Package 'BenfordTests'

Statistical Tests for Benford's Law

Description

This package contains several specialized statistical tests and support functions for determining if numerical data could conform to Benford's law.

Details

Package:	BenfordTests
Type:	Package
Version:	1.2.0
Date:	2015-07-18
License:	GPL-3

BenfordTests is the implementation of eight goodness-of-fit (GOF) tests to assess if data conforms to Benford's law.
Tests include:
Pearson $\chi^2$ statistic (Pearson, 1900)
Kolmogorov-Smirnov D statistic (Kolmogorov, 1933)
Freedman's modification of Watson's $U^2$ statistic (Freedman, 1981; Watson, 1961)
Chebyshev distance m statistic (Leemis, 2000)
Euclidean distance d statistic (Cho and Gaines, 2007)
Judge-Schechter mean deviation $a^*$ statistic (Judge and Schechter, 2009)
Joenssen's $J_P^2$ statistic, a Shapiro-Francia type correlation test (Shapiro and Francia, 1972)
Joint Digit Test $T^2$ statistic, a Hotelling type test (Hotelling, 1931)

All tests may be performed using more than one leading digit. All tests simulate the specific p-values required for statistical inference, while p-values for the $\chi^2$, D, $a^*$, and $T^2$ statistics may also be determined using their asymptotic distributions.

Author(s)

Dieter William Joenssen

Maintainer: Dieter William Joenssen <[email protected]>

References

Benford, F. (1938) The Law of Anomalous Numbers. Proceedings of the American Philosophical Society. 78, 551–572.

Cho, W.K.T. and Gaines, B.J. (2007) Breaking the (Benford) Law: Statistical Fraud Detection in Campaign Finance. The American Statistician. 61, 218–223.

Freedman, L.S. (1981) Watson's Un2 Statistic for a Discrete Distribution. Biometrika. 68, 708–711.

Joenssen, D.W. (2013) Two Digit Testing for Benford's Law. Proceedings of the ISI World Statistics Congress, 59th Session in Hong Kong. [available under http://www.statistics.gov.hk/wsc/CPS021-P2-S.pdf]

Judge, G. and Schechter, L. (2009) Detecting Problems in Survey Data using Benford's Law. Journal of Human Resources. 44, 1–24.

Kolmogorov, A.N. (1933) Sulla determinazione empirica di una legge di distibuzione. Giornale dell'Istituto Italiano degli Attuari. 4, 83–91.

Leemis, L.M., Schmeiser, B.W. and Evans, D.L. (2000) Survival Distributions Satisfying Benford's law. The American Statistician. 54, 236–241.

Newcomb, S. (1881) Note on the Frequency of Use of the Different Digits in Natural Numbers. American Journal of Mathematics. 4, 39–40.

Pearson, K. (1900) On the Criterion that a Given System of Deviations from the Probable in the Case of a Correlated System of Variables is Such that it can be Reasonably Supposed to have Arisen from Random Sampling. Philosophical Magazine Series 5. 50, 157–175.

Shapiro, S.S. and Francia, R.S. (1972) An Approximate Analysis of Variance Test for Normality. Journal of the American Statistical Association. 67, 215–216.

Watson, G.S. (1961) Goodness-of-Fit Tests on a Circle. Biometrika. 48, 109–114.

Hotelling, H. (1931). The generalization of Student's ratio. Annals of Mathematical Statistics. 2, 360–378.

Examples

#Set the random seed to an arbitrary number
set.seed(421)
#Create a sample satisfying Benford's law
X<-rbenf(n=20)
#Look at sample
X
#Look at the first digits of the sample
signifd(X)

#Perform a Chi-squared Test on the sample's first digits using defaults
chisq.benftest(X)
#p-value = 0.648

---

Pearson's Chi-squared Goodness-of-Fit Test for Benford's Law

Description

chisq.benftest takes any numerical vector reduces the sample to the specified number of significant digits and performs Pearson's chi-square goodness-of-fit test to assert if the data conforms to Benford's law.

Usage

chisq.benftest(x = NULL, digits = 1, pvalmethod = "asymptotic", pvalsims = 10000)

Arguments

x	A numeric vector.
digits	An integer determining the number of first digits to use for testing, i.e. 1 for only the first, 2 for the first two etc.
pvalmethod	Method used for calculating the p-value. Either "asymptotic" or "simulate".
pvalsims	An integer specifying the number of replicates to use if pvalmethod = "simulate".

