From f6f4b876d6e37c8a3d4283494ed410603358bef9 Mon Sep 17 00:00:00 2001 From: Koen Derks Date: Thu, 26 Nov 2020 09:36:03 +0100 Subject: [PATCH] Ready for 0.5.0 --- .Rbuildignore | 2 +- .gitignore | 3 +- .travis.yml | 42 -- DESCRIPTION | 2 +- Meta/vignette.rds | Bin 0 -> 407 bytes R/auditPrior.R | 7 +- R/evaluation.R | 23 +- R/methods.R | 8 + R/planning.R | 2 +- R/report.R | 2 +- README.Rmd | 17 +- README.md | 17 +- doc/jfa.R | 43 ++ doc/jfa.Rmd | 149 +++++ doc/jfa.html | 643 ++++++++++++++++++++ doc/v1auditWorkflow.R | 66 +++ doc/v1auditWorkflow.Rmd | 155 +++++ doc/v1auditWorkflow.html | 569 ++++++++++++++++++ doc/v2priorDistributions.R | 75 +++ doc/v2priorDistributions.Rmd | 165 ++++++ doc/v2priorDistributions.html | 658 +++++++++++++++++++++ doc/v3estimation.R | 4 + doc/v3estimation.Rmd | 32 + doc/v3estimation.html | 321 ++++++++++ doc/v4testing.R | 12 + doc/v4testing.Rmd | 74 +++ doc/v4testing.html | 529 +++++++++++++++++ inst/rmd/report.Rmd | 2 +- man/evaluation.Rd | 5 +- man/figures/cheatsheet/cheatsheet.pdf | Bin 0 -> 258691 bytes man/figures/cheatsheet/cheatsheet.png | Bin 0 -> 729897 bytes man/figures/cheatsheet/cheatsheet.pptx | Bin 0 -> 253354 bytes man/figures/readme/downloads/downloads.svg | 151 ++--- man/figures/readme/worldmap/downloads.csv | 648 ++++++++++++++++++++ man/figures/readme/worldmap/worldmap.svg | 98 +-- man/manual/jfa_0.5.0.pdf | Bin 212696 -> 212651 bytes man/planning.Rd | 2 +- man/report.Rd | 2 +- tests/testthat/test-function-evaluation.R | 12 + vignettes/jfa.Rmd | 2 +- vignettes/v3estimation.Rmd | 6 +- 41 files changed, 4360 insertions(+), 188 deletions(-) delete mode 100644 .travis.yml create mode 100644 Meta/vignette.rds create mode 100644 doc/jfa.R create mode 100644 doc/jfa.Rmd create mode 100644 doc/jfa.html create mode 100644 doc/v1auditWorkflow.R create mode 100644 doc/v1auditWorkflow.Rmd create mode 100644 doc/v1auditWorkflow.html create mode 100644 doc/v2priorDistributions.R create mode 100644 doc/v2priorDistributions.Rmd create mode 100644 doc/v2priorDistributions.html create mode 100644 doc/v3estimation.R create mode 100644 doc/v3estimation.Rmd create mode 100644 doc/v3estimation.html create mode 100644 doc/v4testing.R create mode 100644 doc/v4testing.Rmd create mode 100644 doc/v4testing.html create mode 100644 man/figures/cheatsheet/cheatsheet.pdf create mode 100644 man/figures/cheatsheet/cheatsheet.png create mode 100644 man/figures/cheatsheet/cheatsheet.pptx diff --git a/.Rbuildignore b/.Rbuildignore index c61ac1bb4..a29410813 100644 --- a/.Rbuildignore +++ b/.Rbuildignore @@ -1,4 +1,4 @@ -^\.travis\.yml$ +^.yml$ ^cran-comments.md$ ^codecov.yml$ ^.github$ diff --git a/.gitignore b/.gitignore index 073074d29..13c468a05 100644 --- a/.gitignore +++ b/.gitignore @@ -1,3 +1,2 @@ -doc -Meta docs +Meta diff --git a/.travis.yml b/.travis.yml deleted file mode 100644 index 4055f7186..000000000 --- a/.travis.yml +++ /dev/null @@ -1,42 +0,0 @@ -# R for travis: see documentation at https://docs.travis-ci.com/user/languages/r - -# Select branches -branches: - only: - - master - - development - -# Use R language -language: r - -# Do not treat warnings as errors -warnings_are_errors: false - -# Test for multiple R-versions, one from bioconductor -r: - - release - # - oldrel - # - devel - # - bioc-devel - -# Test for multiple operating systems -os: - - linux - - osx - -cache: packages - -# Include r packages -r_packages: - - covr - - testthat - -# Turn of email notifications -notifications: - email: - on_success: false - on_failure: false - -# Only report coverage after build is successful -after_success: - - Rscript -e 'covr::codecov()' diff --git a/DESCRIPTION b/DESCRIPTION index 2d84e8a9f..adbe641ed 100644 --- a/DESCRIPTION +++ b/DESCRIPTION @@ -1,7 +1,7 @@ Package: jfa Title: Bayesian and Classical Audit Sampling Version: 0.5.0 -Date: 2020-11-07 +Date: 2021-01-04 Authors@R: person(given = "Koen", family = "Derks", diff --git a/Meta/vignette.rds b/Meta/vignette.rds new file mode 100644 index 0000000000000000000000000000000000000000..aedda8269ce1052ec98247b308edaa1fe21ff41f GIT binary patch literal 407 zcmV;I0cidoiwFP!000002F+B-PQx$|O`E10Dk>^MATHqtDzW?lr5v~*R0*!8aj7A8 zB72gGTmKu0GI8zTgx=u5!H#F~yytnNJoP=#YkGd$_nJN){Sj_$Tti%e*YbM!?Yu6? z_<={p9OdUEix@m6@@)|(D<^cEU!;;Ha>5iy_M8DrgmU7;{E{kQoD>1aKgh432#6Q$ ztitczQV10g3DoAF&XzRPBtk`a8so)qW$2icvulI~nazu;Aqh3FFmj05S~rFfrU91a zOd%LhP6gO!&9>xz#;Z)bMWdF$V`CrIJLSYqcDBk%IH>f>3G9D2OM9yv0xecqqb`*R z9PJjWM5xGQb#O!~#z2E^!5b#UNAN#(xiIA$^`nf@r#Xo;;wY+b8b#8W2!WKXZO6q_ z^mkS4)<&s*>Gw1O-TePvuGMtp2n69&8H~1Zwx%qmHetq~H2nL?)EpajemWDHQV}VO zdPm=}(P&qJdJ^ZJG>?h0jI6UC5g_A*#QSJ{egW4J+Tq{?002$Y B!?gea literal 0 HcmV?d00001 diff --git a/R/auditPrior.R b/R/auditPrior.R index 95fa50081..04130d97c 100644 --- a/R/auditPrior.R +++ b/R/auditPrior.R @@ -129,8 +129,11 @@ auditPrior <- function(confidence = 0.95, likelihood = "binomial", method = "non if(likelihood == "hypergeometric" && (is.null(N) || N <= 0)) stop("The hypergeometric likelihood requires that you specify a positive value for the populatin size N.") - if(expectedError >= 1 || expectedError < 0) - stop("The expected errors must be entered as a proportion.") + if(expectedError < 0) + stop("The expected errors must be zero or larger than zero.") + + if(expectedError >= 1 && method != "none") + stop("The expected errors must be entered as a proportion to use this prior construction method.") # Create the prior distribution depending on the specified method if(method == "none"){ diff --git a/R/evaluation.R b/R/evaluation.R index eb9bec1d3..d3cdbaaeb 100644 --- a/R/evaluation.R +++ b/R/evaluation.R @@ -1,6 +1,6 @@ #' Evaluation of Audit Samples using Confidence / Credible Bounds #' -#' @description This function takes a data frame (using \code{sample}, \code{bookValue}, and \code{auditValues}) or summary statistics (using \code{nSumstats} and \code{kSumstats}) and evaluates the audit sample according to the specified method. The returned object is of class \code{jfaEvaluation} and can be used with associated \code{print()} and \code{plot()} methods. +#' @description This function takes a data frame (using \code{sample}, \code{bookValues}, and \code{auditValues}) or summary statistics (using \code{nSumstats} and \code{kSumstats}) and evaluates the audit sample according to the specified method. The returned object is of class \code{jfaEvaluation} and can be used with associated \code{print()} and \code{plot()} methods. #' #' For more details on how to use this function see the package vignette: #' \code{vignette("jfa", package = "jfa")} @@ -14,7 +14,7 @@ #' csA = 1, csB = 3, csMu = 0.5) #' #' @param confidence the required confidence level for the bound. Default is 0.95 for 95\% confidence. -#' @param method the method that is used to evaluate the sample. This can be either one of \code{poisson}, \code{binomial}, \code{hypergeometric}, \code{stringer}, \code{stringer-meikle}, \code{stringer-lta}, \code{stringer-pvz}, \code{rohrbach}, \code{moment}, \code{direct}, \code{difference}, \code{quotient}, or \code{regression}. +#' @param method the method that is used to evaluate the sample. This can be either one of \code{poisson}, \code{binomial}, \code{hypergeometric}, \code{mpus}, \code{stringer}, \code{stringer-meikle}, \code{stringer-lta}, \code{stringer-pvz}, \code{rohrbach}, \code{moment}, \code{direct}, \code{difference}, \code{quotient}, or \code{regression}. #' @param N an integer specifying the total number of units (transactions or monetary units) in the population. #' @param sample a data frame containing at least a column of Ist values and a column of Soll (true) values. #' @param bookValues a character specifying the column name for the Ist values in the sample. @@ -40,6 +40,7 @@ #' \item{\code{poisson}: The confidence bound taken from the Poisson distribution. If combined with \code{prior = TRUE}, performs Bayesian evaluation using a \emph{gamma} prior and posterior.} #' \item{\code{binomial}: The confidence bound taken from the binomial distribution. If combined with \code{prior = TRUE}, performs Bayesian evaluation using a \emph{beta} prior and posterior.} #' \item{\code{hypergeometric}: The confidence bound taken from the hypergeometric distribution. If combined with \code{prior = TRUE}, performs Bayesian evaluation using a \emph{beta-binomial} prior and posterior.} +#' \item{\code{mpu}}: Mean per unit estimator using the observed sample taints. #' \item{\code{stringer}: The Stringer bound (Stringer, 1963).} #' \item{\code{stringer-meikle}: Stringer bound with Meikle's correction for understatements (Meikle, 1972).} #' \item{\code{stringer-lta}: Stringer bound with LTA correction for understatements (Leslie, Teitlebaum, and Anderson, 1979).} @@ -187,13 +188,13 @@ evaluation <- function(confidence = 0.95, method = "binomial", N = NULL, if(!is.null(minPrecision) && minPrecision == 0) stop("The minimum required precision cannot be zero.") - if(!(method %in% c("poisson", "binomial", "hypergeometric", "stringer", "stringer-meikle", "stringer-lta", "stringer-pvz", "rohrbach", "moment", "coxsnell", "direct", "difference", "quotient", "regression")) || length(method) != 1) + if(!(method %in% c("poisson", "binomial", "hypergeometric", "stringer", "stringer-meikle", "stringer-lta", "stringer-pvz", "rohrbach", "moment", "coxsnell", "direct", "difference", "quotient", "regression", "mpu")) || length(method) != 1) stop("Specify a valid method for the evaluation.") if(!is.null(counts) && any(counts < 1)) stop("When specified, your 'counts' must all be equal to, or larger than, 1.") - if(((class(prior) == "logical" && prior == TRUE) || class(prior) == "jfaPrior") && method %in% c("stringer", "stringer-meikle", "stringer-lta", "stringer-pvz", "rohrbach", "moment", "direct", "difference", "quotient", "regression")) + if(((class(prior) == "logical" && prior == TRUE) || class(prior) == "jfaPrior") && method %in% c("stringer", "stringer-meikle", "stringer-lta", "stringer-pvz", "rohrbach", "moment", "direct", "difference", "quotient", "regression", "mpu")) stop("To use a prior distribution, you must use either the poisson, the binomial, or the hypergeometric method.") if((class(prior) == "logical" && prior == TRUE) && kPrior < 0 || nPrior < 0) @@ -208,7 +209,7 @@ evaluation <- function(confidence = 0.95, method = "binomial", N = NULL, stop("Specify one value for nSumstat and kSumstat") if(kSumstats > nSumstats) stop("The sum of the errors is higher than the sample size") - if(method %in% c("stringer", "stringer-meikle", "stringer-lta", "stringer-pvz", "coxsnell", "rohrbach", "moment", "direct", "difference", "quotient", "regression")) + if(method %in% c("stringer", "stringer-meikle", "stringer-lta", "stringer-pvz", "coxsnell", "rohrbach", "moment", "direct", "difference", "quotient", "regression", "mpu")) stop("The selected method requires raw observations, and does not accomodate summary statistics") n <- nSumstats @@ -220,10 +221,11 @@ evaluation <- function(confidence = 0.95, method = "binomial", N = NULL, if(is.null(bookValues) || is.null(auditValues) || length(bookValues) != 1 || length(auditValues) != 1) stop("Specify a valid book value column name and a valid audit value column name when using a sample") - sample <- stats::na.omit(sample) - n <- nrow(sample) - if(n == 0) + missingValues <- unique(c(which(is.na(sample[, bookValues])), which(is.na(sample[, auditValues])))) + if(length(missingValues) == nrow(sample)) stop("Your sample has 0 rows after removing missing values.") + sample <- stats::na.omit(sample) + n <- nrow(sample) if(!is.null(counts)) n <- sum(counts) bv <- sample[, bookValues] @@ -319,6 +321,11 @@ evaluation <- function(confidence = 0.95, method = "binomial", N = NULL, bound <- out[["confBound"]] mle <- out[["mle"]] precision <- out[["precision"]] + } else if(method == "mpu"){ + out <- .mpuMethod(taints, confidence, n) + bound <- out[["confBound"]] + mle <- out[["mle"]] + precision <- out[["precision"]] } else if(method == "direct"){ out <- .directMethod(bv, av, confidence, N, n, populationBookValue) mle <- out[["pointEstimate"]] diff --git a/R/methods.R b/R/methods.R index 6c56ec159..15d1b4019 100644 --- a/R/methods.R +++ b/R/methods.R @@ -116,6 +116,14 @@ return(result) } +.mpuMethod <- function(taints, confidence, n){ + result <- list() + result[["confBound"]] <- mean(taints) + stats::qnorm(p = confidence) * (stats::sd(taints) / sqrt(n)) + result[["mle"]] <- sum(taints) / n + result[["precision"]] <- result[["confBound"]] - result[["mle"]] + return(result) +} + .directMethod <- function(bookValues, auditValues, confidence, N = NULL, n, populationBookValue = NULL, correction = FALSE){ if(is.null(N)) stop("The direct method requires that you specify the population size N") diff --git a/R/planning.R b/R/planning.R index f5ecaae0f..a9d9c106d 100644 --- a/R/planning.R +++ b/R/planning.R @@ -1,6 +1,6 @@ #' Frequentist and Bayesian Planning for Audit Sampling #' -#' @description This function calculates the required sample size for an audit, based on the poisson, binomial, or hypergeometric likelihood. A prior can be specified to perform Bayesian planning. The returned object is of class \code{jfaPlanning} and can be used with associated \code{print()} and \code{plot()} methods. +#' @description This function calculates the required sample size for an audit, based on the Poisson, binomial, or hypergeometric likelihood. A prior can be specified to perform Bayesian planning. The returned object is of class \code{jfaPlanning} and can be used with associated \code{print()} and \code{plot()} methods. #' #' For more details on how to use this function see the package vignette: #' \code{vignette("jfa", package = "jfa")} diff --git a/R/report.R b/R/report.R index bf84defd2..9d1d59124 100644 --- a/R/report.R +++ b/R/report.R @@ -9,7 +9,7 @@ #' #' @param object an object of class 'jfaEvaluation' as returned by the \code{evaluation()} function. #' @param file a string that gives the desired name of the file (e.g. \code{"report.html"}). The report is created in your current working directory. -#' @param format can be either one of \code{"html_document"} or \code{"pdf_document"} (required MikTex). +#' @param format can be either one of \code{"html_document"} or \code{"pdf_document"} (compiling to pdf requires MikTex). #' #' @return A html or pdf report containing the results of the evaluation. #' diff --git a/README.Rmd b/README.Rmd index 894cfb38b..4b048355d 100644 --- a/README.Rmd +++ b/README.Rmd @@ -30,6 +30,7 @@ knitr::opts_chunk$set( * [Getting started](#getting-started) * [Benchmarks](#benchmarks) * [Contributing](#contributing) +* [Cheatsheet](#cheatsheet) * [Functions](#functions) * [References](#references) * [Package statistics](#package-statistics) @@ -40,7 +41,7 @@ The `jfa` package can be used to set up the entire audit sampling workflow. banner

