A Comprehensive Simulation Study of Estimation Methods for the Rasch Model
Abstract
:1. Introduction
2. Rasch Model
3. Estimation Methods for the Rasch Model
3.1. Joint Maximum Likelihood (JML) Estimation
3.1.1. JML with Bias Correction (JMLM and JMLW)
3.1.2. Penalized JML (PJML)
3.1.3. JML with Adjustment (JML)
3.2. Conditional Maximum Likelihood (CML) Estimation
3.3. Marginal Maximum Likelihood (MML) Estimation
3.3.1. MML with Normality Assumption (MMLN)
3.3.2. MML with Multinomial Distribution (MMLMN)
3.3.3. MML with Log-Linear Smoothing (MMLLS)
3.3.4. MML with Located Latent Classes (MMLLC)
3.4. Limited Information Estimation Methods
3.4.1. Pairwise Marginal Maximum Likelihood (PMML)
3.4.2. Pairwise Conditional Maximum Likelihood (PCML)
3.4.3. Minimum Chi-Square Method (MINCHI)
3.4.4. Row Averaging Method (RA)
3.4.5. Eigenvector Method (EVM)
3.4.6. Log-Linear by Linear Association Models (LLLA)
4. Simulation Study
4.1. Purpose
4.2. Design
- NO: A normal distribution () with zero mean and a standard deviation of one
- Chi: A scaled chi-squared distribution with one degree of freedom
- UN: A uniform distribution on the interval (i.e., )
- BE: A scaled U-shaped beta distribution with shape and scale parameters of 0.5; that is,
- SM: A symmetric mixture distribution with , and
- AM: An asymmetric mixture distribution with , and
- LC2: A discrete distribution with points −2.0, 0.5 and corresponding probabilities 0.20 and 0.80
- LC3: A discrete distribution with points −0.790, 1.033, 2.248 and corresponding probabilities 0.60, 0.35, and 0.05
4.3. Analysis Models
4.4. Outcome Measures
4.5. Results
5. Discussion
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Abbreviations
CML | conditional maximum likelihood |
EVM | eigenvector method |
IRT | item response theory |
JML | joint maximum likelihood |
JML | JML with adjustment |
JMLM | JML with maximum likelihood ability estimator |
JMLW | JML with Warm’s maximum likelihood ability estimator |
LIM | limited information methods |
LLLA | log-linear by linear association method |
MAB | mean absolute bias |
MCAR | missing completely at random |
MML | marginal maximum likelihood |
MMLLLC | MML with located latent classes |
MMLLS | MML with log-linear smoothing |
MMLMN | MML with multinomial distribution |
MMLN | MML with normal distribution |
MINCHI | minimum chi-square estimation |
PJML | penalized JML |
PMML | pairwise MML |
PCML | pairwise CML |
RA | row-averaging method |
RM | Rasch model |
RMSE | root mean square error |
Appendix A. Item Parameters Used in the Simulation Study
Appendix B. Additional Results for Simulation Study
Method | Bias | Relative RMSE | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
NO | AM | UN | BE | LC2 | LC3 | NO | AM | UN | BE | LC2 | LC3 | |
MMLN | 0.002 | 0.005 | 0.007 | 0.010 | 0.013 | 0.004 | 100.7 | 101.5 | 102.0 | 102.8 | 105.8 | 101.1 |
MMLLS(3) | 0.003 | 0.002 | 0.002 | 0.004 | 0.004 | 0.004 | 100.3 | 100.4 | 100.9 | 101.3 | 104.7 | 101.1 |
MMLLS(4) | 0.003 | 0.003 | 0.004 | 0.004 | 0.004 | 0.005 | 100.3 | 100.4 | 100.3 | 100.4 | 105.7 | 100.0 |
