Package 'jmuOutlier'

Title:	Permutation Tests for Nonparametric Statistics
Description:	Performs a permutation test on the difference between two location parameters, a permutation correlation test, a permutation F-test, the Siegel-Tukey test, a ratio mean deviance test. Also performs some graphing techniques, such as for confidence intervals, vector addition, and Fourier analysis; and includes functions related to the Laplace (double exponential) and triangular distributions. Performs power calculations for the binomial test.
Authors:	Steven T. Garren
Maintainer:	Steven T. Garren <[email protected]>
License:	GPL-3
Version:	2.2
Built:	2024-12-06 06:45:51 UTC
Source:	CRAN

Help Index

Permutation Tests for Nonparametric Statistics
Allows Abbreviations of Character Data
Student's t-Confidence Interval with Finite Population Correction
Coin Toss
Laplace (Double Exponential) Density Function
Triangular Density Function
Determining and Graphing Fourier Approximation
Latin Square
Line Graph Plotting
Permutation Test on Correlation
Permutation Test on the F-statistic
Permutation Test
Laplace (Double Exponential) Cumulative Distribution Function
Confidence Interval Plot
Plotting Two Empirical Cumulative Distribution Functions
Plotting Vector Addition
Power Calculations for Exact Binomial Test
Triangular Cumulative Distribution Function
Laplace (Double Exponential) Quantile Function
Triangular Quantile Function
Confidence Intervals on Quantiles
Reads Tables from Author's Website
Laplace (Double Exponential) Random Generation
Ratio Mean Deviance Test
Triangular Random Generation
Scans Data from Author's Website
Generating van der Waerden and Exponential Scores
Siegel-Tukey Test
Truncated Histograms

Permutation Tests for Nonparametric Statistics

Description

Performs a permutation test on the difference between two location parameters, a permutation correlation test, a permutation F-test, the Siegel-Tukey test, a ratio mean deviance test. Also performs some graphing techniques, such as for confidence intervals, vector addition, and Fourier analysis; and includes functions related to the Laplace (double exponential) and triangular distributions. Performs power calculations for the binomial test.

Details

(I) Permutation tests

perm.cor.test performs a permutation test based on Pearson and Spearman correlations.
perm.f.test performs a permutation F-test and a one-way analysis of variance F-test.
perm.test performs one-sample and two-sample permutation tests on vectors of data.
rmd.test performs a permutation test based on the estimated RMD, the ratio of the mean of the absolute value of the deviances, using two datasets.
siegel.test performs the Siegel-Tukey test using two datasets.

(II) Confidence intervals

CI.t.test produces two-sided confidence intervals on population mean, allowing for a finite population correction.
quantileCI produces exact confidence intervals on quantiles corresponding to the stated probabilities, based on the binomial test.

(III) Graphs

coin.toss illustrates the Law of Large Numbers for proportions.
fourier determines the Fourier approximation for any function on domain $(0, 2\pi)$ and then graphs both the function and the approximation.
lineGraph constructs a line graph on a vector of numerical observations.
plotCI plots multiple confidence intervals on the same graph, and determines the proportion of confidence intervals containing the true population mean.
plotEcdf graphs one or two empirical cumulative distribution functions on the same plot.
plotVector plots one or two 2-dimensional vectors along with their vector sum.
truncHist produces a truncated histogram, which may be useful if data contain some extreme outliers.

(IV) Laplace (double exponential) and symmetric triangular distributions

dlaplace, plaplace, qlaplace, and rlaplace give the density, the distribution function, the quantile function, and random deviates, respectively, of the Laplace distribution.
dtriang, ptriang, qtriang, and rtriang give the density, the distribution function, the quantile function, and random deviates, respectively, of the triangular distribution.

(V) Reading datasets

read.table2 reads table of data from author's website.
scan2 scans data from author's website.

(VI) Additional functions

abbreviation determines if one character variable is an abbreviation among a selection of other character variables.
latin generates a Latin square.
power.binom.test computes the power of the binomial test of a simple null hypothesis about a population median.
score generates van der Waerden scores (i.e., normal quantiles) and exponential (similar to Savage) scores.

Author(s)

Steven T. Garren, James Madison University, Harrisonburg, Virginia, USA

References

Higgins, J. J. (2004) Introduction to Modern Nonparametric Statistics.

Examples

print( x <- rtriang(20,50) ) 

perm.test( x, mu=25, stat=median )

quantileCI( x, c(0.25, 0.5, 0.75)  )

power.binom.test( 20, 0.05, "less", 47, plaplace, 45.2, 3.7 )

fourier (function(x){ (x-pi)^3 }, 4 )
print( x <- rtriang(20,50) ) 

perm.test( x, mu=25, stat=median )

quantileCI( x, c(0.25, 0.5, 0.75)  )

power.binom.test( 20, 0.05, "less", 47, plaplace, 45.2, 3.7 )

fourier (function(x){ (x-pi)^3 }, 4 )

Allows Abbreviations of Character Data

Description

Determines if one character variable is an abbreviation among a section of other character variables.

Usage

abbreviation(x, choices)
abbreviation(x, choices)

Arguments

`x`	A character string, and consists of some or all letters in a value in `choices` or may equal `choices`.
`choices`	A vector of character strings.

Details

The function abbreviation returns a value in choices specified by x, which may be an abbreviation. If no such abbreviation exists, then the original value of x is returned.

Value

The value in choices, which can be abbreviated by x.

Author(s)

Steven T. Garren, James Madison University, Harrisonburg, Virginia, USA

Examples

choices = c("two.sided", "less", "greater")

abbreviation( "two", choices ) 

abbreviation( "l", choices ) 

abbreviation( "gr", choices ) 

abbreviation( "greater", choices ) 

abbreviation( "Not in choices", choices )
choices = c("two.sided", "less", "greater")

abbreviation( "two", choices ) 

abbreviation( "l", choices ) 

abbreviation( "gr", choices ) 

abbreviation( "greater", choices ) 

abbreviation( "Not in choices", choices )

Student's t-Confidence Interval with Finite Population Correction

Description

Performs two-sided confidence interval on population mean, allowing for a finite population correction.