-For complete documentation of the package, see the [package website](https://koenderks.github.io/jfa/) or the [package manual](https://cran.r-project.org/web/packages/jfa/jfa.pdf). +For complete documentation of the package, see the [package website](https://koenderks.github.io/jfa/) or the [package manual](https://cran.r-project.org/package=jfa/jfa.pdf). ### Authors @@ -102,6 +103,14 @@ The package vignettes contain explanations about the functionality of `jfa` illu If you are willing to contribute to the improvement of the package by adding a benchmark, please check out the Wiki page on [how to contribute a benchmark to jfa](https://github.com/koenderks/jfa/wiki/Benchmarks). If you are willing to contribute to the improvement of the package by adding a new statistical method, please check the Wiki page on [how to contribute a new method to jfa](https://github.com/koenderks/jfa/wiki/Methods). +## Cheatsheet + +The cheatsheet can help you get started with the `jfa` package and its workflow. You can download a pdf version [here](https://github.com/koenderks/jfa/raw/master/man/figures/cheatsheet/cheatsheet.pdf). + +

+ cheatsheet +

+ ## Functions Below is a list of the available functions in the current version of `jfa`, sorted by their occurrence in the standard audit sampling workflow. @@ -213,7 +222,7 @@ The `report()` function takes an object of class `jfaEvaluation` as returned by `report(object = NULL, file = NULL, format = "html_document")` -For an example report, see the following [link](https://github.com/koenderks/jfa/tree/master/man/figures/readme/report/report.pdf). +For an example report, see the following [link](https://github.com/koenderks/jfa/raw/master/man/figures/readme/report/report.pdf). ## References @@ -244,8 +253,8 @@ xBreaks <- plotData[["date"]] yBreaks <- pretty(c(0, plotData[["count"]], max(plotData[["count"]]) + 200), n = 6) # Releases -releases <- c("2020-01-01", "2020-08-01", "2020-09-01", "2020-11-01") -releaseLabs <- c("v0.1.0", "v0.2.0", "v0.3.0", "v0.4.0") +releases <- c("2020-01-01", "2020-08-01", "2020-09-01", "2020-11-01", "2021-01-01") +releaseLabs <- c("v0.1.0", "v0.2.0", "v0.3.0", "v0.4.0", "v0.5.0") p <- ggplot2::ggplot(plotData, ggplot2::aes(x = date, y = count)) + ggplot2::geom_bar(stat = "identity", fill = rgb(65, 104, 195, maxColorValue = 255), diff --git a/README.md b/README.md index d653c4e40..23b743874 100644 --- a/README.md +++ b/README.md @@ -29,6 +29,7 @@ prior probability distribution for use in these functions. - [Getting started](#getting-started) - [Benchmarks](#benchmarks) - [Contributing](#contributing) + - [Cheatsheet](#cheatsheet) - [Functions](#functions) - [References](#references) - [Package statistics](#package-statistics) @@ -44,7 +45,7 @@ workflow. For complete documentation of the package, see the [package website](https://koenderks.github.io/jfa/) or the [package -manual](https://cran.r-project.org/web/packages/jfa/jfa.pdf). +manual](https://cran.r-project.org/package=jfa/jfa.pdf). ### Authors @@ -126,6 +127,18 @@ adding a new statistical method, please check the Wiki page on [how to contribute a new method to jfa](https://github.com/koenderks/jfa/wiki/Methods). +## Cheatsheet + +The cheatsheet can help you get started with the `jfa` package and its +workflow. You can download a pdf version +[here](https://github.com/koenderks/jfa/raw/master/man/figures/cheatsheet/cheatsheet.pdf). + +

+ +cheatsheet + +

+ ## Functions Below is a list of the available functions in the current version of @@ -275,7 +288,7 @@ interpretation, and saves the report to your local computer. `report(object = NULL, file = NULL, format = "html_document")` For an example report, see the following -[link](https://github.com/koenderks/jfa/tree/master/man/figures/readme/report/report.pdf). +[link](https://github.com/koenderks/jfa/raw/master/man/figures/readme/report/report.pdf). ## References diff --git a/doc/jfa.R b/doc/jfa.R new file mode 100644 index 000000000..f8a161d4d --- /dev/null +++ b/doc/jfa.R @@ -0,0 +1,43 @@ +## ---- eval=FALSE-------------------------------------------------------------- +# install.packages("jfa") + +## ---- eval=FALSE-------------------------------------------------------------- +# devtools::install_github("koenderks/jfa") + +## ----------------------------------------------------------------------------- +library(jfa) + +data("BuildIt") +BuildIt <- BuildIt[, c("ID", "bookValue")] # Let's remove the auditValue column for this example +head(BuildIt, n = 10) + +## ----------------------------------------------------------------------------- +planning(confidence = 0.95, expectedError = 0, likelihood = "poisson", N = 3500, materiality = 0.05) + +## ----------------------------------------------------------------------------- +planning(confidence = 0.95, expectedError = 0, likelihood = "poisson", N = 3500, minPrecision = 0.02) + +## ----------------------------------------------------------------------------- +selection(population = BuildIt, sampleSize = 150, units = "records", algorithm = "random") + +## ----------------------------------------------------------------------------- +selection(population = BuildIt, sampleSize = 150, units = "mus", algorithm = "interval", + bookValues = "bookValue") + +## ----------------------------------------------------------------------------- +result <- selection(population = BuildIt, sampleSize = 150, units = "mus", algorithm = "interval", + bookValues = "bookValue") + +sample <- result$sample +head(sample, n = 10) + +## ----------------------------------------------------------------------------- +evaluation(confidence = 0.95, method = "binomial", N = 3500, nSumstats = 60, kSumstats = 1, materiality = 0.05) + +## ----------------------------------------------------------------------------- +sample$auditValue <- sample$bookValue + +## ----------------------------------------------------------------------------- +evaluation(confidence = 0.95, method = "stringer", N = 3500, materiality = 0.05, + sample = sample, bookValues = "bookValue", auditValues = "auditValue", counts = sample$count) + diff --git a/doc/jfa.Rmd b/doc/jfa.Rmd new file mode 100644 index 000000000..0d6f6d56e --- /dev/null +++ b/doc/jfa.Rmd @@ -0,0 +1,149 @@ +--- +title: "Get started" +author: Koen Derks +date: "last modified: 07-11-2020" +output: + rmarkdown::html_vignette: + toc: true +vignette: > + %\VignetteIndexEntry{Get started} + %\VignetteEngine{knitr::rmarkdown} + %\VignetteDepends{jfa} + %\VignetteKeywords{audit, evaluation, jfa, planning, sampling} + %\VignettePackage{jfa} + %\VignetteEncoding{UTF-8} +--- + +## Overview + +The `jfa` package allows you to plan, select, and evaluate an audit sample using classical and Bayesian statistics. + +Below it is explained how to install and use the `jfa` package. + +## Installation + +Install `jfa` from CRAN + +```{r, eval=FALSE} +install.packages("jfa") +``` + +or from GitHub + +```{r, eval=FALSE} +devtools::install_github("koenderks/jfa") +``` + +## Load data + +We can try out some examples with the `BuildIt` data set that is included in the package. It includes a population of 3500 transactions from a fictional construction company BuildIt, which we can use to illustrate statistical audit sampling using the `jfa` package (for more info, see ?BuildIt). + +```{r} +library(jfa) + +data("BuildIt") +BuildIt <- BuildIt[, c("ID", "bookValue")] # Let's remove the auditValue column for this example +head(BuildIt, n = 10) +``` + +Because this data set contains the *Ist* values of the transactions, we need to consider each monetary unit in the population as a possible unit of inference. + +For an example of the `jfa`'s audit sampling workflow see [The audit sampling workflow](https://koenderks.github.io/jfa/articles/v1auditWorkflow.html). + +## Using `planning()`: The basics + +Planning a sample using the `planning()` function requires that you have knowledge of the goal of the analysis and the statistical distribution of your data (`poisson`, `binomial`, or `hypergeometric`). + +### Testing against a performance materiality + +Let's take the goal of testing against a performance materiality with the `poisson` distribution as an example. + +Analysis goal: *Plan a sample such that, when zero misstatements are found in the sample, you can be 95% confident that the total misstatement in the population is lower than 5% of its total value.* + +Planning a sample with this goal can be done using the code below (specifically using the `materiality` argument). As you can see, the required sample size for this goal is 60 monetary units. + +```{r} +planning(confidence = 0.95, expectedError = 0, likelihood = "poisson", N = 3500, materiality = 0.05) +``` + +### Obtaining a minimum required precision + +The goal of the analysis can also involve obtaining a minimum precision about the estimate of the misstatement. + +Analysis goal: *Plan a sample such that, when zero misstatements are found in the sample, you can be 95% confident that the inaccuracy of your estimate is at most 2%.* + +Planning a sample with this goal can be done using the code below (specifically using the `minPrecision` argument). As you can see, the required sample size for this goal is 150 monetary units. + +```{r} +planning(confidence = 0.95, expectedError = 0, likelihood = "poisson", N = 3500, minPrecision = 0.02) +``` + +## Using `selection()`: The basics + +Selecting a sample using the `selection()` function requires that you have knowledge of the sampling units. Transactions can be selected using *record sampling* (also called attribute sampling) with `units = "records"`, or using *monetary unit sampling* with `units = "mus"`. + +It also requires knowledge of the sampling algorithm. Transactions can be selected using *random sampling* with `algorithm = "random"`, using *cell sampling* with `algorithm = "cell"`, or using fixed interval sampling (also known as systematic sampling) with `algorithm = "interval"`. + +### Record sampling + +For example, the code below samples 60 monetary units from the `BuildIt` data set using a *random record sampling* scheme. + +```{r} +selection(population = BuildIt, sampleSize = 150, units = "records", algorithm = "random") +``` + +### Monetary unit sampling + +As another example, the code below samples 150 monetary units from the `BuildIt` data set using a *fixed interval monetary unit sampling* scheme. + +```{r} +selection(population = BuildIt, sampleSize = 150, units = "mus", algorithm = "interval", + bookValues = "bookValue") +``` + +### Extracting the sample + +The selected sample is saved in the object that is returned by the `selection()` function and can be extracted via `$sample`. + +```{r} +result <- selection(population = BuildIt, sampleSize = 150, units = "mus", algorithm = "interval", + bookValues = "bookValue") + +sample <- result$sample +head(sample, n = 10) +``` + +## Using `evaluation()`: The basics + +After executing the audit and annotating the transactions in the sample with their *Soll* values, you can evaluate whether you have achieved your analysis goal via the `evaluation()` function. The function can be used with summary statistics from the sample, or with an annotated sample as input. + +For more details on how to use this function see the package vignettes: [Estimating the misstatement](https://koenderks.github.io/jfa/articles/v3estimation.html) and [Testing the misstatement](https://koenderks.github.io/jfa/articles/v4testing.html) + +### Summary statistics from the sample + +Suppose that in 60 transactions, you have found 1 misstatement. Using `nSumstats = 60` and `kSumstats = 1` you can specify the outcomes of the sample in the `evaluation()` function. Do not forget to specify your analysis goal using the `materiality` or `minPrecision` arguments. + +```{r} +evaluation(confidence = 0.95, method = "binomial", N = 3500, nSumstats = 60, kSumstats = 1, materiality = 0.05) +``` + +### Annotated sample + +Suppose that you have audited the transactions in the sample and have found no deviations from the *ist* values. + +```{r} +sample$auditValue <- sample$bookValue +``` + +You can evaluate the annotated sample using the `sample`, `bookValues`, `auditValues`, and `counts` arguments. The code below evaluates the analysis goal using the popular Stringer bound. You can find more information about which methods are implemented on the [home page](https://koenderks.github.io/jfa/). + +```{r} +evaluation(confidence = 0.95, method = "stringer", N = 3500, materiality = 0.05, + sample = sample, bookValues = "bookValue", auditValues = "auditValue", counts = sample$count) +``` + +## Using `auditPrior()`: Prior probability distributions + +The `auditPrior()` function allows you to perform the workflow as discussed above in a Bayesian fashion. Using a prior distribution is fairly simple, you only have to insert the returned object from the `auditPrior()` function as an argument for the `prior` argument in the `planning()` and `evaluation()` functions. + +For more information about how to create a prior distribution, see the vignettes [Constructing a prior distribution](https://koenderks.github.io/jfa/articles/v2priorDistributions.html). \ No newline at end of file diff --git a/doc/jfa.html b/doc/jfa.html new file mode 100644 index 000000000..c087851eb --- /dev/null +++ b/doc/jfa.html @@ -0,0 +1,643 @@ + + + + + + + + + + + + + + + +Get started + + + + + + + + + + + + + + + + + + + + + + + + + + +

