La statistique d’écart studentisée
Objectifs. Introduire une statistique qui est rarement utilisée pour elle-même, mais prend part à de nombreux tests post-hoc dans les ANOVA.
Prérequis. Test t de Student ; ANOVA
Résumé. La statistique d'écart studentisée q s'obtient en procédant à une sorte de test t portant sur les deux valeurs les plus extrêmes (haute et basse) d'un ensemble de moyennes. Une table des valeurs critiques de q en fonction du nombre de comparaisons impliquées dans l'ANOVA globale et le nombre de degrés de libertés de la comparaison est fournie.
2. Une table des valeurs critiques de q au seuil de .05
Cette table a été réalisée par nos soins en utilisant la fonction "qtukey" du logiciel de statistiques R, et en faisant varier de 3 à 10 le nombre r de groupes impliqués dans la comparaison, et de 2 à 100 le nombre de degrés de libertés de la comparaison.
Ainsi par exemple pour une comparaison des deux moyennes les plus extrêmes parmi 6 ( r =6) groupes de 25 sujets chacun, la valeur que q devra dépasser pour que la comparaison soit significative est 4.358.
Nous n'avons pas indiqué les valeurs pour le cas r =2 car alors il n'y a pas véritablement lieu d'utiliser la statistique d'écart studentisée.
| ddl | r=3 | r=4 | r=5 | r=6 | r=7 | r=8 | r=9 | r=10 |
| 2 | 8.331 | 9.799 | 10.881 | 11.734 | 12.435 | 13.028 | 13.542 | 13.994 |
| 3 | 5.910 | 6.825 | 7.502 | 8.037 | 8.478 | 8.852 | 9.177 | 9.462 |
| 4 | 5.040 | 5.757 | 6.287 | 6.706 | 7.053 | 7.347 | 7.602 | 7.826 |
| 5 | 4.602 | 5.218 | 5.673 | 6.033 | 6.330 | 6.582 | 6.801 | 6.995 |
| 6 | 4.339 | 4.896 | 5.305 | 5.628 | 5.895 | 6.122 | 6.319 | 6.493 |
| 7 | 4.165 | 4.681 | 5.060 | 5.359 | 5.606 | 5.815 | 5.997 | 6.158 |
| 8 | 4.041 | 4.529 | 4.886 | 5.167 | 5.399 | 5.596 | 5.767 | 5.918 |
| 9 | 3.948 | 4.415 | 4.755 | 5.024 | 5.244 | 5.432 | 5.595 | 5.738 |
| 10 | 3.877 | 4.327 | 4.654 | 4.912 | 5.124 | 5.304 | 5.460 | 5.598 |
| 11 | 3.820 | 4.256 | 4.574 | 4.823 | 5.028 | 5.202 | 5.353 | 5.486 |
| 12 | 3.773 | 4.199 | 4.508 | 4.750 | 4.950 | 5.119 | 5.265 | 5.395 |
| 13 | 3.734 | 4.151 | 4.453 | 4.690 | 4.884 | 5.049 | 5.192 | 5.318 |
| 14 | 3.701 | 4.111 | 4.407 | 4.639 | 4.829 | 4.990 | 5.130 | 5.253 |
| 15 | 3.673 | 4.076 | 4.367 | 4.595 | 4.782 | 4.940 | 5.077 | 5.198 |
| 16 | 3.649 | 4.046 | 4.333 | 4.557 | 4.741 | 4.896 | 5.031 | 5.150 |
| 17 | 3.628 | 4.020 | 4.303 | 4.524 | 4.705 | 4.858 | 4.991 | 5.108 |
| 18 | 3.609 | 3.997 | 4.276 | 4.494 | 4.673 | 4.824 | 4.955 | 5.071 |
| 19 | 3.593 | 3.977 | 4.253 | 4.468 | 4.645 | 4.794 | 4.924 | 5.037 |
| 20 | 3.578 | 3.958 | 4.232 | 4.445 | 4.620 | 4.768 | 4.895 | 5.008 |
| 21 | 3.565 | 3.942 | 4.213 | 4.424 | 4.597 | 4.743 | 4.870 | 4.981 |
| 22 | 3.553 | 3.927 | 4.196 | 4.405 | 4.577 | 4.722 | 4.847 | 4.957 |