Usage

CI.t.test(x, conf.level = 0.95, fpc = 1)
CI.t.test(x, conf.level = 0.95, fpc = 1)

Arguments

`x`	A nonempty numeric vector of data values.
`conf.level`	Confidence level of the interval, and should be between 0 and 1.
`fpc`	The finite population correction, and should be between 0 and 1.

Details

The fpc is typically defined as $1-n/N$ , where n is the sample size, and N is the population size, for simple random sampling without replacement. When sampling with replacement, set fpc=1 (default).

Value

A confidence interval for the population mean.

Note

The definition of fpc is based on the textbook by Scheaffer, Mendenhall, Ott, Gerow (2012), chapter 4.

Author(s)

Steven T. Garren, James Madison University, Harrisonburg, Virginia, USA

References

Scheaffer, R. L., Mendenhall, W., Ott, R. L., Gerow, K. G. (2012) Elementary Survey Sampling, 7th edition.

Examples

# Sample 43 observations from a population of 200 numbers, and compute the 95% confidence interval.
pop = sqrt(1:200) ; x1 = sample( pop, 43 ) ; print(sort(x1))

CI.t.test( x1, fpc = 1-length(x1)/length(pop) )

# Sample 14 observations from a Normal(mean=50, sd=5) distribution,
#    and compute the 90% confidence interval.
x2 = rnorm( 14, 50, 5 ) ; print(sort(x2)) 

CI.t.test( x2, 0.9 )
# Sample 43 observations from a population of 200 numbers, and compute the 95% confidence interval.
pop = sqrt(1:200) ; x1 = sample( pop, 43 ) ; print(sort(x1))

CI.t.test( x1, fpc = 1-length(x1)/length(pop) )

# Sample 14 observations from a Normal(mean=50, sd=5) distribution,
#    and compute the 90% confidence interval.
x2 = rnorm( 14, 50, 5 ) ; print(sort(x2)) 

CI.t.test( x2, 0.9 )

Coin Toss

Description

Graphs a simulation of the sample proportion of heads.

Usage

 coin.toss(n, p=0.5, burn.in=0, log.scale=FALSE, col=c("black","red"), ...) coin.toss(n, p=0.5, burn.in=0, log.scale=FALSE, col=c("black","red"), ...)

Arguments

`n`	An integer denoting the number of times the coin is tossed.
`p`	The probability of heads, which must be between 0 and 1.
`burn.in`	An integer denoting the number of initial coin tosses which should be omitted from the graph.
`log.scale`	Logical; indicating whether or not the x-axis should have a logarithmic scale.
`col`	A vector of two colors, where the first color is used for the graph of the sample proportions, and the second color is used for the horizontal line occurring at the value `p`.
`...`	Optional arguments to be passed to the `plot` function (see `par`).

Details

This function coin.toss illustrates the Law of Large Numbers for proportions, by simulating cumulative sample proportions. Using nonzero burn.in typically reveals greater precision in the graph as the number of coin tosses increases.

Author(s)

Steven T. Garren, James Madison University, Harrisonburg, Virginia, USA

Examples

par( mfrow=c(2,2) )
coin.toss( 600, 0.5 )
coin.toss( 3e4, 0.4, )
coin.toss( 3e4, 0.7, 1000, col=c("hotpink","turquoise") )
coin.toss( 7e4, 0.3, 1000, TRUE, col=c("purple","green") )
par( mfrow=c(1,1) )
par( mfrow=c(2,2) )
coin.toss( 600, 0.5 )
coin.toss( 3e4, 0.4, )
coin.toss( 3e4, 0.7, 1000, col=c("hotpink","turquoise") )
coin.toss( 7e4, 0.3, 1000, TRUE, col=c("purple","green") )
par( mfrow=c(1,1) )

Laplace (Double Exponential) Density Function

Description

Laplace (double exponential) density with mean equal to mean and standard deviation equal to sd.

Usage

dlaplace(x, mean = 0, sd = 1)

dlaplace(x, mean = 0, sd = 1)

Arguments

`x`	Vector of quantiles.
`mean`	Population mean.
`sd`	Population standard deviation.

Details

The Laplace distribution has density $e^{-|x-\mu| \sqrt{2} / \sigma} / (\sigma \sqrt{2}),$ where $\mu$ is the mean of the distribution and $\sigma$ is the standard deviation.

Value

dlaplace gives the density.

Note

The formulas computed within dlaplace are based on the textbook by Higgins (2004).

Author(s)

Steven T. Garren, James Madison University, Harrisonburg, Virginia, USA

References

Higgins, J. J. (2004) Introduction to Modern Nonparametric Statistics.

Examples

 dlaplace( seq( 20, 80, length.out=11 ), 50, 10 ) 


dlaplace( seq( 20, 80, length.out=11 ), 50, 10 )

Triangular Density Function

Description

Symmetric triangular density with endpoints equal to min and max.

Usage

dtriang(x, min = 0, max = 1)
dtriang(x, min = 0, max = 1)

Arguments

`x`	Vector of quantiles.
`min`	Left endpoint of the triangular distribution.
`max`	Right endpoint of the triangular distribution.

Details

The triangular distribution has density $4 (x-a) / (b-a)^2$ for $a \le x \le \mu$ , and $4 (b-x) / (b-a)^2$ for $\mu < x \le b$ , where $a$ and $b$ are the endpoints, and the mean of the distribution is $\mu = (a+b) / 2$ .

Value

dtriang gives the density.

Author(s)

Steven T. Garren, James Madison University, Harrisonburg, Virginia, USA

Examples

dtriang( seq( 100, 200, length.out=11 ), 100, 200 )dtriang( seq( 100, 200, length.out=11 ), 100, 200 )

Determining and Graphing Fourier Approximation

Description

The Fourier approximation is determined for any function on domain $(0, 2\pi)$ and then graphed.

Usage

fourier(f, order = 3, ...)
fourier(f, order = 3, ...)

Arguments

`f`	The function to be approximated by Fourier analysis.
`order`	Integer; the order of the Fourier transformation.
`...`	Optional arguments to be passed to the `plot` function (see `par`).

Details

The numerical output consists of $a_0/2, a_1, ..., a_n, b_1, ..., b_2.$ The equation is (constant) $+ a_1 cos(x) + ... + a_n cos(n x) + b_1 sin(x) + ... + b_n sin(n x).$

Value

`constant`	The constant term.
`cosine.coefficients`	The coefficients for the cosine terms.
`sine.coefficients`	The coefficients for the sine terms.