Get started

+

Koen Derks

+

last modified: 07-11-2020

+ + + + +
+

Overview

+

The jfa package allows you to plan, select, and evaluate an audit sample using classical and Bayesian statistics.

+

Below it is explained how to install and use the jfa package.

+
+
+

Installation

+

Install jfa from CRAN

+
install.packages("jfa")
+

or from GitHub

+
devtools::install_github("koenderks/jfa")
+
+
+

Load data

+

We can try out some examples with the BuildIt data set that is included in the package. It includes a population of 3500 transactions from a fictional construction company BuildIt, which we can use to illustrate statistical audit sampling using the jfa package (for more info, see ?BuildIt).

+
library(jfa)
+
+data("BuildIt")
+BuildIt <- BuildIt[, c("ID", "bookValue")] # Let's remove the auditValue column for this example
+head(BuildIt, n = 10)
+
##       ID bookValue
+## 1  82884    242.61
+## 2  25064    642.99
+## 3  81235    628.53
+## 4  71769    431.87
+## 5  55080    620.88
+## 6  93224    501.76
+## 7  24331    466.01
+## 8  81460    295.20
+## 9  14608    216.48
+## 10 79064    243.43
+

Because this data set contains the Ist values of the transactions, we need to consider each monetary unit in the population as a possible unit of inference.

+

For an example of the jfa’s audit sampling workflow see The audit sampling workflow.

+
+
+

Using planning(): The basics

+

Planning a sample using the planning() function requires that you have knowledge of the goal of the analysis and the statistical distribution of your data (poisson, binomial, or hypergeometric).

+
+

Testing against a performance materiality

+

Let’s take the goal of testing against a performance materiality with the poisson distribution as an example.

+

Analysis goal: Plan a sample such that, when zero misstatements are found in the sample, you can be 95% confident that the total misstatement in the population is lower than 5% of its total value.

+

Planning a sample with this goal can be done using the code below (specifically using the materiality argument). As you can see, the required sample size for this goal is 60 monetary units.

+
planning(confidence = 0.95, expectedError = 0, likelihood = "poisson", N = 3500, materiality = 0.05)
+
## # ------------------------------------------------------------
+## #              jfa Planning Summary (Frequentist)
+## # ------------------------------------------------------------     
+## # Input:
+## # 
+## # Confidence:              95% 
+## # Materiality:             5% 
+## # Minimum precision:       Not specified 
+## # Likelihood:              poisson 
+## # Expected sample errors:  0 
+## # ------------------------------------------------------------
+## # Output:
+## #
+## # Sample size:             60 
+## # ------------------------------------------------------------
+## # Statistics:
+## #
+## # Expected upper bound:    4.99% 
+## # Expected precision:      4.99% 
+## # ------------------------------------------------------------
+
+
+

Obtaining a minimum required precision

+

The goal of the analysis can also involve obtaining a minimum precision about the estimate of the misstatement.

+

Analysis goal: Plan a sample such that, when zero misstatements are found in the sample, you can be 95% confident that the inaccuracy of your estimate is at most 2%.

+

Planning a sample with this goal can be done using the code below (specifically using the minPrecision argument). As you can see, the required sample size for this goal is 150 monetary units.

+
planning(confidence = 0.95, expectedError = 0, likelihood = "poisson", N = 3500, minPrecision = 0.02)
+
## # ------------------------------------------------------------
+## #              jfa Planning Summary (Frequentist)
+## # ------------------------------------------------------------     
+## # Input:
+## # 
+## # Confidence:              95% 
+## # Materiality:             Not specified 
+## # Minimum precision:       2% 
+## # Likelihood:              poisson 
+## # Expected sample errors:  0 
+## # ------------------------------------------------------------
+## # Output:
+## #
+## # Sample size:             150 
+## # ------------------------------------------------------------
+## # Statistics:
+## #
+## # Expected upper bound:    2% 
+## # Expected precision:      2% 
+## # ------------------------------------------------------------
+
+
+
+

Using selection(): The basics

+

Selecting a sample using the selection() function requires that you have knowledge of the sampling units. Transactions can be selected using record sampling (also called attribute sampling) with units = "records", or using monetary unit sampling with units = "mus".

+

It also requires knowledge of the sampling algorithm. Transactions can be selected using random sampling with algorithm = "random", using cell sampling with algorithm = "cell", or using fixed interval sampling (also known as systematic sampling) with algorithm = "interval".

+
+

Record sampling

+

For example, the code below samples 60 monetary units from the BuildIt data set using a random record sampling scheme.

+
selection(population = BuildIt, sampleSize = 150, units = "records", algorithm = "random")
+
## # ------------------------------------------------------------
+## #                  jfa Selection Summary
+## # ------------------------------------------------------------
+## # Input:
+## #       
+## # Population size:         3500 
+## # Requested sample size:   150 
+## # Sampling units:          Records 
+## # Algorithm:               Random sampling   
+## # ------------------------------------------------------------ 
+## # Output:
+## # 
+## # Obtained sample size:    150 
+## # ------------------------------------------------------------
+## # Statistics:
+## #
+## # Proportion n/N:          0.04 
+## # ------------------------------------------------------------
+
+
+

Monetary unit sampling

+

As another example, the code below samples 150 monetary units from the BuildIt data set using a fixed interval monetary unit sampling scheme.

+
selection(population = BuildIt, sampleSize = 150, units = "mus", algorithm = "interval", 
+          bookValues = "bookValue")
+
## # ------------------------------------------------------------
+## #                  jfa Selection Summary
+## # ------------------------------------------------------------
+## # Input:
+## #      
+## # Population size:         3500 
+## # Requested sample size:   150 
+## # Sampling units:          Monetary units 
+## # Algorithm:               Fixed interval sampling 
+## # Interval:                9354.81
+## # Starting point:          1  
+## # ------------------------------------------------------------ 
+## # Output:
+## #
+## # Obtained sample size:    150 
+## # ------------------------------------------------------------
+## # Statistics:
+## #
+## # Proportion n/N:          0.04 
+## # Percentage of value:     4.25% 
+## # ------------------------------------------------------------
+
+
+

Extracting the sample

+

The selected sample is saved in the object that is returned by the selection() function and can be extracted via $sample.

+
result <- selection(population = BuildIt, sampleSize = 150, units = "mus", algorithm = "interval", 
+                    bookValues = "bookValue")
+
+sample <- result$sample
+head(sample, n = 10)
+
##    rowNumber count    ID bookValue
+## 1          1     1 30568     14.47
+## 2         24     1  1469     72.57
+## 3         50     1 58378     94.17
+## 4         73     1 64917    103.37
+## 5         96     1 50703    110.76
+## 6        123     1 91342    118.20
+## 7        146     1 63567    125.21
+## 8        167     1 10883    132.14
+## 9        195     1 94854    141.08
+## 10       218     1 66118    147.14
+
+
+
+

Using evaluation(): The basics

+

After executing the audit and annotating the transactions in the sample with their Soll values, you can evaluate whether you have achieved your analysis goal via the evaluation() function. The function can be used with summary statistics from the sample, or with an annotated sample as input.

+

For more details on how to use this function see the package vignettes: Estimating the misstatement and Testing the misstatement

+
+

Summary statistics from the sample

+

Suppose that in 60 transactions, you have found 1 misstatement. Using nSumstats = 60 and kSumstats = 1 you can specify the outcomes of the sample in the evaluation() function. Do not forget to specify your analysis goal using the materiality or minPrecision arguments.

+
evaluation(confidence = 0.95, method = "binomial", N = 3500, nSumstats = 60, kSumstats = 1, materiality = 0.05)
+
## # ------------------------------------------------------------ 
+## #             jfa Evaluation Summary (Frequentist)
+## # ------------------------------------------------------------ 
+## # Input: 
+## #
+## # Confidence:               95%   
+## # Materiality:              5% 
+## # Minium precision:         Not specified 
+## # Sample size:              60 
+## # Sample errors:            1 
+## # Sum of taints:            1 
+## # Method:                   binomial 
+## # ------------------------------------------------------------
+## # Output:
+## #
+## # Most likely error:        1.67% 
+## # Upper bound:              7.66% 
+## # Precision:                6% 
+## # Conclusion:               Do not approve population 
+## # ------------------------------------------------------------
+
+
+

Annotated sample

+

Suppose that you have audited the transactions in the sample and have found no deviations from the ist values.

+
sample$auditValue <- sample$bookValue
+

You can evaluate the annotated sample using the sample, bookValues, auditValues, and counts arguments. The code below evaluates the analysis goal using the popular Stringer bound. You can find more information about which methods are implemented on the home page.

+
evaluation(confidence = 0.95, method = "stringer", N = 3500, materiality = 0.05,
+           sample = sample, bookValues = "bookValue", auditValues = "auditValue", counts = sample$count)
+
## # ------------------------------------------------------------ 
+## #             jfa Evaluation Summary (Frequentist)
+## # ------------------------------------------------------------ 
+## # Input: 
+## #
+## # Confidence:               95%   
+## # Materiality:              5% 
+## # Minium precision:         Not specified 
+## # Sample size:              150 
+## # Sample errors:            0 
+## # Sum of taints:            0 
+## # Method:                   stringer 
+## # ------------------------------------------------------------
+## # Output:
+## #
+## # Most likely error:        0% 
+## # Upper bound:              1.98% 
+## # Precision:                1.98% 
+## # Conclusion:               Approve population 
+## # ------------------------------------------------------------
+
+
+
+

Using auditPrior(): Prior probability distributions

+

The auditPrior() function allows you to perform the workflow as discussed above in a Bayesian fashion. Using a prior distribution is fairly simple, you only have to insert the returned object from the auditPrior() function as an argument for the prior argument in the planning() and evaluation() functions.

+

For more information about how to create a prior distribution, see the vignettes Constructing a prior distribution.

