Acta Psychologica Sinica ›› 2022, Vol. 54 ›› Issue (11): 1416-1432.doi: 10.3724/SP.J.1041.2022.01416
• Reports of Empirical Studies • Previous Articles
Received:
2021-06-10
Published:
2022-11-25
Online:
2022-09-08
Contact:
ZHAN Peida
E-mail:pdzhan@gmail.com
Supported by:
ZHAN Peida. (2022). Joint-cross-loading multimodal cognitive diagnostic modeling incorporating visual fixation counts. Acta Psychologica Sinica, 54(11), 1416-1432.
Figure 1. Schematic diagram of multimodal joint modeling (taking RA and RT as an example). Note. θ = latent ability; τ = latent processing speed; α = latent attribute; Y = response accuracy; T= response time; I = number of items; K = number of attributes; IRT = item response theory.
Figure 2. Schematic Diagram of Joint-Hierarchical and Joint-Cross-Loading Multimodal Cognitive Diagnosis Modeling. Note. θ = latent ability; τ = latent processing speed; ε = latent visual engagement; α = latent attribute; Y = response accuracy; T = response time; V = fixation count; I = number of items; K = number of attribute.
φi | λi | |
---|---|---|
> 0 | < 0 | |
> 0 | Much key information and low cognitive load requirements: If f(θn, αn, qi) ↑, then RT↓, FC↑; If f(θn, αn, qi) ↓, then RT↑, FC↓; | Less key information and low cognitive load requirements: If f(θn, αn, qi) ↑, then RT↓, FC↓; If f(θn, αn, qi) ↓, then RT↑, FC↑; |
< 0 | Much key information and high cognitive load requirements: If f(θn, αn, qi) ↑, then RT↑, FC↑; If f(θn, αn, qi) ↓, then RT↓, FC↓; | Less key information and high cognitive load requirements: If f(θn, αn, qi) ↑, then RT↑, FC↓; If f(θn, αn, qi) ↓, then RT↓, FC↑; |
Table 1 The item information that the positive and negative values of φi and λi parameters in C-MCDM may reflect.
φi | λi | |
---|---|---|
> 0 | < 0 | |
> 0 | Much key information and low cognitive load requirements: If f(θn, αn, qi) ↑, then RT↓, FC↑; If f(θn, αn, qi) ↓, then RT↑, FC↓; | Less key information and low cognitive load requirements: If f(θn, αn, qi) ↑, then RT↓, FC↓; If f(θn, αn, qi) ↓, then RT↑, FC↑; |
< 0 | Much key information and high cognitive load requirements: If f(θn, αn, qi) ↑, then RT↑, FC↑; If f(θn, αn, qi) ↓, then RT↓, FC↓; | Less key information and high cognitive load requirements: If f(θn, αn, qi) ↑, then RT↑, FC↓; If f(θn, αn, qi) ↓, then RT↓, FC↑; |
θ | τ | ε | Corresponding Inference | Possible Cause or Behavior |
---|---|---|---|---|
+ | + | + | Fluency + Focuser | Able to extract key information from items comprehensively |
+ | + | - | Fluency + Non-focuser | Able to eliminate distractions and extract rare key information from items |
+ | - | + | Reflective + Focuser | Deliberate on possible solutions based on the key information extracted |
+ | - | - | Reflective + Non-focuser | Able to eliminate distractions and deliberate based on a few key information |
- | + | + | Impulsive + Focuser | Quickly extract some information and quickly validate the solution |
- | + | - | Impulsive + Non-focuser | Under-extracting information and tending to make rapid-guessing |
- | - | + | Non-fluency + Focuser | Motivated to respond but has the low cognitive ability |
- | - | - | Non-fluency + Non-focuser | Unmotivated |
Table 2 Comprehensive categories of 8 cognitive characteristics and possible causes or behaviors (Zhan et al., 2022).
θ | τ | ε | Corresponding Inference | Possible Cause or Behavior |
---|---|---|---|---|
+ | + | + | Fluency + Focuser | Able to extract key information from items comprehensively |
+ | + | - | Fluency + Non-focuser | Able to eliminate distractions and extract rare key information from items |
+ | - | + | Reflective + Focuser | Deliberate on possible solutions based on the key information extracted |
+ | - | - | Reflective + Non-focuser | Able to eliminate distractions and deliberate based on a few key information |
- | + | + | Impulsive + Focuser | Quickly extract some information and quickly validate the solution |
- | + | - | Impulsive + Non-focuser | Under-extracting information and tending to make rapid-guessing |
- | - | + | Non-fluency + Focuser | Motivated to respond but has the low cognitive ability |
- | - | - | Non-fluency + Non-focuser | Unmotivated |
Model | -2LL | DIC | ppp_RA | ppp_RT | ppp_FC |
---|---|---|---|---|---|
S-MCDM | 11104.13 | 11890.70 | 0.42 | 0.52 | 0.78 |
H-MCDM | 11040.62 | 11804.00 | 0.42 | 0.51 | 0.57 |
C-MCDM-θ | 10275.27 | 11542.78 | 0.43 | 0.51 | 0.68 |
C-MCDM-D | 10127.21 | 10549.14 | 0.40 | 0.52 | 0.64 |
C-MCDM-C | 10009.65 | 10434.95 | 0.36 | 0.51 | 0.67 |
Table 3 Model-Data Fitting in Empirical Data Analysis.
Model | -2LL | DIC | ppp_RA | ppp_RT | ppp_FC |
---|---|---|---|---|---|
S-MCDM | 11104.13 | 11890.70 | 0.42 | 0.52 | 0.78 |
H-MCDM | 11040.62 | 11804.00 | 0.42 | 0.51 | 0.57 |
C-MCDM-θ | 10275.27 | 11542.78 | 0.43 | 0.51 | 0.68 |
C-MCDM-D | 10127.21 | 10549.14 | 0.40 | 0.52 | 0.64 |
C-MCDM-C | 10009.65 | 10434.95 | 0.36 | 0.51 | 0.67 |
Item | Attribute Pattern | Response Accuracy | Response Time | Fixation counts | φ | λ | |||
---|---|---|---|---|---|---|---|---|---|
g | s | ξ | ω | m | d | ||||
1 | (1010) | 0.71 (0.23) | 0.06 (0.04) | 2.89 (0.04) | 3.78 (0.340) | 4.23 (0.04) | 0.08 (0.011) | 0.25 (0.05) | -0.22 (0.04) |
2 | (1100) | 0.60 (0.22) | 0.05 (0.04) | 3.43 (0.05) | 2.56 (0.20) | 4.85 (0.04) | 0.03 (0.002) | -0.07 (0.06) | 0.13 (0.04) |
3 | (1100) | 0.33 (0.13) | 0.38 (0.11) | 2.60 (0.06) | 2.65 (0.25) | 3.87 (0.05) | 0.07 (0.010) | 0.43 (0.07) | -0.40 (0.05) |
4 | (0100) | 0.31 (0.21) | 0.18 (0.08) | 3.34 (0.05) | 2.56 (0.20) | 4.68 (0.04) | 0.03 (0.003) | -0.11 (0.06) | 0.15 (0.04) |
5 | (0010) | 0.49 (0.20) | 0.25 (0.07) | 3.13 (0.07) | 1.83 (0.14) | 4.50 (0.05) | 0.03 (0.002) | 0.21 (0.08) | -0.18 (0.06) |
6 | (0100) | 0.09 (0.04) | 0.86 (0.07) | 3.08 (0.08) | 3.14 (0.42) | 4.42 (0.07) | 0.05 (0.012) | -0.65 (0.08) | 0.61 (0.07) |
7 | (0001) | 0.43 (0.19) | 0.22 (0.11) | 3.74 (0.06) | 1.98 (0.16) | 5.06 (0.05) | 0.02 (0.001) | -0.22 (0.07) | 0.19 (0.06) |
8 | (1001) | 0.15 (0.08) | 0.61 (0.16) | 3.92 (0.06) | 2.88 (0.29) | 5.18 (0.06) | 0.02 (0.002) | -0.43 (0.07) | 0.41 (0.06) |
9 | (0001) | 0.44 (0.13) | 0.36 (0.10) | 3.41 (0.10) | 1.22 (0.10) | 4.82 (0.07) | 0.01 (0.001) | -0.07 (0.12) | 0.17 (0.08) |
10 | (1001) | 0.11 (0.06) | 0.73 (0.15) | 3.80 (0.12) | 1.13 (0.10) | 5.18 (0.09) | 0.01 (0.001) | 0.31 (0.14) | -0.29 (0.10) |
Table 4 Estimates of Item Parameters of C-MCDM-θ Model in Empirical Data Analysis.