Note

The formulas computed within fourier are based on the textbook by Larson (2013).

Author(s)

Steven T. Garren, James Madison University, Harrisonburg, Virginia, USA

References

Larson, R. (2013) Elementary Linear Algebra, 7th edition.

Examples

par( mfrow=c(2,2) )
fourier( function(x){ exp(-x)*(x-pi) }, 4 )
fourier( function(x){ exp(-x) }, 7 )
fourier( function(x){ (x-pi) }, 5 )
fourier( function(x){ (x-pi)^2 } )
par( mfrow=c(1,1) )
par( mfrow=c(2,2) )
fourier( function(x){ exp(-x)*(x-pi) }, 4 )
fourier( function(x){ exp(-x) }, 7 )
fourier( function(x){ (x-pi) }, 5 )
fourier( function(x){ (x-pi)^2 } )
par( mfrow=c(1,1) )

Latin Square

Description

Generates a Latin square, which is either standard or based on randomized rows and columns.

Usage

 latin(n, random = TRUE) latin(n, random = TRUE)

Arguments

`n`	An integer between 2 and 26, inclusively, denoting the number of treatment groups.
`random`	Logical; if `TRUE` (default), a Latin square with randomized rows and columns is produced. If `FALSE`, a standard non-random Latin square is produced.

Details

The Latin square is produced in matrix format with treatments labeled as A, B, C, etc.

Author(s)

Steven T. Garren, James Madison University, Harrisonburg, Virginia, USA

Examples

latin( 5, random=FALSE )
latin( 6 ) # Default is random=TRUE
latin( 5, random=FALSE )
latin( 6 ) # Default is random=TRUE

Line Graph Plotting

Description

Constructs a line graph.

Usage

lineGraph(x, freq = TRUE, prob = NULL, col = "red", ...)
lineGraph(x, freq = TRUE, prob = NULL, col = "red", ...)

Arguments

`x`	Vector of numerical observations to be graphed.
`freq`	Logical; if `freq` is `FALSE` or `prob` sums to 1, then relative frequencies are graphed; otherwise, frequencies are graphed.
`prob`	Vector of the probabilities or weights on `x`, and does not need to sum to one. If `prob` is `NULL`, then all `x` values are equally weighted. Also, see `freq`.
`col`	The color of the plotted lines. Type `colors()` for selections.
`...`	Optional arguments to `plot`.

Author(s)

Steven T. Garren, James Madison University, Harrisonburg, Virginia, USA

Examples

par( mfrow=c(2,2) )
lineGraph( c( rep(6,4), rep(9,7), rep(3,5), 5, 8, 8 ) )
lineGraph( c( rep(6,4), rep(9,7), rep(3,5), 5, 8, 8 ), FALSE, col="purple" )
lineGraph( 11:14, , c( 12, 9, 17, 5 ), col="blue" )
lineGraph( 0:10, FALSE, dbinom(0:10,10,0.4), col="darkgreen", 
   main="Binomial(n=10,p=0.4) probabilities" )
par( mfrow=c(1,1) )par( mfrow=c(2,2) )
lineGraph( c( rep(6,4), rep(9,7), rep(3,5), 5, 8, 8 ) )
lineGraph( c( rep(6,4), rep(9,7), rep(3,5), 5, 8, 8 ), FALSE, col="purple" )
lineGraph( 11:14, , c( 12, 9, 17, 5 ), col="blue" )
lineGraph( 0:10, FALSE, dbinom(0:10,10,0.4), col="darkgreen", 
   main="Binomial(n=10,p=0.4) probabilities" )
par( mfrow=c(1,1) )

Permutation Test on Correlation

Description

A permutation test is performed based on Pearson and Spearman correlations.

Usage

perm.cor.test(x, y = NULL, alternative = c("two.sided", "less", "greater"), 
              method = c("pearson", "spearman"), num.sim = 20000)
perm.cor.test(x, y = NULL, alternative = c("two.sided", "less", "greater"), 
              method = c("pearson", "spearman"), num.sim = 20000)

Arguments

`x`	Numeric vector of design variable if `y` is not `NULL`, or N by 2 data frame or matrix of design and response variables if `y` is `NULL`.
`y`	Numeric vector of response variable, and should be `NULL` if `x` is an N by 2 data frame or matrix.
`alternative`	A character string specifying the alternative hypothesis, and must be one of `"two.sided"` (default), `"greater"` or `"less"`. Only the initial letter needs to be specified.
`method`	A character string specifying the type of correlation, and must be one of `"pearson"` (default) or `"spearman"`. Only the initial letter needs to be specified.
`num.sim`	The number of simulations generated.

Details

The p-value is estimated by randomly generating the permutations, and is hence not exact. The larger the value of num.sim the more precise the estimate of the p-value, but also the greater the computing time. Thus, the p-value is not based on asymptotic approximation. The output states more details about the permutation test, such as the values of method and num.sim.

Value

`alternative`	Same as the input.
`p.value`	The p-value of the permutation test.

Note

The formulas computed within perm.cor.test are based on the textbook by Higgins (2004).

Author(s)

Steven T. Garren, James Madison University, Harrisonburg, Virginia, USA

References

Higgins, J. J. (2004) Introduction to Modern Nonparametric Statistics.

Examples

perm.cor.test( c( 4, 6, 8, 11 ), c( 19, 44, 15, 13 ), "less", "pearson" ) 
perm.cor.test( c( 4, 6, 8, 11 ), c( 19, 44, 15, 13 ), "less", "spearman" )
perm.cor.test( c( 4, 6, 8, 11 ), c( 19, 44, 15, 13 ), "less", "pearson" ) 
perm.cor.test( c( 4, 6, 8, 11 ), c( 19, 44, 15, 13 ), "less", "spearman" )

Permutation Test on the F-statistic

Description

A permutation F-test is performed, and a one-way analysis of variance F-test is performed.

Usage

perm.f.test(response, treatment = NULL, num.sim = 20000)
perm.f.test(response, treatment = NULL, num.sim = 20000)

Arguments

`response`	Numeric vector of responses if treatment is not `NULL`. If `treatment` is `NULL`, then `response` must be an N by 2 data frame or matrix, such that the first column represents response and the second column represents treatment.
`treatment`	Vector of treatments, which need not be numerical. If `response` is an N by 2 data frame or matrix, then `treatment` should be set to `NULL`.
`num.sim`	The number of simulations performed. If `num.sim` is smaller than one, then the permutation test is not performed.

