Goodness-of-fit Test Results
The data below includes all the results from the chi square test applied to the special logistic-J, the general logistic-J, and the log-series distributions. The column labeled “ndf” represents the test scores at the degrees of freedom appropriate to each particular biosurvey. The column labeled “10df” represents the corresponding scores adjusted to 10 degrees of freedom to make direct comparison between test results possible. The adjusted scores correspond to exactly the same p-values as thier raw counterparts.
A glance down the various “10df” columns will reveal that the scores vary wildly, which is precisely what we expect in carrying out chi square tests. The fact that sometimes the log-series does better than either of the logistic-J distributions is simply part of the natural variation in the data being revealed by the chi square test. Any approximately J-shaped curve would thus “win” some of the time. The only relevant outcome is the long-term behaviour of the average scores.
| |
Logistic-J (special) |
Logistic-J (general) |
log-series |
| |
ndf |
10df |
ndf |
10df |
ndf |
10df |
| 1 |
18.686/18 |
10.355 |
18.667/18 |
10.341 |
26.079/17 |
17.066 |
| 2 |
14.614/8 |
17.341 |
5.219/8 |
6.907 |
18.347/8 |
21.343 |
| 3 |
3.063/3 |
10.625 |
4.637/3 |
13.397 |
2.802/3 |
10.144 |
| 4 |
2.783/5 |
6.906 |
0.277/4 |
2.468 |
0.397/4 |
2.926 |
| 5 |
9.804/12 |
7.963 |
5.728/12 |
4.371 |
13.063/10 |
13.063 |
| 6 |
3.481/4 |
9.538 |
3.876/5 |
8.603 |
3.923/5 |
8.676 |
| 7 |
1.031/3 |
6.196 |
1.490/3 |
7.364 |
4.643/4 |
11.375 |
| 8 |
0.490/3 |
4.491 |
0.462/3 |
4.378 |
1.719/4 |
6.294 |
| 9 |
0.759/4 |
4.067 |
1.039/5 |
3.705 |
3.521/7 |
5.771 |
| 10 |
5.927/6 |
10.087 |
7.633/6 |
12.282 |
4.778/6 |
8.558 |
| 11 |
0.094/3 |
2.364 |
0.202/3 |
3.153 |
0.472/4 |
3.201 |
| 12 |
6.198/5 |
11.946 |
4.563/5 |
9.654 |
2.718/6 |
5.634 |
| 13 |
5.809/9 |
6.640 |
4.441/9 |
5.160 |
14.111/10 |
14.111 |
| 14 |
42.187/26 |
20.677 |
41.547/28 |
18.453 |
46.972/28 |
22.277 |
| 15 |
5.290/3 |
14.445 |
5.214/3 |
14.323 |
5.689/5 |
11.288 |
| 16 |
5.006/3 |
13.909 |
4.864/3 |
13.761 |
4.637/4 |
11.366 |
| 17 |
5.572/8 |
8.457 |
7.017/8 |
8.967 |
2.452/9 |
2.968 |
| 18 |
2.067/3 |
8.667 |
1.732/3 |
7.910 |
5.471/5 |
10.921 |
| 19 |
4.723/5 |
9.866 |
6.095/5 |
11.801 |
10.365/7 |
14.081 |
| 20 |
2.105/4 |
7.069 |
2.253/4 |
7.337 |
12.796/6 |
18.546 |
| 21 |
12.527/13 |
9.507 |
18.872/14 |
14.078 |
17.087/15 |
11.598 |
| 22 |
