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 compare the following two hypotheses:
\[H_-:\theta<\theta_{max}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, H_+:\theta\geq\theta_{max}\].
The evaluation() function allows you to make a statement about the credibility of these two hypotheses after inspecting a sample. The output for testing as discussed in this vignette is only displayed when you enter a value for materiality argument.
This will be added in a future version of jfa.
Bayesian hypothesis testing uses the Bayes factor, \(BF_{-+}\) or \(BF_{+-}\), to make a statement about the evidence provided by the sample in support for one of the two hypotheses \(H_-\) or \(H_+\). The subscript The Bayes factor denotes which hypothesis it favors. By default, the evaluation() function returns the value for \(BF_{-+}\).
As an example of how to interpret the Bayes factor, the value of \(BF_{-+} = 10\) (provided by the evaluation() function) can be interpreted as: the data are 10 times more likely to have occurred under the hypothesis \(H_-:\theta<\theta_{max}\) than under the hypothesis \(H_+:\theta\geq\theta_{max}\). \(BF_{-+} > 1\) indicates evidence for \(H_-\), while \(BF_{-+} < 1\) indicates evidence for \(H_+\).
| \(BF_{-+}\) | Strength of evidence |
|---|---|
| \(< 0.01\) | Extreme evidence for \(H_+\) |
| \(0.01 - 0.033\) | Very strong evidence for \(H_+\) |
| \(0.033 - 0.10\) | Strong evidence for \(H_+\) |
| \(0.10 - 0.33\) | Moderate evidence for \(H_+\) |
| \(0.33 - 1\) | Anecdotal evidence for \(H_+\) |
| \(1\) | No evidence for \(H_-\) or \(H_+\) |
| \(1 - 3\) | Anecdotal evidence for \(H_-\) |
| \(3 - 10\) | Moderate evidence for \(H_-\) |
| \(10 - 30\) | Strong evidence for \(H_-\) |
| \(30 - 100\) | Very strong evidence for \(H_-\) |
| \(> 100\) | Extreme evidence for \(H_-\) |
As an example, consider that an auditor wants to verify whether the population contains less than 5 percent misstatement, implying the hypotheses \(H_-:\theta<0.05\) and \(H_+:\theta\geq0.05\). They have taken a sample of 40 items, of which 1 contained an error. The prior distribution is assumed to be a non-informative \(beta(1,1)\) prior.
The output below shows that \(BF_{-+}=1.508\), implying that there is anecdotal evidence for \(H_-\), the hypothesis that the population contains misstatements lower than 5 percent of the population.
prior <- auditPrior(materiality = 0.05, method = "default", likelihood = "binomial")
stage4 <- evaluation(materiality = 0.05, x = 1, n = 40, prior = prior)
summary(stage4)##
## Bayesian Audit Sample Evaluation Summary
##
## Options:
## Confidence level: 0.95
## Materiality: 0.05
## Materiality: 0.05
## Hypotheses: H0: T > 0.05 vs. H1: T < 0.05
## Method: binomial
## Prior distribution: beta(a = 1, ß = 1)
##
## Data:
## Sample size: 40
## Number of errors: 1
## Sum of taints: 1
##
## Results:
## Posterior distribution: beta(a = 2, ß = 40)
## Most likely error: 0.025
## 95 percent credible interval: [0, 0.11055]
## Precision: 0.085553
## BF10: 30.282In 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 (such as method = 'default') with respect to the misstatement parameter \(\theta\), the Bayes factor overestimates the evidence in favor of \(H_-\). 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 \(H_-\) and \(H_+\).
The output below shows that \(BF_{-+}=3.08\), implying that there is anecdotal evidence for \(H_-\), the hypothesis that the population contains misstatements lower than 5 percent of the population.
prior <- auditPrior(materiality = 0.05, method = "impartial", likelihood = "binomial")
stage4 <- evaluation(materiality = 0.05, x = 1, n = 40, prior = prior)
summary(stage4)##
## Bayesian Audit Sample Evaluation Summary
##
## Options:
## Confidence level: 0.95
## Materiality: 0.05
## Materiality: 0.05
## Hypotheses: H0: T > 0.05 vs. H1: T < 0.05
## Method: binomial
## Prior distribution: beta(a = 1, ß = 13.513)
##
## Data:
## Sample size: 40
## Number of errors: 1
## Sum of taints: 1
##
## Results:
## Posterior distribution: beta(a = 2, ß = 52.513)
## Most likely error: 0.019043
## 95 percent credible interval: [0, 0.085607]
## Precision: 0.066564
## BF10: 3.0778