Background
Methods
Log-linear non-uniform association models
Log-linear modelling and parameters interpretation
Uniform Association (UA) and Non-Uniform Association (NUA) models
Power estimation of tests in NUA models
Simulation of I × I contingency tables from the NUA models
Definition of alternative hypotheses
Hypothesis | Association parameters | Distinguishability patterns |
---|---|---|
H
0
| All association parameters are equal | |
β
1,2 = β
2,3 = β
3,4 = β
4,5 = log(3) | 1 ---- 2 ---- 3 ---- 4 ---- 5 | |
1 association parameter is different
| ||
β
1,2 ≠ β
2,3 = β
3,4 = β
4,5 = log(3) | 1 - 2 ---- 3 ---- 4 ---- 5 1-------- 2 - 3 - 4 ---- 5 | |
β
2,3 ≠ β
1,2 = β
3,4 = β
4,5 = log(3) | 1 ---- 2 - 3 ---- 4 ---- 5 1 -- 2-------- 3 - 4 -- 5 | |
Β
3,4 ≠ β
1,2 = β
2,3 = β
4,5 = log(3) | 1------ 2 -- 3 - 4 -- 5 1 -- 2 - 3 ------ 4 - 5 | |
Β
4,5 ≠ β
1,2 = β
2,3 = β
3,4 = log(3) | 1 ---- 2 ---- 3 ---- 4 - 5 1 - 2 -- 3 - 4 ---------- 5 | |
2 association parameters are different
| ||
β
1,2 = β
2,3 ≠ β
3,4 = β
4,5 = log(3) | 1-2 - 3------4------5 1------2------ 3-4-5 | |
β
1,2 = β
4,3 ≠ β
2,3 = β
3,4 = log(3) | 1--2---- 3---- 4--5 1 ---- 2 - 3 - 4 ---- 5 | |
All association parameters are different
| ||
β
1,2 ≠ β
2,3 ≠ β
3,4 ≠ β
4,5
| 1 - 2 ---- 3 -- 4 ------ 5 |
Distribution of marginal probabilities
Description | |||||
---|---|---|---|---|---|
.20 | .20 | .20 | .20 | .20 | Homogeneous distribution |
.05 | .24 | .24 | .24 | .23 | Few counts in first category |
.24 | .05 | .24 | .24 | .23 | Few counts in intermediate category |
.24 | .24 | .05 | .24 | .23 | Few counts in central category |
.05 | .30 | .30 | .30 | .05 | Few counts in extreme categories |
.05 | .05 | .30 | .30 | .30 | Few counts in the first two adjacent categories |
.05 | .15 | .40 | .30 | .10 | Heterogeneous distribution |
Power and Type I error estimation
Results
N= | 50 | 100 | 150 | 200 | 250 | 50 | 100 | 150 | 200 | 250 | ||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
β
12
| OR | DD |
a. |
d. | ||||||||||
.00 | 1 | .00 | .34 | .57 | .74 |
.85
|
.92
| .21 | .30 | .43 | .54 | .63 | ||
.69 | 2 | .50 | .10 | .12 | .16 | .19 | .21 | .10 | .10 | .10 | .11 | .13 | ||
1.10 | 3 | .67 | .07 | .06 | .06 | .05 | .05 | .09 | .10 | .07 | .06 | .06 | ||
1.39 | 4 | .75 | .08 | .08 | .10 | .11 | .12 | .11 | .13 | .11 | .10 | .10 | ||
1.61 | 5 | .80 | .11 | .14 | .17 | .21 | .26 | .14 | .17 | .16 | .16 | .18 | ||
