 percent credible interval: [0, 0.046107]
## Precision: 0.036107
## BF₁₀: 515.86In audit sampling, the Bayes factor is dependent on the prior
distribution for \(\theta\). As a rule
of thumb, when the prior distribution is very uninformative (as with
method = 'default') with respect to \(\theta\), the Bayes factor tends to
overquantify the evidence in favor of \(H_1\). You can mitigate this dependency
using method = "impartial" in the auditPrior()
function, which constructs a prior distribution that is impartial with
respect to the hypotheses \(H_1\) and
\(H_0\).
The output below shows that \(BF_{10}=47\), implying that there is strong evidence for \(H_1\), the hypothesis that the population contains misstatements lower than 5 percent of the population. Since the two priors both resulted in convincing Bayes factors, the results are robust to the choice of prior distribution.
prior <- auditPrior(materiality = 0.05, method = "impartial", likelihood = "binomial")
result_bayesian <- evaluation(materiality = 0.05, x = 1, n = 100, prior = prior)## Warning in evaluation(materiality = 0.05, x = 1, n = 100, prior = prior): using
## 'method = binomial' from 'prior'summary(result_bayesian)##
## Bayesian Audit Sample Evaluation Summary
##
## Options:
## Confidence level: 0.95
## Materiality: 0.05
## Materiality: 0.05
## Hypotheses: H₀: Θ > 0.05 vs. H₁: Θ < 0.05
## Method: binomial
## Prior distribution: beta(α = 1, β = 13.513)
##
## Data:
## Sample size: 100
## Number of errors: 1
## Sum of taints: 1
##
## Results:
## Posterior distribution: beta(α = 2, β = 112.513)
## Most likely error: 0.0088878
## 95 percent credible interval: [0, 0.041108]
## Precision: 0.03222
## BF₁₀: 47.435Derks, K., de Swart, J., van Batenburg, P., Wagenmakers, E.-J., and 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.
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.