Details

A $\chi^2$ goodness-of-fit test is performed on signifd(x,digits) versus pbenf(digits). Specifically:

$\chi^2 = n\cdot\displaystyle\sum_{i=10^{k-1}}^{10^k-1}\frac{\left(f_i^o - f_i^e\right)^2}{f_i^e}$

where $f_i^o$ denotes the observed frequency of digits $i$, and $f_i^e$ denotes the expected frequency of digits $i$. x is a numeric vector of arbitrary length. Values of x should be continuous, as dictated by theory, but may also be integers. digits should be chosen so that signifd(x,digits) is not influenced by previous rounding.

Value

A list with class "htest" containing the following components:

statistic	the value of the $\chi^2$ test statistic
p.value	the p-value for the test
method	a character string indicating the type of test performed
data.name	a character string giving the name of the data

Author(s)

Dieter William Joenssen [email protected]

References

Benford, F. (1938) The Law of Anomalous Numbers. Proceedings of the American Philosophical Society. 78, 551–572.

Joenssen, D.W. (2013) Two Digit Testing for Benford's Law. Proceedings of the ISI World Statistics Congress, 59th Session in Hong Kong. [available under http://www.statistics.gov.hk/wsc/CPS021-P2-S.pdf]

Pearson, K. (1900) On the Criterion that a Given System of Deviations from the Probable in the Case of a Correlated System of Variables is Such that it can be Reasonably Supposed to have Arisen from Random Sampling. Philosophical Magazine Series 5. 50, 157–175.

See Also

pbenf, simulateH0

Examples

#Set the random seed to an arbitrary number
set.seed(421)
#Create a sample satisfying Benford's law
X<-rbenf(n=20)
#Perform a Chi-squared Test on the sample's 
#first digits using defaults but determine
#the p-value by simulation
chisq.benftest(X,pvalmethod ="simulate")
#p-value = 0.6401

---

Euclidean Distance Test for Benford's Law

Description

edist.benftest takes any numerical vector reduces the sample to the specified number of significant digits and performs a goodness-of-fit test based on the Euclidean distance between the first digits' distribution and Benford's distribution to assert if the data conforms to Benford's law.

Usage

edist.benftest(x = NULL, digits = 1, pvalmethod = "simulate", pvalsims = 10000)

Arguments

x	A numeric vector.
digits	An integer determining the number of first digits to use for testing, i.e. 1 for only the first, 2 for the first two etc.
pvalmethod	Method used for calculating the p-value. Currently only "simulate" is available.
pvalsims	An integer specifying the number of replicates used if pvalmethod = "simulate".

Details

A statistical test is performed utilizing the Euclidean distance between signifd(x,digits) and pbenf(digits). Specifically:

$d = \sqrt{n}\cdot \sqrt{\displaystyle\sum_{i=10^{k-1}}^{10^k-1}\left(f_i^o - f_i^e\right)^2}$

where $f_i^o$ denotes the observed frequency of digits $i$, and $f_i^e$ denotes the expected frequency of digits $i$. x is a numeric vector of arbitrary length. Values of x should be continuous, as dictated by theory, but may also be integers. digits should be chosen so that signifd(x,digits) is not influenced by previous rounding.

Value

A list with class "htest" containing the following components:

statistic	the value of the Euclidean distance test statistic
p.value	the p-value for the test
method	a character string indicating the type of test performed
data.name	a character string giving the name of the data

Author(s)

Dieter William Joenssen [email protected]

References

Benford, F. (1938) The Law of Anomalous Numbers. Proceedings of the American Philosophical Society. 78, 551–572.

Cho, W.K.T. and Gaines, B.J. (2007) Breaking the (Benford) Law: Statistical Fraud Detection in Campaign Finance. The American Statistician. 61, 218–223.