+
+ + + + + + + + + + + diff --git a/doc/v1auditWorkflow.R b/doc/v1auditWorkflow.R new file mode 100644 index 000000000..14ee1c023 --- /dev/null +++ b/doc/v1auditWorkflow.R @@ -0,0 +1,66 @@ +## ----------------------------------------------------------------------------- +library(jfa) +data("BuildIt") + +## ----------------------------------------------------------------------------- +# Specify the materiality, confidence, and expected errors. +materiality <- 0.05 # 5% +confidence <- 0.95 # 95% +expectedError <- 0.025 # 2.5% + +## ----------------------------------------------------------------------------- +# Specify the inherent risk (ir) and control risk (cr). +ir <- 1 # 100% +cr <- 0.6 # 60% + +## ----------------------------------------------------------------------------- +# Adjust the required confidence for a frequentist analysis. +adjustedConfidence <- 1 - ((1 - confidence) / (ir * cr)) + +## ----------------------------------------------------------------------------- +# Step 0: Create a prior distribution according to the audit risk model. +priorResult <- auditPrior(confidence = confidence, likelihood = "binomial", method = "arm", expectedError = expectedError, materiality = materiality, ir = ir, cr = cr) + +## ----------------------------------------------------------------------------- +print(priorResult) + +## ----fig.align="center", fig.height=4, fig.width=6---------------------------- +plot(priorResult) + +## ----------------------------------------------------------------------------- +# Step 1: Calculate the required sample size. +planningResult <- planning(confidence = confidence, expectedError = expectedError, materiality = materiality, prior = priorResult) + +## ----------------------------------------------------------------------------- +print(planningResult) + +## ----fig.align="center", fig.height=4, fig.width=6---------------------------- +plot(planningResult) + +## ----------------------------------------------------------------------------- +# Step 2: Draw a sample from the financial statements. +samplingResult <- selection(population = BuildIt, sampleSize = planningResult, units = "mus", bookValues = "bookValue", seed = 999) + +## ----------------------------------------------------------------------------- +print(samplingResult) + +## ----------------------------------------------------------------------------- +# Step 3: Isolate the sample for execution of the audit. +sample <- samplingResult$sample + +# To write the sample to a .csv file: +# write.csv(x = sample, file = "auditSample.csv", row.names = FALSE) + +# To load annotated sample back into R: +# sample <- read.csv(file = "auditSample.csv") + +## ----------------------------------------------------------------------------- +# Step 4: Evaluate the sample. +evaluationResult <- evaluation(sample = sample, materiality = materiality, prior = priorResult, bookValues = "bookValue", auditValues = "auditValue") + +## ----------------------------------------------------------------------------- +print(evaluationResult) + +## ----fig.align="center", fig.height=4, fig.width=6---------------------------- +plot(evaluationResult) + diff --git a/doc/v1auditWorkflow.Rmd b/doc/v1auditWorkflow.Rmd new file mode 100644 index 000000000..4561fd51d --- /dev/null +++ b/doc/v1auditWorkflow.Rmd @@ -0,0 +1,155 @@ +--- +title: "The audit sampling workflow" +author: Koen Derks +date: "last modified: 07-11-2020" +output: + rmarkdown::html_vignette: + toc: true +vignette: > + %\VignetteIndexEntry{The audit sampling workflow} + %\VignetteEngine{knitr::rmarkdown} + %\VignetteDepends{jfa} + %\VignetteKeywords{audit, workflow, sampling} + %\VignettePackage{jfa} + %\VignetteEncoding{UTF-8} +--- + +## Introduction and scenario + +This vignette accompanies the `jfa` R package and aims to show how it facilitates auditors in their standard audit sampling workflow (hereafter "audit workflow"). In this example of the audit workflow, we will consider the case of BuildIt. BuildIt is a fictional construction company in the United States that is being audited by Laura, an external auditor for a fictional audit firm. Throughout the year, BuildIt has recorded every transaction they have made in their financial statements. Laura's job as an auditor is to make a judgment about the fairness of these financial statements. In other words, Laura needs to either approve or not approve BuildIt's financial statements. To not approve the financial statements would mean that, as a whole, the financial statements contain errors that are considered *material*. This means that the errors in the financial statements are large enough that they might influence the decision of someone relying on the financial statements. Since BuildIt is a small company, their financial statements only consist of 3500 transactions that each have a corresponding recorded book value. Before assessing the details in the financial statements, Laura already tested BuildIt's computer systems that processed these transactions and found that they were quite reliable. + +In order to draw a conclusion about the fairness of BuildIt's recorded transactions, Laura separates her audit workflow into four stages. First, she will plan the size of the subset she needs to inspect from the financial statements to make a well-substantiated inference about them as a whole. Second, she will select the required subset from the financial statements. Third, she will inspect the selected subset and determines the audit value (true value) of the transactions it contains. Fourth, she will use the information from her inspected subset to make an inference about the financial statements as a whole. To start off this workflow, Laura first loads BuildIt's financial statements in R. + +```{r} +library(jfa) +data("BuildIt") +``` + +## Setting up the audit + +In statistical terms, Laura wants to make a statement that, with 95% confidence, the maximum error in the financial statements is lower than what is considered *material*. She therefore determines that *materiality*, the maximum tolerable error in the financial statements, as 5%. Based on last year's audit at BuildIt, where the maximum error turned out to be 2.5%, she expects at most 2.5% errors in the sample that she will inspect. Laura can therefore re-formulate her statistical statement as that she wants to conclude that, when 2.5% errors are found in her sample, she can conclude with 95% confidence, that the misstatement in the population is lower than the materiality of 5%. Below, Laura defines the materiality, confidence, and expected errors. + +```{r} +# Specify the materiality, confidence, and expected errors. +materiality <- 0.05 # 5% +confidence <- 0.95 # 95% +expectedError <- 0.025 # 2.5% +``` + +Many audits are performed according to the *audit risk model (ARM)*, which determines that the uncertainty about Laura's statement as a whole (1 - her confidence) is a factor of three terms: the inherent risk, the control risk, and the detection risk. Inherent risk is the risk posed by an error in BuildIt's financial statement that could be material, before consideration of any related control systems (e.g., computer systems). Control risk is the risk that a material misstatement is not prevented or detected by BuildIt's internal control systems. Detection risk is the risk that Laura will fail to find material misstatements that exist in an BuildIt's financial statements. The *ARM* is practically useful because for a given level of audit risk, the tolerable detection risk bears an inverse relation to the other two risks. The *ARM* is useful for Laura because it enables her to incorporate prior knowledge on BuildIt's organization to increase the required risk that she will fail to find material misstatements. According to the *ARM*, the audit risk will then be retained. + +$$ \text{Audit risk} = \text{Inherent risk} \,\times\, \text{Control risk} \,\times\, \text{Detection risk}$$ + +Usually the auditor judges inherent risk and control risk on a three-point scale consisting of low, medium, and high. Different audit firms handle different standard percentages for these categories. Laura's firm defines the probabilities of low, medium, and high respectively as 50%, 60%, and 100%. Because Laura performed testing of BuildIt's computer systems, she assesses the control risk as medium (60%). + +```{r} +# Specify the inherent risk (ir) and control risk (cr). +ir <- 1 # 100% +cr <- 0.6 # 60% +``` + +## Stage 1: Planning an audit sample + +Laura can choose to either perform a frequentist analysis, where she uses the increased detection risk as her level of uncertainty, or perform a Bayesian analysis, where she captures the information in the control risk in a prior distribution. For this example, we will show how Laura performs a Bayesian analysis. A frequentist analysis can easily be done through the following functions by setting `prior = FALSE`. In a frequentist audit, Laura immediately starts at step 1 and uses the value `adjustedConfidence` as her new value for `confidence`. + +```{r} +# Adjust the required confidence for a frequentist analysis. +adjustedConfidence <- 1 - ((1 - confidence) / (ir * cr)) +``` + +In a Bayesian audit, Laura starts at step 0 by defining the prior distribution that corresponds to her assessment of the control risk. She assumes the likelihood for a sample of $n$ observations, in which $k$ were in error, to be $\text{binomial}(n, k)$. Using the `auditPrior()` function, she can create a prior distribution that incorporates the information in the risk assessments from the *ARM*. For more information on how this is done, see Derks et al. (2019). + +```{r} +# Step 0: Create a prior distribution according to the audit risk model. +priorResult <- auditPrior(confidence = confidence, likelihood = "binomial", method = "arm", expectedError = expectedError, materiality = materiality, ir = ir, cr = cr) +``` + +Laura can inspect the resulting prior distribution with the `print()` function. + +```{r} +print(priorResult) +``` + +The prior distribution can be shown by using the `plot()` function. + +```{r fig.align="center", fig.height=4, fig.width=6} +plot(priorResult) +``` + +Now that the prior distribution is specified, Laura can calculate the required sample size for her desired statement by using the `planning()` function. She uses the `priorResult` object as input for the `planning()` function to use her prior distribution. + +```{r} +# Step 1: Calculate the required sample size. +planningResult <- planning(confidence = confidence, expectedError = expectedError, materiality = materiality, prior = priorResult) +``` + +Laura can then inspect the result from her planning procedure by using the `print()` function. Her result tells her that, given her prior distribution she needs to audit a sample of 169 transactions so that, when at most 4.23 errors are found, she can conclude with 95% confidence that the maximum error in BuildIt's financial statements is lower the materiality of 5%. + +```{r} +print(planningResult) +``` + +Laura 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 Laura actually observed a sample of 169 transactions, from which 4.223 were in error. + +```{r fig.align="center", fig.height=4, fig.width=6} +plot(planningResult) +``` + +## Stage 2: Selecting a sample + +Laura is now ready to select the required 169 transactions from the financial statements. She can choose to do this according to one of two statistical methods. In *record sampling* (`units = "records"`), inclusion probabilities are assigned on the transaction level, treating transactions with a high value and a low value the same, a transaction of $5,000 is equally likely to be selected as a transaction of $1,000. In *monetary unit sampling* (`units = "mus"`), inclusion probabilities are assigned on the level of individual monetary units (e.g., a dollar). When a dollar is selected to be in the sample, the transaction that includes that dollar is selected. This favors higher transactions, as a transaction of \$5,000 is five times more likely to be selected than a transaction of \$1,000. + +Laura chooses to use *monetary unit sampling*, as she wants to include more high-valued transactions. The `selection()` function allows her to sample from the financial statements. She uses the `planningResult` object as an input for the `selection()` function. + +```{r} +# Step 2: Draw a sample from the financial statements. +samplingResult <- selection(population = BuildIt, sampleSize = planningResult, units = "mus", bookValues = "bookValue", seed = 999) +``` + +Laura can inspect the outcomes of her sampling procedure by using the `print()` function. + +```{r} +print(samplingResult) +``` + +## Stage 3: Executing the audit + +The selected sample can be isolated by indexing the `sample` object from the sampling result. Now Laura can execute her audit by annotating the sample with their audit value (for exampling by writing the sample to a *.csv* file using `write.csv()`. She can then load her annotated sample back into R for further evaluation. + +```{r} +# Step 3: Isolate the sample for execution of the audit. +sample <- samplingResult$sample + +# To write the sample to a .csv file: +# write.csv(x = sample, file = "auditSample.csv", row.names = FALSE) + +# To load annotated sample back into R: +# sample <- read.csv(file = "auditSample.csv") +``` + +For this example, the audit values of the sample are already included in the `auditValue` column of the dataset . + +## Stage 4: Evaluating the sample + +Using her annotated sample, Laura can perform her inference with the `evaluation()` function. By passing the `priorResult` object to the function, she automatically sets `method = "binomial"` to be consistent with her prior distribution. + +```{r} +# Step 4: Evaluate the sample. +evaluationResult <- evaluation(sample = sample, materiality = materiality, prior = priorResult, bookValues = "bookValue", auditValues = "auditValue") +``` + +Laura can inspect the outcomes of her inference by using the `print()` function. Her resulting upper bound is 3.518%, which is lower than the materiality of 5%. The output tells Laura the correct conclusion immediately. + +```{r} +print(evaluationResult) +``` + +She can inspect the prior and posterior distribution by using the `plot()` function. The shaded area quantifies the area under the posterior distribution that contains 95% of the probability, which ends at 3.518%. Therefore, Laura can state that there is a 95% probability that the misstatement in BuildIt's statements is lower than 3.518%. + +```{r fig.align="center", fig.height=4, fig.width=6} +plot(evaluationResult) +``` + +## Conclusion + +Since the 95% upper confidence bound on the misstatement in BuildIt's financial statements is lower than the materiality, Laura can conclude that the financial statements as a whole do not contain material misstatement. Therefore, she can approve BuildIt's financial statements. diff --git a/doc/v1auditWorkflow.html b/doc/v1auditWorkflow.html new file mode 100644 index 000000000..7be808438 --- /dev/null +++ b/doc/v1auditWorkflow.html @@ -0,0 +1,569 @@ + + + + + + + + + + + + + + + +The audit sampling workflow + + + + + + + + + + + + + + + + + + + + + + + + + + +

The audit sampling workflow

+

Koen Derks

+

last modified: 07-11-2020

+ + + + +
+

Introduction and scenario

+

This vignette accompanies the jfa R package and aims to show how it facilitates auditors in their standard audit sampling workflow (hereafter “audit workflow”). In this example of the audit workflow, we will consider the case of BuildIt. BuildIt is a fictional construction company in the United States that is being audited by Laura, an external auditor for a fictional audit firm. Throughout the year, BuildIt has recorded every transaction they have made in their financial statements. Laura’s job as an auditor is to make a judgment about the fairness of these financial statements. In other words, Laura needs to either approve or not approve BuildIt’s financial statements. To not approve the financial statements would mean that, as a whole, the financial statements contain errors that are considered material. This means that the errors in the financial statements are large enough that they might influence the decision of someone relying on the financial statements. Since BuildIt is a small company, their financial statements only consist of 3500 transactions that each have a corresponding recorded book value. Before assessing the details in the financial statements, Laura already tested BuildIt’s computer systems that processed these transactions and found that they were quite reliable.