10.001/9 |
11.088 |
9.860/9 |
10.939 |
8.864/9 |
9.891 |
| 23 |
3.493/8 |
4.869 |
4.295/8 |
5.819 |
7.120/9 |
8.039 |
| 24 |
5.292/6 |
9.265 |
6.441/7 |
9.455 |
5.541/8 |
7.276 |
| 25 |
24.001/17 |
15.370 |
23.359/17 |
14.855 |
18.421/19 |
9.403 |
| |
|
––––––– |
|
––––––– |
|
––––––– |
| |
|
241.707 |
|
229.481 |
|
265.816 |
| |
|
(9.67) |
|
(9.18) |
|
(10.63) |
|
26
|
2.937/5
|
7.145
|
2.720/5
|
6.808
|
11.826/6
|
19.240
|
|
27
|
8.824/9
|
9.848
|
8.944/9
|
9.975
|
30.380/11
|
28.725
|
|
28
|
20.040/18
|
11.373
|
8.636/18
|
3.489
|
33.425/18
|
22.004
|
|
29
|
3.267/6
|
6.456
|
2.806/6
|
5.766
|
3.827/7
|
6.174
|
|
30
|
49.296/41
|
13.984
|
39.637/41
|
9.496
|
57.258/38
|
19.199
|
|
31
|
6.044/5
|
11.729
|
7.346/5
|
13.498
|
4.291/5
|
9.248
|
|
32
|
36.104/20
|
22.036
|
37.262/20
|
22.965
|
36.607/22
|
20.351
|
|
33
|
3.164/4
|
8.992
|
3.123/4
|
8.917
|
3.642/5
|
8.212
|
|
34
|
13.157/10
|
13.157
|
14.254/10
|
14.254
|
13.366/10
|
13.366
|
|
35
|
7.506/3
|
17.849
|
5.719/3
|
15.133
|
7.886/4
|
16.132
|
|
36
|
2.622/3
|
9.813
|
2.373/3
|
9.355
|
2.174/4
|
7.194
|
|
37
|
10.517/5
|
17.607
|
9.870/5
|
16.789
|
13.105/6
|
18.908
|
|
38
|
8.647/4
|
17.165
|
9.308/4
|
18.063
|
13.759/5
|
21.604
|
|
39
|
16.479/10
|
16.479
|
16.993/10
|
16.993
|
15.413/11
|
14.196
|
|
40
|
27.104/14
|
21.372
|
28.325/14
|
22.473
|
59.643/15
|
46.615
|
|
41
|
9.352/12
|
7.557
|
9.546/12
|
7.731
|
13.325/14
|
9.330
|
|
42
|
4.212/6
|
7.779
|
8.242/6
|
13.041
|
8.166/8
|
10.259
|
|
43
|
1.291/4
|
5.361
|
0.707/5
|
2.938
|
1.025/5
|
3.675
|
|
44
|
8.605/4
|
17.108
|
8.970/4
|
17.604
|
8.820/6
|
13.750
|
|
45
|
0.556/4
|
3.446
|
0.737/4
|
4.009
|
3.223/6
|
6.390
|
|
46
|
0.629/3
|
4.998
|
0.244/3
|
3.391
|
1.619/5
|
4.880
|
|
47
|
3.495/4
|
9.560
|
3.497/4
|
9.597
|
55.147/8
|
55.147
|
|
48
|
26.368/7
|
29.588
|
26.402/7
|
29.588
|
75.212/16
|
61.360
|
|
49
|
17.018/11
|
15.740
|
19.770/11
|
18.399
|
12.555/11
|
11.455
|
|
50
|
4.797/5
|
9.970
|
5.122/5
|
10.428
|
6.979/7
|
10.096
|
|
|
|
–––––––
|
|
–––––––
|
|
–––––––
|
|
|
|
316.112
|
|
310.700
|
|
457.510
|
|
|
|
(12.64)
|
|
(12.430)
|
|
(18.30)
|
|
|
|
|
|
|
|
|
|
51
|
1.026/4
|
4.767
|
0.954/4
|
4.578
|
6.979/7
|
10.096
|
|
52
|
13.362/7
|
17.508
|
14.666/7
|
18.977
|
13.085/9
|
14.319
|
|
53
|
4.135/3
|
12.592
|
4.020/3
|
12.386
|
10.056/5
|
17.024
|
|
54
|
2.689/4
|
8.129
|
2.831/4
|
8.387
|
23.658/7
|
28.895
|
|
55
|
13.769/14
|
9.707
|
14.359/15
|
9.358
|
17.597/17
|
10.315
|
|
56
|
8.246/4
|