Item | Attribute Pattern | Response Accuracy | Response Time | Fixation counts | φ | λ | |||
---|---|---|---|---|---|---|---|---|---|
g | s | ξ | ω | m | d | ||||
1 | (1010) | 0.71 (0.23) | 0.06 (0.04) | 2.89 (0.04) | 3.78 (0.340) | 4.23 (0.04) | 0.08 (0.011) | 0.25 (0.05) | -0.22 (0.04) |
2 | (1100) | 0.60 (0.22) | 0.05 (0.04) | 3.43 (0.05) | 2.56 (0.20) | 4.85 (0.04) | 0.03 (0.002) | -0.07 (0.06) | 0.13 (0.04) |
3 | (1100) | 0.33 (0.13) | 0.38 (0.11) | 2.60 (0.06) | 2.65 (0.25) | 3.87 (0.05) | 0.07 (0.010) | 0.43 (0.07) | -0.40 (0.05) |
4 | (0100) | 0.31 (0.21) | 0.18 (0.08) | 3.34 (0.05) | 2.56 (0.20) | 4.68 (0.04) | 0.03 (0.003) | -0.11 (0.06) | 0.15 (0.04) |
5 | (0010) | 0.49 (0.20) | 0.25 (0.07) | 3.13 (0.07) | 1.83 (0.14) | 4.50 (0.05) | 0.03 (0.002) | 0.21 (0.08) | -0.18 (0.06) |
6 | (0100) | 0.09 (0.04) | 0.86 (0.07) | 3.08 (0.08) | 3.14 (0.42) | 4.42 (0.07) | 0.05 (0.012) | -0.65 (0.08) | 0.61 (0.07) |
7 | (0001) | 0.43 (0.19) | 0.22 (0.11) | 3.74 (0.06) | 1.98 (0.16) | 5.06 (0.05) | 0.02 (0.001) | -0.22 (0.07) | 0.19 (0.06) |
8 | (1001) | 0.15 (0.08) | 0.61 (0.16) | 3.92 (0.06) | 2.88 (0.29) | 5.18 (0.06) | 0.02 (0.002) | -0.43 (0.07) | 0.41 (0.06) |
9 | (0001) | 0.44 (0.13) | 0.36 (0.10) | 3.41 (0.10) | 1.22 (0.10) | 4.82 (0.07) | 0.01 (0.001) | -0.07 (0.12) | 0.17 (0.08) |
10 | (1001) | 0.11 (0.06) | 0.73 (0.15) | 3.80 (0.12) | 1.13 (0.10) | 5.18 (0.09) | 0.01 (0.001) | 0.31 (0.14) | -0.29 (0.10) |
Participants | α1 | α2 | α3 | α4 | θ | τ | ε | Cognitive Characteristics Inference |
---|---|---|---|---|---|---|---|---|
5 | 1 | 1 | 1 | 1 | 0.12 (0.25) | -0.17 (0.11) | 0.20 (0.09) | Reflective + Focuser |
9 | 1 | 1 | 1 | 1 | 0.31 (0.23) | 0.05 (0.11) | 0.06 (0.08) | Fluency + Focuser |
34 | 1 | 0 | 1 | 0 | -0.19 (0.26) | 0.20 (0.11) | -0.06 (0.08) | Impulsive + Non-focuser |
65 | 1 | 1 | 1 | 1 | 0.25 (0.24) | 0.20 (0.11) | -0.22 (0.09) | Fluency + Non-focuser |
67 | 1 | 0 | 1 | 0 | -1.52 (0.27) | -0.18 (0.11) | 0.10 (0.09) | Non-fluency + Focuser |
76 | 1 | 1 | 1 | 0 | 0.56 (0.25) | -0.01 (0.11) | -0.12 (0.09) | Reflective + Non-focuser |
Table 5 Examples of individual cognitive structure diagnosis and other cognitive characteristics inference in empirical data.
Participants | α1 | α2 | α3 | α4 | θ | τ | ε | Cognitive Characteristics Inference |
---|---|---|---|---|---|---|---|---|
5 | 1 | 1 | 1 | 1 | 0.12 (0.25) | -0.17 (0.11) | 0.20 (0.09) | Reflective + Focuser |
9 | 1 | 1 | 1 | 1 | 0.31 (0.23) | 0.05 (0.11) | 0.06 (0.08) | Fluency + Focuser |
34 | 1 | 0 | 1 | 0 | -0.19 (0.26) | 0.20 (0.11) | -0.06 (0.08) | Impulsive + Non-focuser |
65 | 1 | 1 | 1 | 1 | 0.25 (0.24) | 0.20 (0.11) | -0.22 (0.09) | Fluency + Non-focuser |
67 | 1 | 0 | 1 | 0 | -1.52 (0.27) | -0.18 (0.11) | 0.10 (0.09) | Non-fluency + Focuser |
76 | 1 | 1 | 1 | 0 | 0.56 (0.25) | -0.01 (0.11) | -0.12 (0.09) | Reflective + Non-focuser |
Figure 7. Attribute (pattern) correct classification rate of three C-MCDMs in Simulation Study 1. Note. N = sample size; I = test length; CL = cross-loading; ACCR = attribute correct classification rate; PCCR = attribute pattern correct classification rate.
Figure 8. Recovery of latent ability, latent processing speed, and latent visual engagement of three C-MCDM in simulation study 1. Note. N = sample size; I = test length; CL = cross-loading; θ = high-level latent ability; τ = latent processing speed; ε = latent visual engagement; RMSE = root mean square error; Cor = the correlation coefficient between the estimated value and the true value.
Data generation model | Data analysis model | DIC |
---|---|---|
C-MCDM-θ | H-MCDM | 186481.5 |
C-MCDM-θ | 181029.5 | |
C-MCDM-D | H-MCDM | 183774.8 |
C-MCDM-D | 181720.3 | |
C-MCDM-C | H-MCDM | 189775.6 |
C-MCDM-C | 183881.5 | |
H-MCDM | H-MCDM | 180286.2 |
C-MCDM-θ | 180395.0 | |
C-MCDM-D | 180347.8 | |
C-MCDM-C | 180351.9 |
Table 6 Model-data fitting in simulation Study 2.
Data generation model | Data analysis model | DIC |
---|---|---|
C-MCDM-θ | H-MCDM | 186481.5 |
C-MCDM-θ | 181029.5 | |
C-MCDM-D | H-MCDM | 183774.8 |
C-MCDM-D | 181720.3 | |
C-MCDM-C | H-MCDM | 189775.6 |
C-MCDM-C | 183881.5 | |
H-MCDM | H-MCDM | 180286.2 |
C-MCDM-θ | 180395.0 | |
C-MCDM-D | 180347.8 | |
C-MCDM-C | 180351.9 |
Data generation model | Data analysis model | ACCR | PCCR | ||||
---|---|---|---|---|---|---|---|
A1 | A2 | A3 | A4 | A5 | |||
C-MCDM-θ | H-MCDM | 0.957 | 0.966 | 0.967 | 0.975 | 0.977 | 0.862 |
C-MCDM-θ | 0.961 | 0.969 | 0.969 | 0.977 | 0.979 | 0.874 | |
C-MCDM-D | H-MCDM | 0.953 | 0.966 | 0.966 | 0.971 | 0.982 | 0.860 |
C-MCDM-D | 0.960 | 0.970 | 0.984 | 0.980 | 0.987 | 0.894 | |
C-MCDM-C | H-MCDM | 0.955 | 0.961 | 0.969 | 0.970 | 0.980 | 0.858 |
C-MCDM-C | 0.991 | 0.990 | 0.988 | 0.990 | 0.995 | 0.958 | |
H-MCDM | H-MCDM | 0.960 | 0.958 | 0.967 | 0.977 | 0.977 | 0.863 |
C-MCDM-θ | 0.960 | 0.957 | 0.967 | 0.977 | 0.977 | 0.863 | |
C-MCDM-D | 0.957 | 0.958 | 0.967 | 0.977 | 0.976 | 0.861 | |
C-MCDM-C | 0.957 | 0.957 | 0.966 | 0.976 | 0.976 | 0.859 |
Table 7 The attribute (pattern) correct classification rate in simulation Study 2.
