A statistical method for the analysis of errors on the Farnsworth-Munsell 100-hue test is introduced. The extent of asymmetry of errors are summarized by two indices, I1 and I2, derived from Fourier analysis of the error scores for the individual caps. The second index, I2, describes the (bipolar) color axis; the first index, I1, describes (monopolar) asymmetry of performance. The present analysis differs from previous approaches based on Fourier analyis of the errors in two ways. (1) A procedure is introduced which corrects the indices I1 and I2 for the biases that result from the segmentation of the test into four boxes. (2) Statistics for the significance of I1 and I2 are derived by a Monte Carlo procedure, which properly handles the complex interdependence of individual error scores for each cap.
Cap arrangements are generated by applying pairwise swaps of
adjacent
caps. The statistics for each average number of swaps per tray
(f)
are calculated from 10000 independent arrangements.
The statistics for the last line of the table (infinite number of
swaps
per tray) were calculated from 10000 completely random cap
arrangements.
s.d., standard deviation; c.v., coefficient of variation
(s.d./mean).
swaps/ mean s.d. c.v. ..................critical values....................
tray total 0.001 0.010 0.025 0.050 0.500 0.950 0.975 0.990 0.999
(f) error
1 15.2 7.4 0.491 0 0 4 4 16 29 33 37 45
2 29.3 10.1 0.346 4 8 12 12 28 49 53 57 65
4 54.0 13.3 0.247 20 24 28 32 52 77 81 89 101
6 75.7 15.6 0.207 32 40 48 52 76 105 109 117 133
10 112.2 19.3 0.172 56 68 76 80 112 145 153 161 177
20 181.5 26.2 0.145 108 124 132 140 180 225 237 245 269
40 277.0 35.8 0.129 176 196 208 220 276 337 353 369 397
60 348.2 43.0 0.124 228 256 268 280 348 425 437 457 493
100 454.3 53.5 0.118 304 340 356 372 452 549 569 589 645
200 633.2 68.7 0.109 432 484 504 524 632 753 773 805 869
300 749.7 79.6 0.106 512 576 600 624 748 832 913 945 1013
400 834.6 84.4 0.101 592 644 672 700 832 977 1005 1037 1093
600 952.4 91.3 0.096 696 740 776 804 952 1105 1133 1173 1237
1000 1085.2 93.1 0.086 792 872 904 936 1084 1241 1269 1305 1373
1500 1151.1 88.9 0.077 876 940 976 1004 1152 1297 1321 1353 1425
inf. 1204.2 84.7 0.070 948 1004 1036 1060 1208 1341 1369 1397 1465
|I1| and |I2| are the ratio of the first and second Fourier components of the error scores to the zeroth Fourier component. The statistics for each degree of randomization are calculated from 10000 independent arrangements, as in Table 1. Harmonic indices are measured with or without the correction for finite boxes.
(corrected for finite boxes) mean swaps/ .................critical values.................. total tray first harmonic second harmonic error (f) 0.500 0.950 0.990 0.999 0.500 0.950 0.990 0.999 15.2 1 0.450 0.997 0.997 0.997 0.439 0.987 0.987 0.987 29.3 2 0.304 0.693 0.914 0.997 0.300 0.699 0.899 0.987 54.0 4 0.213 0.464 0.590 0.749 0.208 0.460 0.583 0.742 75.7 6 0.177 0.377 0.477 0.624 0.174 0.377 0.484 0.606 112.2 10 0.148 0.311 0.391 0.484 0.143 0.303 0.385 0.485 181.5 20 0.122 0.255 0.312 0.380 0.116 0.248 0.317 0.415 277.0 40 0.108 0.224 0.270 0.327 0.098 0.215 0.268 0.342 348.2 60 0.101 0.211 0.263 0.322 0.088 0.196 0.251 0.310 454.3 100 0.092 0.194 0.239 0.305 0.078 0.176 0.226 0.281 633.2 200 0.083 0.174 0.214 0.264 0.062 0.146 0.187 0.237 749.7 300 0.076 0.159 0.198 0.238 0.055 0.130 0.170 0.225 834.6 400 0.072 0.154 0.190 0.236 0.050 0.120 0.156 0.198 952.4 600 0.065 0.136 0.172 0.207 0.044 0.109 0.144 0.183 1085.2 1000 0.057 0.121 0.151 0.187 0.042 0.098 0.128 0.162 1151.1 1500 0.052 0.112 0.141 0.171 0.042 0.094 0.121 0.157 1204.2 inf. 0.048 0.101 0.128 0.166 0.045 0.097 0.124 0.152 (uncorrected for finite boxes) mean swaps/ .................critical values.................. total tray first harmonic second harmonic error (f) 0.500 0.950 0.990 0.999 0.500 0.950 0.990 0.999 15.2 1 0.439 0.997 0.997 0.997 0.435 0.987 0.987 0.987 29.3 2 0.299 0.685 0.913 0.997 0.298 0.680 0.895 0.987 54.0 4 0.208 0.455 0.586 0.740 0.207 0.450 0.566 0.708 75.7 6 0.175 0.374 0.468 0.619 0.174 0.367 0.471 0.590 112.2 10 0.144 0.304 0.385 0.469 0.143 0.297 0.366 0.460 181.5 20 0.120 0.249 0.308 0.370 0.116 0.241 0.300 0.377 277.0 40 0.105 0.220 0.268 0.329 0.099 0.207 0.257 0.313 348.2 60 0.099 0.208 0.257 0.306 0.090 0.190 0.233 0.282 454.3 100 0.093 0.192 0.238 0.298 0.080 0.171 0.212 0.265 633.2 200 0.084 0.176 0.215 0.260 0.064 0.140 0.178 0.220 749.7 300 0.078 0.164 0.201 0.243 0.057 0.125 0.157 0.207 834.6 400 0.075 0.157 0.195 0.242 0.052 0.115 0.147 0.182 952.4 600 0.068 0.143 0.178 0.218 0.045 0.104 0.134 0.170 1085.2 1000 0.059 0.126 0.155 0.192 0.041 0.093 0.118 0.151 1151.1 1500 0.054 0.115 0.147 0.177 0.041 0.090 0.115 0.145 1204.2 inf. 0.049 0.105 0.132 0.164 0.043 0.093 0.116 0.143