MMLMN(5) | 0.017 | 0.021 | 0.022 | 0.026 | 0.030 | 0.016 | 102.8 | 105.2 | 105.4 | 108.6 | 130.5 | 102.5 |
MMLMN(7) | 0.003 | 0.004 | 0.004 | 0.004 | 0.028 | 0.004 | 100.3 | 100.5 | 100.1 | 100.1 | 113.1 | 100.1 |
MMLMN(11) | 0.004 | 0.004 | 0.005 | 0.003 | 0.003 | 0.005 | 100.4 | 100.4 | 100.2 | 100.2 | 100.0 | 101.2 |
MMLMN(15) | 0.004 | 0.003 | 0.004 | 0.004 | 0.003 | 0.004 | 100.4 | 100.5 | 100.4 | 100.6 | 106.0 | 100.8 |
MMLLC(2) | 0.032 | 0.026 | 0.023 | 0.019 | 0.005 | 0.028 | 104.4 | 102.4 | 101.6 | 100.8 | 102.5 | 102.9 |
MMLLC(3) | 0.015 | 0.013 | 0.013 | 0.012 | 0.004 | 0.016 | 100.3 | 100.1 | 100.1 | 100.1 | 102.6 | 100.3 |
MMLLC(4) | 0.011 | 0.009 | 0.009 | 0.008 | 0.003 | 0.012 | 100.0 | 100.0 | 100.0 | 100.0 | 102.6 | 100.0 |
MMLLC(5) | 0.010 | 0.008 | 0.008 | 0.007 | 0.004 | 0.011 | 100.0 | 100.1 | 100.1 | 100.0 | 102.4 | 100.1 |
CML | 0.002 | 0.003 | 0.002 | 0.002 | 0.002 | 0.003 | 100.8 | 101.0 | 100.9 | 101.0 | 103.5 | 100.8 |
JMLM | 0.015 | 0.014 | 0.014 | 0.013 | 0.015 | 0.014 | 104.3 | 103.9 | 104.0 | 103.7 | 106.2 | 104.2 |
JMLW | 0.052 | 0.047 | 0.052 | 0.052 | 0.035 | 0.060 | 115.9 | 113.6 | 116.6 | 117.1 | 110.2 | 120.6 |
PJML(1.0) | 0.045 | 0.035 | 0.045 | 0.044 | 0.009 | 0.063 | 112.1 | 107.7 | 112.7 | 113.0 | 103.7 | 123.1 |
PJML(1.5) | 0.002 | 0.009 | 0.005 | 0.008 | 0.033 | 0.013 | 100.3 | 101.9 | 100.9 | 101.4 | 114.9 | 100.4 |
PJML(2.0) | 0.030 | 0.037 | 0.032 | 0.032 | 0.053 | 0.021 | 112.0 | 114.9 | 112.1 | 112.6 | 129.9 | 107.0 |
JML(0.1) | 0.047 | 0.049 | 0.048 | 0.048 | 0.052 | 0.044 | 122.2 | 122.6 | 122.1 | 122.4 | 128.7 | 120.1 |
JML(0.2) | 0.027 | 0.027 | 0.024 | 0.022 | 0.030 | 0.025 | 104.4 | 102.9 | 102.3 | 102.8 | 107.8 | 100.6 |
JML(0.24) | 0.036 | 0.035 | 0.032 | 0.032 | 0.034 | 0.038 | 107.4 | 105.4 | 104.6 | 105.0 | 108.4 | 104.6 |
JML(0.3) | 0.060 | 0.058 | 0.057 | 0.056 | 0.053 | 0.063 | 120.5 | 117.8 | 117.6 | 117.5 | 117.5 | 120.0 |
JML(0.4) | 0.101 | 0.099 | 0.097 | 0.096 | 0.088 | 0.105 | 153.4 | 150.0 | 149.2 | 149.9 | 146.3 | 153.7 |
JML(0.5) | 0.139 | 0.135 | 0.137 | 0.135 | 0.124 | 0.144 | 189.9 | 184.7 | 186.5 | 186.5 | 179.7 | 191.9 |
PMML | 0.002 | 0.005 | 0.007 | 0.011 | 0.012 | 0.004 | 100.8 | 101.3 | 101.9 | 102.7 | 105.3 | 101.3 |
PCML | 0.002 | 0.003 | 0.003 | 0.003 | 0.002 | 0.003 | 103.2 | 102.7 | 103.0 | 102.6 | 105.1 | 102.6 |
LLLA | 0.004 | 0.007 | 0.010 | 0.014 | 0.014 | 0.008 | 101.1 | 102.0 | 102.9 | 103.9 | 105.8 | 102.2 |
MINCHI | 0.002 | 0.002 | 0.002 | 0.002 | 0.002 | 0.003 | 103.0 | 102.5 | 102.8 | 102.4 | 104.9 | 102.4 |
EVM(2) | 0.002 | 0.003 | 0.002 | 0.003 | 0.002 | 0.003 | 104.4 | 103.5 | 104.1 | 103.4 | 106.0 | 103.6 |
EVM(3) | 0.002 | 0.003 | 0.002 | 0.003 | 0.002 | 0.003 | 104.3 | 103.5 | 104.1 | 103.4 | 105.9 | 103.6 |
RA(1) | 0.003 | 0.004 | 0.004 | 0.004 | 0.004 | 0.004 | 104.8 | 104.1 | 104.7 | 104.1 | 106.4 | 104.3 |
RA(2) | 0.002 | 0.003 | 0.002 | 0.003 | 0.002 | 0.003 | 104.4 | 103.5 | 104.1 | 103.4 | 106.0 | 103.6 |
RA(3) | 0.002 | 0.003 | 0.002 | 0.003 | 0.002 | 0.003 | 104.3 | 103.5 | 104.1 | 103.4 | 105.9 | 103.6 |