1.79 | 6 | .83 | .15 | .20 | .27 | .35 | .42 | .15 | .21 | .21 | .23 | .25 | ||
1.95 | 7 | .86 | .18 | .27 | .38 | .46 | .55 | .17 | .25 | .26 | .29 | .33 | ||
2.08 | 8 | .87 | .22 | .34 | .45 | .56 | .67 | .18 | .29 | .31 | .34 | .40 | ||
2.20 | 9 | .88 | .24 | .39 | .54 | .66 | .76 | .20 | .30 | .35 | .40 | .48 | ||
2.30 | 10 | .90 | .28 | .43 | .60 | .73 |
.82
| .21 | .35 | .40 | .44 | .52 | ||
2.48 | 12 | .92 | .33 | .53 | .70 |
.83
|
.89
| .23 | .39 | .45 | .52 | .61 | ||
2.64 | 14 | .93 | .38 | .61 | .78 |
.89
|
.95
| .26 | .43 | .52 | .58 | .67 | ||
2.77 | 16 | .94 | .42 | .67 |
.83
|
.92
|
.97
| .26 | .47 | .56 | .63 | .73 | ||
β
12, β
23
| OR | DD |
b. |
e. | ||||||||||
.00 | 1 | .00 | .73 |
.95
|
.99
|
1
|
1
| .59 |
.84
|
.96
|
.99
|
1
| ||
.69 | 2 | .50 | .14 | .21 | .28 | .35 | .42 | .13 | .17 | .24 | .29 | .37 | ||
1.10 | 3 | .67 | .06 | .06 | .05 | .05 | .05 | .09 | .07 | .06 | .06 | .06 | ||
1.39 | 4 | .75 | .09 | .10 | .13 | .15 | .18 | .12 | .13 | .14 | .15 | .17 | ||
1.61 | 5 | .80 | .13 | .19 | .26 | .33 | .39 | .17 | .20 | .26 | .31 | .36 | ||
1.79 | 6 | .83 | .19 | .29 | .40 | .50 | .59 | .21 | .27 | .36 | .43 | .51 | ||
1.95 | 7 | .86 | .22 | .37 | .51 | .64 | .74 | .26 | .34 | .46 | .55 | .66 | ||
2.08 | 8 | .87 | .27 | .47 | .62 | .74 |
.82
| .28 | .39 | .54 | .64 | .74 | ||
2.20 | 9 | .88 | .32 | .53 | .69 |
.81
|
.89
| .32 | .45 | .61 | .71 | .81
| ||
2.30 | 10 | .90 | .35 | .57 | .76 |
.87
|
.92
| .35 | .49 | .66 | .77 |
.85
| ||
2.48 | 12 | .92 | .41 | .67 |
.84
|
.92
|
.97
| .40 | .56 | .74 |
.84
|
.92
| ||
2.64 | 14 | .93 | .46 | .74 |
.89
|
.96
|
.99
| .44 | .61 | .79 |
.88
|
.95
| ||
2.77 | 16 | .94 | .50 | .79 |
.93
|
.97
|
.99
| .46 | .66 |
.84
|
.92
|
.97
| ||
β
12, β
45
| OR | DD |
c. |
f. | ||||||||||
.00 | 1 | .00 | .37 | .64 |
.80
|
.90
|
.95
| .18 | .32 | .45 | .57 | .67 | ||
.69 | 2 | .50 | .10 | .13 | .17 | .21 | .25 | .09 | .10 | .11 | .13 | .15 | ||
1.10 | 3 | .67 | .06 | .06 | .05 | .05 | .05 | .08 | .06 | .06 | .05 | .05 | ||
1.39 | 4 | .75 | .08 | .08 | .10 | .12 | .14 | .09 | .09 | .09 | .09 | .10 | ||
1.61 | 5 | .80 | .11 | .16 | .21 | .27 | .31 | .12 | .13 | .16 | .18 | .22 | ||
1.79 | 6 | .83 | .15 | .24 | .33 | .42 | .49 | .15 | .19 | .24 | .30 | .35 | ||
1.95 | 7 | .86 | .20 | .32 | .44 | .55 | .66 | .18 | .25 | .32 | .41 | .47 | ||