Details

The one-way analysis of variance F-test is performed, regardless of the value of num.sim. The permutation F-test is performed whenever num.sim is at least 1. The p-value of the permutation F-test is estimated by randomly generating the permutations, and is hence not exact. The larger the value of num.sim the more precise the estimate of the p-value of the permutation F-test, but also the greater the computing time. Thus, the p-value of the permutation F-test is not based on asymptotic approximation.

Value

The output consists of results from calling aov and from the permutation F-test.

Note

The formulas computed within perm.f.test are based on the textbook by Higgins (2004).

Author(s)

Steven T. Garren, James Madison University, Harrisonburg, Virginia, USA

References

Higgins, J. J. (2004) Introduction to Modern Nonparametric Statistics.

Examples

perm.f.test( c( 14,6,5,2,54,7,9,15,11,13,12 ), rep( c("I","II","III"), c(4,4,3) ) )
perm.f.test( c( 14,6,5,2,54,7,9,15,11,13,12 ), rep( c("I","II","III"), c(4,4,3) ) )

Permutation Test

Description

Performs one-sample and two-sample permutation tests on vectors of data.

Usage

perm.test(x, y = NULL, alternative = c("two.sided", "less", "greater"), mu = 0, 
      paired = FALSE, all.perms = TRUE, num.sim = 20000, plot = FALSE, stat = mean, ...)
perm.test(x, y = NULL, alternative = c("two.sided", "less", "greater"), mu = 0, 
      paired = FALSE, all.perms = TRUE, num.sim = 20000, plot = FALSE, stat = mean, ...)

Arguments

`x`	A (non-empty) numeric vector of data values.
`y`	An optional numeric vector data values.
`alternative`	A character string specifying the alternative hypothesis, and must be one of `"two.sided"` (default), `"greater"` or `"less"`. Only the initial letter needs to be specified.
`mu`	A number indicating the null value of the location parameter (or the difference in location parameters if performing a two-sample test).
`paired`	Logical, indicating whether or not a two-sample test should be paired, and is ignored for a one-sample test.
`all.perms`	Logical. The exact p-value is attempted when `all.perms` (i.e., all permutations) is `TRUE` (default), and is simulated when `all.perms` is `FALSE` or when computing an exact p-value requires more than `num.sim` calculations.
`num.sim`	The upper limit on the number of permutations generated.
`plot`	Logical. If `TRUE`, then plot the histogram of the permutation distribution; otherwise, list the p-value.
`stat`	Function, naming the test statistic, such as `mean` and `median`.
`...`	Optional arguments to `stat`; and is the second argument to `stat` when unspecified. For example, if `stat` equals `mean`, then the second argument `trim` denotes the fraction (0 to 0.5) of observations to be trimmed from each end of `x` and `y` before the mean is computed.

Details

A paired test using data x and nonNULL y is equivalent to a one-sample test using data x-y. The output states more details about the permutation test, such as one-sample or two-sample, and whether or not the p.value calculated was based on all permutations.

Value

`alternative`	Same as the input.
`mu`	Same as the input.
`p.value`	The p-value of the permutation test.

Note

The formulas computed within perm.test are based on the textbook by Higgins (2004).

Author(s)

Steven T. Garren, James Madison University, Harrisonburg, Virginia, USA

References

Higgins, J. J. (2004) Introduction to Modern Nonparametric Statistics.

Examples

# One-sample test

print( x <- rnorm(10,0.5) ) 

perm.test( x, stat=median )

# Two-sample unpaired test

print( y <- rnorm(13,1) )

perm.test( x, y )
# One-sample test

print( x <- rnorm(10,0.5) ) 

perm.test( x, stat=median )

# Two-sample unpaired test

print( y <- rnorm(13,1) )

perm.test( x, y )

Laplace (Double Exponential) Cumulative Distribution Function

Description

Laplace (double exponential) cumulative distribution function with mean equal to mean and standard deviation equal to sd.

Usage

plaplace(q, mean = 0, sd = 1, lower.tail = TRUE)
plaplace(q, mean = 0, sd = 1, lower.tail = TRUE)

Arguments

`q`	Vector of quantiles.
`mean`	Population mean.
`sd`	Population standard deviation.
`lower.tail`	Logical; if `TRUE` (default), probabilities are $P[X \le x]$ ; otherwise, $P[X > x]$ .

Details

The Laplace distribution has density $e^{-|x-\mu| \sqrt{2} / \sigma} / (\sigma \sqrt{2}),$ where $\mu$ is the mean of the distribution and $\sigma$ is the standard deviation.

Value

plaplace gives the distribution function.

Note

The formulas computed within plaplace are based on the textbook by Higgins (2004).

Author(s)

Steven T. Garren, James Madison University, Harrisonburg, Virginia, USA

References

Higgins, J. J. (2004) Introduction to Modern Nonparametric Statistics.

Examples

plaplace( seq( 20, 80, length.out=11 ), 50, 10 )

plaplace( seq( 20, 80, length.out=11 ), 50, 10, FALSE )
plaplace( seq( 20, 80, length.out=11 ), 50, 10 )

plaplace( seq( 20, 80, length.out=11 ), 50, 10, FALSE )

Confidence Interval Plot

Description

Plots multiple confidence intervals on the same graph, and determines the proportion of confidence intervals containing the true population mean.

Usage

plotCI(CI, mu = NULL, plot.midpoints = TRUE, 
       col = c("black", "red", "darkgreen", "purple"))
plotCI(CI, mu = NULL, plot.midpoints = TRUE, 
       col = c("black", "red", "darkgreen", "purple"))

Arguments

`CI`	N by 2 matrix or 2 by N matrix consisting of N two-sided confidence intervals.
`mu`	Numeric; the population mean, and is `NULL` if unknown.
`plot.midpoints`	Logical; plots the midpoints of the confidence intervals if `TRUE` (default); otherwise, does not plot the midpoints.
`col`	A vector of size four, specifying the colors of the line representing population mean, confidence intervals not containing the population mean, confidence intervals containing the population mean, and the sample means, respectively.