Method | Bias | Relative RMSE | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
NO | AM | UN | BE | LC2 | LC3 | NO | AM | UN | BE | LC2 | LC3 | |
MMLN | 0.014 | 0.021 | 0.020 | 0.018 | 0.070 | 0.032 | 108.6 | 104.0 | 107.0 | 108.0 | 116.1 | 109.0 |
MMLLS(3) | 0.012 | 0.013 | 0.020 | 0.017 | 0.045 | 0.023 | 108.6 | 103.6 | 107.3 | 108.1 | 115.9 | 109.1 |
MMLLS(4) | 0.011 | 0.011 | 0.017 | 0.014 | 0.024 | 0.015 | 108.5 | 103.4 | 106.3 | 106.7 | 110.1 | 106.8 |
MMLMN(5) | 0.011 | 0.011 | 0.020 | 0.020 | 0.037 | 0.019 | 107.8 | 102.7 | 106.8 | 108.5 | 108.3 | 108.9 |
MMLMN(7) | 0.015 | 0.014 | 0.022 | 0.021 | 0.038 | 0.022 | 109.0 | 104.0 | 107.6 | 109.0 | 109.1 | 110.7 |
MMLMN(11) | 0.014 | 0.014 | 0.020 | 0.017 | 0.023 | 0.019 | 109.0 | 104.0 | 106.7 | 107.2 | 107.4 | 108.3 |
MMLMN(15) | 0.015 | 0.015 | 0.019 | 0.016 | 0.026 | 0.017 | 109.0 | 104.0 | 106.8 | 107.1 | 110.7 | 107.2 |
MMLLC(2) | 0.029 | 0.028 | 0.014 | 0.011 | 0.011 | 0.010 | 106.2 | 101.1 | 103.1 | 104.5 | 106.9 | 104.4 |
MMLLC(3) | 0.007 | 0.008 | 0.013 | 0.009 | 0.020 | 0.010 | 107.9 | 103.0 | 105.8 | 106.3 | 108.5 | 106.1 |
MMLLC(4) | 0.010 | 0.012 | 0.016 | 0.012 | 0.023 | 0.012 | 108.5 | 103.5 | 106.3 | 106.6 | 109.0 | 106.7 |
MMLLC(5) | 0.011 | 0.013 | 0.016 | 0.012 | 0.022 | 0.013 | 108.7 | 103.7 | 106.4 | 106.8 | 108.9 | 106.8 |
CML | 0.014 | 0.015 | 0.017 | 0.013 | 0.016 | 0.013 | 109.0 | 104.1 | 106.5 | 106.8 | 108.6 | 106.9 |
JMLM | 0.093 | 0.095 | 0.097 | 0.094 | 0.096 | 0.091 | 128.8 | 123.7 | 127.3 | 126.7 | 128.5 | 126.8 |
JMLW | 0.046 | 0.048 | 0.052 | 0.049 | 0.050 | 0.047 | 115.4 | 110.2 | 114.0 | 114.3 | 114.9 | 113.8 |
PJML(1.0) | 0.099 | 0.098 | 0.103 | 0.107 | 0.110 | 0.107 | 116.9 | 112.5 | 115.4 | 117.9 | 121.1 | 118.4 |
PJML(1.5) | 0.009 | 0.016 | 0.014 | 0.013 | 0.054 | 0.025 | 106.4 | 101.8 | 104.0 | 105.0 | 110.5 | 105.6 |
PJML(2.0) | 0.084 | 0.085 | 0.083 | 0.079 | 0.095 | 0.082 | 122.5 | 117.7 | 120.2 | 120.0 | 124.0 | 120.4 |
JML(0.1) | 0.160 | 0.162 | 0.163 | 0.160 | 0.162 | 0.158 | 150.0 | 145.1 | 148.8 | 148.0 | 149.0 | 148.1 |
JML(0.2) | 0.053 | 0.051 | 0.050 | 0.050 | 0.047 | 0.047 | 107.0 | 106.5 | 106.5 | 106.9 | 105.1 | 106.4 |
JML(0.24) | 0.034 | 0.035 | 0.032 | 0.027 | 0.043 | 0.029 | 100.0 | 100.0 | 100.0 | 100.0 | 100.0 | 100.0 |
JML(0.3) | 0.049 | 0.053 | 0.049 | 0.050 | 0.066 | 0.056 | 102.7 | 100.8 | 101.3 | 101.8 | 104.0 | 102.5 |
JML(0.4) | 0.132 | 0.133 | 0.135 | 0.136 | 0.146 | 0.140 | 123.4 | 122.5 | 123.8 | 123.9 | 126.0 | 125.2 |
JML(0.5) | 0.213 | 0.211 | 0.213 | 0.216 | 0.218 | 0.217 | 156.9 | 152.5 | 154.1 | 156.6 | 155.8 | 157.6 |
PMML | 0.014 | 0.022 | 0.020 | 0.019 | 0.081 | 0.035 | 108.7 | 104.2 | 107.1 | 108.1 | 118.9 | 109.7 |
PCML | 0.017 | 0.018 | 0.021 | 0.016 | 0.019 | 0.016 | 115.2 | 109.8 | 112.8 | 113.2 | 114.4 | 112.3 |
LLLA | 0.013 | 0.021 | 0.020 | 0.018 | 0.073 | 0.032 | 108.4 | 103.9 | 106.9 | 107.9 | 116.5 | 109.0 |
MINCHI | 0.008 | 0.008 | 0.006 | 0.010 | 0.008 | 0.010 | 111.4 | 106.2 | 108.8 | 109.6 | 110.6 | 108.8 |
EVM(2) | 0.020 | 0.021 | 0.024 | 0.019 | 0.022 | 0.018 | 122.9 | 117.2 | 120.9 | 121.4 | 122.6 | 119.9 |