+

In order to draw a conclusion about the fairness of BuildIt’s recorded transactions, Laura separates her audit workflow into four stages. First, she will plan the size of the subset she needs to inspect from the financial statements to make a well-substantiated inference about them as a whole. Second, she will select the required subset from the financial statements. Third, she will inspect the selected subset and determines the audit value (true value) of the transactions it contains. Fourth, she will use the information from her inspected subset to make an inference about the financial statements as a whole. To start off this workflow, Laura first loads BuildIt’s financial statements in R.

+
library(jfa)
+data("BuildIt")
+
+
+

Setting up the audit

+

In statistical terms, Laura wants to make a statement that, with 95% confidence, the maximum error in the financial statements is lower than what is considered material. She therefore determines that materiality, the maximum tolerable error in the financial statements, as 5%. Based on last year’s audit at BuildIt, where the maximum error turned out to be 2.5%, she expects at most 2.5% errors in the sample that she will inspect. Laura can therefore re-formulate her statistical statement as that she wants to conclude that, when 2.5% errors are found in her sample, she can conclude with 95% confidence, that the misstatement in the population is lower than the materiality of 5%. Below, Laura defines the materiality, confidence, and expected errors.

+
# Specify the materiality, confidence, and expected errors.
+materiality   <- 0.05   # 5%
+confidence    <- 0.95   # 95%
+expectedError <- 0.025  # 2.5%
+

Many audits are performed according to the audit risk model (ARM), which determines that the uncertainty about Laura’s statement as a whole (1 - her confidence) is a factor of three terms: the inherent risk, the control risk, and the detection risk. Inherent risk is the risk posed by an error in BuildIt’s financial statement that could be material, before consideration of any related control systems (e.g., computer systems). Control risk is the risk that a material misstatement is not prevented or detected by BuildIt’s internal control systems. Detection risk is the risk that Laura will fail to find material misstatements that exist in an BuildIt’s financial statements. The ARM is practically useful because for a given level of audit risk, the tolerable detection risk bears an inverse relation to the other two risks. The ARM is useful for Laura because it enables her to incorporate prior knowledge on BuildIt’s organization to increase the required risk that she will fail to find material misstatements. According to the ARM, the audit risk will then be retained.

+

\[ \text{Audit risk} = \text{Inherent risk} \,\times\, \text{Control risk} \,\times\, \text{Detection risk}\]

+

Usually the auditor judges inherent risk and control risk on a three-point scale consisting of low, medium, and high. Different audit firms handle different standard percentages for these categories. Laura’s firm defines the probabilities of low, medium, and high respectively as 50%, 60%, and 100%. Because Laura performed testing of BuildIt’s computer systems, she assesses the control risk as medium (60%).

+
# Specify the inherent risk (ir) and control risk (cr).
+ir <- 1     # 100%
+cr <- 0.6   # 60%
+
+
+

Stage 1: Planning an audit sample

+

Laura can choose to either perform a frequentist analysis, where she uses the increased detection risk as her level of uncertainty, or perform a Bayesian analysis, where she captures the information in the control risk in a prior distribution. For this example, we will show how Laura performs a Bayesian analysis. A frequentist analysis can easily be done through the following functions by setting prior = FALSE. In a frequentist audit, Laura immediately starts at step 1 and uses the value adjustedConfidence as her new value for confidence.

+
# Adjust the required confidence for a frequentist analysis.
+adjustedConfidence <- 1 - ((1 - confidence) / (ir * cr))
+

In a Bayesian audit, Laura starts at step 0 by defining the prior distribution that corresponds to her assessment of the control risk. She assumes the likelihood for a sample of \(n\) observations, in which \(k\) were in error, to be \(\text{binomial}(n, k)\). Using the auditPrior() function, she can create a prior distribution that incorporates the information in the risk assessments from the ARM. For more information on how this is done, see Derks et al. (2019).

+
# Step 0: Create a prior distribution according to the audit risk model.
+priorResult <- auditPrior(confidence = confidence, likelihood = "binomial", method = "arm", expectedError = expectedError, materiality = materiality, ir = ir, cr = cr)
+

Laura can inspect the resulting prior distribution with the print() function.

+
print(priorResult)
+
## # ------------------------------------------------------------
+## #         jfa Prior Distribution Summary (Bayesian)
+## # ------------------------------------------------------------
+## # Input:
+## #
+## # Confidence:              95%    
+## # Expected sample errors:  2.5%       
+## # Likelihood:              binomial 
+## # Specifics:               Inherent risk = 1; Internal control risk = 0.6; Detection risk = 0.08 
+## # ------------------------------------------------------------
+## # Output: 
+## #
+## # Prior distribution:      beta(a = 2.275, ß = 50.725) 
+## # Implicit sample size:    51 
+## # Implicit errors:         1.27 
+## # ------------------------------------------------------------
+## # Statistics: 
+## #
+## # Upper bound:             0.1 
+## # Precision:               0.07 
+## # Mode:                    0.02 
+## # Mean:                    0.04 
+## # Median:                  0.04 
+## # ------------------------------------------------------------
+

The prior distribution can be shown by using the plot() function.

+
plot(priorResult)
+

+

Now that the prior distribution is specified, Laura can calculate the required sample size for her desired statement by using the planning() function. She uses the priorResult object as input for the planning() function to use her prior distribution.

+
# Step 1: Calculate the required sample size.
+planningResult <- planning(confidence = confidence, expectedError = expectedError, materiality = materiality, prior = priorResult)
+

Laura can then inspect the result from her planning procedure by using the print() function. Her result tells her that, given her prior distribution she needs to audit a sample of 169 transactions so that, when at most 4.23 errors are found, she can conclude with 95% confidence that the maximum error in BuildIt’s financial statements is lower the materiality of 5%.

+
print(planningResult)
+
## # ------------------------------------------------------------
+## #              jfa Planning Summary (Bayesian)
+## # ------------------------------------------------------------
+## # Input:
+## # 
+## # Confidence:              95%      
+## # Materiality:             5% 
+## # Minimum precision:       Not specified 
+## # Likelihood:              binomial 
+## # Prior distribution:      beta(a = 2.275, ß = 50.725) 
+## # Expected sample errors:  4.23 
+## # ------------------------------------------------------------
+## # Output:
+## #
+## # Sample size:             169 
+## # Posterior distribution:  beta(a = 6.5, ß = 215.5) 
+## # ------------------------------------------------------------
+## # Statistics:
+## #
+## # Expected upper bound:    4.99% 
+## # Expected precision:      2.49% 
+## # Expected Bayes factor-+: 9.32 
+## # ------------------------------------------------------------
+

Laura 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 Laura actually observed a sample of 169 transactions, from which 4.223 were in error.

+
plot(planningResult)
+

+
+
+

Stage 2: Selecting a sample

+

Laura is now ready to select the required 169 transactions from the financial statements. She can choose to do this according to one of two statistical methods. In record sampling (units = "records"), inclusion probabilities are assigned on the transaction level, treating transactions with a high value and a low value the same, a transaction of $5,000 is equally likely to be selected as a transaction of $1,000. In monetary unit sampling (units = "mus"), inclusion probabilities are assigned on the level of individual monetary units (e.g., a dollar). When a dollar is selected to be in the sample, the transaction that includes that dollar is selected. This favors higher transactions, as a transaction of $5,000 is five times more likely to be selected than a transaction of $1,000.

+

Laura chooses to use monetary unit sampling, as she wants to include more high-valued transactions. The selection() function allows her to sample from the financial statements. She uses the planningResult object as an input for the selection() function.

+
# Step 2: Draw a sample from the financial statements.
+samplingResult <- selection(population = BuildIt, sampleSize = planningResult, units = "mus", bookValues = "bookValue", seed = 999)
+

Laura can inspect the outcomes of her sampling procedure by using the print() function.

+
print(samplingResult)
+
## # ------------------------------------------------------------
+## #                  jfa Selection Summary
+## # ------------------------------------------------------------
+## # Input:
+## #      
+## # Population size:         3500 
+## # Requested sample size:   169 
+## # Sampling units:          Monetary units 
+## # Algorithm:               Random sampling   
+## # ------------------------------------------------------------ 
+## # Output:
+## #
+## # Obtained sample size:    169 
+## # ------------------------------------------------------------
+## # Statistics:
+## #
+## # Proportion n/N:          0.05 
+## # Percentage of value:     4.65% 
+## # ------------------------------------------------------------
+
+
+

Stage 3: Executing the audit

+

The selected sample can be isolated by indexing the sample object from the sampling result. Now Laura can execute her audit by annotating the sample with their audit value (for exampling by writing the sample to a .csv file using write.csv(). She can then load her annotated sample back into R for further evaluation.

+
# Step 3: Isolate the sample for execution of the audit.
+sample <- samplingResult$sample
+
+# To write the sample to a .csv file:
+# write.csv(x = sample, file = "auditSample.csv", row.names = FALSE)
+
+# To load annotated sample back into R:
+# sample <- read.csv(file = "auditSample.csv")
+

For this example, the audit values of the sample are already included in the auditValue column of the dataset .

+
+
+

Stage 4: Evaluating the sample

+

Using her annotated sample, Laura can perform her inference with the evaluation() function. By passing the priorResult object to the function, she automatically sets method = "binomial" to be consistent with her prior distribution.

+
# Step 4: Evaluate the sample.
+evaluationResult <- evaluation(sample = sample, materiality = materiality, prior = priorResult, bookValues = "bookValue", auditValues = "auditValue")
+

Laura can inspect the outcomes of her inference by using the print() function. Her resulting upper bound is 3.518%, which is lower than the materiality of 5%. The output tells Laura the correct conclusion immediately.

+
print(evaluationResult)
+
## # ------------------------------------------------------------ 
+## #             jfa Evaluation Summary (Bayesian)
+## # ------------------------------------------------------------
+## # Input: 
+## #
+## # Confidence:               95%   
+## # Materiality:              5% 
+## # Minium precision:         Not specified 
+## # Sample size:              169 
+## # Sample errors:            3 
+## # Sum of taints:            1.8  
+## # Method:                   binomial 
+## # Prior distribution:       beta(a = 2.275, ß = 50.725) 
+## # ------------------------------------------------------------
+## # Output:
+## #
+## # Posterior distribution:   beta(a = 4.075, ß = 217.925) 
+## # Most likely error:        1.39% 
+## # Upper bound:              3.52% 
+## # Precision:                2.13% 
+## # Bayes factor-+:           108.94 
+## # Conclusion:               Approve population 
+## # ------------------------------------------------------------
+

She can inspect the prior and posterior distribution by using the plot() function. The shaded area quantifies the area under the posterior distribution that contains 95% of the probability, which ends at 3.518%. Therefore, Laura can state that there is a 95% probability that the misstatement in BuildIt’s statements is lower than 3.518%.

+
plot(evaluationResult)
+

+
+
+

Conclusion

+

Since the 95% upper confidence bound on the misstatement in BuildIt’s financial statements is lower than the materiality, Laura can conclude that the financial statements as a whole do not contain material misstatement. Therefore, she can approve BuildIt’s financial statements.