Details

The title of the graph states the proportion of the confidence intervals containing the true population mean, when the population mean is not NULL.

Author(s)

Steven T. Garren, James Madison University, Harrisonburg, Virginia, USA

Examples

# Plot fifty 90% confidence intervals, each based on 13 observations from a 
#    Normal( mean=70, sd=10 ) distribution.

plotCI( replicate( 50, t.test( rnorm( 13, 70, 10 ), conf.level=0.9 )$conf.int ), 70 )
# Plot fifty 90% confidence intervals, each based on 13 observations from a 
#    Normal( mean=70, sd=10 ) distribution.

plotCI( replicate( 50, t.test( rnorm( 13, 70, 10 ), conf.level=0.9 )$conf.int ), 70 )

Plotting Two Empirical Cumulative Distribution Functions

Description

Graphs one or two empirical cumulative distribution functions on the same plot.

Usage

plotEcdf(x, y = NULL, col = c("black", "red"))
plotEcdf(x, y = NULL, col = c("black", "red"))

Arguments

`x`	Vector of numerical observations whose empirical cdf is to be graphed.
`y`	Optional vector of observations whose empirical cdf is to be graphed.
`col`	Scalar or vector of length two, specifying the colors of the two empirical distribution functions. The two colors correspond to `x` and `y`, respectively, and preferably should differ. Type `colors()` for selections.

Author(s)

Steven T. Garren, James Madison University, Harrisonburg, Virginia, USA

Examples

par( mfrow=c(2,2) )
plotEcdf( c(2,4,9,6), c(1,7,11,3,8) )
plotEcdf( c(2,4,9,6), c(1,7,11,3), col=c("navyblue", "orange") ) 
plotEcdf( c(11,5,3), c(3,7,9), col=c("tomato","darkgreen") ) 
plotEcdf( c(15,19,11,4,6), col="purple" ) 
par( mfrow=c(1,1) )
par( mfrow=c(2,2) )
plotEcdf( c(2,4,9,6), c(1,7,11,3,8) )
plotEcdf( c(2,4,9,6), c(1,7,11,3), col=c("navyblue", "orange") ) 
plotEcdf( c(11,5,3), c(3,7,9), col=c("tomato","darkgreen") ) 
plotEcdf( c(15,19,11,4,6), col="purple" ) 
par( mfrow=c(1,1) )

Plotting Vector Addition

Description

Plots one or two 2-dimensional vectors along with their vector sum.

Usage

plotVector(x1, y1, x2 = NULL, y2 = NULL, add.vectors = TRUE, 
           col = c("black", "red", "darkgreen", "purple"), lwd = 8, font = 2, 
           font.lab = 2, las = 1, cex.lab = 1.3, cex.axis = 2, usr = NULL, ...)
plotVector(x1, y1, x2 = NULL, y2 = NULL, add.vectors = TRUE, 
           col = c("black", "red", "darkgreen", "purple"), lwd = 8, font = 2, 
           font.lab = 2, las = 1, cex.lab = 1.3, cex.axis = 2, usr = NULL, ...)

Arguments

`x1`	Value on the x-axis of the first vector.
`y1`	Value on the y-axis of the first vector.
`x2`	Value on the x-axis of the second vector.
`y2`	Value on the y-axis of the second vector.
`add.vectors`	Logical; if `TRUE` (default), display the vector sum.
`col`	A vector of size four, specifying the colors of the first vector, the second vector, the vector sum, and parallel lines, respectively. Type `colors()` for selections.
`lwd`	The line width of the vectors.
`font`	An integer specifying which font to use for text.
`font.lab`	The font to be used for `x` and `y` labels.
`las`	Numeric in (0,1,2,3); the style of axis labels.
`cex.lab`	The magnification to be used for `x` and `y` labels.
`cex.axis`	The magnification to be used for axis annotation.
`usr`	A vector of the form `c(x1, x2, y1, y2)` giving the extremes of the user coordinates of the plotting region.
`...`	Optional arguments to be passed to the `plot` function (see `par`).

Author(s)

Steven T. Garren, James Madison University, Harrisonburg, Virginia, USA

Examples

par( mfrow=c(2,2) )

# Vectors (2,8) and (4,-3) and their vector sum.
     plotVector( 2, 8, 4, -3 ) 
     
# Colinear vectors (-3,6) and (-1,2).
     plotVector( -3, 6, -1, 2, add=FALSE, col=c("red","black") )

# Colinear vectors (-1,2) and (3,-6).
     plotVector( -1, 2, 3, -6, add=FALSE )

# Vectors (2,3) and (5,-4)
     plotVector( 2, 3, 5, -4, add=FALSE, usr=c( -5, 5, -4, 7) )

par( mfrow=c(1,1) )
par( mfrow=c(2,2) )

# Vectors (2,8) and (4,-3) and their vector sum.
     plotVector( 2, 8, 4, -3 ) 
     
# Colinear vectors (-3,6) and (-1,2).
     plotVector( -3, 6, -1, 2, add=FALSE, col=c("red","black") )

# Colinear vectors (-1,2) and (3,-6).
     plotVector( -1, 2, 3, -6, add=FALSE )

# Vectors (2,3) and (5,-4)
     plotVector( 2, 3, 5, -4, add=FALSE, usr=c( -5, 5, -4, 7) )

par( mfrow=c(1,1) )

Power Calculations for Exact Binomial Test

Description

Compute the power of the binomial test of a simple null hypothesis about a population median.

Usage

power.binom.test(n, alpha = 0.05, alternative = c("two.sided", "less", "greater"), 
                 null.median, alt.pdist, ...)
power.binom.test(n, alpha = 0.05, alternative = c("two.sided", "less", "greater"), 
                 null.median, alt.pdist, ...)

Arguments

`n`	The sample size.
`alpha`	Probability of Type I error.
`alternative`	A character string specifying the alternative hypothesis, and must be one of `"two.sided"` (default), `"greater"` or `"less"`. Only the initial letter needs to be specified.
`null.median`	The population median under the null hypothesis.
`alt.pdist`	Name of the cumulative distribution function under the alternative distribution. Some options include `pnorm, pexp, pcauchy, plaplace, pt, pchisq, pf, ptriang, punif, pbinom, pgeom, ppois.`
`...`	Optional arguments to `alt.pdist`, excluding the first argument of `alt.pdist`. See the examples below.