2.08 | 8 | .87 | .25 | .41 | .55 | .67 | .77 | .21 | .30 | .40 | .50 | .59 | ||
2.20 | 9 | .88 | .28 | .47 | .63 | .76 |
.84
| .23 | .35 | .47 | .58 | .67 | ||
2.30 | 10 | .90 | .32 | .53 | .71 |
.82
|
.90
| .27 | .40 | .53 | .65 | .74 | ||
2.48 | 12 | .92 | .38 | .64 |
.81
|
.90
|
.95
| .32 | .18 | .64 | .76 |
.84
| ||
2.64 | 14 | .93 | .45 | .71 |
.87
|
.95
|
.98
| .37 | .56 | .73 |
.84
|
.90
| ||
2.77 | 16 | .94 | .49 | .77 |
.91
|
.97
|
.99
| .40 | .61 | .78 |
.89
|
.94
|
N= | 50 | 100 | 150 | 200 | 250 |
H
1
11
| 50 | 100 | 150 | 200 | 250 | |||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
β
12
| OR | DD |
a. |
d. | ||||||||||
.00 | 1 | .00 | .22 | .38 | .51 | .59 | .72 | .14 | .16 | .21 | .26 | .32 | ||
.69 | 2 | .50 | .06 | .05 | .05 | .05 | .06 | .07 | .07 | .07 | .06 | .06 | ||
1.10 | 3 | .67 | .11 | .13 | .17 | .22 | .26 | .10 | .10 | .10 | .10 | .14 | ||
1.39 | 4 | .75 | .16 | .25 | .41 | .48 | .57 | .13 | .15 | .21 | .25 | .28 | ||
1.61 | 5 | .80 | .26 | .41 | .56 | .67 | .79 | .17 | .22 | .29 | .36 | .43 | ||
1.79 | 6 | .83 | .33 | .52 | .70 |
.82
|
.88
| .22 | .30 | .39 | .47 | .55 | ||
1.95 | 7 | .86 | .38 | .63 | .78 |
.92
|
.94
| .25 | .34 | .46 | .57 | .66 | ||
2.08 | 8 | .87 | .43 | .72 |
.85
|
.94
|
.98
| .30 | .41 | .56 | .62 | .74 | ||
2.20 | 9 | .88 | .47 | .76 |
.90
|
.97
|
.99
| .34 | .43 | .58 | .70 | .78 | ||
2.30 | 10 | .90 | .52 | .79 |
.94
|
.98
|
.99
| .38 | .51 | .67 | .74 |
.86
| ||
2.48 | 12 | .92 | .58 |
.85
|
.96
|
.99
|
1
| .39 | .55 | .71 |
.81
|
.91
| ||
2.64 | 14 | .93 | .64 |
.90
|
.97
|
1
|
1
| .41 | .58 | .78 |
.86
|
.93
| ||
2.77 | 16 | .94 | .69 |
.95
|
.99
|
1
|
1
| .46 | .62 |
.84
|
.88
|
.97
| ||
β
12, β
23
| OR | DD |
b. |
e. | ||||||||||
.00 | 1 | .00 | .43 | .74 |
.87
|
.96
|
.97
| .34 | .52 | .73 |
.86
|
.92
| ||
.69 | 2 | .50 | .06 | .06 | .05 | .05 | .06 | .08 | .07 | .06 | .06 | .05 | ||
1.10 | 3 | .67 | .12 | .20 | .28 | .37 | .44 | .16 | .19 | .28 | .35 | .40 | ||
1.39 | 4 | .75 | .24 | .41 | .57 | .66 | .78 | .28 | .41 | .54 | .62 | .74 | ||
1.61 | 5 | .80 | .34 | .57 | .78 |
.86
|
.93
| .36 | .52 | .71 |
.81
|
.89
| ||
1.79 | 6 | .83 | .42 | .69 |
.86
|
.95
|
.97
| .40 | .62 |
.81
|
.89
|
.94
| ||
1.95 | 7 | .86 | .51 | .78 |
.90
|
.97
|
.98
| .48 | .71 |
.89
|
.94
|
.98
| ||
2.08 | 8 | .87 | .53 |
.85
|
.95
|
.98
|
1
| .54 | .76 |
.93
|
.96
|
.99
| ||
2.20 | 9 | .88 | .62 |
.88
|
.97
|
1
|
1
