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Table 2 Sorted p-values and associated \(\updelta{\text{s}}\), Rmax and breakpoints

From: A non-parametric maximum for number of selected features: objective optima for FDR and significance threshold with application to ordinal survey analysis

Column: 2 3 4 5 6 7 8 9 10
Set p-value of the set (alpha if this set is rejected) Frequency in set Less than cumulative frequency hypotheses in rejected sets Cumulative expected number of false discoveries (CEFD) if this set is rejected Expected false discovery contribution of this set FDR Expected cumulative number of true discoveries (CETD) if this set is rejected Expected true discovery contribution of this set \({\varvec{\updelta}}\) = CETD − CEFD
1 0 7 7 0 0 0 7 7 7
2 0.000001 1 8 0.001044 0.001044 0.000131 7.998956 0.998956 7.997912
3 0.000005 1 9 0.00522 0.004176 0.00058 8.99478 0.995824 8.98956
4 0.000008 1 10 0.008352 0.003132 0.000835 9.991648 0.996868 9.983296
5 0.000009 1 11 0.009396 0.001044 0.000854 10.9906 0.998956 10.98121
6 0.000015 1 12 0.01566 0.006264 0.001305 11.98434 0.993736 11.96868
7 0.000021 1 13 0.021924 0.006264 0.001686 12.97808 0.993736 12.95615
8 0.000022 1 14 0.022968 0.001044 0.001641 13.97703 0.998956 13.95406
9 0.000034 1 15 0.035496 0.012528 0.002366 14.9645 0.987472 14.92901
10 0.000037 1 16 0.038628 0.003132 0.002414 15.96137 0.996868 15.92274
11 0.0001 1 17 0.1044 0.065772 0.006141 16.8956 0.934228 16.7912
34 0.00183 1 40 1.91052 0.061596 0.047763 38.08948 0.938404 36.17896
35 0.001841 1 41 1.922004 0.011484 0.046878 39.078 0.988516 37.15599
36 0.001944 1 R @ FDR 0.05 42 2.029536 0.107532 0.048322 39.97046 0.892468 37.94093
37 0.002081 1 43 2.172564 0.143028 0.050525 40.82744 0.856972 38.65487
38 0.00213 1 44 2.22372 0.051156 0.050539 41.77628 0.948844 39.55256
39 0.002369 1 45 2.473236 0.249516 0.054961 42.52676 0.750484 40.05353
104 0.010835 1 110 11.31174 0.363312 0.102834 98.68826 0.636688 87.37652
105 0.010966 1 R @ Break Point 111 11.4485 0.136764 0.10314 99.5515 0.863236 88.10299
106 0.011041 1 112 11.5268 0.0783 0.102918 100.4732 0.9217 88.94639
107 0.01112 1 113 11.60928 0.082476 0.102737 101.3907 0.917524 89.78144
108 0.011198 1 114 11.69071 0.081432 0.10255 102.3093 0.918568 90.61858
109 0.011256 1 115 11.75126 0.060552 0.102185 103.2487 0.939448 91.49747
110 0.01126 1 116 11.75544 0.004176 0.10134 104.2446 0.995824 92.48912
111 0.012177 1 117 12.71279 0.957348 0.108656 104.2872 0.042652 91.57442
112 0.012225 1 R @ Break Point 118 12.7629 0.050112 0.10816 105.2371 0.949888 92.4742
113 0.014141 1 119 14.7632 2.000304 0.124061 104.2368 -1.0003 89.47359
114 0.014273 1 120 14.90101 0.137808 0.124175 105.099 0.862192 90.19798
115 0.014334 1 121 14.9647 0.063684 0.123675 106.0353 0.936316 91.07061
174 0.038832 1 181 40.54061 1.092024 0.223981 140.4594 -0.09202 99.91878
175 0.039077 1 182 40.79639 0.25578 0.224156 141.2036 0.74422 100.4072
176 0.039224 1 183 40.94986 0.153468 0.22377 142.0501 0.846532 101.1003
177 0.0393 1 R max 184 41.0292 0.079344 0.222985 142.9708 0.920656 101.9416
178 0.04194 1 185 43.78536 2.75616 0.236678 141.2146 -1.75616 97.42928
179 0.042014 1 186 43.86262 0.077256 0.235821 142.1374 0.922744 98.27477
180 0.042584 1 187 44.4577 0.59508 0.237742 142.5423 0.40492 98.08461
181 0.042642 1 188 44.51825 0.060552 0.236799 143.4818 0.939448 98.9635
  1. If we set p-value threshold at 0.001944, we reject 42 null hypotheses, and we can report FDR less than 0.05
  2. If we set p-value threshold at 0.010966, we reject 111 null hypotheses, and we can report FDR less than 0.11
  3. If we set p-value threshold at 0.012225, we reject 118 null hypotheses, and we can report FDR less than 0.11
  4. If we set p-value threshold at 0.0393, we reject 184 null hypotheses, and we can report FDR less than 0.23