Value

Power of the test.

Author(s)

Steven T. Garren, James Madison University, Harrisonburg, Virginia, USA

References

Higgins, J. J. (2004) Introduction to Modern Nonparametric Statistics.

Examples

# Alternative distribution is Normal( mean=55.7, sd=2.5 ).
power.binom.test( 30, 0.05, "greater", 55, pnorm, 55.7, 2.5 )

# Alternative distribution is Laplace( mean=55.7, sd=2.5 ).
power.binom.test( 30, 0.05, "greater", 55, plaplace, 55.7, 2.5 )
# Alternative distribution is Normal( mean=55.7, sd=2.5 ).
power.binom.test( 30, 0.05, "greater", 55, pnorm, 55.7, 2.5 )

# Alternative distribution is Laplace( mean=55.7, sd=2.5 ).
power.binom.test( 30, 0.05, "greater", 55, plaplace, 55.7, 2.5 )

Triangular Cumulative Distribution Function

Description

Triangular cumulative distribution function with endpoints equal to min and max.

Usage

ptriang(q, min = 0, max = 1, lower.tail = TRUE)
ptriang(q, min = 0, max = 1, lower.tail = TRUE)

Arguments

`q`	Vector of quantiles.
`min`	Left endpoint of the triangular distribution.
`max`	Right endpoint of the triangular distribution.
`lower.tail`	Logical; if `TRUE` (default), probabilities are $P[X \le x]$ ; otherwise, $P[X > x]$ .

Details

Value

ptriang gives the distribution function.

Author(s)

Steven T. Garren, James Madison University, Harrisonburg, Virginia, USA

Examples

ptriang( seq( 100, 200, length.out=11 ), 100, 200 )

ptriang( seq( 100, 200, length.out=11 ), 100, 200, FALSE ) ptriang( seq( 100, 200, length.out=11 ), 100, 200 )

ptriang( seq( 100, 200, length.out=11 ), 100, 200, FALSE )

Laplace (Double Exponential) Quantile Function

Description

Laplace (double exponential) quantile function with mean equal to mean and standard deviation equal to sd.

Usage

qlaplace(p, mean = 0, sd = 1, lower.tail = TRUE)
qlaplace(p, mean = 0, sd = 1, lower.tail = TRUE)

Arguments

`p`	Vector of probabilities.
`mean`	Population mean.
`sd`	Population standard deviation.
`lower.tail`	Logical; if `TRUE` (default), probabilities are $P[X \le x]$ ; otherwise, $P[X > x]$ .

Details

The Laplace distribution has density $e^{-|x-\mu| \sqrt{2} / \sigma} / (\sigma \sqrt{2}),$ where $\mu$ is the mean of the distribution and $\sigma$ is the standard deviation.

Value

qlaplace gives the quantile function.

Note

The formulas computed within qlaplace are based on the textbook by Higgins (2004).

Author(s)

Steven T. Garren, James Madison University, Harrisonburg, Virginia, USA

References

Higgins, J. J. (2004) Introduction to Modern Nonparametric Statistics.

Examples

# 5th, 15th, 25th, ..., 95th percentiles from a Laplace( 50, 10 ) distribution.

qlaplace( seq( 0.05, 0.95, length.out=11 ), 50, 10 )
# 5th, 15th, 25th, ..., 95th percentiles from a Laplace( 50, 10 ) distribution.

qlaplace( seq( 0.05, 0.95, length.out=11 ), 50, 10 )

Triangular Quantile Function

Description

Symmetric triangular density with endpoints equal to min and max.

Usage

 qtriang(p, min = 0, max = 1) qtriang(p, min = 0, max = 1)

Arguments

`p`	Vector of probabilities.
`min`	Left endpoint of the triangular distribution.
`max`	Right endpoint of the triangular distribution.

Details

Value

qtriang gives the quantile function.

Author(s)

Steven T. Garren, James Madison University, Harrisonburg, Virginia, USA

Examples

# 5th, 15th, 25th, ..., 95th percentiles from a Triangular( 100, 200 ) distribution.

qtriang( seq( 0.05, 0.95, length.out=11 ), 100, 200 )
# 5th, 15th, 25th, ..., 95th percentiles from a Triangular( 100, 200 ) distribution.

qtriang( seq( 0.05, 0.95, length.out=11 ), 100, 200 )

Confidence Intervals on Quantiles

Description

Produces exact confidence intervals on quantiles corresponding to the stated probabilities, based on the binomial test.

Usage

quantileCI(x, probs = 0.5, conf.level = 0.95)quantileCI(x, probs = 0.5, conf.level = 0.95)

Arguments

`x`	Numeric vector of observations.
`probs`	Numeric vector of cumulative probabilities between 0 and 1.
`conf.level`	Confidence level of the interval.

Details

If probs=0.5 (default), then a confidence interval on the population median is produced.

Value

Confidence interval for each quantile based on probs.

Author(s)

Steven T. Garren, James Madison University, Harrisonburg, Virginia, USA

References

Higgins, J. J. (2004) Introduction to Modern Nonparametric Statistics.

Examples

# Sample 20 observations from an Exponential distribution with mean=10.

print( sort( x <- rexp( 20, 0.1 ) ) ) 

# Construct 90% confidence intervals on the 25th, 50th, and 75th percentiles.  

quantileCI( x, c( 0.25, 0.5, 0.75 ), 0.9 ) # Sample 20 observations from an Exponential distribution with mean=10.

print( sort( x <- rexp( 20, 0.1 ) ) ) 

# Construct 90% confidence intervals on the 25th, 50th, and 75th percentiles.  

quantileCI( x, c( 0.25, 0.5, 0.75 ), 0.9 )

Reads Tables from Author's Website

Description

Performs read.table of dataset without typing the URL.

Usage

 read.table2(file.name, course.num=course.number, na.strings=".", ...) read.table2(file.name, course.num=course.number, na.strings=".", ...)

Arguments

`file.name`	The file name in character format without the URL.
`course.num`	The course number in character or numeric format, where `course.number` is a global variable.
`na.strings`	Character vector. Elements of this vector are to be interpreted as missing `NA` values.
`...`	Optional arguments to be passed to the `read.table` function.