16.621
|
8.556/4
|
17.042
|
5.485/5
|
10.940
|
|
57
|
1.456/2
|
9.503
|
1.844/2
|
10.401
|
1.115/3
|
6.447
|
|
58
|
4.276/4
|
10.735
|
1.353/4
|
5.496
|
2.495/5
|
6.421
|
|
59
|
27.640/16
|
19.454
|
24.625/16
|
16.908
|
24.090/16
|
16.449
|
|
60
|
2.425/4
|
7.650
|
2.176/4
|
7.196
|
4.148/5
|
9.026
|
|
61
|
0.311/3
|
3.732
|
0.531/2
|
6.509
|
3.097/3
|
10.687
|
|
62
|
9.664/10
|
9.664
|
9.258/10
|
9.258
|
7.894/11
|
7.030
|
|
63
|
1.662/4
|
6.169
|
4.900/4
|
11.782
|
3.106/5
|
7.407
|
|
64
|
2.213/4
|
7.265
|
3.281/4
|
9.204
|
3.249/5
|
7.629
|
|
65
|
2.763/5
|
6.875
|
2.773/5
|
6.890
|
0.882/7
|
1.922
|
|
66
|
2.275/3
|
9.136
|
3.306/4
|
9.250
|
2.644/5
|
6.683
|
|
67
|
5.419/4
|
12.597
|
0.123/3
|
2.614
|
0.611/4
|
3.635
|
|
68
|
10.126/8
|
12.455
|
9.088/8
|
11.287
|
6.792/10
|
6.792
|
|
69
|
2.145/6
|
4.771
|
3.129/6
|
6.250
|
2.369/7
|
4.220
|
|
70
|
0.454/4
|
3.135
|
4.520/4
|
11.181
|
1.602/5
|
4.849
|
|
71
|
3.790/3
|
11.963
|
4.151/3
|
12.617
|
7.724/4
|
15.907
|
|
72
|
14.027/10
|
14.027
|
7.373/9
|
8.308
|
8.957/11
|
8.034
|
|
73
|
3.844/4
|
10.112
|
2.563/4
|
7.900
|
2.779/5
|
6.899
|
|
74
|
0.276/5
|
1.445
|
0.308/5
|
1.615
|
13.787/8
|
16.447
|
|
75
|
1.210/4
|
5.184
|
2.962/5
|
7.184
|
5.216/6
|
9.160
|
|
|
|
–––––––
|
|
––––––
|
|
–––––––
|
|
|
|
235.196
|
|
232.278
|
|
247.233
|
|
|
|
(9.41)
|
|
(9.29)
|
|
(9.89)
|
|
|
|
|
|
|
|
|
|
76
|
10.517/5
|
17.607
|
11.500/6
|
17.006
|
14.406/8
|
17.116
|
|
77
|
2.063/4
|
6.992
|
6.185/3
|
15.881
|
3.690/5
|
8.314
|
|
78
|
2.478/4
|
7.746
|
2.410/4
|
7.622
|
3.608/5
|
8.187
|
|
79
|
10.761/5
|
17.916
|
12.266/5
|
19.783
|
11.149/6
|
16.588
|
|
80
|
5.013/6
|
8.881
|
4.821/6
|
8.617
|
5.668/7
|
8.498
|
|
81
|
3.908/3
|
12.180
|
4.040/3
|
12.423
|
15.432/5
|
23.621
|
|
82
|
11.040/3
|
22.793
|
11.101/3
|
22.876
|
33.165/5
|
29.588*
|
|
83
|
1.631/3
|
8.111
|
1.631/3
|
7.682
|
7.835/4
|
16.063
|
|
84
|
13.102/5
|
20.809
|
13.252/5
|
20.991
|
28.489/7
|
29.588*
|
|
85
|
0.635/1
|
10.007
|
0.635/1
|
10.007
|
33.533/5
|
29.588*
|
|
86
|
5.164/5
|
10.488
|
4.974/5
|
10.220
|
13.957/7
|
18.182
|
|
87
|
11.153/8
|
13.571
|
13.456/8
|
16.089
|
12.417/8
|
14.954
|
|
88
|
6.139/6
|
10.359
|
6.475/6
|
10.792
|
5.588/7
|
8.398
|
|
89
|
8.593/6
|
13.471
|
7.645/6
|
12.297
|
9.443/6
|
14.514
|
|
90
|
8.057/6
|
12.814
|
5.020/6
|
8.890
|
6.647/6
|
11.013
|
|
91
|
6.552/6
|
10.891
|
9.625/7
|
13.227
|
4.751/6
|
8.526
|
|
92
|
1.067/3
|
6.303
|
0.363/2
|
5.647
|
2.079/4
|
7.021
|
|
93
|