| 23 | 3.542 | 3.914 | 4.180 | 4.388 | 4.558 | 4.702 | 4.826 | 4.935 |
| 24 | 3.532 | 3.901 | 4.166 | 4.373 | 4.541 | 4.684 | 4.807 | 4.915 |
| 25 | 3.523 | 3.890 | 4.153 | 4.358 | 4.526 | 4.667 | 4.789 | 4.897 |
| 26 | 3.514 | 3.880 | 4.141 | 4.345 | 4.511 | 4.652 | 4.773 | 4.880 |
| 27 | 3.506 | 3.870 | 4.130 | 4.333 | 4.498 | 4.638 | 4.758 | 4.864 |
| 28 | 3.499 | 3.861 | 4.120 | 4.322 | 4.486 | 4.625 | 4.745 | 4.850 |
| 29 | 3.493 | 3.853 | 4.111 | 4.311 | 4.475 | 4.613 | 4.732 | 4.837 |
| 30 | 3.486 | 3.845 | 4.102 | 4.301 | 4.464 | 4.601 | 4.720 | 4.824 |
| 31 | 3.481 | 3.838 | 4.094 | 4.292 | 4.454 | 4.591 | 4.709 | 4.812 |
| 32 | 3.475 | 3.832 | 4.086 | 4.284 | 4.445 | 4.581 | 4.698 | 4.802 |
| 33 | 3.470 | 3.825 | 4.079 | 4.276 | 4.436 | 4.572 | 4.689 | 4.791 |
| 34 | 3.465 | 3.820 | 4.072 | 4.268 | 4.428 | 4.563 | 4.680 | 4.782 |
| 35 | 3.461 | 3.814 | 4.066 | 4.261 | 4.421 | 4.555 | 4.671 | 4.773 |
| 36 | 3.457 | 3.809 | 4.060 | 4.255 | 4.414 | 4.547 | 4.663 | 4.764 |
| 37 | 3.453 | 3.804 | 4.054 | 4.249 | 4.407 | 4.540 | 4.655 | 4.756 |
| 38 | 3.449 | 3.799 | 4.049 | 4.243 | 4.400 | 4.533 | 4.648 | 4.749 |
| 39 | 3.445 | 3.795 | 4.044 | 4.237 | 4.394 | 4.527 | 4.641 | 4.741 |
| 40 | 3.442 | 3.791 | 4.039 | 4.232 | 4.388 | 4.521 | 4.634 | 4.735 |
| 41 | 3.439 | 3.787 | 4.035 | 4.227 | 4.383 | 4.515 | 4.628 | 4.728 |
| 42 | 3.436 | 3.783 | 4.030 | 4.222 | 4.378 | 4.509 | 4.622 | 4.722 |
| 43 | 3.433 | 3.779 | 4.026 | 4.217 | 4.373 | 4.504 | 4.617 | 4.716 |
| 44 | 3.430 | 3.776 | 4.022 | 4.213 | 4.368 | 4.499 | 4.611 | 4.710 |
| 45 | 3.428 | 3.773 | 4.018 | 4.209 | 4.364 | 4.494 | 4.606 | 4.705 |
| 46 | 3.425 | 3.770 | 4.015 | 4.205 | 4.359 | 4.489 | 4.601 | 4.700 |
| 47 | 3.423 | 3.767 | 4.011 | 4.201 | 4.355 | 4.485 | 4.597 | 4.695 |
| 48 | 3.420 | 3.764 | 4.008 | 4.197 | 4.351 | 4.481 | 4.592 | 4.690 |
| 49 | 3.418 | 3.761 | 4.005 | 4.194 | 4.347 | 4.477 | 4.588 | 4.686 |
| 50 | 3.416 | 3.758 | 4.002 | 4.190 | 4.344 | 4.473 | 4.584 | 4.681 |
| 51 | 3.414 | 3.756 | 3.999 | 4.187 | 4.340 | 4.469 | 4.580 | 4.677 |
| 52 | 3.412 | 3.753 | 3.996 | 4.184 | 4.337 | 4.465 | 4.576 | 4.673 |
| 53 | 3.410 | 3.751 | 3.994 | 4.181 | 4.334 | 4.462 | 4.572 | 4.669 |
| 54 | 3.408 | 3.749 | 3.991 | 4.178 | 4.331 | 4.459 | 4.569 | 4.666 |
| 55 | 3.406 | 3.747 | 3.989 | 4.176 | 4.328 | 4.455 | 4.566 | 4.662 |
| 56 | 3.405 | 3.745 | 3.986 | 4.173 | 4.325 | 4.452 | 4.562 | 4.659 |
| 57 | 3.403 | 3.743 | 3.984 | 4.170 | 4.322 | 4.449 | 4.559 | 4.656 |
| 58 | 3.402 | 3.741 | 3.982 | 4.168 | 4.319 | 4.447 | 4.556 | 4.652 |
| 59 | 3.400 | 3.739 | 3.979 | 4.165 | 4.317 | 4.444 | 4.553 | 4.649 |
| 60 | 3.399 | 3.737 | 3.977 | 4.163 | 4.314 | 4.441 | 4.550 | 4.646 |
| 61 | 3.397 | 3.735 | 3.975 | 4.161 | 4.312 | 4.438 | 4.548 | 4.643 |