Morrow, J. (2010) Benford's Law, Families of Distributions and a Test Basis. [available under http://www.johnmorrow.info/projects/benford/benfordMain.pdf]

See Also

pbenf, simulateH0

Examples

#Set the random seed to an arbitrary number
set.seed(421)
#Create a sample satisfying Benford's law
X<-rbenf(n=20)
#Perform a Euclidean Distance Test on the
#sample's first digits using defaults
edist.benftest(X,pvalmethod ="simulate")
#p-value = 0.6085

---

A Hotelling T-square Type Test for Benford's Law

Description

jointdigit.benftest takes any numerical vector reduces the sample to the specified number of significant digits and performs a Hotelling T-square type goodness-of-fit test to assert if the data conforms to Benford's law.

Usage

jointdigit.benftest(x = NULL, digits = 1, eigenvalues="all", tol = 1e-15, 
					pvalmethod = "asymptotic", pvalsims = 10000)

Arguments

x	A numeric vector.
digits	An integer determining the number of first digits to use for testing, i.e. 1 for only the first, 2 for the first two etc.
eigenvalues	How are the eigenvalues, which are used in testing, selected.
tol	Tolerance in detecting values that are essentially zero.
pvalmethod	Method used for calculating the p-value. Currently only "asymptotic" is available.
pvalsims	An integer specifying the number of replicates used if pvalmethod = "simulate".

Details

A Hotelling $T^2$ type goodness-of-fit test is performed on signifd(x,digits) versus pbenf(digits). x is a numeric vector of arbitrary length. argument: eigenvalues can be defined as:

• numeric, a vector containing which eigenvalues should be used

• string length = 1, eigenvalue selection scheme:

  • "all", use all non-zero eigenvalues

  • "kaiser", use all eigenvalues larger than the mean of all non-zero eigenvalues

Values of x should be continuous, as dictated by theory, but may also be integers. digits should be chosen so that signifd(x,digits) is not influenced by previous rounding.

Value

A list with class "htest" containing the following components:

statistic	the value of the $T^2$ test statistic
p.value	the p-value for the test
method	a character string indicating the type of test performed
data.name	a character string giving the name of the data
eigenvalues_tested	a vector containing the index numbers of the eigenvalues used in testing.
eigen_val_vect	the eigen values and vectors of the null distribution. computed using eigen.

Author(s)

Dieter William Joenssen [email protected]

References

Benford, F. (1938) The Law of Anomalous Numbers. Proceedings of the American Philosophical Society. 78, 551–572.

Hotelling, H. (1931). The generalization of Student's ratio. Annals of Mathematical Statistics. 2, 360–378.

See Also

pbenf

Examples

#Set the random seed to an arbitrary number
set.seed(421)
#Create a sample satisfying Benford's law
X<-rbenf(n=20)
#Perform  Test
#on the sample's first digits using defaults
jointdigit.benftest(X)
#p-value = 0.648
#Perform  Test
#using only the two largest eigenvalues
jointdigit.benftest(x=X,eigenvalues=1:2)
#p-value = 0.5176
#Perform  Test
#using the kaiser selection criterion
jointdigit.benftest(x=X,eigenvalues="kaiser")
#p-value = 0.682

---

Joenssen's JP-square Test for Benford's Law

Description

jpsq.benftest takes any numerical vector reduces the sample to the specified number of significant digits and performs a goodness-of-fit test based on the correlation between the first digits' distribution and Benford's distribution to assert if the data conforms to Benford's law.

Usage

jpsq.benftest(x = NULL, digits = 1, pvalmethod = "simulate", pvalsims = 10000)

Arguments

x	A numeric vector.
digits	An integer determining the number of first digits to use for testing, i.e. 1 for only the first, 2 for the first two etc.
pvalmethod	Method used for calculating the p-value. Currently only "simulate" is available.
pvalsims	An integer specifying the number of replicates used if pvalmethod = "simulate".

Details

A statistical test is performed utilizing the sign-preserved squared correlation between
signifd(x,digits) and pbenf(digits). Specifically:

$J_P^2=sgn\left(cor\left(f^o, f^e\right)\right)\cdot cor\left(f^o, f^e\right) ^2$

where $f^o$ denotes the observed frequencies and $f^e$ denotes the expected frequency of digits
$10^{k-1},10^{k-1}+1,\ldots,10^k-1$. x is a numeric vector of arbitrary length. Values of x should be continuous, as dictated by theory, but may also be integers. digits should be chosen so that signifd(x,digits) is not influenced by previous rounding.