| .55 |
.80
|
.94
|
.97
|
.99
| ||
2.30 | 10 | .90 | .64 |
.90
|
.97
|
1
|
1
| .57 |
.85
|
.96
|
.98
|
1
| ||
2.48 | 12 | .92 | .69 |
.93
|
.99
|
1
|
1
| .65 |
.86
|
.98
|
.99
|
1
| ||
2.64 | 14 | .93 | .73 |
.96
|
.99
|
1
|
1
| .67 |
.87
|
.97
|
1
|
1
| ||
2.77 | 16 | .94 | .77 |
.98
|
.99
|
1
|
1
| .71 |
.93
|
.99
|
1
|
1
| ||
β
12, β
45
| OR | DD |
c. |
f. | ||||||||||
.00 | 1 | .00 | .20 | .37 | .52 | .63 | .74 | .10 | .16 | .27 | .32 | .37 | ||
.69 | 2 | .50 | .07 | .05 | .05 | .05 | .04 | .06 | .07 | .05 | .05 | .05 | ||
1.10 | 3 | .67 | .11 | .12 | .16 | .23 | .26 | .10 | .10 | .13 | .14 | .18 | ||
1.39 | 4 | .75 | .19 | .32 | .44 | .52 | .61 | .15 | .21 | .28 | .34 | .38 | ||
1.61 | 5 | .80 | .26 | .46 | .62 | .74 |
.82
| .21 | .32 | .42 | .52 | .60 | ||
1.79 | 6 | .83 | .35 | .58 | .75 |
.88
|
.94
| .25 | .42 | .54 | .71 | .77 | ||
1.95 | 7 | .86 | .45 | .68 |
.85
|
.93
|
.97
| .31 | .49 | .65 | .79 |
.84
| ||
2.08 | 8 | .87 | .44 | .74 |
.92
|
.96
|
1
| .36 | .55 | .73 |
.86
|
.91
| ||
2.20 | 9 | .88 | .55 |
.84
|
.94
|
.98
|
.99
| .41 | .61 | .78 |
.89
|
.94
| ||
2.30 | 10 | .90 | .58 |
.87
|
.96
|
.99
|
1
| .47 | .67 |
.85
|
.92
|
.96
| ||
2.48 | 12 | .92 | .66 |
.91
|
.99
|
.99
|
1
| .52 | .77 |
.91
|
.97
|
.98
| ||
2.64 | 14 | .93 | .70 |
.94
|
.98
|
1
|
1
| .53 |
.82
|
.93
|
.98
|
1
| ||
2.77 | 16 | .94 | .74 |
.95
|
.99
|
1
|
1
| .58 |
.85
|
.96
|
.98
|
1
|
N= | 50 | 100 | 150 | 200 | 250 |
H
1
21
| 50 | 100 | 150 | 200 | 250 | |||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
β
12
| OR | DD |
a. |
d. | ||||||||||
.00 | 1 | .00 | .40 | .66 |
.82
|
.92
|
.97
| .21 | .31 | .43 | .50 | .65 | ||
.69 | 2 | .50 | .16 | .22 | .31 | .35 | .45 | .11 | .12 | .16 | .18 | .24 | ||
1.10 | 3 | .67 | .07 | .09 | .08 | .12 | .12 | .09 | .08 | .07 | .08 | .08 | ||
1.39 | 4 | .75 | .06 | .05 | .05 | .04 | .06 | .08 | .07 | .07 | .06 | .05 | ||
1.61 | 5 | .80 | .08 | .06 | .06 | .08 | .08 | .08 | .08 | .08 | .07 | .06 | ||
1.79 | 6 | .83 | .09 | .10 | .12 | .14 | .16 | .07 | .09 | .10 | .10 | .11 | ||
1.95 | 7 | .86 | .11 | .11 | .17 | .22 | .25 | .12 | .12 | .12 | .13 | .15 | ||
2.08 | 8 | .87 | .13 | .17 | .24 | .31 | .36 | .11 | .14 | .15 | .18 | .23 | ||
2.20 | 9 | .88 | .15 | .21 | .30 | .37 | .45 | .13 | .17 | .20 | .23 | .27 | ||
2.30 | 10 | .90 | .18 | .24 | .34 | .43 | .52 | .14 | .19 | .25 | .24 | .30 | ||