| 62 | 3.396 | 3.734 | 3.973 | 4.159 | 4.309 | 4.436 | 4.545 | 4.641 |
| 63 | 3.395 | 3.732 | 3.972 | 4.157 | 4.307 | 4.434 | 4.542 | 4.638 |
| 64 | 3.393 | 3.730 | 3.970 | 4.155 | 4.305 | 4.431 | 4.540 | 4.635 |
| 65 | 3.392 | 3.729 | 3.968 | 4.153 | 4.303 | 4.429 | 4.538 | 4.633 |
| 66 | 3.391 | 3.727 | 3.966 | 4.151 | 4.301 | 4.427 | 4.535 | 4.630 |
| 67 | 3.390 | 3.726 | 3.965 | 4.149 | 4.299 | 4.425 | 4.533 | 4.628 |
| 68 | 3.389 | 3.725 | 3.963 | 4.147 | 4.297 | 4.423 | 4.531 | 4.626 |
| 69 | 3.387 | 3.723 | 3.962 | 4.146 | 4.295 | 4.421 | 4.529 | 4.624 |
| 70 | 3.386 | 3.722 | 3.960 | 4.144 | 4.293 | 4.419 | 4.527 | 4.621 |
| 71 | 3.385 | 3.721 | 3.959 | 4.142 | 4.291 | 4.417 | 4.525 | 4.619 |
| 72 | 3.384 | 3.719 | 3.957 | 4.141 | 4.290 | 4.415 | 4.523 | 4.617 |
| 73 | 3.383 | 3.718 | 3.956 | 4.139 | 4.288 | 4.413 | 4.521 | 4.615 |
| 74 | 3.382 | 3.717 | 3.954 | 4.138 | 4.286 | 4.411 | 4.519 | 4.613 |
| 75 | 3.382 | 3.716 | 3.953 | 4.136 | 4.285 | 4.410 | 4.517 | 4.611 |
| 76 | 3.381 | 3.715 | 3.952 | 4.135 | 4.283 | 4.408 | 4.515 | 4.610 |
| 77 | 3.380 | 3.714 | 3.951 | 4.133 | 4.282 | 4.406 | 4.514 | 4.608 |
| 78 | 3.379 | 3.713 | 3.949 | 4.132 | 4.280 | 4.405 | 4.512 | 4.606 |
| 79 | 3.378 | 3.712 | 3.948 | 4.131 | 4.279 | 4.403 | 4.510 | 4.604 |
| 80 | 3.377 | 3.711 | 3.947 | 4.129 | 4.277 | 4.402 | 4.509 | 4.603 |
| 81 | 3.377 | 3.710 | 3.946 | 4.128 | 4.276 | 4.400 | 4.507 | 4.601 |
| 82 | 3.376 | 3.709 | 3.945 | 4.127 | 4.275 | 4.399 | 4.506 | 4.600 |
| 83 | 3.375 | 3.708 | 3.944 | 4.126 | 4.274 | 4.398 | 4.504 | 4.598 |
| 84 | 3.374 | 3.707 | 3.943 | 4.125 | 4.272 | 4.396 | 4.503 | 4.597 |
| 85 | 3.374 | 3.706 | 3.942 | 4.123 | 4.271 | 4.395 | 4.502 | 4.595 |
| 86 | 3.373 | 3.705 | 3.941 | 4.122 | 4.270 | 4.394 | 4.500 | 4.594 |
| 87 | 3.372 | 3.704 | 3.940 | 4.121 | 4.269 | 4.392 | 4.499 | 4.592 |
| 88 | 3.371 | 3.704 | 3.939 | 4.120 | 4.268 | 4.391 | 4.498 | 4.591 |
| 89 | 3.371 | 3.703 | 3.938 | 4.119 | 4.266 | 4.390 | 4.496 | 4.590 |
| 90 | 3.370 | 3.702 | 3.937 | 4.118 | 4.265 | 4.389 | 4.495 | 4.588 |
| 91 | 3.370 | 3.701 | 3.936 | 4.117 | 4.264 | 4.388 | 4.494 | 4.587 |
| 92 | 3.369 | 3.700 | 3.935 | 4.116 | 4.263 | 4.387 | 4.493 | 4.586 |
| 93 | 3.368 | 3.700 | 3.934 | 4.115 | 4.262 | 4.386 | 4.492 | 4.585 |
| 94 | 3.368 | 3.699 | 3.934 | 4.114 | 4.261 | 4.384 | 4.490 | 4.583 |
| 95 | 3.367 | 3.698 | 3.933 | 4.114 | 4.260 | 4.383 | 4.489 | 4.582 |
| 96 | 3.367 | 3.698 | 3.932 | 4.113 | 4.259 | 4.382 | 4.488 | 4.581 |
| 97 | 3.366 | 3.697 | 3.931 | 4.112 | 4.258 | 4.381 | 4.487 | 4.580 |
| 98 | 3.366 | 3.696 | 3.930 | 4.111 | 4.257 | 4.380 | 4.486 | 4.579 |
| 99 | 3.365 | 3.696 | 3.930 | 4.110 | 4.257 | 4.379 | 4.485 | 4.578 |
| 100 | 3.365 | 3.695 | 3.929 | 4.109 | 4.256 | 4.379 | 4.484 | 4.577 |