+
+ + + + + + + + + + + diff --git a/doc/v2priorDistributions.R b/doc/v2priorDistributions.R new file mode 100644 index 000000000..585edb809 --- /dev/null +++ b/doc/v2priorDistributions.R @@ -0,0 +1,75 @@ +## ----setup, include=FALSE----------------------------------------------------- +knitr::opts_chunk$set(echo = TRUE, fig.height=4, fig.width=6) +library(jfa) + +## ----------------------------------------------------------------------------- +confidence <- 0.95 # 95% confidence +likelihood <- "binomial" # Binomial likelihood +materiality <- 0.05 # Performance materiality of 5% +expectedError <- 0 # Zero errors expected in sample + +## ----------------------------------------------------------------------------- +prior1 <- auditPrior(confidence = confidence, likelihood = likelihood, expectedError = expectedError, + materiality = materiality, method = "none") +prior1 + +## ----------------------------------------------------------------------------- +plot(prior1) + +## ----------------------------------------------------------------------------- +prior2 <- auditPrior(confidence = confidence, likelihood = likelihood, expectedError = expectedError, + materiality = materiality, method = "arm", ir = 0.9, cr = 0.6) +prior2 + +## ----------------------------------------------------------------------------- +plot(prior2) + +## ----------------------------------------------------------------------------- +prior3 <- auditPrior(confidence = confidence, likelihood = likelihood, expectedError = expectedError, + materiality = materiality, method = "median") +prior3 + +## ----------------------------------------------------------------------------- +plot(prior3) + +## ----------------------------------------------------------------------------- +prior4 <- auditPrior(confidence = confidence, likelihood = likelihood, expectedError = expectedError, + materiality = materiality, method = "hypotheses", pHmin = 0.6) +prior4 + +## ----------------------------------------------------------------------------- +plot(prior4) + +## ----------------------------------------------------------------------------- +prior5 <- auditPrior(confidence = confidence, likelihood = likelihood, expectedError = expectedError, + materiality = materiality, method = "sample", sampleN = 30, sampleK = 0) +prior5 + +## ----------------------------------------------------------------------------- +plot(prior5) + +## ----------------------------------------------------------------------------- +prior6 <- auditPrior(confidence = confidence, likelihood = likelihood, expectedError = expectedError, + materiality = materiality, method = "factor", sampleN = 58, sampleK = 0, factor = 0.7) +prior6 + +## ----------------------------------------------------------------------------- +plot(prior6) + +## ---- collapse = TRUE, echo = FALSE------------------------------------------- +s1 <- planning(confidence = confidence, expectedError = expectedError, likelihood = likelihood, + materiality = materiality, prior = prior1)$sampleSize +s2 <- planning(confidence = confidence, expectedError = expectedError, likelihood = likelihood, + materiality = materiality, prior = prior2)$sampleSize +s3 <- planning(confidence = confidence, expectedError = expectedError, likelihood = likelihood, + materiality = materiality, prior = prior3)$sampleSize +s4 <- planning(confidence = confidence, expectedError = expectedError, likelihood = likelihood, + materiality = materiality, prior = prior4)$sampleSize +s5 <- planning(confidence = confidence, expectedError = expectedError, likelihood = likelihood, + materiality = materiality, prior = prior5)$sampleSize +s6 <- planning(confidence = confidence, expectedError = expectedError, likelihood = likelihood, + materiality = materiality, prior = prior6)$sampleSize + +knitr::kable(data.frame("none" = s1, "arm" = s2, "median" = s3, "hypotheses" = s4, + "sample" = s5, "factor" = s6, row.names = "Required sample size")) + diff --git a/doc/v2priorDistributions.Rmd b/doc/v2priorDistributions.Rmd new file mode 100644 index 000000000..e00e1b06b --- /dev/null +++ b/doc/v2priorDistributions.Rmd @@ -0,0 +1,165 @@ +--- +title: "Constructing a prior distribution" +author: Koen Derks +date: "last modified: 07-11-2020" +output: + rmarkdown::html_vignette: + toc: true +vignette: > + %\VignetteIndexEntry{Constructing a prior distribution} + %\VignetteEngine{knitr::rmarkdown} + %\VignetteDepends{jfa} + %\VignetteKeywords{audit, evaluation, jfa, planning, prior} + %\VignettePackage{jfa} + %\VignetteEncoding{UTF-8} +--- + +```{r setup, include=FALSE} +knitr::opts_chunk$set(echo = TRUE, fig.height=4, fig.width=6) +library(jfa) +``` + +## Bayesian statistics + +Bayesian statistics allows you to incorporate existing information into the statistical analysis and revise this information using the information from the sample, possibly increasing your efficiency in the sampling procedure. For example, when you have information that indicates a low-risk profile for your client, you may require less evidence from the audit sampling procedures. Be aware that all information that you incorporate into the statistical analysis should be justified. + +### Prior probability distributions + +Bayesian statistics incorporates existing information into the sampling procedure using a prior probability distribution that reflects your current knowledge about the misstatement in the population. The prior distribution is created using existing information, and is therefore generally created before the planning stage in the procedure. + +## Prior construction methods + +An important question is how to incorporate various kinds of existing information into the prior distribution. `jfa` offers six methods to create a prior distribution. These methods are explained below. + +First, let's set some default options for the confidence, performance materiality, the likelihood, and the expected errors. + +```{r} +confidence <- 0.95 # 95% confidence +likelihood <- "binomial" # Binomial likelihood +materiality <- 0.05 # Performance materiality of 5% +expectedError <- 0 # Zero errors expected in sample +``` + + +### [Default] No explicit information (`method = "none"`) + +You can refrain from incorporating explicit information in the prior distribution, and still use Bayesian statistics, using `method = none`. As an example, the code below incorporates no explicit information into a prior distribution + +```{r} +prior1 <- auditPrior(confidence = confidence, likelihood = likelihood, expectedError = expectedError, + materiality = materiality, method = "none") +prior1 +``` + +You can visually inspect the prior distribution using the `plot()` function. + +```{r} +plot(prior1) +``` + +### Audit Risk Model (`method = "arm"`) + +You can incorporate the risk assessments from the Audit Risk Model (inherent risk and internal control risk) using `method = "arm` in combination with the `ir` and `cr` arguments. As an example, the code below incorporates the information that the inherent risk is equal to 90% and that the internal control risk is equal to 60% into a prior distribution. + +```{r} +prior2 <- auditPrior(confidence = confidence, likelihood = likelihood, expectedError = expectedError, + materiality = materiality, method = "arm", ir = 0.9, cr = 0.6) +prior2 +``` + +You can visually inspect the prior distribution using the `plot()` function. + +```{r} +plot(prior2) +``` + +### Equal prior probabilities (`method = "median"`) + +You can incorporate the assumption that tolerable misstatement is equally likely as intolerable misstatement using `method = "median"`. As an example, the code below incorporates this assumption into a prior distribution. + +*Note: This method requires that you specify a value for the `materiality`.* + +```{r} +prior3 <- auditPrior(confidence = confidence, likelihood = likelihood, expectedError = expectedError, + materiality = materiality, method = "median") +prior3 +``` + +You can visually inspect the prior distribution using the `plot()` function. + +```{r} +plot(prior3) +``` + +### Custom prior probabilities (`method = "hypotheses"`) + +You can assign custom probabilities to the hypothesis of tolerable misstatement (using `pHmin`) and/or the hypotheses of intolerable misstatement (using `pHplus`) in combination with `method = "hypotheses`. As an example, the code below incorporates the information that the hypothesis of tolerable misstatement has a probability of 60% into a prior distribution. + +*Note: This method requires that you specify a value for the `materiality` and that the expected errors are zero.* + +```{r} +prior4 <- auditPrior(confidence = confidence, likelihood = likelihood, expectedError = expectedError, + materiality = materiality, method = "hypotheses", pHmin = 0.6) +prior4 +``` + +You can visually inspect the prior distribution using the `plot()` function. + +```{r} +plot(prior4) +``` + +### Earlier sample (`method = "sample"`) + +You can incorporate information from an earlier sample into the prior distribution using `method = "sample` in combination with `sampleN` and `sampleK`. As an example, the code below incorporates the information from an earlier sample of 30 transactions, in which 0 misstatements were found, into a prior distribution. + +```{r} +prior5 <- auditPrior(confidence = confidence, likelihood = likelihood, expectedError = expectedError, + materiality = materiality, method = "sample", sampleN = 30, sampleK = 0) +prior5 +``` + +You can visually inspect the prior distribution using the `plot()` function. + +```{r} +plot(prior5) +``` + +### Weighted earlier sample (`method = "factor"`) + +You can incorporate information from last years results, weighted by a factor, into the prior distribution using `method = "factor` in combination with `sampleN` and `sampleK`. As an example, the code below incorporates the information from a last years results (a sample of 58 transactions in which 0 misstatements were found), weighted by a factor 0.7, into a prior distribution. + +```{r} +prior6 <- auditPrior(confidence = confidence, likelihood = likelihood, expectedError = expectedError, + materiality = materiality, method = "factor", sampleN = 58, sampleK = 0, factor = 0.7) +prior6 +``` + +You can visually inspect the prior distribution using the `plot()` function. + +```{r} +plot(prior6) +``` + +## Comparing efficiency through required sample sizes + +To illustrate how the prior distribution can facilitate a more efficient audit, the table below lists the required sample sizes for the created priors after calling `planning(confidence = confidence, expectedError = expectedError, likelihood = likelihood, materiality = materiality, prior = X)`, where `X` is the object returned by the `auditPrior()` function. + +```{r, collapse = TRUE, echo = FALSE} +s1 <- planning(confidence = confidence, expectedError = expectedError, likelihood = likelihood, + materiality = materiality, prior = prior1)$sampleSize +s2 <- planning(confidence = confidence, expectedError = expectedError, likelihood = likelihood, + materiality = materiality, prior = prior2)$sampleSize +s3 <- planning(confidence = confidence, expectedError = expectedError, likelihood = likelihood, + materiality = materiality, prior = prior3)$sampleSize +s4 <- planning(confidence = confidence, expectedError = expectedError, likelihood = likelihood, + materiality = materiality, prior = prior4)$sampleSize +s5 <- planning(confidence = confidence, expectedError = expectedError, likelihood = likelihood, + materiality = materiality, prior = prior5)$sampleSize +s6 <- planning(confidence = confidence, expectedError = expectedError, likelihood = likelihood, + materiality = materiality, prior = prior6)$sampleSize + +knitr::kable(data.frame("none" = s1, "arm" = s2, "median" = s3, "hypotheses" = s4, + "sample" = s5, "factor" = s6, row.names = "Required sample size")) +``` + diff --git a/doc/v2priorDistributions.html b/doc/v2priorDistributions.html new file mode 100644 index 000000000..5b40e5f15 --- /dev/null +++ b/doc/v2priorDistributions.html @@ -0,0 +1,658 @@ + + + + + + + + + + + + + + + +Constructing a prior distribution + + + + + + + + + + + + + + + + + + + + + + + + + + +

Constructing a prior distribution

+

Koen Derks

+

last modified: 07-11-2020

+ + + + +
+

Bayesian statistics

+

Bayesian statistics allows you to incorporate existing information into the statistical analysis and revise this information using the information from the sample, possibly increasing your efficiency in the sampling procedure. For example, when you have information that indicates a low-risk profile for your client, you may require less evidence from the audit sampling procedures. Be aware that all information that you incorporate into the statistical analysis should be justified.

+
+

Prior probability distributions

+

Bayesian statistics incorporates existing information into the sampling procedure using a prior probability distribution that reflects your current knowledge about the misstatement in the population. The prior distribution is created using existing information, and is therefore generally created before the planning stage in the procedure.

+
+
+
+

Prior construction methods

+

An important question is how to incorporate various kinds of existing information into the prior distribution. jfa offers six methods to create a prior distribution. These methods are explained below.

+

First, let’s set some default options for the confidence, performance materiality, the likelihood, and the expected errors.

+
confidence <- 0.95        # 95% confidence
+likelihood <- "binomial"  # Binomial likelihood
+materiality <- 0.05       # Performance materiality of 5%
+expectedError <- 0        # Zero errors expected in sample
+
+

[Default] No explicit information (method = "none")

+

You can refrain from incorporating explicit information in the prior distribution, and still use Bayesian statistics, using method = none. As an example, the code below incorporates no explicit information into a prior distribution

+
prior1 <- auditPrior(confidence = confidence, likelihood = likelihood, expectedError = expectedError,
+                     materiality = materiality, method = "none")
+prior1
+
## # ------------------------------------------------------------
+## #         jfa Prior Distribution Summary (Bayesian)
+## # ------------------------------------------------------------
+## # Input:
+## #
+## # Confidence:              95%    
+## # Expected sample errors:  0%       
+## # Likelihood:              binomial 
+## # Specifics:               None 
+## # ------------------------------------------------------------
+## # Output: 
+## #
+## # Prior distribution:      beta(a = 1, ß = 1) 
+## # Implicit sample size:    0 
+## # Implicit errors:         0 
+## # ------------------------------------------------------------
+## # Statistics: 
+## #
+## # Upper bound:             0.95 
+## # Precision:               NaN 
+## # Mode:                    NaN 
+## # Mean:                    0.5 
+## # Median:                  0.5 
+## # ------------------------------------------------------------
+

You can visually inspect the prior distribution using the plot() function.

+
plot(prior1)
+

+
+
+

Audit Risk Model (method = "arm")

+

You can incorporate the risk assessments from the Audit Risk Model (inherent risk and internal control risk) using method = "arm in combination with the ir and cr arguments. As an example, the code below incorporates the information that the inherent risk is equal to 90% and that the internal control risk is equal to 60% into a prior distribution.

+
prior2 <- auditPrior(confidence = confidence, likelihood = likelihood, expectedError = expectedError,
+                     materiality = materiality, method = "arm", ir = 0.9, cr = 0.6)
+prior2
+
## # ------------------------------------------------------------
+## #         jfa Prior Distribution Summary (Bayesian)
+## # ------------------------------------------------------------
+## # Input:
+## #
+## # Confidence:              95%    
+## # Expected sample errors:  0%       
+## # Likelihood:              binomial 
+## # Specifics:               Inherent risk = 0.9; Internal control risk = 0.6; Detection risk = 0.09 
+## # ------------------------------------------------------------
+## # Output: 
+## #
+## # Prior distribution:      beta(a = 1, ß = 13) 
+## # Implicit sample size:    12 
+## # Implicit errors:         0 
+## # ------------------------------------------------------------
+## # Statistics: 
+## #
+## # Upper bound:             0.21 
+## # Precision:               0.21 
+## # Mode:                    0 
+## # Mean:                    0.07 
+## # Median:                  0.05 
+## # ------------------------------------------------------------
+

You can visually inspect the prior distribution using the plot() function.