EVM(3) | 0.020 | 0.021 | 0.024 | 0.019 | 0.022 | 0.018 | 123.0 | 117.3 | 121.1 | 121.5 | 122.7 | 120.0 |
RA(1) | 0.026 | 0.028 | 0.027 | 0.026 | 0.027 | 0.026 | 115.8 | 111.3 | 114.0 | 114.8 | 114.6 | 114.2 |
RA(2) | 0.020 | 0.021 | 0.024 | 0.018 | 0.022 | 0.018 | 122.9 | 117.2 | 120.9 | 121.4 | 122.6 | 119.9 |
RA(3) | 0.020 | 0.021 | 0.024 | 0.019 | 0.022 | 0.018 | 123.0 | 117.3 | 121.1 | 121.5 | 122.7 | 120.0 |
Method | Bias | Relative RMSE | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
NO | AM | UN | BE | LC2 | LC3 | NO | AM | UN | BE | LC2 | LC3 | |
MMLN | 0.003 | 0.009 | 0.008 | 0.010 | 0.040 | 0.018 | 101.2 | 101.4 | 102.3 | 105.1 | 112.3 | 103.8 |
MMLLS(3) | 0.003 | 0.003 | 0.007 | 0.009 | 0.035 | 0.011 | 100.4 | 100.2 | 101.6 | 104.7 | 110.6 | 101.9 |
MMLLS(4) | 0.003 | 0.005 | 0.004 | 0.004 | 0.008 | 0.007 | 100.3 | 100.2 | 100.5 | 102.5 | 101.5 | 100.9 |
MMLMN(5) | 0.010 | 0.006 | 0.006 | 0.005 | 0.024 | 0.016 | 100.4 | 100.0 | 100.5 | 103.6 | 107.6 | 102.2 |
MMLMN(7) | 0.003 | 0.004 | 0.006 | 0.005 | 0.024 | 0.016 | 100.3 | 100.2 | 100.5 | 103.6 | 107.5 | 102.8 |
MMLMN(11) | 0.003 | 0.004 | 0.005 | 0.005 | 0.009 | 0.013 | 100.3 | 100.2 | 100.4 | 102.4 | 100.9 | 101.3 |
MMLMN(15) | 0.002 | 0.004 | 0.004 | 0.003 | 0.008 | 0.007 | 100.4 | 100.2 | 100.5 | 102.7 | 101.7 | 100.0 |
MMLLC(2) | 0.083 | 0.082 | 0.065 | 0.052 | 0.016 | 0.040 | 136.7 | 135.7 | 123.3 | 117.0 | 101.4 | 108.5 |
MMLLC(3) | 0.031 | 0.030 | 0.024 | 0.021 | 0.008 | 0.019 | 105.5 | 104.9 | 103.1 | 104.3 | 100.6 | 101.8 |
MMLLC(4) | 0.013 | 0.012 | 0.014 | 0.013 | 0.006 | 0.009 | 100.9 | 100.7 | 101.2 | 103.2 | 100.6 | 100.0 |
MMLLC(5) | 0.008 | 0.007 | 0.009 | 0.009 | 0.007 | 0.010 | 100.6 | 100.3 | 100.8 | 102.9 | 100.5 | 100.2 |
CML | 0.003 | 0.003 | 0.005 | 0.005 | 0.004 | 0.002 | 101.3 | 101.0 | 101.4 | 103.7 | 101.3 | 100.6 |
JMLM | 0.024 | 0.025 | 0.025 | 0.026 | 0.025 | 0.023 | 105.9 | 105.8 | 106.5 | 109.1 | 106.1 | 105.0 |
JMLW | 0.015 | 0.015 | 0.016 | 0.018 | 0.016 | 0.014 | 103.1 | 102.9 | 103.7 | 106.3 | 103.1 | 102.5 |
PJML(1.0) | 0.052 | 0.051 | 0.056 | 0.055 | 0.062 | 0.058 | 115.1 | 114.8 | 117.4 | 118.8 | 122.8 | 118.8 |
PJML(1.5) | 0.007 | 0.008 | 0.008 | 0.008 | 0.024 | 0.011 | 101.4 | 101.3 | 102.0 | 104.6 | 105.2 | 101.9 |
PJML(2.0) | 0.035 | 0.036 | 0.034 | 0.036 | 0.039 | 0.033 | 110.6 | 110.5 | 111.2 | 114.0 | 111.7 | 109.7 |
JML(0.1) | 0.050 | 0.051 | 0.051 | 0.052 | 0.050 | 0.049 | 117.8 | 117.8 | 118.9 | 121.5 | 117.7 | 116.5 |
JML(0.2) | 0.019 | 0.019 | 0.018 | 0.020 | 0.020 | 0.020 | 102.7 | 104.2 | 102.5 | 103.0 | 102.3 | 103.3 |
JML(0.24) | 0.013 | 0.012 | 0.013 | 0.011 | 0.015 | 0.014 | 100.0 | 101.4 | 100.0 | 100.0 | 100.0 | 100.5 |
JML(0.3) | 0.016 | 0.016 | 0.015 | 0.013 | 0.021 | 0.016 | 100.1 | 100.8 | 100.1 | 100.7 | 100.7 | 100.0 |
JML(0.4) | 0.039 | 0.040 | 0.041 | 0.039 | 0.044 | 0.041 | 109.3 | 109.9 | 109.9 | 108.3 | 110.1 | 108.9 |
JML(0.5) | 0.067 | 0.067 | 0.068 | 0.067 | 0.072 | 0.070 | 125.6 | 125.5 | 125.6 | 126.2 | 126.6 | 126.0 |
