Table 4.
Bias detection with the model difference procedure.
| Cond. | Biased itemsa | Unbiased itemsb | ||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Conv. | Δ DIC | TP | Conv. | Δ DIC | FP | |||||||||
| Q05 | Q50 | Q95 | Strict | Liberal | Q05 | Q50 | Q95 | Strict | Liberal | |||||
| CONTINUOUS LATENT VIOLATOR | ||||||||||||||
| ρ(T,V) = 0.0 | No bias | 1 | – | – | – | – | – | – | 0.93 | −10 | 0 | 10 | 0.102 | 0.007 |
| Uniform | 2 | 0.59 | −100 | −60 | −40 | 1.000 | 1.000 | 0.86 | −10 | 0 | 10 | 0.255 | 0.042 | |
| Nonuniform | 3 | 0.93 | −110 | −70 | −30 | 1.000 | 0.989 | 0.92 | −10 | 0 | 10 | 0.219 | 0.032 | |
| Both | 4 | 0.65 | −188 | −120 | −82 | 1.000 | 1.000 | 0.88 | −20 | 0 | 10 | 0.305 | 0.068 | |
| ρ(T,V) = 0.5 | No bias | 5 | – | – | – | – | – | – | 0.94 | −10 | 0 | 10 | 0.115 | 0.012 |
| Uniform | 6 | 0.93 | −80 | −40 | −20 | 1.000 | 0.978 | 0.93 | −10 | 0 | 10 | 0.263 | 0.045 | |
| Nonuniform | 7 | 0.96 | −140 | −80 | −40 | 1.000 | 1.000 | 0.95 | −10 | 0 | 10 | 0.213 | 0.023 | |
| Both | 8 | 0.92 | −190 | −120 | −70 | 1.000 | 1.000 | 0.94 | −20 | 0 | 10 | 0.383 | 0.097 | |
| CONTINUOUS OBSERVED VIOLATOR | ||||||||||||||
| ρ(T,V) = 0.0 | No bias | 9 | – | – | – | – | – | – | 0.99 | −4 | 1 | 5 | 0.232 | 0.002 |
| Uniform | 10 | 0.91 | −100 | −67 | −47 | 1.000 | 1.000 | 0.99 | −9 | 0 | 5 | 0.485 | 0.028 | |
| Nonuniform | 11 | 0.92 | −107 | −67 | −40 | 1.000 | 1.000 | 0.99 | −9 | 0 | 4 | 0.451 | 0.032 | |
| Both | 12 | 0.92 | −173 | −137 | −92 | 1.000 | 1.000 | 0.99 | −12 | −1 | 4 | 0.578 | 0.073 | |
| ρ(T,V) = 0.5 | No bias | 13 | – | – | – | – | – | – | 0.77 | −5 | 2 | 5 | 0.274 | 0.009 |
| Uniform | 14 | 0.52 | −90 | −50 | −27 | 1.000 | 1.000 | 0.61 | −13 | −1 | 4 | 0.529 | 0.082 | |
| Nonuniform | 15 | 0.73 | −121 | −81 | −47 | 1.000 | 1.000 | 0.76 | −10 | 0 | 4 | 0.479 | 0.037 | |
| Both | 16 | 0.49 | −167 | −125 | −94 | 1.000 | 1.000 | 0.64 | −16 | −3 | 4 | 0.672 | 0.172 | |
| DICHOTOMIZED OBSERVED VIOLATOR | ||||||||||||||
| ρ(T,V) = 0.0 | No bias | 17 | – | – | – | – | – | – | 1.00 | −4 | 1 | 5 | 0.239 | 0.008 |
| Uniform | 18 | 0.99 | −66 | −42 | −23 | 1.000 | 1.000 | 1.00 | −9 | 0 | 4 | 0.390 | 0.032 | |
| Nonuniform | 19 | 1.00 | −71 | −43 | −20 | 1.000 | 0.980 | 0.99 | −7 | 0 | 4 | 0.401 | 0.028 | |
| Both | 20 | 0.98 | −115 | −84 | −51 | 1.000 | 1.000 | 0.99 | −10 | 0 | 4 | 0.462 | 0.044 | |
| ρ(T,V) = 0.5 | No bias | 21 | – | – | – | – | – | – | 0.94 | −5 | 1 | 5 | 0.266 | 0.005 |
| Uniform | 22 | 0.70 | −53 | −25 | −10 | 1.000 | 0.943 | 0.86 | −8 | 0 | 4 | 0.456 | 0.019 | |
| Nonuniform | 23 | 0.93 | −79 | −47 | −21 | 1.000 | 1.000 | 0.93 | −8 | 1 | 4 | 0.392 | 0.017 | |
| Both | 24 | 0.66 | −116 | −81 | −55 | 1.000 | 1.000 | 0.83 | −13 | −1 | 4 | 0.550 | 0.099 | |
Cond., Condition; Conv., proportion of converged solutions; Δ DIC denotes the difference in DIC between the reference model and the competing model;
a
Quantile DIC difference values (i.e., 05, 50, 95), and proportions of true positives (TP) are calculated over the converged solutions [of 1 (biased item) × 100 (replicates) = 100 solutions];
b
Quantile DIC values (i.e., 05, 50, 95), and proportions of false positives (FP) are calculated over the converged solutions, which are 6 (non-biased items) × 100 (replicates) = 600 solutions in Conditions 1, 5, 9, 13, 17, and 21, and 5 (non-biased items) × 100 (replicates) = 500 solutions in all other conditions.