Value

A list with class "htest" containing the following components:

statistic	the value of the $J_P^2$ test statistic
p.value	the p-value for the test
method	a character string indicating the type of test performed
data.name	a character string giving the name of the data

Author(s)

Dieter William Joenssen [email protected]

References

Benford, F. (1938) The Law of Anomalous Numbers. Proceedings of the American Philosophical Society. 78, 551–572.

Joenssen, D.W. (2013) A New Test for Benford's Distribution. In: Abstract-Proceedings of the 3rd Joint Statistical Meeting DAGStat, March 18-22, 2013; Freiburg, Germany.

Joenssen, D.W. (2013) Two Digit Testing for Benford's Law. Proceedings of the ISI World Statistics Congress, 59th Session in Hong Kong. [available under http://www.statistics.gov.hk/wsc/CPS021-P2-S.pdf]

Shapiro, S.S. and Francia, R.S. (1972) An Approximate Analysis of Variance Test for Normality. Journal of the American Statistical Association. 67, 215–216.

See Also

pbenf, simulateH0

Examples

#Set the random seed to an arbitrary number
set.seed(421)
#Create a sample satisfying Benford's law
X<-rbenf(n=20)
#Perform Joenssen's \emph{JP-square} Test
#on the sample's first digits using defaults
jpsq.benftest(X)
#p-value = 0.3241

---

Kolmogorov-Smirnov Test for Benford's Law

Description

ks.benftest takes any numerical vector reduces the sample to the specified number of significant digits and performs the Kolmogorov-Smirnov goodness-of-fit test to assert if the data conforms to Benford's law.

Usage

ks.benftest(x = NULL, digits = 1, pvalmethod = "simulate", pvalsims = 10000)

Arguments

x	A numeric vector.
digits	An integer determining the number of first digits to use for testing, i.e. 1 for only the first, 2 for the first two etc.
pvalmethod	Method used for calculating the p-value. Currently only "simulate" is available.
pvalsims	An integer specifying the number of replicates used if pvalmethod = "simulate".

Details

A Kolmogorov-Smirnov test is performed between signifd(x,digits) and pbenf(digits). Specifically:

$D = \sup\limits_{i=10^{k-1},\ldots,10^k-1} \left| \displaystyle\sum_{j=1}^{i} ( f_j^o - f_j^e ) \right|\cdot \sqrt{n}$

where $f_i^o$ denotes the observed frequency of digits $i$, and $f_i^e$ denotes the expected frequency of digits $i$. x is a numeric vector of arbitrary length. Values of x should be continuous, as dictated by theory, but may also be integers. digits should be chosen so that signifd(x,digits) is not influenced by previous rounding.

Value

A list with class "htest" containing the following components:

statistic	the value of the Kolmogorov-Smirnov D test statistic
p.value	the p-value for the test
method	a character string indicating the type of test performed
data.name	a character string giving the name of the data

Author(s)

Dieter William Joenssen [email protected]

References

Benford, F. (1938) The Law of Anomalous Numbers. Proceedings of the American Philosophical Society. 78, 551–572.

Joenssen, D.W. (2013) Two Digit Testing for Benford's Law. Proceedings of the ISI World Statistics Congress, 59th Session in Hong Kong. [available under http://www.statistics.gov.hk/wsc/CPS021-P2-S.pdf]

Kolmogorov, A.N. (1933) Sulla determinazione empirica di una legge di distibuzione. Giornale dell'Istituto Italiano degli Attuari. 4, 83–91.

See Also

pbenf, simulateH0

Examples

#Set the random seed to an arbitrary number
set.seed(421)
#Create a sample satisfying Benford's law
X<-rbenf(n=20)
#Perform a Kolmogorov-Smirnov Test on the
#sample's first digits using defaults
ks.benftest(X)
#0.7483

---

Chebyshev Distance Test (maximum norm) for Benford's Law

Description

mdist.benftest takes any numerical vector reduces the sample to the specified number of significant digits and performs a goodness-of-fit test based on the Chebyshev distance between the first digits' distribution and Benford's distribution to assert if the data conforms to Benford's law.

