Table 2
Simulation: estimates, standard errors and mean square errors for μ1 and β1.
| Case | Model | μ1 | Est. | SE | MSE | β10 | Est. | SE | MSE | β11 | Est. | SE | MSE |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 | PMM | 10 | 10.000 | 0.134 | 0.018 | −3 | −2.913 | 0.093 | 0.016 | −1 | −0.919 | 0.126 | 0.022 |
| survival | DT-HMM | 10 | 9.996 | 0.131 | 0.017 | −3 | −1.290 | 0.107 | 2.935 | −1 | −1.087 | 0.161 | 0.034 |
 | CT-HMM | 10 | 9.902 | 0.138 | 0.029 | −3 | −3.072 | 0.098 | 0.015 | −1 | −1.028 | 0.141 | 0.021 |
| PH-HMM | 10 | 9.884 | 0.140 | 0.033 | −3 | −3.138 | 0.104 | 0.030 | −1 | −1.071 | 0.149 | 0.027 | |
| 1.2 | PMM | 10 | 10.009 | 0.135 | 0.018 | −3 | −2.109 | 0.250 | 0.856 | −1 | −1.162 | 0.517 | 0.293 |
| survival | DT-HMM | 10 | 10.004 | 0.129 | 0.017 | −3 | −3.704 | 0.396 | 0.651 | −1 | −1.086 | 0.554 | 0.313 |
 | CT-HMM | 10 | 9.999 | 0.130 | 0.017 | −3 | −3.058 | 0.395 | 0.159 | −1 | −1.079 | 0.550 | 0.309 |
| PH-HMM | 10 | 10.010 | 0.128 | 0.017 | −3 | −2.952 | 0.366 | 0.136 | −1 | −1.054 | 0.541 | 0.295 | |
| 1.3 | PMM | 10 | 9.997 | 0.128 | 0.016 | −3 | −2.609 | 0.178 | 0.184 | −1 | −0.772 | 0.241 | 0.110 |
| logistic | DT-HMM | 10 | 9.996 | 0.122 | 0.015 | −3 | −3.007 | 0.218 | 0.048 | −1 | −1.029 | 0.320 | 0.103 |
| CT-HMM | 10 | 9.996 | 0.122 | 0.015 | −3 | −3.072 | 0.208 | 0.048 | −1 | −0.970 | 0.298 | 0.089 | |
| PH-HMM | 10 | 9.996 | 0.122 | 0.015 | −3 | −3.072 | 0.208 | 0.048 | −1 | −0.970 | 0.298 | 0.089 | |
| 2.1 | PMM | 10 | 9.995 | 0.128 | 0.016 | −2 | −1.991 | 0.127 | 0.016 | −5 | −3.757 | 0.475 | 1.769 |
| survival | DT-HMM | 10 | 9.992 | 0.126 | 0.016 | −2 | −0.134 | 0.170 | 3.509 | −5 | −5.616 | 0.561 | 0.693 |
 | CT-HMM | 10 | 9.835 | 0.145 | 0.048 | −2 | −2.250 | 0.104 | 0.073 | −5 | −3.348 | 0.364 | 2.860 |
| PH-HMM | 10 | 9.960 | 0.128 | 0.018 | −2 | −2.123 | 0.107 | 0.027 | −5 | −4.356 | 0.360 | 0.544 | |
| 2.2 | PMM | 10 | 9.973 | 0.322 | 0.104 | −2 | −1.512 | 0.271 | 0.312 | −5 | −2.785 | 0.481 | 5.137 |
| survival | DT-HMM | 10 | 10.000 | 0.127 | 0.016 | −2 | −2.663 | 0.452 | 0.644 | −5 | −5.608 | 1.509 | 2.643 |
 | CT-HMM | 10 | 9.989 | 0.129 | 0.017 | −2 | −2.120 | 0.402 | 0.176 | −5 | −5.425 | 1.083 | 1.350 |
| PH-HMM | 10 | 10.006 | 0.127 | 0.016 | −2 | −1.924 | 0.341 | 0.122 | −5 | −4.802 | 0.911 | 0.868 | |
| 2.3 | PMM | 10 | 9.994 | 0.132 | 0.017 | −2 | −2.014 | 0.148 | 0.022 | −5 | −2.628 | 0.248 | 5.689 |
| logistic | DT-HMM | 10 | 9.990 | 0.129 | 0.017 | −2 | −2.004 | 0.235 | 0.055 | −5 | −5.151 | 0.736 | 0.564 |
| CT-HMM | 10 | 9.988 | 0.129 | 0.017 | −2 | −2.339 | 0.163 | 0.141 | −5 | −3.434 | 0.314 | 2.552 | |
| PH-HMM | 10 | 9.990 | 0.129 | 0.017 | −2 | −2.327 | 0.162 | 0.133 | −5 | −3.392 | 0.309 | 2.680 | |
| 3.1 | PMM | 5 | 5.024 | 0.120 | 0.015 | −3 | −2.514 | 0.070 | 0.241 | −1 | −0.571 | 0.099 | 0.194 |
| survival | DT-HMM | 5 | 5.028 | 0.107 | 0.012 | −3 | −1.309 | 0.148 | 2.880 | −1 | −1.104 | 0.213 | 0.056 |
