Nonparametric Bootstrap to estimate the overlapping area

Description

Resampling via non-parametric bootstrap to estimate the overlapping area between two or more kernel density estimations from empirical data.

Usage

boot.overlap( x, B = 1000, pairsOverlap = FALSE, ... )

Arguments

Details

If the list x contains more than two elements (i.e., more than two distributions) it computes the bootstrap overlapping measure between all the q paired distributions. For example, if x contains three elements then q = 3; if x contains four elements then q = 6.

Value

It returns a list containing the following components:

Note

Call function overlap.

Thanks to Jeremy Vollen for suggestions.

Author(s)

Massimiliano Pastore

References

Pastore, M. (2018). Overlapping: a R package for Estimating Overlapping in Empirical Distributions. The Journal of Open Source Software, 3 (32), 1023. doi: 10.21105/joss.01023

Pastore, M., Calcagnì, A. (2019). Measuring Distribution Similarities Between Samples: A Distribution-Free Overlapping Index. Frontiers in Psychology, 10:1089. doi: 10.3389/fpsyg.2019.01089

Examples

set.seed(20150605)
x <- list(X1=rnorm(100), X2=rt(50,8), X3=rchisq(80,2))

## bootstrapping
out <- boot.overlap( x, B = 10 )
out$OVboot_stats

# bootstrap quantile intervals
apply( out$OVboot_dist, 2, quantile, probs = c(.05, .9) )

# plot of bootstrap distributions
Y <- stack( data.frame( out$OVboot_dist ))
ggplot( Y, aes( values )) + facet_wrap( ~ind ) + geom_density()

Final plot

Description

Graphical representation of the estimated densities along with the overlapping area.

Usage

final.plot( x, pairs = FALSE, boundaries = NULL )

Arguments

Details

It requires the package ggplot2.

Note

The output plot can be customized using the ggplot2 rules, see example below.

Author(s)

Massimiliano Pastore

Examples

set.seed(20150605)
x <- list(X1=rnorm(100),X2=rt(50,8),X3=rchisq(80,2))
final.plot(x)
final.plot(x, pairs = TRUE)

# customizing plot
final.plot(x) + scale_fill_brewer() + scale_color_brewer()
final.plot(x) + theme(text=element_text(size=15)) 

Estimate the overlapping measure.

Description

It returns the overlapped estimated area between two or more kernel density estimations from empirical data. The overlapping measure can be computed either as the integral of the minimum between two densities (type = "1") or as the proportion of overlapping area between two densities (type = "2"). In the last case, the integral of the minimum between two densities is divided by the integral of the maximum of the two densities.

Usage

overlap( x, nbins = 1024, type = c( "1", "2" ), 
    pairsOverlap = TRUE, plot = FALSE, boundaries = NULL, 
    get_xpoints = FALSE, ... )

Arguments

Details

When dealing with two densities: type = "1" corresponds to the integral of the minimum between the two densities; type = "2" corresponds to the proportion of the overlapped area over the total area.

If the list x contains more than two elements (i.e. more than two distributions) it computes both the multiple and the pairwise overlapping among all distributions.

If plot = TRUE all the overlapped areas are plotted. It requires ggplot2.

The optional vector boundaries has to contain two numbers for the empirical minimum and maximum of the overlapped area. See examples below.

Value

It returns a list containing the following components:

Note

Call function ovmult.

Author(s)

Massimiliano Pastore, Pierfrancesco Alaimo Di Loro, Marco Mingione

References

Pastore, M. (2018). Overlapping: a R package for Estimating Overlapping in Empirical Distributions. The Journal of Open Source Software, 3 (32), 1023. doi: 10.21105/joss.01023

Pastore, M., Calcagnì, A. (2019). Measuring Distribution Similarities Between Samples: A Distribution-Free Overlapping Index. Frontiers in Psychology, 10:1089. doi: 10.3389/fpsyg.2019.01089

Examples

set.seed(20150605)
x <- list(X1=rnorm(100), X2=rt(50,8), X3=rchisq(80,2))
overlap(x, plot=TRUE)

# including boundaries
x <- list(X1=runif(100), X2=runif(100,.5,1))
overlap(x, plot=TRUE, boundaries=c(.5,1))

x <- list(X1=runif(100), X2=runif(50), X3=runif(30))
overlap(x, plot=TRUE, boundaries=c(.1,.9))

# changing kernel
overlap(x, plot=TRUE, kernel="rectangular")

# normalized overlap
N <- 1e5
x <- list(X1=runif(N),X2=runif(N,.5))
overlap(x)
overlap(x, type = "2")

Multiple overlapping estimation

Description

It gives the overlap area between two or more kernel density estimations from empirical data.

Usage

ovmult( x, nbins = 1024, type = c( "1", "2" ), 
    boundaries = NULL, get_xpoints = FALSE, ... )

Arguments

Details

If the list x contains more than two elements (i.e. more than two distributions) it computes multiple overlap measures.

The optional vector boundaries has to contain two numbers for the empirical minimum and maximum of the overlapped area. See examples below.

Value

It returns the value of overlapped area.

Note

Called from the function overlap.

Author(s)

Pierfrancesco Alaimo Di Loro, Marco Mingione, Massimiliano Pastore

Examples

set.seed(20150605)
x <- list(X1=rnorm(100), X2=rt(50,8), X3=rchisq(80,2))
ovmult(x)
ovmult(x, normalized = TRUE)

# including boundaries
x <- list(X1=runif(100), X2=runif(100,.5,1))
ovmult(x, boundaries=c( 0, .8 ))

x <- list(X1=runif(100), X2=runif(50), X3=runif(30))
ovmult(x, boundaries=c( .2, .8 ))

# changing kernel
ovmult(x, kernel="rectangular")

Paired permutation

Description

Perform a random permutation of the data list.

Usage

perm.pairs( x )

Arguments

Value

It returns a list with paired elements of x randomly permuted.

Note

Internal function called by perm.test.

Author(s)

Massimiliano Pastore

Examples

set.seed(20150605)
x <- list(X1=rnorm(10), X2=rt(15,8))
perm.pairs( x )

x <- list(X1=rnorm(10), X2=rt(15,8), X3=rchisq(12,3))
perm.pairs( x )

Permutation test on the (non-)overlapping area

Description

Perform a permutation test on the overlapping index.

Usage

perm.test( x, B = 1000, 
          return.distribution = FALSE, ... )

Arguments

Details

It performs a permutation test of the null hypothesis that there is no difference between the two distributions, i.e. the overlapping index (\eta) is one, or the non-overlapping index (1-\eta = \zeta) is zero.

Value

It returns a list containing the following components:

Warning

Currently, it only runs the permutation test on two groups at a time. If x contains more than 2 elements, it performs all paired permutation tests.

Note

Call function overlap.

Author(s)

Massimiliano Pastore

References

Perugini, A., Calignano, G., Nucci, M., Finos, L., & Pastore, M. (2024, December 30). How do my distributions differ? Significance testing for the Overlapping Index using Permutation Test. doi: 10.31219/osf.io/8h4fe

Examples

set.seed(20150605)
x <- list(X1=rnorm(100), X2=rt(50,8))

## not run: this example take several minutes
## permutation test
# out <- perm.test( x, return.distribution = TRUE )
# out$pval
# plot( density( out$Zperm ) )
# abline( v = out$Zobs ) 

x <- list(X1=rnorm(100), X2=rt(50,8), X3=rchisq(75,3))
# out <- perm.test( x )
# out$pval