Details

The datasets are available on the author's website, http://educ.jmu.edu/~garrenst. The global variable course.number may be entered as the value of the second argument, course.num, in function read.table2.

Author(s)

Steven T. Garren, James Madison University, Harrisonburg, Virginia, USA

Examples

# The following two commands, when uncommented, are equivalent.

# read.table2( "ex6.1.txt", 321, header=TRUE )

# read.table( "http://educ.jmu.edu/~garrenst/math321.dir/datasets/ex6.1.txt", header=TRUE )# The following two commands, when uncommented, are equivalent.

# read.table2( "ex6.1.txt", 321, header=TRUE )

# read.table( "http://educ.jmu.edu/~garrenst/math321.dir/datasets/ex6.1.txt", header=TRUE )

Laplace (Double Exponential) Random Generation

Description

Laplace (double exponential) random generation with mean equal to mean and standard deviation equal to sd.

Usage

rlaplace(n, mean = 0, sd = 1)
rlaplace(n, mean = 0, sd = 1)

Arguments

`n`	Number of observations. If `length(n)>1`, the length is taken to be the number required.
`mean`	Population mean.
`sd`	Population standard deviation.

Details

The Laplace distribution has density $e^{-|x-\mu| \sqrt{2} / \sigma} / (\sigma \sqrt{2}),$ where $\mu$ is the mean of the distribution and $\sigma$ is the standard deviation.

Value

rlaplace generates random deviates.

Note

The formulas computed within rlaplace are based on the textbook by Higgins (2004).

Author(s)

Steven T. Garren, James Madison University, Harrisonburg, Virginia, USA

References

Higgins, J. J. (2004) Introduction to Modern Nonparametric Statistics.

Examples

# 20 random variates from a Laplace( 50, 10 ) distribution.

rlaplace( 20, 50, 10 )
# 20 random variates from a Laplace( 50, 10 ) distribution.

rlaplace( 20, 50, 10 )

Ratio Mean Deviance Test

Description

A permutation test is performed based on the estimated RMD, the ratio of the mean of the absolute value of the deviances, for data x and y.

Usage

rmd.test(x, y, alternative = c("two.sided", "less", "greater"), all.perms = TRUE, 
         num.sim = 20000)
rmd.test(x, y, alternative = c("two.sided", "less", "greater"), all.perms = TRUE, 
         num.sim = 20000)

Arguments

`x`	Numeric vector of data values.
`y`	Numeric vector of data values.
`alternative`	A character string specifying the alternative hypothesis, and must be one of `"two.sided"` (default), `"greater"` or `"less"`. Only the initial letter needs to be specified.
`all.perms`	Logical. The exact p-value is attempted when `all.perms` (i.e., all permutations) is `TRUE` (default), and is simulated when `all.perms` is `FALSE` or when computing an exact p-value requires more than `num.sim` calculations.
`num.sim`	The upper limit on the number of permutations generated.

Value

`alternative`	Same as the input.
`rmd.hat`	The value of the RMD test statistic.
`p.value`	The p-value of the test.

Note

The formulas computed within rmd.test are based on the textbook by Higgins (2004).

Author(s)

Steven T. Garren, James Madison University, Harrisonburg, Virginia, USA

References

Higgins, J. J. (2004) Introduction to Modern Nonparametric Statistics.

Examples

rmd.test( c(13, 34, 2, 19, 49, 63), c(17, 29, 22) )
rmd.test( c(13, 34, 2, 19, 49, 63), c(17, 29, 22), "greater" )
rmd.test( c(13, 34, 2, 19, 49, 63), c(17, 29, 22) )
rmd.test( c(13, 34, 2, 19, 49, 63), c(17, 29, 22), "greater" )

Triangular Random Generation

Description

Symmetric triangular random generation with endpoints equal to min and max.

Usage

rtriang(n, min = 0, max = 1)
rtriang(n, min = 0, max = 1)

Arguments

`n`	Number of observations. If `length(n)>1`, the length is taken to be the number required.
`min`	Left endpoint of the triangular distribution.
`max`	Right endpoint of the triangular distribution.

Details

Value

rtriang generates random deviates.

Author(s)

Steven T. Garren, James Madison University, Harrisonburg, Virginia, USA

Examples

# 20 random variates from a Triangular( 100, 200 ) distribution.

rtriang( 20, 100, 200 )  # 20 random variates from a Triangular( 100, 200 ) distribution.

rtriang( 20, 100, 200 )

Scans Data from Author's Website

Description

Performs scan of dataset without typing the URL.

Usage

 scan2(file.name, course.num=course.number, na.strings=".", comment.char="#", ...) scan2(file.name, course.num=course.number, na.strings=".", comment.char="#", ...)

Arguments

`file.name`	The file name in character format without the URL.
`course.num`	The course number in character or numeric format, where `course.number` is a global variable.
`na.strings`	Character vector. Elements of this vector are to be interpreted as missing `NA` values.
`comment.char`	Single character or empty string, denoting beginning of comment. Use "" to turn off the interpretation of comments altogether.
`...`	Optional arguments to be passed to the `scan` function.

Details

Author(s)

Steven T. Garren, James Madison University, Harrisonburg, Virginia, USA

Examples

# The following two commands, when uncommented, are equivalent.

# scan2( "exercise2.7.txt", 324 )

# scan( "http://educ.jmu.edu/~garrenst/math324.dir/datasets/exercise2.7.txt", comment.char="#" )# The following two commands, when uncommented, are equivalent.

# scan2( "exercise2.7.txt", 324 )

# scan( "http://educ.jmu.edu/~garrenst/math324.dir/datasets/exercise2.7.txt", comment.char="#" )

Generating van der Waerden and Exponential Scores

Description

Generates van der Waerden scores (i.e., normal quantiles) and exponential (similar to Savage) scores, for combined data x and y.

Usage

 score(x, y = NULL, expon = FALSE) score(x, y = NULL, expon = FALSE)

Arguments

`x`	A positive integer equal to the number of desired scores when `y` is `NULL`, or `x` is a vector of observations.
`y`	An optional vector of observations, typically used with two-sample tests.
`expon`	Logical; if `FALSE` (default), van der Waerden scores are computed, even for ties. If `TRUE`, Exponential scores are computed, and interpolation is used for ties.

Details

The scored values for x are the output, when y is NULL.