PMML | 0.049 | 0.045 | 0.105 | 0.112 | 0.079 | 0.097 | 161.5 | 158.1 | 210.7 | 217.0 | 182.8 | 202.4 |
PCML | 0.004 | 0.004 | 0.005 | 0.006 | 0.004 | 0.003 | 104.6 | 104.5 | 104.5 | 106.7 | 104.2 | 103.9 |
LLLA | 0.003 | 0.010 | 0.008 | 0.010 | 0.042 | 0.019 | 101.0 | 101.3 | 102.2 | 105.0 | 113.2 | 103.9 |
MINCHI | 0.004 | 0.004 | 0.004 | 0.002 | 0.003 | 0.005 | 103.9 | 103.8 | 103.7 | 105.8 | 103.4 | 103.4 |
EVM(2) | 0.004 | 0.004 | 0.005 | 0.006 | 0.004 | 0.003 | 109.2 | 109.2 | 109.1 | 111.3 | 108.4 | 108.6 |
EVM(3) | 0.004 | 0.004 | 0.005 | 0.006 | 0.004 | 0.003 | 109.2 | 109.2 | 109.2 | 111.3 | 108.5 | 108.7 |
RA(1) | 0.020 | 0.020 | 0.020 | 0.022 | 0.019 | 0.019 | 117.1 | 117.0 | 117.2 | 120.0 | 115.5 | 116.5 |
RA(2) | 0.004 | 0.004 | 0.005 | 0.006 | 0.004 | 0.003 | 109.2 | 109.2 | 109.1 | 111.3 | 108.4 | 108.6 |
RA(3) | 0.004 | 0.004 | 0.005 | 0.006 | 0.004 | 0.003 | 109.2 | 109.2 | 109.2 | 111.3 | 108.5 | 108.7 |
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Source | Bias | Rel. RMSE |
---|---|---|
N | 0.2 | 0.9 |
I | 5.3 | 5.5 |
Skew | 0.2 | 0.1 |
Range | 6.3 | 6.5 |
Meth | 51.6 | 29.3 |
Dist | 0.4 | 0.1 |
N×I | 0.0 | 0.6 |
N× Skew | 0.0 | 0.0 |
N× Range | 0.0 | 0.4 |
N× Meth | 0.7 | 9.2 |
N× Dist | 0.0 | 0.1 |
I× Skew | 0.0 | 0.1 |
I× Range | 0.7 | 1.4 |
I× Meth | 17.5 | 21.3 |
I× Dist | 0.0 | 0.0 |
Skew× Range | 0.4 | 0.4 |
Skew× Meth | 0.4 | 0.6 |
Skew× Dist | 0.0 | 0.0 |
Range× Meth | 7.9 | 7.3 |
Range× Dist | 0.1 | 0.1 |
Meth× Dist | 1.6 | 0.5 |
N×I× Skew | 0.0 | 0.0 |
N×I× Range | 0.0 | 0.0 |
N×I× Meth | 0.0 | 4.4 |
N×I× Dist | 0.0 | 0.0 |
N× Skew× Range | 0.0 | 0.0 |
N× Skew× Meth | 0.1 | 0.3 |
N× Skew× Dist | 0.0 | 0.0 |
N× Range× Meth | 0.1 | 1.4 |
N× Range× Dist | 0.0 | 0.0 |
N× Meth× Dist | 0.1 | 0.2 |
I× Skew× Range | 0.1 | 0.1 |
I× Skew× Meth | 0.2 | 0.3 |
I× Skew× Dist | 0.0 | 0.0 |
I× Range× Meth | 2.7 | 4.5 |
I× Range× Dist | 0.0 | 0.0 |
I× Meth× Dist | 0.2 | 0.1 |
Skew× Range× Meth | 1.0 | 0.7 |
Skew× Range× Dist | 0.0 | 0.0 |
Skew× Meth× Dist | 0.1 | 0.1 |
Range× Meth× Dist | 1.1 | 0.7 |
Residual | 1.0 | 3.0 |
Method | Bias | Relative RMSE | ||||||||
---|---|---|---|---|---|---|---|---|---|---|
Rk | %Acc | Med | Q90 | MAD | Rk | %Acc | Med | Q90 | MAD | |
MMLN | 17 | 81.3 | 0.013 | 0.040 | 0.009 | 14 | 74.5 | 104.2 | 109.9 | 3.1 |
MMLLS(3) | 14 | 90.1 | 0.008 | 0.023 | 0.007 | 11 | 81.3 | 103.5 | 109.1 | 3.0 |
MMLLS(4) | 3 | 100.0 | 0.006 | 0.014 | 0.003 | 5 | 93.8 | 102.4 | 106.2 | 2.5 |
MMLMN(5) | 21 | 62.0 | 0.016 | 0.080 | 0.015 | 18 | 68.8 | 104.4 | 125.4 | 4.2 |
MMLMN(7) | 15 | 85.9 | 0.009 | 0.029 | 0.007 | 10 | 81.8 | 103.4 | 108.4 | 3.2 |
MMLMN(11) | 12 | 96.9 | 0.006 | 0.016 | 0.004 | 6 | 91.7 | 102.5 | 106.3 | 2.5 |
MMLMN(15) | 9 | 99.5 | 0.006 | 0.016 | 0.004 | 7 | 91.7 | 102.6 | 106.7 | 2.5 |
MMLLC(2) | 26 | 40.1 | 0.028 | 0.061 | 0.020 | 17 | 72.4 | 104.2 | 116.5 | 4.0 |
MMLLC(3) | 13 | 94.3 | 0.009 | 0.022 | 0.006 | 3 | 94.8 | 102.6 | 105.7 | 2.5 |
MMLLC(4) | 6 | 100.0 | 0.007 | 0.015 | 0.004 | 2 | 94.8 | 102.1 | 106.0 | 2.2 |
MMLLC(5) | 4 | 100.0 | 0.007 | 0.014 | 0.004 | 4 | 94.3 | 102.3 | 106.3 | 2.3 |