Data generation model | Data analysis model | ACCR | PCCR | ||||
---|---|---|---|---|---|---|---|
A1 | A2 | A3 | A4 | A5 | |||
C-MCDM-θ | H-MCDM | 0.957 | 0.966 | 0.967 | 0.975 | 0.977 | 0.862 |
C-MCDM-θ | 0.961 | 0.969 | 0.969 | 0.977 | 0.979 | 0.874 | |
C-MCDM-D | H-MCDM | 0.953 | 0.966 | 0.966 | 0.971 | 0.982 | 0.860 |
C-MCDM-D | 0.960 | 0.970 | 0.984 | 0.980 | 0.987 | 0.894 | |
C-MCDM-C | H-MCDM | 0.955 | 0.961 | 0.969 | 0.970 | 0.980 | 0.858 |
C-MCDM-C | 0.991 | 0.990 | 0.988 | 0.990 | 0.995 | 0.958 | |
H-MCDM | H-MCDM | 0.960 | 0.958 | 0.967 | 0.977 | 0.977 | 0.863 |
C-MCDM-θ | 0.960 | 0.957 | 0.967 | 0.977 | 0.977 | 0.863 | |
C-MCDM-D | 0.957 | 0.958 | 0.967 | 0.977 | 0.976 | 0.861 | |
C-MCDM-C | 0.957 | 0.957 | 0.966 | 0.976 | 0.976 | 0.859 |
Figure S1. Attribute (pattern) correct classification rate of the three models under prior distributions with different levels of information. Note. N = sample size; I = test length; CL = cross-loading; ACCR = attribute correct classification rate; PCCR = attribute pattern correct classification rate.
Figure S2. The recovery of the latent ability, latent processing speed, and latent visual engagement of the three models under prior distributions with different levels of information. Note. N = sample size; I = test length; CL = cross-loading; θ = higher-order latent ability; τ = latent processing speed; ε = latent visual engagement; RMSE = root mean square error; Cor = the correlation coefficient between the estimated value and the true value.
Model | Amount of information contained in the prior distribution | g | s | ζ | ω | m | d | φ | λ | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Bias | RMSE | Cor | Bias | RMSE | Cor | Bias | RMSE | Cor | Bias | RMSE | Cor | Bias | RMSE | Cor | Bias | RMSE | Cor | Bias | RMSE | Bias | RMSE | ||
C-MCDM-θ | Low | 0.016 | 0.051 | 0.875 | 0.055 | 0.086 | 0.952 | -0.063 | 0.088 | 0.993 | -0.332 | 0.355 | 0.878 | -0.067 | 0.095 | 0.992 | 0.001 | 0.004 | 0.985 | -0.042 | 0.112 | 0.049 | 0.134 |
Medium | 0.014 | 0.050 | 0.875 | 0.056 | 0.086 | 0.952 | 0.002 | 0.062 | 0.993 | -0.015 | 0.121 | 0.869 | 0.000 | 0.067 | 0.992 | 0.000 | 0.004 | 0.985 | 0.152 | 0.182 | -0.148 | 0.188 | |
High | -0.010 | 0.046 | 0.882 | -0.022 | 0.078 | 0.954 | -0.002 | 0.061 | 0.993 | 0.003 | 0.100 | 0.890 | -0.003 | 0.065 | 0.992 | 0.000 | 0.003 | 0.985 | -0.023 | 0.100 | 0.026 | 0.108 | |
C-MCDM-D | Low | 0.021 | 0.049 | 0.914 | 0.084 | 0.127 | 0.930 | -0.113 | 0.257 | 0.913 | -0.233 | 0.258 | 0.822 | -0.123 | 0.227 | 0.912 | 0.000 | 0.003 | 0.982 | -0.066 | 0.218 | 0.085 | 0.195 |
Medium | 0.014 | 0.045 | 0.923 | 0.033 | 0.069 | 0.959 | 0.005 | 0.086 | 0.986 | -0.013 | 0.124 | 0.825 | 0.009 | 0.073 | 0.989 | 0.000 | 0.003 | 0.982 | 0.029 | 0.152 | -0.023 | 0.132 | |
High | -0.007 | 0.041 | 0.930 | -0.028 | 0.080 | 0.956 | -0.004 | 0.085 | 0.986 | 0.013 | 0.116 | 0.826 | 0.003 | 0.072 | 0.989 | 0.000 | 0.003 | 0.982 | -0.012 | 0.150 | 0.006 | 0.129 | |
C-MCDM-C | Low | 0.009 | 0.041 | 0.928 | 0.042 | 0.080 | 0.941 | -0.036 | 0.122 | 0.977 | -0.334 | 0.356 | 0.869 | -0.066 | 0.166 | 0.952 | 0.001 | 0.003 | 0.987 | 0.001 | 0.096 | 0.018 | 0.128 |
Medium | 0.009 | 0.040 | 0.929 | 0.035 | 0.073 | 0.947 | 0.009 | 0.106 | 0.981 | -0.021 | 0.121 | 0.864 | -0.001 | 0.100 | 0.981 | 0.000 | 0.003 | 0.987 | 0.014 | 0.096 | -0.012 | 0.092 | |
High | -0.009 | 0.040 | 0.931 | -0.024 | 0.073 | 0.949 | -0.003 | 0.103 | 0.981 | 0.004 | 0.103 | 0.874 | -0.011 | 0.098 | 0.981 | 0.000 | 0.003 | 0.987 | -0.006 | 0.095 | 0.004 | 0.091 |
Table S1 The recovery of the item parameters of the three models under the prior distributions of different levels of information.