Value

`x`	Scored values for `x`, when `y` is not `NULL`.
`y`	Scored values for `y`, when `y` is not `NULL`.

Note

The formulas computed within score are based on the textbook by Higgins (2004).

Author(s)

Steven T. Garren, James Madison University, Harrisonburg, Virginia, USA

References

Higgins, J. J. (2004) Introduction to Modern Nonparametric Statistics.

Examples

score( 10 )

score( 15, expon=TRUE )

score( c(4,7,6,22,13), c(15,16,7) )  # Two samples, including a tie. 
score( 10 )

score( 15, expon=TRUE )

score( c(4,7,6,22,13), c(15,16,7) )  # Two samples, including a tie.

Siegel-Tukey Test

Description

Performs the Siegel-Tukey test on data x and y, where ties are handled by averaging ranks, not by asymptotic approximations.

Usage

siegel.test(x, y, alternative = c("two.sided", "less", "greater"), reverse = FALSE, 
            all.perms = TRUE, num.sim = 20000)
siegel.test(x, y, alternative = c("two.sided", "less", "greater"), reverse = FALSE, 
            all.perms = TRUE, num.sim = 20000)

Arguments

`x`	Numeric vector of data values.
`y`	Numeric vector of data values.
`alternative`	A character string specifying the alternative hypothesis, and must be one of `"two.sided"` (default), `"greater"` or `"less"`. Only the initial letter needs to be specified.
`reverse`	Logical; If `FALSE` (default), then assign rank 1 to the smallest observation. If `TRUE`, then assign rank 1 to the largest observation.
`all.perms`	Logical. The exact p-value is attempted when `all.perms` (i.e., all permutations) is `TRUE` (default), and is simulated when `all.perms` is `FALSE` or when computing an exact p-value requires more than `num.sim` calculations.
`num.sim`	The upper limit on the number of permutations generated.

Details

Since the logical value of reverse may affect the p-value, yet neither logical value of reverse is preferred over the other, one should consider using ansari.test instead.

Value

`alternative`	Same as the input.
`rank.x`	The Siegel-Tukey ranks of the data `x`.
`rank.y`	The Siegel-Tukey ranks of the data `y`.
`p.value`	The p-value of the test.

Note

The formulas computed within siegel.test are based on the textbook by Higgins (2004).

Author(s)

Steven T. Garren, James Madison University, Harrisonburg, Virginia, USA

References

Higgins, J. J. (2004) Introduction to Modern Nonparametric Statistics.

Examples

# The same data are used in the following two commands.

siegel.test( c(13, 34, 2, 19, 49, 63), c(17, 29, 22) )
siegel.test( c(13, 34, 2, 19, 49, 63), c(17, 29, 22), reverse=TRUE )
# The same data are used in the following two commands.

siegel.test( c(13, 34, 2, 19, 49, 63), c(17, 29, 22) )
siegel.test( c(13, 34, 2, 19, 49, 63), c(17, 29, 22), reverse=TRUE )

Truncated Histograms

Description

Produces a truncated histogram.

Usage

truncHist(x, xmin = NULL, xmax = NULL, trim = 0.025, main = NULL, xlab = "x", ...)
truncHist(x, xmin = NULL, xmax = NULL, trim = 0.025, main = NULL, xlab = "x", ...)

Arguments

`x`	Vector of numerical observations.
`xmin`	Minimum numerical value to be shown in graph.
`xmax`	Maximum numerical value to be shown in graph.
`trim`	The fraction (0 to 0.5) of observations to be trimmed from each end of `x` before the histogram is constructed.
`main`	An overall title for the histogram.
`xlab`	A title for the x-axis.
`...`	Optional arguments to `hist`.

Details

truncHist may be useful if data contain some extreme outliers.

Author(s)

Steven T. Garren, James Madison University, Harrisonburg, Virginia, USA

Examples

x1 = sort(rnorm(1000)) ;   c( head(x1), tail(x1)) 

x2 = sort(rnorm(1000)) ;   c( head(x2), tail(x2)) 

y1 = sort(rcauchy(1000)) ; c( head(y1), tail(y1)) 

y2 = sort(rcauchy(1000)) ; c( head(y2), tail(y2)) 

par( mfrow=c(2,2) )
truncHist(x1, main="Normal data; first simulation",  xlab="x1")
truncHist(x2, main="Normal data; second simulation", xlab="x2")
truncHist(y1, main="Cauchy data; first simulation",  xlab="y1")
truncHist(y2, main="Cauchy data; second simulation", xlab="y2")
par( mfrow=c(1,1) )
x1 = sort(rnorm(1000)) ;   c( head(x1), tail(x1)) 

x2 = sort(rnorm(1000)) ;   c( head(x2), tail(x2)) 

y1 = sort(rcauchy(1000)) ; c( head(y1), tail(y1)) 

y2 = sort(rcauchy(1000)) ; c( head(y2), tail(y2)) 

par( mfrow=c(2,2) )
truncHist(x1, main="Normal data; first simulation",  xlab="x1")
truncHist(x2, main="Normal data; second simulation", xlab="x2")
truncHist(y1, main="Cauchy data; first simulation",  xlab="y1")
truncHist(y2, main="Cauchy data; second simulation", xlab="y2")
par( mfrow=c(1,1) )

Package 'jmuOutlier'

Help Index

Permutation Tests for Nonparametric Statistics

Description

Details

Author(s)

References

See Also

Examples

Allows Abbreviations of Character Data

Description

Usage

Arguments

Details

Value

Author(s)

Examples

Student's t-Confidence Interval with Finite Population Correction

Description

Usage

Arguments

Details

Value

Note

Author(s)

References

See Also

Examples

Coin Toss

Description

Usage

Arguments

Details

Author(s)

Examples

Laplace (Double Exponential) Density Function

Description

Usage

Arguments

Details

Value

Note

Author(s)

References

See Also

Examples

Triangular Density Function

Description

Usage

Arguments

Details

Value

Author(s)

See Also

Examples

Determining and Graphing Fourier Approximation

Description

Usage

Arguments

Details

Value

Note

Author(s)

References

Examples

Latin Square

Description

Usage

Arguments

Details

Author(s)

See Also

Examples

Line Graph Plotting

Description

Usage

Arguments

Author(s)

See Also

Examples