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 pretest or comparison) group X 
y 
A vector or 1 column matrix with n_y values from (treatment or posttest) 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) , 
$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) , 
$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) , 
$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 withinchanges, 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 deltawithin, 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 withincasesuccess 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 withinpair changes, i.e.: PS(Y>X)=\#(y_i>x_j)/(n_y n_x) , 
$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 deltabetween, 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 deltabetween 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, 485503.
Kraemer, H.C. & Kupfer, D.J. (2006). Size of Treatment Effects and Their Importance to Clinical Research and Practice. Biological Psychiatry, 59, 990996.
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, 221223.
Vargha, A., & Delaney, H. D. (1998). The KruskalWallis test and stochastic homogeneity. Journal of Educational and Behavioral Statistics, 23, 170192.
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, 101132.
See Also
dm, orddom
Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97  ## 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, 201223.
> 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)

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