Model | Amount of information contained in the prior distribution | g | s | ζ | ω | m | d | φ | λ | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Bias | RMSE | Cor | Bias | RMSE | Cor | Bias | RMSE | Cor | Bias | RMSE | Cor | Bias | RMSE | Cor | Bias | RMSE | Cor | Bias | RMSE | Bias | RMSE | ||
C-MCDM-θ | Low | 0.016 | 0.051 | 0.875 | 0.055 | 0.086 | 0.952 | -0.063 | 0.088 | 0.993 | -0.332 | 0.355 | 0.878 | -0.067 | 0.095 | 0.992 | 0.001 | 0.004 | 0.985 | -0.042 | 0.112 | 0.049 | 0.134 |
Medium | 0.014 | 0.050 | 0.875 | 0.056 | 0.086 | 0.952 | 0.002 | 0.062 | 0.993 | -0.015 | 0.121 | 0.869 | 0.000 | 0.067 | 0.992 | 0.000 | 0.004 | 0.985 | 0.152 | 0.182 | -0.148 | 0.188 | |
High | -0.010 | 0.046 | 0.882 | -0.022 | 0.078 | 0.954 | -0.002 | 0.061 | 0.993 | 0.003 | 0.100 | 0.890 | -0.003 | 0.065 | 0.992 | 0.000 | 0.003 | 0.985 | -0.023 | 0.100 | 0.026 | 0.108 | |
C-MCDM-D | Low | 0.021 | 0.049 | 0.914 | 0.084 | 0.127 | 0.930 | -0.113 | 0.257 | 0.913 | -0.233 | 0.258 | 0.822 | -0.123 | 0.227 | 0.912 | 0.000 | 0.003 | 0.982 | -0.066 | 0.218 | 0.085 | 0.195 |
Medium | 0.014 | 0.045 | 0.923 | 0.033 | 0.069 | 0.959 | 0.005 | 0.086 | 0.986 | -0.013 | 0.124 | 0.825 | 0.009 | 0.073 | 0.989 | 0.000 | 0.003 | 0.982 | 0.029 | 0.152 | -0.023 | 0.132 | |
High | -0.007 | 0.041 | 0.930 | -0.028 | 0.080 | 0.956 | -0.004 | 0.085 | 0.986 | 0.013 | 0.116 | 0.826 | 0.003 | 0.072 | 0.989 | 0.000 | 0.003 | 0.982 | -0.012 | 0.150 | 0.006 | 0.129 | |
C-MCDM-C | Low | 0.009 | 0.041 | 0.928 | 0.042 | 0.080 | 0.941 | -0.036 | 0.122 | 0.977 | -0.334 | 0.356 | 0.869 | -0.066 | 0.166 | 0.952 | 0.001 | 0.003 | 0.987 | 0.001 | 0.096 | 0.018 | 0.128 |
Medium | 0.009 | 0.040 | 0.929 | 0.035 | 0.073 | 0.947 | 0.009 | 0.106 | 0.981 | -0.021 | 0.121 | 0.864 | -0.001 | 0.100 | 0.981 | 0.000 | 0.003 | 0.987 | 0.014 | 0.096 | -0.012 | 0.092 | |
High | -0.009 | 0.040 | 0.931 | -0.024 | 0.073 | 0.949 | -0.003 | 0.103 | 0.981 | 0.004 | 0.103 | 0.874 | -0.011 | 0.098 | 0.981 | 0.000 | 0.003 | 0.987 | -0.006 | 0.095 | 0.004 | 0.091 |
N | I | CL | g | s | ζ | ω | m | d | φ | λ | |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Bias | RMSE | Cor | Bias | RMSE | Cor | Bias | RMSE | Cor | Bias | RMSE | Cor | Bias | RMSE | Cor | Bias | RMSE | Cor | Bias | RMSE | Bias | RMSE | ||||
100 | 15 | 0.0 | 0.013 | 0.054 | 0.852 | 0.047 | 0.080 | 0.950 | -0.001 | 0.066 | 0.992 | -0.010 | 0.118 | 0.868 | 0.001 | 0.058 | 0.993 | 0.000 | 0.004 | 0.987 | 0.003 | 0.104 | 0.012 | 0.106 | |
0.2 | 0.016 | 0.050 | 0.894 | 0.046 | 0.084 | 0.937 | 0.002 | 0.064 | 0.992 | -0.015 | 0.127 | 0.857 | -0.006 | 0.065 | 0.992 | 0.001 | 0.004 | 0.984 | 0.032 | 0.111 | -0.036 | 0.112 | |||
0.5 | 0.014 | 0.050 | 0.875 | 0.056 | 0.086 | 0.952 | 0.002 | 0.062 | 0.993 | -0.015 | 0.121 | 0.869 | 0.000 | 0.067 | 0.992 | 0.000 | 0.004 | 0.985 | 0.152 | 0.182 | -0.148 | 0.188 | |||
30 | 0.0 | 0.008 | 0.043 | 0.927 | 0.031 | 0.065 | 0.960 | 0.000 | 0.064 | 0.992 | -0.008 | 0.116 | 0.878 | -0.004 | 0.062 | 0.993 | 0.000 | 0.004 | 0.984 | -0.013 | 0.130 | 0.003 | 0.126 | ||
0.2 | 0.007 | 0.041 | 0.934 | 0.034 | 0.069 | 0.958 | -0.002 | 0.064 | 0.992 | -0.015 | 0.114 | 0.878 | -0.006 | 0.057 | 0.994 | 0.000 | 0.004 | 0.986 | 0.080 | 0.151 | -0.095 | 0.151 | |||
0.5 | 0.009 | 0.040 | 0.941 | 0.032 | 0.068 | 0.956 | 0.000 | 0.066 | 0.992 | -0.008 | 0.112 | 0.872 | 0.000 | 0.058 | 0.994 | 0.000 | 0.003 | 0.987 | 0.270 | 0.299 | -0.249 | 0.283 | |||
500 | 15 | 0.0 | 0.004 | 0.024 | 0.968 | 0.010 | 0.033 | 0.989 | 0.002 | 0.027 | 0.999 | 0.000 | 0.052 | 0.973 | 0.001 | 0.027 | 0.999 | 0.000 | 0.002 | 0.997 | 0.004 | 0.043 | 0.004 | 0.043 | |
0.2 | 0.002 | 0.025 | 0.970 | 0.008 | 0.031 | 0.989 | 0.001 | 0.029 | 0.998 | -0.002 | 0.053 | 0.969 | -0.001 | 0.026 | 0.999 | 0.000 | 0.002 | 0.997 | 0.010 | 0.046 | -0.010 | 0.044 | |||
0.5 | 0.003 | 0.025 | 0.965 | 0.013 | 0.035 | 0.988 | -0.002 | 0.029 | 0.998 | 0.000 | 0.052 | 0.970 | -0.001 | 0.027 | 0.999 | 0.000 | 0.002 | 0.997 | 0.035 | 0.055 | -0.030 | 0.052 | |||
30 | 0.0 | 0.002 | 0.020 | 0.984 | 0.007 | 0.028 | 0.991 | 0.003 | 0.029 | 0.998 | -0.002 | 0.053 | 0.971 | 0.000 | 0.027 | 0.999 | 0.000 | 0.002 | 0.997 | 0.000 | 0.042 | -0.004 | 0.038 | ||
0.2 | 0.002 | 0.019 | 0.985 | 0.007 | 0.029 | 0.991 | 0.000 | 0.028 | 0.999 | 0.000 | 0.051 | 0.972 | -0.002 | 0.027 | 0.999 | 0.000 | 0.002 | 0.997 | -0.001 | 0.042 | -0.004 | 0.045 | |||
0.5 | 0.003 | 0.019 | 0.986 | 0.006 | 0.027 | 0.991 | 0.000 | 0.029 | 0.998 | -0.004 | 0.054 | 0.970 | 0.001 | 0.028 | 0.999 | 0.000 | 0.002 | 0.997 | 0.040 | 0.057 | -0.037 | 0.054 |
Table S2 The recovery of item parameters of C-MCDM-θ model in simulation study 1.