CML | 2 | 100.0 | 0.006 | 0.015 | 0.004 | 8 | 91.1 | 103.0 | 106.8 | 2.3 |
JMLM | 23 | 54.2 | 0.021 | 0.132 | 0.022 | 21 | 56.8 | 105.9 | 145.6 | 5.2 |
JMLW | 27 | 35.4 | 0.035 | 0.077 | 0.025 | 22 | 53.1 | 106.4 | 125.2 | 6.8 |
PJML(1.0) | 30 | 22.4 | 0.048 | 0.111 | 0.031 | 23 | 45.8 | 108.1 | 134.2 | 9.1 |
PJML(1.5) | 16 | 84.9 | 0.010 | 0.032 | 0.007 | 9 | 85.9 | 103.4 | 107.6 | 3.0 |
PJML(2.0) | 28 | 27.1 | 0.038 | 0.085 | 0.024 | 30 | 26.0 | 111.1 | 129.9 | 6.7 |
JML(0.1) | 29 | 25.0 | 0.053 | 0.174 | 0.036 | 32 | 23.4 | 116.0 | 179.4 | 13.9 |
JML(0.2) | 24 | 53.6 | 0.024 | 0.052 | 0.017 | 13 | 79.2 | 103.4 | 109.5 | 2.5 |
JML(0.24) | 22 | 56.3 | 0.023 | 0.040 | 0.015 | 1 | 95.3 | 101.1 | 104.8 | 1.6 |
JML(0.3) | 25 | 46.4 | 0.036 | 0.077 | 0.030 | 16 | 72.4 | 101.3 | 119.4 | 1.9 |
JML(0.4) | 31 | 3.1 | 0.065 | 0.166 | 0.050 | 28 | 41.7 | 109.6 | 161.1 | 14.3 |
JML(0.5) | 32 | 0.0 | 0.101 | 0.248 | 0.069 | 31 | 25.0 | 125.5 | 215.3 | 33.3 |
PMML | 20 | 70.3 | 0.015 | 0.067 | 0.011 | 20 | 59.9 | 105.7 | 120.9 | 4.5 |
PCML | 5 | 100.0 | 0.007 | 0.017 | 0.004 | 19 | 62.0 | 106.1 | 111.0 | 2.9 |
LLLA | 19 | 80.7 | 0.013 | 0.042 | 0.007 | 15 | 74.5 | 104.2 | 110.4 | 2.8 |
MINCHI | 1 | 100.0 | 0.005 | 0.012 | 0.003 | 12 | 80.7 | 104.9 | 108.6 | 2.2 |
EVM(2) | 11 | 98.4 | 0.007 | 0.019 | 0.004 | 25 | 41.7 | 108.9 | 117.8 | 6.6 |
EVM(3) | 8 | 100.0 | 0.007 | 0.019 | 0.004 | 27 | 41.7 | 108.8 | 118.1 | 6.6 |
RA(1) | 18 | 81.3 | 0.019 | 0.028 | 0.010 | 29 | 34.4 | 110.0 | 123.0 | 6.1 |
RA(2) | 10 | 99.5 | 0.007 | 0.019 | 0.004 | 24 | 41.7 | 108.9 | 117.8 | 6.6 |
RA(3) | 7 | 100.0 | 0.007 | 0.019 | 0.004 | 26 | 41.7 | 108.8 | 118.1 | 6.6 |
Method | Bias | Relative RMSE | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
NO | AM | UN | BE | LC2 | LC3 | NO | AM | UN | BE | LC2 | LC3 | |
MMLN | 0.004 | 0.013 | 0.010 | 0.012 | 0.063 | 0.030 | 102.2 | 102.4 | 102.3 | 103.6 | 128.9 | 108.8 |
MMLLS(3) | 0.003 | 0.006 | 0.007 | 0.010 | 0.029 | 0.015 | 101.6 | 101.0 | 101.5 | 102.7 | 112.2 | 103.9 |
MMLLS(4) | 0.003 | 0.005 | 0.004 | 0.004 | 0.009 | 0.004 | 101.6 | 101.0 | 100.4 | 101.0 | 104.6 | 100.0 |
MMLMN(5) | 0.008 | 0.005 | 0.005 | 0.005 | 0.023 | 0.011 | 101.0 | 100.4 | 100.6 | 102.4 | 106.2 | 105.3 |
MMLMN(7) | 0.003 | 0.004 | 0.005 | 0.006 | 0.024 | 0.013 | 101.6 | 101.0 | 100.9 | 102.6 | 106.4 | 105.9 |
MMLMN(11) | 0.003 | 0.004 | 0.005 | 0.005 | 0.006 | 0.005 | 101.7 | 101.2 | 100.2 | 100.5 | 100.0 | 101.1 |
MMLMN(15) | 0.002 | 0.004 | 0.003 | 0.003 | 0.009 | 0.003 | 101.7 | 101.2 | 100.6 | 101.2 | 104.8 | 100.0 |
MMLLC(2) | 0.054 | 0.055 | 0.044 | 0.037 | 0.019 | 0.033 | 118.5 | 118.0 | 111.1 | 108.1 | 102.8 | 106.0 |
MMLLC(3) | 0.016 | 0.017 | 0.020 | 0.020 | 0.012 | 0.020 | 102.3 | 101.9 | 102.0 | 102.8 | 102.0 | 102.2 |
MMLLC(4) | 0.009 | 0.011 | 0.012 | 0.011 | 0.007 | 0.008 | 101.7 | 101.2 | 101.0 | 101.4 | 101.8 | 100.5 |
MMLLC(5) | 0.009 | 0.010 | 0.011 | 0.010 | 0.006 | 0.009 | 101.7 | 101.2 | 100.7 | 101.2 | 101.7 | 100.5 |
CML | 0.004 | 0.003 | 0.003 | 0.004 | 0.003 | 0.004 | 102.6 | 101.8 | 101.2 | 101.9 | 102.6 | 101.4 |