5.933/5
|
11.572
|
6.645/5
|
12.574
|
13.534/6
|
19.411
|
|
94
|
4.669/5
|
9.790
|
4.641/5
|
9.750
|
13.988/7
|
18.217
|
|
95
|
5.153/4
|
12.182
|
8.821/4
|
17.402
|
6.526/4
|
14.187
|
|
96
|
38.537/15
|
30.247
|
42.296/15
|
33.721
|
46.288/18
|
33.498
|
|
97
|
5.564/5
|
10.706
|
5.319/5
|
10.706
|
4.256/6
|
7.840
|
|
98
|
0.815/4
|
4.214
|
0.741/4
|
4.020
|
4.002/6
|
7.490
|
|
99
|
12.716/2
|
28.085
|
11.046/2
|
25.802
|
21.048/4
|
29.588*
|
|
100
|
3.078/5
|
7.364
|
3.004/5
|
7.249
|
16.698/7
|
21.242
|
|
|
|
–––––––
|
|
–––––––
|
|
–––––––
|
|
|
|
325.099
|
|
341.274
|
|
421.232
|
|
|
|
(13.00)
|
|
(13.65)
|
|
(16.85)
|
|
101
|
0.779/3
|
5.445
|
0.445/3
|
4.311
|
1.649/4
|
6.141
|
|
102
|
1.592/2
|
9.818
|
1.945/2
|
10.634
|
2.982/3
|
10.476
|
|
103
|
3.207/3
|
10.890
|
2.431/3
|
9.462
|
2.562/3
|
9.703
|
|
104
|
5.161/6
|
9.084
|
5.984/6
|
10.160
|
4.588/6
|
8.296
|
|
105
|
4.288/2
|
15.392
|
4.166/2
|
15.163
|
7.542/4
|
15.646
|
|
106
|
7.924/4
|
16.183
|
6.324/5
|
12.123
|
11.503/5
|
18.841
|
|
107
|
0.067/4
|
0.699
|
0.908/3
|
5.831
|
2.269/4
|
7.366
|
|
108
|
31.654/23
|
15.738
|
27.324/23
|
12.678
|
29.833/24
|
13.653
|
|
109
|
2.342/2
|
11.553
|
2.780/3
|
10.104
|
5.606/5
|
11.111
|
|
110
|
4.696/7
|
7.287
|
3.256/6
|
6.440
|
3.662/6
|
7.026
|
|
111
|
3.274/4
|
9.192
|
3.169/4
|
9.001
|
3.839/6
|
7.266
|
|
112
|
1.584/2
|
9.799
|
0.199/3
|
3.132
|
1.182/4
|
5.123
|
|
113
|
5.621/6
|
9.693
|
7.383/5
|
13.546
|
11.913/8
|
14.402
|
|
114
|
5.069/4
|
12.049
|
5.096/4
|
12.092
|
21.824/5
|
31.476
|
|
115
|
0.957/3
|
5.976
|
3.321/3
|
11.100
|
2.776/4
|
8.287
|
|
116
|
0.486/3
|
4.474
|
0.750/3
|
5.359
|
1.685/3
|
7.804
|
|
117
|
1.666/3
|
7.761
|
2.107/3
|
8.757
|
3.733/3
|
11.856
|
|
118
|
2.138/2
|
11.081
|
1.345/2
|
9.209
|
2.187/3
|
8.938
|
|
119
|
4.692/3
|
13.485
|
4.587/3
|
13.317
|
3.935/4
|
10.256
|
|
120
|
1.056/3
|
6.271
|
4.051/4
|
10.439
|
6.235/4
|
13.769
|
|
121
|
10.023/13
|
7.357
|
10.446/13
|
7.720
|
21.041/11
|
19.631
|
|
122
|
3.344/6
|
6.572
|
3.064/6
|
6.152
|
3.501/7
|
5.745
|
|
123
|
6.110/6
|
10.322
|
6.782/6
|
11.187
|
14.273/8`
|
16.972
|
|
124
|
10.642/10
|
10.642
|
10.714/10
|
10.714
|
16.088/12
|
13.702
|
|
125
|
1.788/2
|
10.271
|
2.422/3
|
9.445
|
6.527/4
|
14.190
|
|
|
|
--------
|
|
---------
|
|
--------
|
|
|
|
237.034
|
|
238.076
|
|
297.676
|
|
|
|
(9.48)
|
|
(9.52)
|
|
(11.91)
|
|
|
|
|
|
|
|
|
|
final sums:
|
|
1355.148
|
|
1351.809
|
|
1689.467
|
|
|
|
(10.84)
|
|
(10.81)
|
|
(13.56)
|