N | I | CL | g | s | ζ | ω | m | d | φ | λ | |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Bias | RMSE | Cor | Bias | RMSE | Cor | Bias | RMSE | Cor | Bias | RMSE | Cor | Bias | RMSE | Cor | Bias | RMSE | Cor | Bias | RMSE | Bias | RMSE | ||||
100 | 15 | 0.0 | 0.013 | 0.054 | 0.852 | 0.047 | 0.080 | 0.950 | -0.001 | 0.066 | 0.992 | -0.010 | 0.118 | 0.868 | 0.001 | 0.058 | 0.993 | 0.000 | 0.004 | 0.987 | 0.003 | 0.104 | 0.012 | 0.106 | |
0.2 | 0.016 | 0.050 | 0.894 | 0.046 | 0.084 | 0.937 | 0.002 | 0.064 | 0.992 | -0.015 | 0.127 | 0.857 | -0.006 | 0.065 | 0.992 | 0.001 | 0.004 | 0.984 | 0.032 | 0.111 | -0.036 | 0.112 | |||
0.5 | 0.014 | 0.050 | 0.875 | 0.056 | 0.086 | 0.952 | 0.002 | 0.062 | 0.993 | -0.015 | 0.121 | 0.869 | 0.000 | 0.067 | 0.992 | 0.000 | 0.004 | 0.985 | 0.152 | 0.182 | -0.148 | 0.188 | |||
30 | 0.0 | 0.008 | 0.043 | 0.927 | 0.031 | 0.065 | 0.960 | 0.000 | 0.064 | 0.992 | -0.008 | 0.116 | 0.878 | -0.004 | 0.062 | 0.993 | 0.000 | 0.004 | 0.984 | -0.013 | 0.130 | 0.003 | 0.126 | ||
0.2 | 0.007 | 0.041 | 0.934 | 0.034 | 0.069 | 0.958 | -0.002 | 0.064 | 0.992 | -0.015 | 0.114 | 0.878 | -0.006 | 0.057 | 0.994 | 0.000 | 0.004 | 0.986 | 0.080 | 0.151 | -0.095 | 0.151 | |||
0.5 | 0.009 | 0.040 | 0.941 | 0.032 | 0.068 | 0.956 | 0.000 | 0.066 | 0.992 | -0.008 | 0.112 | 0.872 | 0.000 | 0.058 | 0.994 | 0.000 | 0.003 | 0.987 | 0.270 | 0.299 | -0.249 | 0.283 | |||
500 | 15 | 0.0 | 0.004 | 0.024 | 0.968 | 0.010 | 0.033 | 0.989 | 0.002 | 0.027 | 0.999 | 0.000 | 0.052 | 0.973 | 0.001 | 0.027 | 0.999 | 0.000 | 0.002 | 0.997 | 0.004 | 0.043 | 0.004 | 0.043 | |
0.2 | 0.002 | 0.025 | 0.970 | 0.008 | 0.031 | 0.989 | 0.001 | 0.029 | 0.998 | -0.002 | 0.053 | 0.969 | -0.001 | 0.026 | 0.999 | 0.000 | 0.002 | 0.997 | 0.010 | 0.046 | -0.010 | 0.044 | |||
0.5 | 0.003 | 0.025 | 0.965 | 0.013 | 0.035 | 0.988 | -0.002 | 0.029 | 0.998 | 0.000 | 0.052 | 0.970 | -0.001 | 0.027 | 0.999 | 0.000 | 0.002 | 0.997 | 0.035 | 0.055 | -0.030 | 0.052 | |||
30 | 0.0 | 0.002 | 0.020 | 0.984 | 0.007 | 0.028 | 0.991 | 0.003 | 0.029 | 0.998 | -0.002 | 0.053 | 0.971 | 0.000 | 0.027 | 0.999 | 0.000 | 0.002 | 0.997 | 0.000 | 0.042 | -0.004 | 0.038 | ||
0.2 | 0.002 | 0.019 | 0.985 | 0.007 | 0.029 | 0.991 | 0.000 | 0.028 | 0.999 | 0.000 | 0.051 | 0.972 | -0.002 | 0.027 | 0.999 | 0.000 | 0.002 | 0.997 | -0.001 | 0.042 | -0.004 | 0.045 | |||
0.5 | 0.003 | 0.019 | 0.986 | 0.006 | 0.027 | 0.991 | 0.000 | 0.029 | 0.998 | -0.004 | 0.054 | 0.970 | 0.001 | 0.028 | 0.999 | 0.000 | 0.002 | 0.997 | 0.040 | 0.057 | -0.037 | 0.054 |
N | I | CL | g | s | ζ | ω | m | d | φ | λ | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Bias | RMSE | Cor | Bias | RMSE | Cor | Bias | RMSE | Cor | Bias | RMSE | Cor | Bias | RMSE | Cor | Bias | RMSE | Cor | Bias | RMSE | Bias | RMSE | |||
100 | 15 | 0.0 | 0.016 | 0.051 | 0.894 | 0.037 | 0.077 | 0.948 | 0.001 | 0.084 | 0.985 | -0.021 | 0.122 | 0.854 | 0.000 | 0.078 | 0.988 | 0.000 | 0.004 | 0.986 | 0.012 | 0.151 | -0.002 | 0.148 |
0.2 | 0.019 | 0.051 | 0.883 | 0.040 | 0.077 | 0.940 | 0.001 | 0.083 | 0.987 | -0.014 | 0.118 | 0.870 | -0.001 | 0.083 | 0.986 | 0.000 | 0.004 | 0.984 | 0.025 | 0.149 | -0.018 | 0.150 | ||
0.5 | 0.014 | 0.045 | 0.923 | 0.033 | 0.069 | 0.959 | 0.005 | 0.086 | 0.986 | -0.013 | 0.124 | 0.825 | 0.009 | 0.073 | 0.989 | 0.000 | 0.003 | 0.982 | 0.029 | 0.152 | -0.023 | 0.132 | ||
30 | 0.0 | 0.013 | 0.044 | 0.925 | 0.031 | 0.068 | 0.954 | -0.001 | 0.080 | 0.987 | -0.020 | 0.114 | 0.871 | 0.000 | 0.075 | 0.988 | 0.000 | 0.004 | 0.987 | -0.007 | 0.140 | -0.001 | 0.135 | |
0.2 | 0.013 | 0.043 | 0.931 | 0.029 | 0.065 | 0.963 | 0.002 | 0.081 | 0.988 | -0.020 | 0.113 | 0.873 | 0.002 | 0.076 | 0.989 | 0.000 | 0.003 | 0.987 | 0.012 | 0.139 | -0.011 | 0.124 | ||
0.5 | 0.010 | 0.039 | 0.939 | 0.028 | 0.067 | 0.954 | 0.009 | 0.081 | 0.987 | -0.026 | 0.120 | 0.865 | 0.007 | 0.073 | 0.990 | 0.000 | 0.003 | 0.987 | 0.039 | 0.146 | -0.035 | 0.136 | ||
500 | 15 | 0.0 | 0.005 | 0.026 | 0.969 | 0.008 | 0.032 | 0.989 | 0.000 | 0.036 | 0.997 | -0.001 | 0.054 | 0.972 | -0.001 | 0.037 | 0.997 | 0.000 | 0.002 | 0.996 | -0.004 | 0.072 | 0.004 | 0.066 |
0.2 | 0.005 | 0.023 | 0.972 | 0.007 | 0.030 | 0.989 | 0.002 | 0.039 | 0.997 | -0.006 | 0.055 | 0.965 | 0.003 | 0.034 | 0.998 | 0.000 | 0.002 | 0.997 | 0.002 | 0.073 | -0.004 | 0.063 | ||
0.5 | 0.003 | 0.021 | 0.979 | 0.009 | 0.029 | 0.992 | 0.001 | 0.037 | 0.997 | -0.006 | 0.055 | 0.971 | 0.002 | 0.037 | 0.997 | 0.000 | 0.002 | 0.997 | 0.008 | 0.066 | -0.011 | 0.065 | ||
30 | 0.0 | 0.003 | 0.019 | 0.984 | 0.007 | 0.030 | 0.989 | 0.001 | 0.038 | 0.997 | -0.002 | 0.053 | 0.970 | 0.003 | 0.034 | 0.998 | 0.000 | 0.002 | 0.997 | 0.001 | 0.065 | -0.003 | 0.065 | |
0.2 | 0.002 | 0.019 | 0.984 | 0.007 | 0.028 | 0.992 | 0.003 | 0.037 | 0.997 | -0.002 | 0.054 | 0.970 | 0.002 | 0.035 | 0.998 | 0.000 | 0.002 | 0.997 | 0.007 | 0.064 | -0.007 | 0.060 | ||
0.5 | 0.001 | 0.018 | 0.987 | 0.006 | 0.027 | 0.991 | 0.001 | 0.037 | 0.997 | -0.002 | 0.053 | 0.972 | 0.002 | 0.035 | 0.998 | 0.000 | 0.001 | 0.997 | 0.006 | 0.064 | -0.007 | 0.058 |
Table S3 The recovery of item parameters of the C-MCDM-D model in simulation study 1.