+
plot(prior2)
+

+
+
+

Equal prior probabilities (method = "median")

+

You can incorporate the assumption that tolerable misstatement is equally likely as intolerable misstatement using method = "median". As an example, the code below incorporates this assumption into a prior distribution.

+

Note: This method requires that you specify a value for the materiality.

+
prior3 <- auditPrior(confidence = confidence, likelihood = likelihood, expectedError = expectedError,
+                     materiality = materiality, method = "median")
+prior3
+
## # ------------------------------------------------------------
+## #         jfa Prior Distribution Summary (Bayesian)
+## # ------------------------------------------------------------
+## # Input:
+## #
+## # Confidence:              95%    
+## # Expected sample errors:  0%       
+## # Likelihood:              binomial 
+## # Specifics:               p(T < 0.05) = p(T > 0.05) = 0.5 
+## # ------------------------------------------------------------
+## # Output: 
+## #
+## # Prior distribution:      beta(a = 1, ß = 13.513) 
+## # Implicit sample size:    12.51 
+## # Implicit errors:         0 
+## # ------------------------------------------------------------
+## # Statistics: 
+## #
+## # Upper bound:             0.2 
+## # Precision:               0.2 
+## # Mode:                    0 
+## # Mean:                    0.07 
+## # Median:                  0.05 
+## # ------------------------------------------------------------
+

You can visually inspect the prior distribution using the plot() function.

+
plot(prior3)
+

+
+
+

Custom prior probabilities (method = "hypotheses")

+

You can assign custom probabilities to the hypothesis of tolerable misstatement (using pHmin) and/or the hypotheses of intolerable misstatement (using pHplus) in combination with method = "hypotheses. As an example, the code below incorporates the information that the hypothesis of tolerable misstatement has a probability of 60% into a prior distribution.

+

Note: This method requires that you specify a value for the materiality and that the expected errors are zero.

+
prior4 <- auditPrior(confidence = confidence, likelihood = likelihood, expectedError = expectedError,
+                     materiality = materiality, method = "hypotheses", pHmin = 0.6)
+prior4
+
## # ------------------------------------------------------------
+## #         jfa Prior Distribution Summary (Bayesian)
+## # ------------------------------------------------------------
+## # Input:
+## #
+## # Confidence:              95%    
+## # Expected sample errors:  0%       
+## # Likelihood:              binomial 
+## # Specifics:               p(T < 0.05) = 0.6; p(T > 0.05) = 0.4 
+## # ------------------------------------------------------------
+## # Output: 
+## #
+## # Prior distribution:      beta(a = 1, ß = 17.864) 
+## # Implicit sample size:    16.86 
+## # Implicit errors:         0 
+## # ------------------------------------------------------------
+## # Statistics: 
+## #
+## # Upper bound:             0.15 
+## # Precision:               0.15 
+## # Mode:                    0 
+## # Mean:                    0.05 
+## # Median:                  0.04 
+## # ------------------------------------------------------------
+

You can visually inspect the prior distribution using the plot() function.

+
plot(prior4)
+

+
+
+

Earlier sample (method = "sample")

+

You can incorporate information from an earlier sample into the prior distribution using method = "sample in combination with sampleN and sampleK. As an example, the code below incorporates the information from an earlier sample of 30 transactions, in which 0 misstatements were found, into a prior distribution.

+
prior5 <- auditPrior(confidence = confidence, likelihood = likelihood, expectedError = expectedError,
+                     materiality = materiality, method = "sample", sampleN = 30, sampleK = 0)
+prior5
+
## # ------------------------------------------------------------
+## #         jfa Prior Distribution Summary (Bayesian)
+## # ------------------------------------------------------------
+## # Input:
+## #
+## # Confidence:              95%    
+## # Expected sample errors:  0%       
+## # Likelihood:              binomial 
+## # Specifics:               Earlier sample of 30 transactions with 0 errors 
+## # ------------------------------------------------------------
+## # Output: 
+## #
+## # Prior distribution:      beta(a = 1, ß = 31) 
+## # Implicit sample size:    30 
+## # Implicit errors:         0 
+## # ------------------------------------------------------------
+## # Statistics: 
+## #
+## # Upper bound:             0.09 
+## # Precision:               0.09 
+## # Mode:                    0 
+## # Mean:                    0.03 
+## # Median:                  0.02 
+## # ------------------------------------------------------------
+

You can visually inspect the prior distribution using the plot() function.

+
plot(prior5)
+

+
+
+

Weighted earlier sample (method = "factor")

+

You can incorporate information from last years results, weighted by a factor, into the prior distribution using method = "factor in combination with sampleN and sampleK. As an example, the code below incorporates the information from a last years results (a sample of 58 transactions in which 0 misstatements were found), weighted by a factor 0.7, into a prior distribution.

+
prior6 <- auditPrior(confidence = confidence, likelihood = likelihood, expectedError = expectedError,
+                     materiality = materiality, method = "factor", sampleN = 58, sampleK = 0, factor = 0.7)
+prior6
+
## # ------------------------------------------------------------
+## #         jfa Prior Distribution Summary (Bayesian)
+## # ------------------------------------------------------------
+## # Input:
+## #
+## # Confidence:              95%    
+## # Expected sample errors:  0%       
+## # Likelihood:              binomial 
+## # Specifics:               Earlier sample of 58 transactions with 0 errors weighted by 0.7 
+## # ------------------------------------------------------------
+## # Output: 
+## #
+## # Prior distribution:      beta(a = 1, ß = 41.6) 
+## # Implicit sample size:    40.6 
+## # Implicit errors:         0 
+## # ------------------------------------------------------------
+## # Statistics: 
+## #
+## # Upper bound:             0.07 
+## # Precision:               0.07 
+## # Mode:                    0 
+## # Mean:                    0.02 
+## # Median:                  0.02 
+## # ------------------------------------------------------------
+

You can visually inspect the prior distribution using the plot() function.

+
plot(prior6)
+

+
+
+
+

Comparing efficiency through required sample sizes

+

To illustrate how the prior distribution can facilitate a more efficient audit, the table below lists the required sample sizes for the created priors after calling planning(confidence = confidence, expectedError = expectedError, likelihood = likelihood, materiality = materiality, prior = X), where X is the object returned by the auditPrior() function.

+ + + + + + + + + + + + + + + + + + + + + + + +
nonearmmedianhypothesessamplefactor
Required sample size584645412817
+
+ + + + + + + + + + + diff --git a/doc/v3estimation.R b/doc/v3estimation.R new file mode 100644 index 000000000..6e1ebe1d3 --- /dev/null +++ b/doc/v3estimation.R @@ -0,0 +1,4 @@ +## ----setup, include=FALSE----------------------------------------------------- +knitr::opts_chunk$set(echo = TRUE, fig.height=4, fig.width=6) +library(jfa) + diff --git a/doc/v3estimation.Rmd b/doc/v3estimation.Rmd new file mode 100644 index 000000000..2029198cc --- /dev/null +++ b/doc/v3estimation.Rmd @@ -0,0 +1,32 @@ +--- +title: "Estimating the misstatement" +author: Koen Derks +date: "last modified: 19-11-2020" +output: + rmarkdown::html_vignette: + toc: true +vignette: > + %\VignetteIndexEntry{Estimating the misstatement} + %\VignetteEngine{knitr::rmarkdown} + %\VignetteDepends{jfa} + %\VignetteKeywords{audit, evaluation, jfa, planning, prior} + %\VignettePackage{jfa} + %\VignetteEncoding{UTF-8} +--- + +```{r setup, include=FALSE} +knitr::opts_chunk$set(echo = TRUE, fig.height=4, fig.width=6) +library(jfa) +``` + +## Estimation in an audit context + +In audit sampling estimation the auditor tries to determine the unknown quantity of the misstatement $\theta$ in the population as best as possible. + +## Frequentist estimation using the likelihood + +This will be featured in a future version of `jfa`. + +## Bayesian estimation using the posterior distribution + +This will be featured in a future version of `jfa`. \ No newline at end of file diff --git a/doc/v3estimation.html b/doc/v3estimation.html new file mode 100644 index 000000000..4c675b827 --- /dev/null +++ b/doc/v3estimation.html @@ -0,0 +1,321 @@ + + + + + + + + + + + + + + + +Estimating the misstatement + + + + + + + + + + + + + + + + + + + + + + + +

Estimating the misstatement

+

Koen Derks

+

last modified: 19-11-2020

+ + + + +
+

Estimation in an audit context

+

In audit sampling estimation the auditor tries to determine the unknown quantity of the misstatement \(\theta\) in the population as best as possible.

+
+
+

Frequentist estimation using the likelihood

+

This will be featured in a future version of jfa.

+
+
+

Bayesian estimation using the posterior distribution

+

This will be featured in a future version of jfa.

+
+ + + + + + + + + + + diff --git a/doc/v4testing.R b/doc/v4testing.R new file mode 100644 index 000000000..64ab7a312 --- /dev/null +++ b/doc/v4testing.R @@ -0,0 +1,12 @@ +## ----setup, include=FALSE----------------------------------------------------- +knitr::opts_chunk$set(echo = TRUE, fig.height=4, fig.width=6) +library(jfa) + +## ----------------------------------------------------------------------------- +prior <- auditPrior(confidence = 0.95, likelihood = "binomial", method = "none", materiality = 0.05) +evaluation(confidence = 0.95, materiality = 0.05, nSumstats = 40, kSumstats = 1, prior = prior) + +## ----------------------------------------------------------------------------- +prior <- auditPrior(confidence = 0.95, likelihood = "binomial", method = "median", materiality = 0.05) +evaluation(confidence = 0.95, materiality = 0.05, nSumstats = 40, kSumstats = 1, prior = prior) + diff --git a/doc/v4testing.Rmd b/doc/v4testing.Rmd new file mode 100644 index 000000000..b662cad49 --- /dev/null +++ b/doc/v4testing.Rmd @@ -0,0 +1,74 @@ +--- +title: "Testing the misstatement" +author: Koen Derks +date: "last modified: 19-11-2020" +output: + rmarkdown::html_vignette: + toc: true +vignette: > + %\VignetteIndexEntry{Testing the misstatement} + %\VignetteEngine{knitr::rmarkdown} + %\VignetteDepends{jfa} + %\VignetteKeywords{audit, evaluation, jfa, planning, prior} + %\VignettePackage{jfa} + %\VignetteEncoding{UTF-8} +--- + +```{r setup, include=FALSE} +knitr::opts_chunk$set(echo = TRUE, fig.height=4, fig.width=6) +library(jfa) +``` + +## Testing in an audit sampling context + +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. + +## Frequentist hypothesis testing using the *p*-value + +This will be added in a future version of `jfa`. + +## Bayesian hypothesis testing using the Bayes factor + +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 `evluation()` 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 $H_-$ | + +### Example + +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 transactions, 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 $BF_{-+}=30.28$, implying that there is very strong evidence for $H_-$, the hypothesis that the population contains misstatements lower than 5 percent of the population. + +```{r} +prior <- auditPrior(confidence = 0.95, likelihood = "binomial", method = "none", materiality = 0.05) +evaluation(confidence = 0.95, materiality = 0.05, nSumstats = 40, kSumstats = 1, prior = prior) +``` + +### Sensitivity to the prior distribution + +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 with respect to the misstatement parameter $\theta$, the Bayes factor overestimates the evidence in favor of $H_-$. You can mitigate this dependency using `method = "median"` 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. + +```{r} +prior <- auditPrior(confidence = 0.95, likelihood = "binomial", method = "median", materiality = 0.05) +evaluation(confidence = 0.95, materiality = 0.05, nSumstats = 40, kSumstats = 1, prior = prior) +``` \ No newline at end of file diff --git a/doc/v4testing.html b/doc/v4testing.html new file mode 100644 index 000000000..6b7184723 --- /dev/null +++ b/doc/v4testing.html @@ -0,0 +1,529 @@ + + + + + + + + + + + + + + + +Testing the misstatement + + + + + + + + + + + + + + + + + + + + + + + + + + +

Testing the misstatement

+

Koen Derks

+

last modified: 19-11-2020

+ + + + +
+

Testing in an audit sampling context

+

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.

+
+
+

Frequentist hypothesis testing using the p-value

+

This will be added in a future version of jfa.

+
+
+

Bayesian hypothesis testing using the Bayes factor

+

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 evluation() 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 \(H_-\)
+
+

Example

+

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 transactions, 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 \(BF_{-+}=30.28\), implying that there is very strong evidence for \(H_-\), the hypothesis that the population contains misstatements lower than 5 percent of the population.

