Dominance Matrix Effect Sizes

Description

Generates simple list of nonparametric ordinal effect size measures such as
-the Probability of Superiority (or discrete case Common Language) effect size,
-the Vargha and Delaney's A (or area under the receiver operating characteristic curve, AUC)
-Cliff's delta (or success rate difference, SRD), and -the number needed to treat (NNT) effect size (based on Cliff's delta value).

Usage

1
dmes(x,y)

Arguments

x

A vector or 1 column matrix with n_x values from (control or pre-test or comparison) group X

y

A vector or 1 column matrix with n_y values from (treatment or post-test) group Y

Details

Based on the dominance matrix created by direct ordinal comparison of values of Y with values of X, an associative list is returned.

Value

$nx

Vector or sample size of x, n_x.

$ny

Vector or sample size of y, n_y

$PSc

Discrete case Common Language CL effect size or Probability of Superiority (PS) of all values of Y over all values of X:

PS(Y>X)=\#(y_i>x_j)/(n_y n_x)

,
where i=\{1, 2, ... , n_y\} and j=\{1, 2, ... , n_x\}. See orddom PS Y>X for details.)

$Ac

Vargha & Delaney's A or Area under the receiver operating characteristics curve (AUC) for all possible comparisons:

A(Y>X)=(\#(y_i>x_j)+.5(\#(y_i=x_j)))/(n_y n_x)

,
where i=\{1, 2, ... , n_y\} and j=\{1, 2, ... , n_x\}. See orddom A Y>X for details.)

$dc

Success rate difference when comparing all values of Y with all values of X:

delta(Y>X)=(\#(y_i>x_j)-\#(y_i<x_j))/(n_y n_x)

,
where i=\{1, 2, ... , n_y\} and j=\{1, 2, ... , n_x\}. See orddom Cliff's delta for independent groups for details.
Note that in the paired samples case with n_y=n_x, $dc does not return the combined estimate, i.e. $dc<>dw+db!

$NNTc

Number needed to treat, based on the success rate difference or \$dc^{-1}. See orddom "NNT" for details.

$PSw

When sample sizes are equal, this value returns the Probability of Superiority (PS) for within-changes, i.e. alle paired values: PS(Y>X)=\#(y_i>x_i)/(n_y n_x), limited to the n_x=n_y paired cases where i=\{1,2,...,n_x=n_y\}. (For unequal sample sizes, this equals $PSc.)

$Aw

When sample sizes are equal, this value returns A for the paired subsample values, i.e. limited to the n_x=n_y paired cases where i=j=\{1,2,...,n_x=n_y\}. (For unequal sample sizes, this equals $Ac.)

$dw

When n_x=n_y, this value returns Cliff's delta-within, i.e. paired comparisons limited to the diagonal of the dominance matrix or those cases where i=j. (For unequal sample sizes, this equals $dc.)

$NNTw

Number needed to treat, based on the within-case-success rate difference or \$dw^{-1}. See orddom NNT within for dependent groups for details.

$PSb

When sample sizes are equal, this gives the Probability of Superiority (PS) for all cases but within-pair changes, i.e.:

PS(Y>X)=\#(y_i>x_j)/(n_y n_x)

,
limited to those cases where i<>j. (For unequal sample sizes, this equals $PSc and $PSw.)

$Ab

When sample sizes are equal, this value returns A for all cases where i<>j. (For unequal sample sizes, this equals $Ac.)

$db

When n_x=n_y, this value returns Cliff's delta-between, i.e. all but the paired comparisons or excepting the diagonal of the dominance matrix. The parameter is calculated by taking only those ordinal comparisons into account where i<>j. (For unequal sample sizes, this equals $dc.)

$NNTb

Number needed to treat, based on Cliff's delta-between or \$db^{-1}. See orddom NNT between for dependent groups for details.

Author(s)

Jens J. Rogmann

References

Delaney, H.D. & Vargha, A. (2002). Comparing Several Robust Tests of Stochastic Equality With Ordinally Scaled Variables and Small to Moderate Sized Samples. Psychological Methods, 7, 485-503.

Kraemer, H.C. & Kupfer, D.J. (2006). Size of Treatment Effects and Their Importance to Clinical Research and Practice. Biological Psychiatry, 59, 990-996.

Ruscio, J. & Mullen, T. (2012). Confidence Intervals for the Probability of Superiority Effect Size Measure and the Area Under a Receiver Operating Characteristic Curve. Multivariate Behavioral Research, 47, 221-223. Vargha, A., & Delaney, H. D. (1998). The Kruskal-Wallis test and stochastic homogeneity. Journal of Educational and Behavioral Statistics, 23, 170-192.

Vargha, A., & Delaney, H. D. (2000). A critique and improvement of the CL common language effect size statistic of McGraw and Wong. Journal of Educational and Behavioral Statistics, 25, 101-132.

See Also

dm, orddom

Examples

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## Not run: 
> #Example from Efron & Tibshirani (1993, Table 2.1, p. 11)
> #cf. Efron, B. & Tibshirani (1993). An Introduction to the Bootstrap. New York/London: Chapman&Hall.
> y<-c(94,197,16,38,99,141,23) # Treatment Group
> x<-c(52,104,146,10,50,31,40,27,46) # Control Group
> dmes(x,y)
$nx
[1] 9

$ny
[1] 7

$PSc
[1] 0.5714286

$Ac
[1] 0.5714286

$dc
[1] 0.1428571

$NNTc
[1] 7

$PSw
[1] 0.5714286

$Aw
[1] 0.5714286

$dw
[1] 0.1428571

$NNTw
[1] 7

$PSb
[1] 0.5714286

$Ab
[1] 0.5714286

$db
[1] 0.1428571

$NNTb
[1] 7

> ############################################################################
> #Example from Ruscio & Mullen (2012, p. 202)
> #Ruscio, J. & Mullen, T. (2012). Confidence Intervals for the Probability of Superiority Effect Size Measure and the Area Under a Receiver Operating Characteristic Curve, Multivariate Behavioral Research, 47, 201-223.
> x <- c(6,7,8,7,9,6,5,4,7,8,7,6,9,5,4) # Treatment Group
> y <- c(4,3,5,3,6,2,2,1,6,7,4,3,2,4,3) # Control Group
> dmes(y,x)
$nx
[1] 15

$ny
[1] 15

$PSc
[1] 0.8444444

$Ac
[1] 0.8844444

$dc
[1] 0.7688889

$NNTc
[1] 1.300578

$PSw
[1] 1

$Aw
[1] 1

$dw
[1] 1

$NNTw
[1] 1

$PSb
[1] 0.8333333

$Ab
[1] 0.8761905

$db
[1] 0.752381

$NNTb
[1] 1.329114

## End(Not run)