Usage

mdist.benftest(x = NULL, digits = 1, pvalmethod = "simulate", pvalsims = 10000)

Arguments

x	A numeric vector.
digits	An integer determining the number of first digits to use for testing, i.e. 1 for only the first, 2 for the first two etc.
pvalmethod	Method used for calculating the p-value. Currently only "simulate" is available.
pvalsims	An integer specifying the number of replicates used if pvalmethod = "simulate".

Details

A statistical test is performed utilizing the Chebyshev distance between signifd(x,digits) and pbenf(digits). Specifically:

$m = \max\limits_{i=10^{k-1},\ldots,10^k-1}\left|f_i^o - f_i^e\right|\cdot\sqrt{n}$

where $f_i^o$ denotes the observed frequency of digits $i$, and $f_i^e$ denotes the expected frequency of digits $i$. x is a numeric vector of arbitrary length. Values of x should be continuous, as dictated by theory, but may also be integers. digits should be chosen so that signifd(x,digits) is not influenced by previous rounding.

Value

A list with class "htest" containing the following components:

statistic	the value of the Chebyshev distance (maximum norm) test statistic
p.value	the p-value for the test
method	a character string indicating the type of test performed
data.name	a character string giving the name of the data

Author(s)

Dieter William Joenssen [email protected]

References

Benford, F. (1938) The Law of Anomalous Numbers. Proceedings of the American Philosophical Society. 78, 551–572.

Leemis, L.M., Schmeiser, B.W. and Evans, D.L. (2000) Survival Distributions Satisfying Benford's law. The American Statistician. 54, 236–241.

Morrow, J. (2010) Benford's Law, Families of Distributions and a Test Basis. [available under http://www.johnmorrow.info/projects/benford/benfordMain.pdf]

See Also

pbenf, simulateH0

Examples

#Set the random seed to an arbitrary number
set.seed(421)
#Create a sample satisfying Benford's law
X<-rbenf(n=20)
#Perform a Chebyshev Distance Test on the
#sample's first digits using defaults
mdist.benftest(X)
#p-value = 0.6421

---

Judge-Schechter Mean Deviation Test for Benford's Law

Description

meandigit.benftest takes any numerical vector reduces the sample to the specified number of significant digits and performs a goodness-of-fit test based on the deviation in means of the first digits' distribution and Benford's distribution to assert if the data conforms to Benford's law.

Usage

meandigit.benftest(x = NULL, digits = 1, pvalmethod = "asymptotic", pvalsims = 10000)

Arguments

x	A numeric vector.
digits	An integer determining the number of first digits to use for testing, i.e. 1 for only the first, 2 for the first two etc.
pvalmethod	Method used for calculating the p-value. Either "asymptotic" or "simulate".
pvalsims	An integer specifying the number of replicates used if pvalmethod = "simulate".

Details

A statistical test is performed utilizing the deviation between the mean digit of signifd(x,digits) and pbenf(digits). Specifically:

$a^*=\frac{|\mu_k^o-\mu_k^e|}{\left(9\cdot10^{k-1}\right)-\mu_k^e}$

where $\mu_k^o$ is the observed mean of the chosen $k$ number of digits, and $\mu_k^e$ is the expected/true mean value for Benford's predictions. $a^*$ conforms asymptotically to a truncated normal distribution under the null-hypothesis, i.e.,

$a^*\sim truncnorm\left(\mu=0,\sigma=\sigma_B,a=0,b=\infty\right)$

x is a numeric vector of arbitrary length. Values of x should be continuous, as dictated by theory, but may also be integers. digits should be chosen so that signifd(x,digits) is not influenced by previous rounding.

Value

A list with class "htest" containing the following components:

statistic	the value of the $a^*$ test statistic
p.value	the p-value for the test
method	a character string indicating the type of test performed
data.name	a character string giving the name of the data

Author(s)

Dieter William Joenssen [email protected]

References

Benford, F. (1938) The Law of Anomalous Numbers. Proceedings of the American Philosophical Society. 78, 551–572.

Judge, G. and Schechter, L. (2009) Detecting Problems in Survey Data using Benford's Law. Journal of Human Resources. 44, 1–24.