N | I | CL | g | s | ζ | ω | m | d | φ | λ | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Bias | RMSE | Cor | Bias | RMSE | Cor | Bias | RMSE | Cor | Bias | RMSE | Cor | Bias | RMSE | Cor | Bias | RMSE | Cor | Bias | RMSE | Bias | RMSE | |||
100 | 15 | 0.0 | 0.016 | 0.051 | 0.894 | 0.037 | 0.077 | 0.948 | 0.001 | 0.084 | 0.985 | -0.021 | 0.122 | 0.854 | 0.000 | 0.078 | 0.988 | 0.000 | 0.004 | 0.986 | 0.012 | 0.151 | -0.002 | 0.148 |
0.2 | 0.019 | 0.051 | 0.883 | 0.040 | 0.077 | 0.940 | 0.001 | 0.083 | 0.987 | -0.014 | 0.118 | 0.870 | -0.001 | 0.083 | 0.986 | 0.000 | 0.004 | 0.984 | 0.025 | 0.149 | -0.018 | 0.150 | ||
0.5 | 0.014 | 0.045 | 0.923 | 0.033 | 0.069 | 0.959 | 0.005 | 0.086 | 0.986 | -0.013 | 0.124 | 0.825 | 0.009 | 0.073 | 0.989 | 0.000 | 0.003 | 0.982 | 0.029 | 0.152 | -0.023 | 0.132 | ||
30 | 0.0 | 0.013 | 0.044 | 0.925 | 0.031 | 0.068 | 0.954 | -0.001 | 0.080 | 0.987 | -0.020 | 0.114 | 0.871 | 0.000 | 0.075 | 0.988 | 0.000 | 0.004 | 0.987 | -0.007 | 0.140 | -0.001 | 0.135 | |
0.2 | 0.013 | 0.043 | 0.931 | 0.029 | 0.065 | 0.963 | 0.002 | 0.081 | 0.988 | -0.020 | 0.113 | 0.873 | 0.002 | 0.076 | 0.989 | 0.000 | 0.003 | 0.987 | 0.012 | 0.139 | -0.011 | 0.124 | ||
0.5 | 0.010 | 0.039 | 0.939 | 0.028 | 0.067 | 0.954 | 0.009 | 0.081 | 0.987 | -0.026 | 0.120 | 0.865 | 0.007 | 0.073 | 0.990 | 0.000 | 0.003 | 0.987 | 0.039 | 0.146 | -0.035 | 0.136 | ||
500 | 15 | 0.0 | 0.005 | 0.026 | 0.969 | 0.008 | 0.032 | 0.989 | 0.000 | 0.036 | 0.997 | -0.001 | 0.054 | 0.972 | -0.001 | 0.037 | 0.997 | 0.000 | 0.002 | 0.996 | -0.004 | 0.072 | 0.004 | 0.066 |
0.2 | 0.005 | 0.023 | 0.972 | 0.007 | 0.030 | 0.989 | 0.002 | 0.039 | 0.997 | -0.006 | 0.055 | 0.965 | 0.003 | 0.034 | 0.998 | 0.000 | 0.002 | 0.997 | 0.002 | 0.073 | -0.004 | 0.063 | ||
0.5 | 0.003 | 0.021 | 0.979 | 0.009 | 0.029 | 0.992 | 0.001 | 0.037 | 0.997 | -0.006 | 0.055 | 0.971 | 0.002 | 0.037 | 0.997 | 0.000 | 0.002 | 0.997 | 0.008 | 0.066 | -0.011 | 0.065 | ||
30 | 0.0 | 0.003 | 0.019 | 0.984 | 0.007 | 0.030 | 0.989 | 0.001 | 0.038 | 0.997 | -0.002 | 0.053 | 0.970 | 0.003 | 0.034 | 0.998 | 0.000 | 0.002 | 0.997 | 0.001 | 0.065 | -0.003 | 0.065 | |
0.2 | 0.002 | 0.019 | 0.984 | 0.007 | 0.028 | 0.992 | 0.003 | 0.037 | 0.997 | -0.002 | 0.054 | 0.970 | 0.002 | 0.035 | 0.998 | 0.000 | 0.002 | 0.997 | 0.007 | 0.064 | -0.007 | 0.060 | ||
0.5 | 0.001 | 0.018 | 0.987 | 0.006 | 0.027 | 0.991 | 0.001 | 0.037 | 0.997 | -0.002 | 0.053 | 0.972 | 0.002 | 0.035 | 0.998 | 0.000 | 0.001 | 0.997 | 0.006 | 0.064 | -0.007 | 0.058 |
N | I | CL | g | s | ζ | ω | m | d | φ | λ | |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Bias | RMSE | Cor | Bias | RMSE | Cor | Bias | RMSE | Cor | Bias | RMSE | Cor | Bias | RMSE | Cor | Bias | RMSE | Cor | Bias | RMSE | Bias | RMSE | ||||
100 | 15 | 0.0 | 0.015 | 0.049 | 0.894 | 0.054 | 0.089 | 0.948 | 0.003 | 0.119 | 0.974 | -0.008 | 0.119 | 0.858 | 0.010 | 0.109 | 0.977 | 0.000 | 0.004 | 0.987 | 0.005 | 0.112 | -0.007 | 0.106 | |
0.2 | 0.014 | 0.046 | 0.916 | 0.045 | 0.079 | 0.951 | 0.001 | 0.119 | 0.974 | -0.029 | 0.120 | 0.837 | -0.007 | 0.113 | 0.976 | 0.000 | 0.003 | 0.987 | 0.009 | 0.108 | -0.008 | 0.112 | |||
0.5 | 0.009 | 0.040 | 0.929 | 0.035 | 0.073 | 0.947 | 0.009 | 0.106 | 0.981 | -0.021 | 0.121 | 0.864 | -0.001 | 0.100 | 0.981 | 0.000 | 0.003 | 0.987 | 0.014 | 0.096 | -0.012 | 0.188 | |||
30 | 0.0 | 0.009 | 0.043 | 0.921 | 0.034 | 0.069 | 0.952 | -0.007 | 0.108 | 0.977 | -0.018 | 0.112 | 0.880 | -0.010 | 0.101 | 0.980 | 0.000 | 0.004 | 0.987 | -0.006 | 0.096 | 0.006 | 0.126 | ||
0.2 | 0.013 | 0.040 | 0.944 | 0.029 | 0.069 | 0.953 | 0.010 | 0.107 | 0.977 | -0.018 | 0.115 | 0.868 | 0.000 | 0.101 | 0.980 | 0.000 | 0.003 | 0.986 | 0.009 | 0.093 | 0.000 | 0.151 | |||
0.5 | 0.012 | 0.041 | 0.935 | 0.029 | 0.065 | 0.958 | 0.017 | 0.104 | 0.980 | -0.019 | 0.117 | 0.873 | 0.010 | 0.094 | 0.983 | 0.000 | 0.002 | 0.987 | 0.021 | 0.094 | -0.018 | 0.283 | |||
500 | 15 | 0.0 | 0.005 | 0.025 | 0.970 | 0.010 | 0.029 | 0.991 | -0.002 | 0.051 | 0.995 | -0.001 | 0.055 | 0.966 | -0.005 | 0.051 | 0.995 | 0.000 | 0.002 | 0.997 | -0.001 | 0.050 | 0.002 | 0.043 | |
0.2 | 0.001 | 0.022 | 0.973 | 0.007 | 0.029 | 0.991 | 0.000 | 0.048 | 0.996 | -0.003 | 0.053 | 0.970 | 0.004 | 0.047 | 0.996 | 0.000 | 0.001 | 0.997 | -0.003 | 0.046 | -0.007 | 0.044 | |||
0.5 | 0.003 | 0.020 | 0.983 | 0.009 | 0.030 | 0.991 | 0.008 | 0.049 | 0.996 | -0.009 | 0.056 | 0.971 | 0.003 | 0.043 | 0.996 | 0.000 | 0.001 | 0.997 | 0.010 | 0.044 | -0.003 | 0.052 | |||
30 | 0.0 | 0.002 | 0.019 | 0.983 | 0.006 | 0.028 | 0.991 | -0.001 | 0.047 | 0.996 | -0.003 | 0.052 | 0.972 | 0.000 | 0.039 | 0.997 | 0.000 | 0.002 | 0.997 | 0.002 | 0.043 | 0.000 | 0.038 | ||
0.2 | 0.002 | 0.018 | 0.987 | 0.006 | 0.027 | 0.991 | 0.004 | 0.047 | 0.996 | -0.005 | 0.054 | 0.971 | -0.001 | 0.043 | 0.996 | 0.000 | 0.001 | 0.997 | 0.006 | 0.041 | -0.001 | 0.045 | |||
0.5 | 0.002 | 0.018 | 0.988 | 0.006 | 0.027 | 0.991 | 0.001 | 0.048 | 0.996 | -0.006 | 0.055 | 0.969 | 0.006 | 0.044 | 0.996 | 0.000 | 0.001 | 0.998 | 0.001 | 0.042 | -0.006 | 0.054 |
Table S4 The recovery of item parameters of the C-MCDM-C model in simulation study 1.
