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 \(H_0\).

```
<- evaluation(materiality = 0.05, x = 1, n = 100)
result_classical 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: 0.040428
```

As 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.

```
<- auditPrior(materiality = 0.05, method = "default", likelihood = "binomial")
prior <- evaluation(materiality = 0.05, x = 1, n = 100, prior = prior) result_bayesian
```

```
## 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: 515.86
```

In 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.

```
<- auditPrior(materiality = 0.05, method = "impartial", likelihood = "binomial")
prior <- evaluation(materiality = 0.05, x = 1, n = 100, prior = prior) result_bayesian
```

```
## 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: 47.435
```

Derks, 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*.