See Also

pbenf, simulateH0

Examples

#Set the random seed to an arbitrary number
set.seed(421)
#Create a sample satisfying Benford's law
X<-rbenf(n=20)
#Perform a Judge-Schechter Mean Deviation Test
#on the sample's first digits using defaults
meandigit.benftest(X)
#p-value = 0.1458

---

Probability Mass Function for Benford's Distribution

Description

Returns the complete probability mass function for Benford's distribution for a given number of first digits.

Usage

pbenf(digits = 1)

Arguments

digits

An integer determining the number of first digits for which the pdf is returned, i.e. 1 for 1:9, 2 for 10:99 etc.

Details

Benford's distribution has the following probability mass function:

$P(d_k)=log_{10}\left(1+ d_k^{-1} \right)$

where $d_k \in \left( 10^{k-1},10^{k-1}+1, \ldots, 10^k-1 \right)$ for any chosen $k$ number of digits.

Value

Returns an object of class "table" containing the expected density of Benford's distribution for the given number of digits.

Author(s)

Dieter William Joenssen [email protected]

References

Benford, F. (1938) The Law of Anomalous Numbers. Proceedings of the American Philosophical Society. 78, 551–572.

Joenssen, D.W. (2013) Two Digit Testing for Benford's Law. Proceedings of the ISI World Statistics Congress, 59th Session in Hong Kong. [available under http://www.statistics.gov.hk/wsc/CPS021-P2-S.pdf]

See Also

qbenf; rbenf

Examples

#show Benford's predictions for the frequencies of the first digit values
pbenf(1)

---

Quantile Function for Benford's Distribution

Description

Returns the complete quantile function for Benford's distribution with a given number of first digits.

Usage

qbenf(digits = 1)

Arguments

digits

An integer determining the number of first digits for which the qdf is returned, i.e. 1 for 1:9, 2 for 10:99 etc.

Value

Returns an object of class "table" containing the expected quantile function of Benford's distribution with a given number of digits.

Author(s)

Dieter William Joenssen [email protected]

References

Benford, F. (1938) The Law of Anomalous Numbers. Proceedings of the American Philosophical Society. 78, 551–572.

See Also

pbenf; rbenf

Examples

qbenf(1)

qbenf(1)==cumsum(pbenf(1))

---

Random Sample Satisfying Benford's Law

Description

Returns a random sample with length n satisfying Benford's law.

Usage

rbenf(n)

Arguments

n	Number of observations.

Details

This distribution has the density:

$f\left(x\right)=\frac{1}{x\cdot ln\left(10\right)} \forall x\in[1,10]$

Value

Returns a random sample with length n satisfying Benford's law.

Author(s)

Dieter William Joenssen [email protected]

References

Benford, F. (1938) The Law of Anomalous Numbers. Proceedings of the American Philosophical Society. 78, 551–572.

See Also

qbenf; pbenf

Examples

#Set the random seed to an arbitrary number
set.seed(421)
#Create a sample satisfying Benford's law
X<-rbenf(n=20)
#Look at sample
X
#should be
# [1] 6.159420 1.396476 5.193371 2.064033 7.001284 5.006184
#7.950332 4.822725 3.386809 1.619609 2.080063 2.242473 1.944697 5.460581
#[15] 6.443031 2.662821 2.079283 3.703353 1.364175 3.354136

---

First Digits Function

Description

Applies the first digits function to each element of a given vector.

Usage

signifd(x = NULL, digits = 1)

Arguments

x	A numeric vector.
digits	An integer determining the number of first digits to use for testing, i.e. 1 for only the first, 2 for the first two etc.

Details

The first digits function can be written as:

$D_k(x) = \lfloor |x| \cdot 10^{\left( -1 \cdot \lfloor log_{10}|x| \rfloor + k -1 \right)}\rfloor$

with $k$ being the number of first digits that should be extracted. x is a numeric vector of arbitrary length. Unlike other solutions, this function will work reliably with all real numbers.

Value

Returns a vector of integers the same length as the input vector x.