N | I | CL | g | s | ζ | ω | m | d | φ | λ | |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Bias | RMSE | Cor | Bias | RMSE | Cor | Bias | RMSE | Cor | Bias | RMSE | Cor | Bias | RMSE | Cor | Bias | RMSE | Cor | Bias | RMSE | Bias | RMSE | ||||
100 | 15 | 0.0 | 0.015 | 0.049 | 0.894 | 0.054 | 0.089 | 0.948 | 0.003 | 0.119 | 0.974 | -0.008 | 0.119 | 0.858 | 0.010 | 0.109 | 0.977 | 0.000 | 0.004 | 0.987 | 0.005 | 0.112 | -0.007 | 0.106 | |
0.2 | 0.014 | 0.046 | 0.916 | 0.045 | 0.079 | 0.951 | 0.001 | 0.119 | 0.974 | -0.029 | 0.120 | 0.837 | -0.007 | 0.113 | 0.976 | 0.000 | 0.003 | 0.987 | 0.009 | 0.108 | -0.008 | 0.112 | |||
0.5 | 0.009 | 0.040 | 0.929 | 0.035 | 0.073 | 0.947 | 0.009 | 0.106 | 0.981 | -0.021 | 0.121 | 0.864 | -0.001 | 0.100 | 0.981 | 0.000 | 0.003 | 0.987 | 0.014 | 0.096 | -0.012 | 0.188 | |||
30 | 0.0 | 0.009 | 0.043 | 0.921 | 0.034 | 0.069 | 0.952 | -0.007 | 0.108 | 0.977 | -0.018 | 0.112 | 0.880 | -0.010 | 0.101 | 0.980 | 0.000 | 0.004 | 0.987 | -0.006 | 0.096 | 0.006 | 0.126 | ||
0.2 | 0.013 | 0.040 | 0.944 | 0.029 | 0.069 | 0.953 | 0.010 | 0.107 | 0.977 | -0.018 | 0.115 | 0.868 | 0.000 | 0.101 | 0.980 | 0.000 | 0.003 | 0.986 | 0.009 | 0.093 | 0.000 | 0.151 | |||
0.5 | 0.012 | 0.041 | 0.935 | 0.029 | 0.065 | 0.958 | 0.017 | 0.104 | 0.980 | -0.019 | 0.117 | 0.873 | 0.010 | 0.094 | 0.983 | 0.000 | 0.002 | 0.987 | 0.021 | 0.094 | -0.018 | 0.283 | |||
500 | 15 | 0.0 | 0.005 | 0.025 | 0.970 | 0.010 | 0.029 | 0.991 | -0.002 | 0.051 | 0.995 | -0.001 | 0.055 | 0.966 | -0.005 | 0.051 | 0.995 | 0.000 | 0.002 | 0.997 | -0.001 | 0.050 | 0.002 | 0.043 | |
0.2 | 0.001 | 0.022 | 0.973 | 0.007 | 0.029 | 0.991 | 0.000 | 0.048 | 0.996 | -0.003 | 0.053 | 0.970 | 0.004 | 0.047 | 0.996 | 0.000 | 0.001 | 0.997 | -0.003 | 0.046 | -0.007 | 0.044 | |||
0.5 | 0.003 | 0.020 | 0.983 | 0.009 | 0.030 | 0.991 | 0.008 | 0.049 | 0.996 | -0.009 | 0.056 | 0.971 | 0.003 | 0.043 | 0.996 | 0.000 | 0.001 | 0.997 | 0.010 | 0.044 | -0.003 | 0.052 | |||
30 | 0.0 | 0.002 | 0.019 | 0.983 | 0.006 | 0.028 | 0.991 | -0.001 | 0.047 | 0.996 | -0.003 | 0.052 | 0.972 | 0.000 | 0.039 | 0.997 | 0.000 | 0.002 | 0.997 | 0.002 | 0.043 | 0.000 | 0.038 | ||
0.2 | 0.002 | 0.018 | 0.987 | 0.006 | 0.027 | 0.991 | 0.004 | 0.047 | 0.996 | -0.005 | 0.054 | 0.971 | -0.001 | 0.043 | 0.996 | 0.000 | 0.001 | 0.997 | 0.006 | 0.041 | -0.001 | 0.045 | |||
0.5 | 0.002 | 0.018 | 0.988 | 0.006 | 0.027 | 0.991 | 0.001 | 0.048 | 0.996 | -0.006 | 0.055 | 0.969 | 0.006 | 0.044 | 0.996 | 0.000 | 0.001 | 0.998 | 0.001 | 0.042 | -0.006 | 0.054 |
Data generation model | Data analysis model | θ | τ | ε | ||||||
---|---|---|---|---|---|---|---|---|---|---|
Bias | RMSE | Cor | Bias | RMSE | Cor | Bias | RMSE | Cor | ||
C-MCDM-θ | H-MCDM | 0.001 | 0.600 | 0.796 | -0.001 | 0.137 | 0.961 | 0.001 | 0.217 | 0.912 |
C-MCDM-θ | 0.001 | 0.239 | 0.972 | -0.001 | 0.109 | 0.976 | 0.001 | 0.098 | 0.980 | |
C-MCDM-D | H-MCDM | 0.001 | 0.625 | 0.775 | 0.000 | 0.110 | 0.975 | 0.000 | 0.104 | 0.978 |
C-MCDM-D | 0.000 | 0.614 | 0.784 | 0.001 | 0.108 | 0.976 | -0.002 | 0.083 | 0.986 | |
C-MCDM-C | H-MCDM | 0.000 | 0.625 | 0.775 | -0.001 | 0.134 | 0.963 | 0.000 | 0.113 | 0.974 |
C-MCDM-C | 0.000 | 0.601 | 0.795 | 0.001 | 0.108 | 0.976 | -0.001 | 0.087 | 0.984 | |
H-MCDM | H-MCDM | 0.000 | 0.571 | 0.817 | 0.000 | 0.108 | 0.976 | 0.000 | 0.082 | 0.986 |
C-MCDM-θ | -0.001 | 0.589 | 0.805 | 0.001 | 0.214 | 0.907 | 0.000 | 0.204 | 0.916 | |
C-MCDM-D | 0.000 | 0.622 | 0.778 | 0.001 | 0.109 | 0.976 | -0.001 | 0.082 | 0.986 | |
C-MCDM-C | 0.000 | 0.622 | 0.779 | 0.000 | 0.110 | 0.975 | 0.000 | 0.083 | 0.986 |
Table S5 The Recovery of latent ability, latent processing speed, and latent visual engagement in simulation study 2.