JMLM | 0.083 | 0.081 | 0.081 | 0.082 | 0.081 | 0.082 | 144.7 | 142.4 | 144.0 | 144.8 | 143.9 | 145.0 |
JMLW | 0.036 | 0.034 | 0.037 | 0.039 | 0.036 | 0.038 | 113.4 | 111.8 | 113.5 | 114.8 | 113.5 | 114.3 |
PJML(1.0) | 0.106 | 0.107 | 0.114 | 0.115 | 0.118 | 0.114 | 154.1 | 154.8 | 161.1 | 161.6 | 167.6 | 162.7 |
PJML(1.5) | 0.002 | 0.010 | 0.009 | 0.011 | 0.047 | 0.024 | 100.0 | 100.0 | 100.0 | 101.0 | 116.6 | 103.9 |
PJML(2.0) | 0.075 | 0.074 | 0.070 | 0.070 | 0.083 | 0.074 | 136.4 | 134.7 | 134.0 | 134.3 | 142.7 | 136.4 |
JML(0.1) | 0.152 | 0.150 | 0.150 | 0.151 | 0.150 | 0.150 | 203.9 | 201.0 | 203.7 | 204.0 | 203.6 | 204.6 |
JML(0.2) | 0.042 | 0.042 | 0.043 | 0.042 | 0.047 | 0.041 | 111.0 | 109.7 | 113.3 | 110.5 | 116.7 | 112.2 |
JML(0.24) | 0.033 | 0.032 | 0.028 | 0.027 | 0.044 | 0.032 | 102.4 | 101.2 | 102.9 | 100.0 | 109.4 | 103.2 |
JML(0.3) | 0.056 | 0.058 | 0.055 | 0.055 | 0.069 | 0.059 | 121.2 | 120.5 | 121.0 | 119.4 | 128.0 | 121.4 |
JML(0.4) | 0.139 | 0.139 | 0.140 | 0.141 | 0.146 | 0.142 | 187.1 | 185.9 | 189.1 | 187.2 | 193.2 | 190.0 |
JML(0.5) | 0.216 | 0.217 | 0.219 | 0.219 | 0.222 | 0.219 | 260.8 | 260.5 | 266.4 | 265.2 | 268.3 | 266.4 |
PMML | 0.004 | 0.015 | 0.010 | 0.013 | 0.073 | 0.034 | 102.3 | 102.8 | 102.4 | 103.8 | 136.3 | 110.8 |
PCML | 0.004 | 0.004 | 0.004 | 0.004 | 0.004 | 0.005 | 107.4 | 107.9 | 106.4 | 106.9 | 107.9 | 106.4 |
LLLA | 0.004 | 0.014 | 0.010 | 0.013 | 0.066 | 0.031 | 102.0 | 102.2 | 102.2 | 103.6 | 130.4 | 109.2 |
MINCHI | 0.003 | 0.004 | 0.004 | 0.003 | 0.003 | 0.002 | 106.5 | 107.1 | 105.7 | 106.1 | 107.1 | 105.4 |
EVM(2) | 0.004 | 0.005 | 0.004 | 0.005 | 0.004 | 0.006 | 113.6 | 114.4 | 113.2 | 113.6 | 114.5 | 113.3 |
EVM(3) | 0.004 | 0.005 | 0.004 | 0.005 | 0.004 | 0.006 | 114.0 | 114.9 | 113.7 | 113.9 | 114.9 | 113.7 |
RA(1) | 0.019 | 0.020 | 0.018 | 0.020 | 0.019 | 0.021 | 126.9 | 128.5 | 127.2 | 127.2 | 127.6 | 128.0 |
RA(2) | 0.004 | 0.005 | 0.004 | 0.005 | 0.004 | 0.006 | 113.6 | 114.4 | 113.2 | 113.6 | 114.5 | 113.3 |
RA(3) | 0.004 | 0.005 | 0.004 | 0.005 | 0.004 | 0.006 | 114.0 | 114.9 | 113.7 | 113.9 | 114.9 | 113.7 |
Method | Bias | Relative RMSE | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
NO | AM | UN | BE | LC2 | LC3 | NO | AM | UN | BE | LC2 | LC3 | |
MMLN | 0.008 | 0.012 | 0.015 | 0.012 | 0.020 | 0.006 | 106.9 | 105.2 | 105.9 | 105.1 | 107.2 | 105.0 |
MMLLS(3) | 0.007 | 0.008 | 0.014 | 0.011 | 0.011 | 0.009 | 106.7 | 104.4 | 105.7 | 105.1 | 105.8 | 105.4 |
MMLLS(4) | 0.007 | 0.008 | 0.011 | 0.008 | 0.015 | 0.010 | 106.7 | 104.4 | 105.1 | 104.3 | 104.7 | 104.6 |
MMLMN(5) | 0.011 | 0.012 | 0.009 | 0.009 | 0.030 | 0.030 | 106.2 | 104.2 | 104.5 | 103.4 | 105.5 | 107.0 |
MMLMN(7) | 0.008 | 0.008 | 0.011 | 0.008 | 0.029 | 0.031 | 106.8 | 104.6 | 105.2 | 104.4 | 106.0 | 107.3 |
MMLMN(11) | 0.008 | 0.008 | 0.011 | 0.007 | 0.027 | 0.014 | 106.7 | 104.5 | 105.1 | 104.3 | 105.5 | 105.6 |
MMLMN(15) | 0.008 | 0.009 | 0.011 | 0.007 | 0.014 | 0.011 | 106.7 | 104.6 | 105.1 | 104.2 | 104.6 | 105.2 |
MMLLC(2) | 0.029 | 0.025 | 0.017 | 0.014 | 0.004 | 0.018 | 103.8 | 102.0 | 102.6 | 102.1 | 104.2 | 102.4 |