Author(s)

Dieter William Joenssen [email protected]

References

Joenssen, D.W. (2013) Two Digit Testing for Benford's Law. Proceedings of the ISI World Statistics Congress, 59th Session in Hong Kong. [available under http://www.statistics.gov.hk/wsc/CPS021-P2-S.pdf]

See Also

chisq.benftest; ks.benftest; usq.benftest; mdist.benftest; edist.benftest; meandigit.benftest; jpsq.benftest

Examples

#Set the random seed to an arbitrary number
set.seed(421)
#Create a sample satisfying Benford's law
X<-rbenf(n=20)
#Look at the first digits of the sample
signifd(X)
#should be:
#[1] 6 1 5 2 7 5 7 4 3 1 2 2 1 5 6 2 2 3 1 3

---

Graphical Analysis of First Significant Digits

Description

signifd.analysis takes any numerical vector reduces the sample to the specified number of significant digits. The (relative) frequencies are then plotted so that a subjective analysis may be performed.

Usage

signifd.analysis(x = NULL, digits = 1, graphical_analysis = TRUE, freq = FALSE, 
alphas = 20, tick_col = "red", ci_col = "darkgreen", ci_lines = c(.05))

Arguments

x	A numeric vector.
digits	An integer determining the number of first digits to use for testing, i.e. 1 for only the first, 2 for the first two etc.
graphical_analysis	Boolean value indicating if results should be plotted.
freq	Boolean value indicating if absolute frequencies should be used.
alphas	Either a vector containing the significance levels([0,1]) that will be shaded, or an integer defining the number of evenly spaced confidence intervals.
tick_col	Color code or name that will be passed to "points" for plotting.
ci_col	Color code or name that will be passed to "polygon" for shading the different confidence intervals. May be more than one color.
ci_lines	Boolean or fractional value(s) indicating significance levels where lines are drawn

Details

Confidence intervals are calculated from the normal distribution with $\mu_i = np_i$ and $\sigma^2 = np_i(1-p_i)$, where i represents the considered digit. Be aware that the normal approximation only holds for "large" n.

Value

A list containing the following components:

summary	the summary printed below the graph, a matrix of digits, their (relative) frequencies and individual p-values
CIs	confidence intervals used for plotting as defined by parameter "ci_lines" or "alphas" if ci_lines==FALSE

Author(s)

Dieter William Joenssen [email protected]

References

Benford, F. (1938) The Law of Anomalous Numbers. Proceedings of the American Philosophical Society. 78, 551–572.

Freedman, L.S. (1981) Watson's Un2 Statistic for a Discrete Distribution. Biometrika. 68, 708–711.

See Also

pbenf

Examples

#Set the random seed to an arbitrary number
set.seed(421)
#Create a sample satisfying Benford's law
X<-rbenf(n=20)
#Analyze the first digits using the the defaults
signifd.analysis(X)
#Turn off plot
signifd.analysis(X,graphical_analysis=FALSE)
#Use absolute frequencies
signifd.analysis(X,graphical_analysis=FALSE,freq=TRUE)
#Use five evenly spaced confidence intervals, no lines
#alphas is used for shadeing
signifd.analysis(X,graphical_analysis=TRUE,alphas=5,freq=TRUE,ci_lines=FALSE)
#Use fifty evenly spaced, gray confidence intervals, blue ticks, and lines at 
#the 1 and 5 percent confidence intervals
signifd.analysis(X,graphical_analysis=TRUE,alphas=50,freq=TRUE,tick_col="blue",
ci_col="gray",ci_lines=c(.01,.05))

---

Sequence of Possible Leading Digits

Description

Returns a vector containing all possible significant digits for a given number of places.

Usage

signifd.seq(digits = 1)

Arguments

digits

An integer determining the number of first digits to be returned, i.e. 1 for 1:9, 2 for 10:99 etc.

Value

Returns an integer vector.

Author(s)

Dieter William Joenssen [email protected]

Examples

signifd.seq(1)
seq(from=1,to=9)==signifd.seq(1)

signifd.seq(2)
seq(from=10,to=99)==signifd.seq(2)

---

Function for Simulating the H0-Distributions needed for BenfordTests

Description

simulateH0 is a wrapper function that calculates the specified test statistic under the null hypothesis a certain number of times.

Usage

simulateH0(teststatistic="chisq", n=10, digits=1, pvalsims=10)

Arguments

teststatistic	Which test statistic should be used: "chisq", "edist", "jpsq", "ks", "mdist", "meandigit", or "usq".
n	Sample size of interest.
digits	An integer determining the number of first digits to use for testing, i.e. 1 for only the first, 2 for the first two etc.
pvalsims	An integer specifying the number of replicates to be used in simulation.

