# Planning statistical audit samples

## Introduction

This vignette illustrates how to use the planning() function from the jfa package to calculate a minimum sample size for audit sampling.

## Required information

First, planning a minimum sample requires knowledge of the conditions that lead to acceptance or rejection of the population (i.e., the sampling objectives). Typically, sampling objectives can be classified into one or both of the following:

• Hypothesis testing: The goal of the sample is to obtain evidence for or against the claim that the misstatement in the population is lower than a given value (i.e., the performance materiality).
• Estimation: The goal of the sample is to obtain an accurate estimate of the misstatement in the population (with a minimum precision).

Second, it is advised to specify the expected (or tolerable) misstatements in the sample. The expected misstatements are the misstatements that you allow in the sample, while still retaining the desired amount of assurance about the population. It is strongly recommended to set the value for the expected misstatements in the sample conservatively to minimize the chance of the observed misstatements in the sample exceeding the expected misstatements, which would imply that insufficient work has been done in the end.

Next to determining the sampling objective(s) and the expected misstatements, it is also important to determine the statistical distribution linking the sample outcomes to the population misstatement. This distribution is called the likelihood (i.e., poisson, binomial, orhypergeometric). All three aforementioned likelihoods are commonly used in an audit sampling context, however, poisson is the default likelihood in jfa because it is the most conservative of the three.

## Planning a sample

To illustrate how the planning() function can be used to calculate a minimum sample size for audit sampling, we will first demonstrate how to set up a sample with the purpose of hypothesis testing and subsequently show how to plan a sample with the purpose of estimation. In both cases, we will tolerate zero misstatements in the sample.

### Hypothesis testing

First, let’s take a look at how you can use the planning() function to calculate the minimum sample size for testing the hypothesis that the misstatement in the population is lower than the performance materiality. In this example the performance materiality is set to 5% of the total population value, meaning that the population cannot contain more than 5% misstatement.

Sampling objective: Calculate a minimum sample size such that, when no misstatements are found in the sample, there is a 95% chance that the misstatement in the population is lower than 5% of the population value.

A minimum sample size for this sampling objective can be calculated by specifying the materiality parameter in the planning() function, see the command below. Next, a summary of the statistical results can be obtained using the summary() function. The result shows that, given zero tolerable errors, the minimum sample size is 60 units.

plan <- planning(materiality = 0.05, expected = 0, conf.level = 0.95)
summary(plan)
##
##  Classical Audit Sample Planning Summary
##
## Options:
##   Confidence level:              0.95
##   Materiality:                   0.05
##   Hypotheses:                    H₀: Θ >= 0.05 vs. H₁: Θ < 0.05
##   Expected:                      0
##   Likelihood:                    poisson
##
## Results:
##   Minimum sample size:           60
##   Tolerable errors:              0
##   Expected most likely error:    0
##   Expected upper bound:          0.049929
##   Expected precision:            0.049929
##   Expected p-value:              0.049787

### Estimation

Next, let’s take a look at how you can use the planning() function to calculate the minimum sample size for estimating the misstatement in the population with a minimum precision. The precision is defined as the difference between the most likely misstatement and the upper confidence bound on the misstatement. For this example, the minimum precision is set to 2% of the population value.

Sampling objective: Calculate a minimum sample size such that, when zero misstatements are found in the sample, there is a 95% chance that the misstatement in the population is at most 2% above the most likely misstatement.

A minimum sample size for this sampling objective can be calculated by specifying the min.precision parameter in the planning() function, see the command below. The result shows that, given zero tolerable errors, the minimum sample size is 150 units.

planning(min.precision = 0.02, expected = 0, conf.level = 0.95)
##
##  Classical Audit Sample Planning
##
## minimum sample size = 150
## sample size obtained in 151 iterations via method 'poisson'

### Bayesian planning

Performing Bayesian planning requires an input for the prior argument in the planning() function. Setting prior = TRUE performs Bayesian planning using a default prior conjugate to the specified likelihood. For example, the command below uses a default gamma(1, 1) prior distribution to plan the sample.

plan <- planning(materiality = 0.05, expected = 0, conf.level = 0.95, prior = TRUE)
summary(plan)
##
##  Bayesian Audit Sample Planning Summary
##
## Options:
##   Confidence level:              0.95
##   Materiality:                   0.05
##   Hypotheses:                    H₀: Θ > 0.05 vs. H₁: Θ < 0.05
##   Expected:                      0
##   Likelihood:                    poisson
##   Prior distribution:            gamma(α = 1, β = 1)
##
## Results:
##   Minimum sample size:           59
##   Tolerable errors:              0
##   Posterior distribution:        gamma(α = 1, β = 60)
##   Expected most likely error:    0
##   Expected upper bound:          0.049929
##   Expected precision:            0.049929
##   Expected BF₁₀:                 372.25

You can inspect how the prior distribution compares to the expected posterior distribution by using the plot() function. The expected posterior distribution is the posterior distribution that would occur if you actually observed the planned sample containing the expected misstatements.

plot(plan)

The input for the prior argument can also be an object created by the auditPrior function. If planning() receives a prior for which there is a conjugate likelihood available, it will inherit the likelihood from the prior. For example, the command below uses a custom beta(1, 10) prior distribution to plan the sample using the binomial likelihood.

prior <- auditPrior(method = "param", likelihood = "binomial", alpha = 1, beta = 10)
planning(materiality = 0.05, expected = 0, conf.level = 0.95, prior = prior)
##
##  Bayesian Audit Sample Planning
##
## minimum sample size = 49
## sample size obtained in 50 iterations via method 'binomial' + 'prior'