MMLLC(3) | 0.006 | 0.004 | 0.008 | 0.005 | 0.006 | 0.004 | 105.2 | 103.4 | 104.4 | 103.7 | 104.7 | 104.1 |
MMLLC(4) | 0.005 | 0.006 | 0.010 | 0.007 | 0.007 | 0.005 | 106.1 | 104.1 | 104.9 | 104.0 | 104.8 | 104.6 |
MMLLC(5) | 0.005 | 0.007 | 0.010 | 0.007 | 0.006 | 0.006 | 106.3 | 104.2 | 104.9 | 104.0 | 104.5 | 104.8 |
CML | 0.008 | 0.009 | 0.012 | 0.008 | 0.009 | 0.008 | 106.9 | 104.8 | 105.3 | 104.4 | 105.6 | 105.1 |
JMLM | 0.010 | 0.010 | 0.012 | 0.009 | 0.010 | 0.010 | 107.1 | 104.8 | 105.4 | 104.5 | 105.4 | 105.3 |
JMLW | 0.007 | 0.005 | 0.007 | 0.007 | 0.004 | 0.007 | 105.2 | 103.1 | 103.5 | 102.7 | 104.0 | 103.3 |
PJML(1.0) | 0.015 | 0.012 | 0.016 | 0.017 | 0.012 | 0.025 | 104.6 | 103.0 | 103.3 | 102.5 | 105.4 | 102.8 |
PJML(1.5) | 0.010 | 0.013 | 0.014 | 0.010 | 0.021 | 0.005 | 107.1 | 105.4 | 105.6 | 104.7 | 107.6 | 104.7 |
PJML(2.0) | 0.020 | 0.022 | 0.023 | 0.019 | 0.027 | 0.017 | 109.2 | 107.3 | 107.6 | 106.6 | 109.0 | 106.8 |
JML(0.1) | 0.022 | 0.021 | 0.023 | 0.021 | 0.023 | 0.021 | 108.9 | 106.5 | 107.2 | 106.3 | 107.3 | 107.0 |
JML(0.2) | 0.011 | 0.011 | 0.011 | 0.011 | 0.015 | 0.008 | 103.5 | 103.8 | 102.9 | 103.5 | 104.4 | 103.1 |
JML(0.24) | 0.011 | 0.011 | 0.010 | 0.010 | 0.013 | 0.008 | 102.4 | 102.7 | 101.9 | 102.6 | 103.4 | 102.0 |
JML(0.3) | 0.015 | 0.013 | 0.014 | 0.015 | 0.015 | 0.015 | 102.2 | 101.1 | 101.1 | 101.2 | 101.7 | 101.2 |
JML(0.4) | 0.026 | 0.026 | 0.028 | 0.032 | 0.027 | 0.030 | 100.0 | 100.3 | 100.0 | 100.9 | 100.9 | 100.0 |
JML(0.5) | 0.045 | 0.042 | 0.041 | 0.044 | 0.038 | 0.046 | 102.5 | 100.0 | 100.3 | 100.0 | 100.0 | 101.1 |
PMML | 0.005 | 0.010 | 0.013 | 0.013 | 0.018 | 0.009 | 107.6 | 105.7 | 107.0 | 106.2 | 106.5 | 105.6 |
PCML | 0.009 | 0.010 | 0.013 | 0.009 | 0.009 | 0.009 | 108.1 | 105.9 | 106.7 | 105.8 | 106.1 | 107.3 |
LLLA | 0.013 | 0.016 | 0.019 | 0.016 | 0.019 | 0.014 | 107.6 | 105.8 | 106.6 | 105.7 | 106.9 | 106.1 |
MINCHI | 0.005 | 0.006 | 0.009 | 0.006 | 0.006 | 0.005 | 107.2 | 104.9 | 105.7 | 104.8 | 105.3 | 106.4 |
EVM(2) | 0.009 | 0.010 | 0.013 | 0.009 | 0.009 | 0.008 | 108.9 | 106.6 | 107.6 | 106.5 | 106.5 | 108.4 |
EVM(3) | 0.009 | 0.010 | 0.013 | 0.009 | 0.009 | 0.009 | 108.9 | 106.6 | 107.6 | 106.5 | 106.4 | 108.4 |
RA(1) | 0.018 | 0.019 | 0.021 | 0.017 | 0.017 | 0.019 | 111.2 | 108.8 | 109.9 | 108.9 | 108.4 | 110.9 |
RA(2) | 0.009 | 0.010 | 0.013 | 0.009 | 0.009 | 0.008 | 108.9 | 106.6 | 107.6 | 106.5 | 106.5 | 108.4 |
RA(3) | 0.009 | 0.010 | 0.013 | 0.009 | 0.009 | 0.009 | 108.9 | 106.6 | 107.6 | 106.5 | 106.4 | 108.4 |
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Robitzsch, A. A Comprehensive Simulation Study of Estimation Methods for the Rasch Model. Stats 2021, 4, 814-836. https://doi.org/10.3390/stats4040048
Robitzsch A. A Comprehensive Simulation Study of Estimation Methods for the Rasch Model. Stats. 2021; 4(4):814-836. https://doi.org/10.3390/stats4040048
Chicago/Turabian StyleRobitzsch, Alexander. 2021. "A Comprehensive Simulation Study of Estimation Methods for the Rasch Model" Stats 4, no. 4: 814-836. https://doi.org/10.3390/stats4040048