Details

Wrapper function that directly outputs the distributions of the specified test statistic under the null hypothesis.

Value

A vector of length equal to "pvalsims".

Author(s)

Dieter William Joenssen [email protected]

References

Benford, F. (1938) The Law of Anomalous Numbers. Proceedings of the American Philosophical Society. 78, 551–572.

Joenssen, D.W. (2013) Two Digit Testing for Benford's Law. Proceedings of the ISI World Statistics Congress, 59th Session in Hong Kong. [available under http://www.statistics.gov.hk/wsc/CPS021-P2-S.pdf]

See Also

pbenf, chisq.benftest, edist.benftest, jpsq.benftest, ks.benftest, mdist.benftest, \ meandigit.benftest, usq.benftest

Examples

#Set the random seed to an arbitrary number
set.seed(421)

#calculate critical value for chisquare test via simulation
quantile(simulateH0(teststatistic="chisq", n=100,digits=1,pvalsims=100000),probs=.95)

#calculate the "real" critical value
qchisq(.95,df=8)

#alternatively look at critical values for the jpsq statistic
#for different sample sizes (notice the low value for pvalsims)
set.seed(421)
apply(sapply((1:9)*10,FUN=simulateH0,teststatistic="jpsq", digits=1, pvalsims=100),
MARGIN=2,FUN=quantile,probs=.05)

---

Freedman-Watson U-square Test for Benford's Law

Description

usq.benftest takes any numerical vector reduces the sample to the specified number of significant digits and performs the Freedman-Watson test for discreet distributions between the first digits' distribution and Benford's distribution to assert if the data conforms to Benford's law.

Usage

usq.benftest(x = NULL, digits = 1, pvalmethod = "simulate", pvalsims = 10000)

Arguments

x	A numeric vector.
digits	An integer determining the number of first digits to use for testing, i.e. 1 for only the first, 2 for the first two etc.
pvalmethod	Method used for calculating the p-value. Currently only "simulate" is available.
pvalsims	An integer specifying the number of replicates used if pvalmethod = "simulate".

Details

A Freedman-Watson test for discreet distributions is performed between signifd(x,digits) and pbenf(digits). Specifically:

$U^2 = \frac{n}{9\cdot 10^{k-1}}\cdot\left[ \displaystyle\sum_{i={10^{k-1}}}^{10^{k}-2}\left( \displaystyle\sum_{j=1}^{i}(f_j^o - f_j^e) \right)^2 - \frac{1}{9\cdot 10^{k-1}}\cdot\left(\displaystyle\sum_{i={10^{k-1}}}^{10^{k}-2}\displaystyle\sum_{j=1}^{i}(f_i^o - f_i^e)\right)^2\right]$

where $f_i^o$ denotes the observed frequency of digits $i$, and $f_i^e$ denotes the expected frequency of digits $i$. x is a numeric vector of arbitrary length. Values of x should be continuous, as dictated by theory, but may also be integers. digits should be chosen so that signifd(x,digits) is not influenced by previous rounding.

Value

A list with class "htest" containing the following components:

statistic	the value of the $U^2$ test statistic
p.value	the p-value for the test
method	a character string indicating the type of test performed
data.name	a character string giving the name of the data

Author(s)

Dieter William Joenssen [email protected]

References

Benford, F. (1938) The Law of Anomalous Numbers. Proceedings of the American Philosophical Society. 78, 551–572.

Freedman, L.S. (1981) Watson's Un2 Statistic for a Discrete Distribution. Biometrika. 68, 708–711.

Joenssen, D.W. (2013) Two Digit Testing for Benford's Law. Proceedings of the ISI World Statistics Congress, 59th Session in Hong Kong. [available under http://www.statistics.gov.hk/wsc/CPS021-P2-S.pdf]

Watson, G.S. (1961) Goodness-of-Fit Tests on a Circle. Biometrika. 48, 109–114.

See Also

pbenf, simulateH0

Examples

#Set the random seed to an arbitrary number
set.seed(421)
#Create a sample satisfying Benford's law
X<-rbenf(n=20)
#Perform Freedman-Watson U-squared Test on
#the sample's first digits using defaults
usq.benftest(X)
#p-value = 0.4847

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