Data generation model | Data analysis model | θ | τ | ε | ||||||
---|---|---|---|---|---|---|---|---|---|---|
Bias | RMSE | Cor | Bias | RMSE | Cor | Bias | RMSE | Cor | ||
C-MCDM-θ | H-MCDM | 0.001 | 0.600 | 0.796 | -0.001 | 0.137 | 0.961 | 0.001 | 0.217 | 0.912 |
C-MCDM-θ | 0.001 | 0.239 | 0.972 | -0.001 | 0.109 | 0.976 | 0.001 | 0.098 | 0.980 | |
C-MCDM-D | H-MCDM | 0.001 | 0.625 | 0.775 | 0.000 | 0.110 | 0.975 | 0.000 | 0.104 | 0.978 |
C-MCDM-D | 0.000 | 0.614 | 0.784 | 0.001 | 0.108 | 0.976 | -0.002 | 0.083 | 0.986 | |
C-MCDM-C | H-MCDM | 0.000 | 0.625 | 0.775 | -0.001 | 0.134 | 0.963 | 0.000 | 0.113 | 0.974 |
C-MCDM-C | 0.000 | 0.601 | 0.795 | 0.001 | 0.108 | 0.976 | -0.001 | 0.087 | 0.984 | |
H-MCDM | H-MCDM | 0.000 | 0.571 | 0.817 | 0.000 | 0.108 | 0.976 | 0.000 | 0.082 | 0.986 |
C-MCDM-θ | -0.001 | 0.589 | 0.805 | 0.001 | 0.214 | 0.907 | 0.000 | 0.204 | 0.916 | |
C-MCDM-D | 0.000 | 0.622 | 0.778 | 0.001 | 0.109 | 0.976 | -0.001 | 0.082 | 0.986 | |
C-MCDM-C | 0.000 | 0.622 | 0.779 | 0.000 | 0.110 | 0.975 | 0.000 | 0.083 | 0.986 |
Data generation model | Data analysis model | g | s | ζ | m | ω | d | φ | λ | ||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Bias | RMSE | Cor | Bias | RMSE | Cor | Bias | RMSE | Cor | Bias | RMSE | Cor | Bias | RMSE | Cor | Bias | RMSE | Cor | Bias | RMSE | Cor | Bias | RMSE | Cor | ||
C-MLDM-θ | H-MLDM | 0.003 | 0.019 | 0.986 | 0.004 | 0.027 | 0.992 | 0.000 | 0.028 | 0.998 | 0.032 | 0.047 | 0.995 | -0.145 | 0.170 | 0.683 | -0.004 | 0.007 | 0.952 | ||||||
C-MLDM-θ | 0.003 | 0.019 | 0.986 | 0.004 | 0.027 | 0.992 | 0.000 | 0.028 | 0.998 | -0.001 | 0.026 | 0.999 | -0.007 | 0.053 | 0.970 | 0.000 | 0.002 | 0.997 | 0.003 | 0.035 | 0.996 | 0.002 | 0.032 | 0.995 | |
C-MLDM-D | H-MLDM | 0.002 | 0.019 | 0.984 | 0.006 | 0.027 | 0.991 | 0.020 | 0.072 | 0.988 | 0.056 | 0.108 | 0.968 | -0.028 | 0.065 | 0.958 | -0.002 | 0.004 | 0.981 | ||||||
C-MLDM-D | 0.002 | 0.019 | 0.984 | 0.005 | 0.027 | 0.991 | 0.002 | 0.036 | 0.997 | 0.001 | 0.036 | 0.997 | -0.004 | 0.054 | 0.971 | 0.000 | 0.002 | 0.997 | 0.003 | 0.062 | 0.969 | 0.004 | 0.062 | 0.982 | |
C-MLDM-C | H-MLDM | 0.003 | 0.020 | 0.984 | 0.005 | 0.028 | 0.991 | 0.139 | 0.292 | 0.821 | 0.138 | 0.281 | 0.789 | -0.120 | 0.148 | 0.763 | -0.003 | 0.006 | 0.967 | ||||||
C-MLDM-C | 0.003 | 0.019 | 0.985 | 0.004 | 0.027 | 0.992 | 0.004 | 0.048 | 0.996 | 0.003 | 0.045 | 0.996 | -0.001 | 0.054 | 0.970 | 0.000 | 0.002 | 0.997 | 0.003 | 0.041 | 0.994 | -0.003 | 0.039 | 0.992 | |
H-MLDM | H-MLDM | 0.003 | 0.021 | 0.983 | 0.007 | 0.028 | 0.991 | 0.000 | 0.028 | 0.998 | -0.002 | 0.028 | 0.999 | -0.002 | 0.052 | 0.972 | 0.000 | 0.002 | 0.998 | ||||||
C-MLDM-θ | 0.003 | 0.021 | 0.983 | 0.007 | 0.028 | 0.991 | 0.000 | 0.028 | 0.998 | -0.002 | 0.028 | 0.999 | -0.001 | 0.052 | 0.972 | 0.000 | 0.002 | 0.997 | |||||||
C-MLDM-D | 0.003 | 0.021 | 0.982 | 0.007 | 0.028 | 0.991 | -0.006 | 0.037 | 0.997 | -0.005 | 0.037 | 0.997 | -0.002 | 0.052 | 0.972 | 0.000 | 0.002 | 0.998 | |||||||
C-MLDM-C | 0.003 | 0.021 | 0.982 | 0.007 | 0.028 | 0.991 | -0.018 | 0.051 | 0.996 | -0.012 | 0.048 | 0.996 | -0.002 | 0.052 | 0.972 | 0.000 | 0.002 | 0.998 |
Table S6 The recovery of item parameters in Simulation Study 2
Data generation model | Data analysis model | g | s | ζ | m | ω | d | φ | λ | ||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Bias | RMSE | Cor | Bias | RMSE | Cor | Bias | RMSE | Cor | Bias | RMSE | Cor | Bias | RMSE | Cor | Bias | RMSE | Cor | Bias | RMSE | Cor | Bias | RMSE | Cor | ||
C-MLDM-θ | H-MLDM | 0.003 | 0.019 | 0.986 | 0.004 | 0.027 | 0.992 | 0.000 | 0.028 | 0.998 | 0.032 | 0.047 | 0.995 | -0.145 | 0.170 | 0.683 | -0.004 | 0.007 | 0.952 | ||||||
C-MLDM-θ | 0.003 | 0.019 | 0.986 | 0.004 | 0.027 | 0.992 | 0.000 | 0.028 | 0.998 | -0.001 | 0.026 | 0.999 | -0.007 | 0.053 | 0.970 | 0.000 | 0.002 | 0.997 | 0.003 | 0.035 | 0.996 | 0.002 | 0.032 | 0.995 | |
C-MLDM-D | H-MLDM | 0.002 | 0.019 | 0.984 | 0.006 | 0.027 | 0.991 | 0.020 | 0.072 | 0.988 | 0.056 | 0.108 | 0.968 | -0.028 | 0.065 | 0.958 | -0.002 | 0.004 | 0.981 | ||||||
C-MLDM-D | 0.002 | 0.019 | 0.984 | 0.005 | 0.027 | 0.991 | 0.002 | 0.036 | 0.997 | 0.001 | 0.036 | 0.997 | -0.004 | 0.054 | 0.971 | 0.000 | 0.002 | 0.997 | 0.003 | 0.062 | 0.969 | 0.004 | 0.062 | 0.982 | |
C-MLDM-C | H-MLDM | 0.003 | 0.020 | 0.984 | 0.005 | 0.028 | 0.991 | 0.139 | 0.292 | 0.821 | 0.138 | 0.281 | 0.789 | -0.120 | 0.148 | 0.763 | -0.003 | 0.006 | 0.967 | ||||||
C-MLDM-C | 0.003 | 0.019 | 0.985 | 0.004 | 0.027 | 0.992 | 0.004 | 0.048 | 0.996 | 0.003 | 0.045 | 0.996 | -0.001 | 0.054 | 0.970 | 0.000 | 0.002 | 0.997 | 0.003 | 0.041 | 0.994 | -0.003 | 0.039 | 0.992 | |
H-MLDM | H-MLDM | 0.003 | 0.021 | 0.983 | 0.007 | 0.028 | 0.991 | 0.000 | 0.028 | 0.998 | -0.002 | 0.028 | 0.999 | -0.002 | 0.052 | 0.972 | 0.000 | 0.002 | 0.998 | ||||||
C-MLDM-θ | 0.003 | 0.021 | 0.983 | 0.007 | 0.028 | 0.991 | 0.000 | 0.028 | 0.998 | -0.002 | 0.028 | 0.999 | -0.001 | 0.052 | 0.972 | 0.000 | 0.002 | 0.997 | |||||||
C-MLDM-D | 0.003 | 0.021 | 0.982 | 0.007 | 0.028 | 0.991 | -0.006 | 0.037 | 0.997 | -0.005 | 0.037 | 0.997 | -0.002 | 0.052 | 0.972 | 0.000 | 0.002 | 0.998 | |||||||
C-MLDM-C | 0.003 | 0.021 | 0.982 | 0.007 | 0.028 | 0.991 | -0.018 | 0.051 | 0.996 | -0.012 | 0.048 | 0.996 | -0.002 | 0.052 | 0.972 | 0.000 | 0.002 | 0.998 |
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