2.48 | 12 | .92 | .21 | .35 | .45 | .58 | .67 | .19 | .21 | .31 | .32 | .40 | ||
2.64 | 14 | .93 | .24 | .38 | .53 | .66 | .77 | .19 | .26 | .33 | .42 | .49 | ||
2.77 | 16 | .94 | .29 | .46 | .61 | .76 |
.84
| .20 | .28 | .39 | .45 | .55 | ||
β
12, β
23
| OR | DD |
b. |
e. | ||||||||||
.00 | 1 | .00 |
.85
|
.99
|
1
|
1
|
1
| .71 |
.92
|
.99
|
1
|
1
| ||
.69 | 2 | .50 | .23 | .43 | .60 | .67 | .76 | .22 | .34 | .50 | .61 | .72 | ||
1.10 | 3 | .67 | .09 | .12 | .13 | .15 | .18 | .11 | .10 | .11 | .11 | .15 | ||
1.39 | 4 | .75 | .05 | .05 | .05 | .06 | .05 | .07 | .06 | .08 | .06 | .05 | ||
1.61 | 5 | .80 | .07 | .08 | .11 | .10 | .10 | .11 | .08 | .10 | .10 | .11 | ||
1.79 | 6 | .83 | .11 | .11 | .15 | .21 | .22 | .16 | .13 | .17 | .17 | .20 | ||
1.95 | 7 | .86 | .14 | .18 | .25 | .28 | .35 | .16 | .19 | .21 | .29 | .30 | ||
2.08 | 8 | .87 | .14 | .22 | .31 | .40 | .45 | .18 | .23 | .27 | .30 | .42 | ||
2.20 | 9 | .88 | .17 | .29 | .41 | .50 | .57 | .23 | .26 | .36 | .39 | .49 | ||
2.30 | 10 | .90 | .23 | .33 | .49 | .56 | .69 | .23 | .30 | .40 | .46 | .53 | ||
2.48 | 12 | .92 | .25 | .41 | .59 | .73 |
.81
| .28 | .33 | .49 | .57 | .67 | ||
2.64 | 14 | .93 | .29 | .51 | .67 | .79 |
.86
| .32 | .42 | .55 | .65 | .76 | ||
2.77 | 16 | .94 | .30 | .56 | .75 |
.86
|
.92
| .35 | .47 | .60 | .71 |
.81
| ||
β
12, β
45
| OR | DD |
c. |
f. | ||||||||||
.00 | 1 | .00 | .45 | .77 |
.90
|
.97
|
.99
| .26 | .47 | .60 | .71 |
.82
| ||
.69 | 2 | .50 | .14 | .26 | .36 | .44 | .50 | .11 | .15 | .21 | .25 | .32 | ||
1.10 | 3 | .67 | .08 | .09 | .10 | .13 | .12 | .08 | .09 | .08 | .09 | .10 | ||
1.39 | 4 | .75 | .05 | .06 | .05 | .05 | .06 | .08 | .06 | .06 | .06 | .06 | ||
1.61 | 5 | .80 | .06 | .07 | .08 | .07 | .09 | .10 | .08 | .09 | .08 | .07 | ||
1.79 | 6 | .83 | .08 | .12 | .14 | .16 | .21 | .12 | .11 | .12 | .14 | .14 | ||
1.95 | 7 | .86 | .12 | .15 | .20 | .24 | .33 | .14 | .14 | .17 | .19 | .23 | ||
2.08 | 8 | .87 | .13 | .22 | .28 | .38 | .44 | .14 | .17 | .22 | .28 | .33 | ||
2.20 | 9 | .88 | .20 | .25 | .36 | .47 | .56 | .17 | .20 | .27 | .35 | .41 | ||
2.30 | 10 | .90 | .20 | .31 | .46 | .55 | .66 | .17 | .22 | .33 | .41 | .49 | ||
2.48 | 12 | .92 | .26 | .38 | .54 | .67 |
.80
| .23 | .31 | .43 | .50 | .60 | ||
2.64 | 14 | .93 | .28 | .50 | .64 | .77 |
.87
| .26 | .42 | .51 | .63 | .73 | ||
2.77 | 16 | .94 | .35 | .55 | .75 |
.87
|
.90
| .29 | .45 | .61 | .70 | .79 |