# dmes: Dominance Matrix Effect Sizes In orddom: Ordinal Dominance Statistics

## 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_idw+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.

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.

 ``` 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, 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) ```