+
prior <- auditPrior(confidence = 0.95, likelihood = "binomial", method = "none", materiality = 0.05)
+evaluation(confidence = 0.95, materiality = 0.05, nSumstats = 40, kSumstats = 1, prior = prior)
+
## # ------------------------------------------------------------ 
+## #             jfa Evaluation Summary (Bayesian)
+## # ------------------------------------------------------------
+## # Input: 
+## #
+## # Confidence:               95%   
+## # Materiality:              5% 
+## # Minium precision:         Not specified 
+## # Sample size:              40 
+## # Sample errors:            1 
+## # Sum of taints:            1  
+## # Method:                   binomial 
+## # Prior distribution:       beta(a = 1, ß = 1) 
+## # ------------------------------------------------------------
+## # Output:
+## #
+## # Posterior distribution:   beta(a = 2, ß = 40) 
+## # Most likely error:        2.38% 
+## # Upper bound:              11.06% 
+## # Precision:                8.67% 
+## # Bayes factor-+:           30.28 
+## # Conclusion:               Do not approve population 
+## # ------------------------------------------------------------
+
+
+

Sensitivity to the prior distribution

+

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 with respect to the misstatement parameter \(\theta\), the Bayes factor overestimates the evidence in favor of \(H_-\). You can mitigate this dependency using method = "median" 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(confidence = 0.95, likelihood = "binomial", method = "median", materiality = 0.05)
+evaluation(confidence = 0.95, materiality = 0.05, nSumstats = 40, kSumstats = 1, prior = prior)
+
## # ------------------------------------------------------------ 
+## #             jfa Evaluation Summary (Bayesian)
+## # ------------------------------------------------------------
+## # Input: 
+## #
+## # Confidence:               95%   
+## # Materiality:              5% 
+## # Minium precision:         Not specified 
+## # Sample size:              40 
+## # Sample errors:            1 
+## # Sum of taints:            1  
+## # Method:                   binomial 
+## # Prior distribution:       beta(a = 1, ß = 13.513) 
+## # ------------------------------------------------------------
+## # Output:
+## #
+## # Posterior distribution:   beta(a = 2, ß = 52.513) 
+## # Most likely error:        1.83% 
+## # Upper bound:              8.56% 
+## # Precision:                6.73% 
+## # Bayes factor-+:           3.08 
+## # Conclusion:               Do not approve population 
+## # ------------------------------------------------------------
+
+
+ + + + + + + + + + + diff --git a/inst/rmd/report.Rmd b/inst/rmd/report.Rmd index 20bb86948..a95b3cacc 100644 --- a/inst/rmd/report.Rmd +++ b/inst/rmd/report.Rmd @@ -137,7 +137,7 @@ if(!(object[["method"]]%in%c("stringer"))){ --- -These results have been calculated using `r switch(object[["method"]], "poisson" = "the Poisson distribution", "binomial" = "the binomial distribution", "the hypergeometric" = "hypergeometric distribution", "stringer" = "the Stringer bound (Bickel, 1992)", "stringer-meikle" = "the Stringer bound with Meikle's correction", "stringer-lta" = "the Stringer bound with LTA correction", "stringer-pvz" = "the modified Stringer bound", "rohrbach" = "Rohrbach's augmented variance bound", "moment" = "the modified moment bound", "coxsnell" = "the Cox and Snell bound", "direct" = "the direct (mean per unit) estimator (Touw & Hoogduin, 2011)", "difference" = "the difference estimator (Touw & Hoogduin, 2011)", "quotient" = "the quotient estimator (Touw & Hoogduin, 2011)", "regression" = "the regression estimator (Touw & Hoogduin, 2011)")`. `r if(object[["method"]] %in% c("difference", "quotient", "direct", "regression")){"The units of inference are individual transactions"} else {"The units of inference are individual monetary units"}`. +These results have been calculated using `r switch(object[["method"]], "poisson" = "the Poisson distribution", "binomial" = "the binomial distribution", "the hypergeometric" = "hypergeometric distribution", "stringer" = "the Stringer bound (Bickel, 1992)", "stringer-meikle" = "the Stringer bound with Meikle's correction", "stringer-lta" = "the Stringer bound with LTA correction", "stringer-pvz" = "the modified Stringer bound", "rohrbach" = "Rohrbach's augmented variance bound", "moment" = "the modified moment bound", "coxsnell" = "the Cox and Snell bound", "direct" = "the direct estimator (Touw & Hoogduin, 2011)", "difference" = "the difference estimator (Touw & Hoogduin, 2011)", "quotient" = "the quotient estimator (Touw & Hoogduin, 2011)", "regression" = "the regression estimator (Touw & Hoogduin, 2011)")`. `r if(object[["method"]] %in% c("difference", "quotient", "direct", "regression")){"The units of inference are individual transactions"} else {"The units of inference are individual monetary units"}`. *Notation:* diff --git a/man/evaluation.Rd b/man/evaluation.Rd index 91c4a360c..cfab2d7dd 100644 --- a/man/evaluation.Rd +++ b/man/evaluation.Rd @@ -15,7 +15,7 @@ evaluation(confidence = 0.95, method = "binomial", N = NULL, \arguments{ \item{confidence}{the required confidence level for the bound. Default is 0.95 for 95\% confidence.} -\item{method}{the method that is used to evaluate the sample. This can be either one of \code{poisson}, \code{binomial}, \code{hypergeometric}, \code{stringer}, \code{stringer-meikle}, \code{stringer-lta}, \code{stringer-pvz}, \code{rohrbach}, \code{moment}, \code{direct}, \code{difference}, \code{quotient}, or \code{regression}.} +\item{method}{the method that is used to evaluate the sample. This can be either one of \code{poisson}, \code{binomial}, \code{hypergeometric}, \code{mpus}, \code{stringer}, \code{stringer-meikle}, \code{stringer-lta}, \code{stringer-pvz}, \code{rohrbach}, \code{moment}, \code{direct}, \code{difference}, \code{quotient}, or \code{regression}.} \item{N}{an integer specifying the total number of units (transactions or monetary units) in the population.} @@ -78,7 +78,7 @@ An object of class \code{jfaEvaluation} containing: \item{data}{a data frame containing the relevant columns from the \code{sample} input.} } \description{ -This function takes a data frame (using \code{sample}, \code{bookValue}, and \code{auditValues}) or summary statistics (using \code{nSumstats} and \code{kSumstats}) and evaluates the audit sample according to the specified method. The returned object is of class \code{jfaEvaluation} and can be used with associated \code{print()} and \code{plot()} methods. +This function takes a data frame (using \code{sample}, \code{bookValues}, and \code{auditValues}) or summary statistics (using \code{nSumstats} and \code{kSumstats}) and evaluates the audit sample according to the specified method. The returned object is of class \code{jfaEvaluation} and can be used with associated \code{print()} and \code{plot()} methods. For more details on how to use this function see the package vignette: \code{vignette("jfa", package = "jfa")} @@ -90,6 +90,7 @@ This section lists the available options for the \code{methods} argument. \item{\code{poisson}: The confidence bound taken from the Poisson distribution. If combined with \code{prior = TRUE}, performs Bayesian evaluation using a \emph{gamma} prior and posterior.} \item{\code{binomial}: The confidence bound taken from the binomial distribution. If combined with \code{prior = TRUE}, performs Bayesian evaluation using a \emph{beta} prior and posterior.} \item{\code{hypergeometric}: The confidence bound taken from the hypergeometric distribution. If combined with \code{prior = TRUE}, performs Bayesian evaluation using a \emph{beta-binomial} prior and posterior.} +\item{\code{mpu}}: Mean per unit estimator using the observed sample taints. \item{\code{stringer}: The Stringer bound (Stringer, 1963).} \item{\code{stringer-meikle}: Stringer bound with Meikle's correction for understatements (Meikle, 1972).} \item{\code{stringer-lta}: Stringer bound with LTA correction for understatements (Leslie, Teitlebaum, and Anderson, 1979).} diff --git a/man/figures/cheatsheet/cheatsheet.pdf b/man/figures/cheatsheet/cheatsheet.pdf new file mode 100644 index 0000000000000000000000000000000000000000..b57c0eb0c3e09b9f448e7e13a4dc6cd71152455c GIT binary patch literal 258691 zcmb@tWmsIxwl<2pyVDRFhsNFA-7Q$-?(S~ELm+5?APMfSL4yV-L4&(n;C8b1K4gFLl+VPDcL%K z&D?DLz?J}ZkP?7{n~#T|9i$52VB-KuL!xXPT$~^kNS@a3JWg&tkQ~4ez{AE3iE zh@gX=EPq!8`1=ECgNygS+mHfVI9r0%%-kH^AuUQeJ9s)exdXUCYHnalTMG|oHvl__ zhzLl=*1-ep29j|w^8o)P0+N?96yoRMljP=O=iuOz;F03z;*#Lum*(T<=8)!;=9CoY 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rzzAtsxmz4J|6dkkj$60LT;0rE+`L>YEXjcFfrgcgmsd_zp6vesIw-HX diff --git a/man/planning.Rd b/man/planning.Rd index 412072fe1..c286ee4c8 100644 --- a/man/planning.Rd +++ b/man/planning.Rd @@ -51,7 +51,7 @@ An object of class \code{jfaPlanning} containing: \item{expectedPosterior}{an object of class \code{jfaPosterior} that represents the expected posterior distribution.} } \description{ -This function calculates the required sample size for an audit, based on the poisson, binomial, or hypergeometric likelihood. A prior can be specified to perform Bayesian planning. The returned object is of class \code{jfaPlanning} and can be used with associated \code{print()} and \code{plot()} methods. +This function calculates the required sample size for an audit, based on the Poisson, binomial, or hypergeometric likelihood. A prior can be specified to perform Bayesian planning. The returned object is of class \code{jfaPlanning} and can be used with associated \code{print()} and \code{plot()} methods. For more details on how to use this function see the package vignette: \code{vignette("jfa", package = "jfa")} diff --git a/man/report.Rd b/man/report.Rd index 659662332..072de2820 100644 --- a/man/report.Rd +++ b/man/report.Rd @@ -11,7 +11,7 @@ report(object = NULL, file = NULL, format = "html_document") \item{file}{a string that gives the desired name of the file (e.g. \code{"report.html"}). The report is created in your current working directory.} -\item{format}{can be either one of \code{"html_document"} or \code{"pdf_document"} (required MikTex).} +\item{format}{can be either one of \code{"html_document"} or \code{"pdf_document"} (compiling to pdf requires MikTex).} } \value{ A html or pdf report containing the results of the evaluation. diff --git a/tests/testthat/test-function-evaluation.R b/tests/testthat/test-function-evaluation.R index affa5aa68..b8aee91c5 100644 --- a/tests/testthat/test-function-evaluation.R +++ b/tests/testthat/test-function-evaluation.R @@ -240,6 +240,18 @@ test_that(desc = "(id: f5-v0.4.0-t1) Bayes factors", { # jfa version 0.5.0 +test_that(desc = "(id: f5-v0.5.0-t1) Test for mpu estimator", { + + sample <- data.frame(ID = 1:100, ist = rnorm(mean = 1000, n = 100)) + sample$soll <- sample$ist + sample$ist[1] <- 120 + sample$soll[1] <- 100 + sample$ist[2] <- 100 + sample$soll[2] <- 120 + jfaEval <- jfa::evaluation(confidence = 0.95, method = "mpu", sample = sample, materiality = 0.1, bookValues = "ist", auditValues = "soll") + expect_equal(jfaEval[["confBound"]], 0.003970126, tolerance = 0.001) +}) + test_that(desc = "(id: f5-v0.5.0-t1) Test for frequentist print function", { set.seed(1) population <- data.frame(ID = sample(1000:100000, size = 1000, replace = FALSE), bookValue = runif(n = 1000, min = 100, max = 500)) diff --git a/vignettes/jfa.Rmd b/vignettes/jfa.Rmd index d59e1f842..0d6f6d56e 100644 --- a/vignettes/jfa.Rmd +++ b/vignettes/jfa.Rmd @@ -135,7 +135,7 @@ Suppose that you have audited the transactions in the sample and have found no d sample$auditValue <- sample$bookValue ``` -You can evaluate the annotated sample using the `sample`, `bookValues`, `auditValues`, and `counts` arguments. The code below evaluates the analysis goal using the popular Stringer bound. You can find more information about which methods are implemented on the [home page](https://koenderks.github.io/jfa). +You can evaluate the annotated sample using the `sample`, `bookValues`, `auditValues`, and `counts` arguments. The code below evaluates the analysis goal using the popular Stringer bound. You can find more information about which methods are implemented on the [home page](https://koenderks.github.io/jfa/). ```{r} evaluation(confidence = 0.95, method = "stringer", N = 3500, materiality = 0.05, diff --git a/vignettes/v3estimation.Rmd b/vignettes/v3estimation.Rmd index b3007b0e1..2029198cc 100644 --- a/vignettes/v3estimation.Rmd +++ b/vignettes/v3estimation.Rmd @@ -25,4 +25,8 @@ In audit sampling estimation the auditor tries to determine the unknown quantity ## Frequentist estimation using the likelihood -## Bayesian estimation using the posterior distribution \ No newline at end of file +This will be featured in a future version of `jfa`. + +## Bayesian estimation using the posterior distribution + +This will be featured in a future version of `jfa`. \ No newline at end of file

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