 | CT-HMM | 5 | 4.826 | 0.120 | 0.044 | −3 | −3.430 | 0.146 | 0.206 | −1 | −1.101 | 0.193 | 0.047 |
PH HMM | 5 | 4.641 | 0.145 | 0.150 | −3 | −4.122 | 0.251 | 1.322 | −1 | −1.421 | 0.317 | 0.277 | |
| 3.2 | PMM | 5 | 4.994 | 0.243 | 0.059 | −3 | −0.689 | 0.194 | 5.378 | −1 | −0.749 | 0.210 | 0.107 |
| survival | DT HMM | 5 | 5.012 | 0.094 | 0.009 | −3 | −3.669 | 0.482 | 0.679 | −1 | −1.135 | 0.618 | 0.400 |
 | CT HMM | 5 | 4.999 | 0.095 | 0.009 | −3 | −3.161 | 0.480 | 0.256 | −1 | −1.133 | 0.612 | 0.392 |
| PH−HMM | 5 | 5.067 | 0.100 | 0.015 | −3 | −2.300 | 0.396 | 0.647 | −1 | −0.895 | 0.676 | 0.467 | |
| 3.3 | PMM | 5 | 5.001 | 0.157 | 0.024 | −3 | −1.454 | 0.113 | 2.404 | −1 | −0.382 | 0.120 | 0.396 |
| logistic | DT HMM | 5 | 5.004 | 0.093 | 0.009 | −3 | −3.011 | 0.253 | 0.064 | −1 | −1.054 | 0.332 | 0.113 |
CT HMM | 5 | 5.001 | 0.093 | 0.009 | −3 | −3.100 | 0.239 | 0.067 | −1 | −1.005 | 0.306 | 0.094 | |
PH HMM | 5 | 5.001 | 0.093 | 0.009 | −3 | −3.099 | 0.239 | 0.067 | −1 | −1.000 | 0.306 | 0.093 | |
| 4.1 | PMM | 5 | 5.002 | 0.153 | 0.023 | −2 | −1.858 | 0.096 | 0.029 | −5 | −2.078 | 0.185 | 8.572 |
| survival | DT HMM | 5 | 5.004 | 0.090 | 0.008 | −2 | −0.135 | 0.221 | 3.525 | −5 | −5.821 | 0.909 | 1.500 |
 | CT−HMM | 5 | 4.295 | 0.207 | 0.539 | −2 | −3.219 | 0.228 | 1.539 | −5 | −1.710 | 0.136 | 10.845 |
| PH-HMM | 5 | 4.850 | 0.129 | 0.039 | −2 | −2.952 | 0.135 | 0.924 | −5 | −2.798 | 0.226 | 4.900 | |
| 4.2 | PMM | 5 | 5.003 | 0.135 | 0.018 | −2 | −0.487 | 0.111 | 2.302 | −5 | −1.234 | 0.137 | 14.205 |
| survival | DT-HMM | 5 | 5.006 | 0.094 | 0.009 | −2 | −2.532 | 0.448 | 0.484 | −5 | −5.597 | 1.840 | 3.737 |
 | CT-HMM | 5 | 4.986 | 0.095 | 0.009 | −2 | −2.268 | 0.488 | 0.310 | −5 | −5.040 | 1.129 | 1.273 |
| PH-HMM | 5 | 5.080 | 0.103 | 0.017 | −2 | −1.243 | 0.236 | 0.628 | −5 | −2.493 | 0.428 | 6.470 | |
| 4.3 | PMM | 5 | 5.002 | 0.162 | 0.026 | −2 | −1.157 | 0.081 | 0.716 | −5 | −1.200 | 0.100 | 14.451 |
| logistic | DT-HMM | 5 | 5.008 | 0.094 | 0.009 | −2 | −1.967 | 0.331 | 0.111 | −5 | −5.195 | 1.091 | 1.225 |
| CT-HMM | 5 | 5.007 | 0.093 | 0.009 | −2 | −2.389 | 0.185 | 0.185 | −5 | −3.246 | 0.303 | 3.167 | |
| PH-HMM | 5 | 5.016 | 0.093 | 0.009 | −2 | −2.328 | 0.177 | 0.139 | −5 | −3.010 | 0.283 | 4.038 |
| Case | Model | μ1 | Est. | SE | MSE | β10 | Est. | SE | MSE | β11 | Est. | SE | MSE |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 | PMM | 10 | 10.000 | 0.134 | 0.018 | −3 | −2.913 | 0.093 | 0.016 | −1 | −0.919 | 0.126 | 0.022 |
| survival | DT-HMM | 10 | 9.996 | 0.131 | 0.017 | −3 | −1.290 | 0.107 | 2.935 | −1 | −1.087 | 0.161 | 0.034 |
| CT-HMM | 10 | 9.902 | 0.138 | 0.029 | −3 | −3.072 | 0.098 | 0.015 | −1 | −1.028 | 0.141 | 0.021 |
| PH-HMM | 10 | 9.884 | 0.140 | 0.033 | −3 | −3.138 | 0.104 | 0.030 | −1 | −1.071 | 0.149 | 0.027 | |
| 1.2 | PMM | 10 | 10.009 | 0.135 | 0.018 | −3 | −2.109 | 0.250 | 0.856 | −1 | −1.162 | 0.517 | 0.293 |
| survival | DT-HMM | 10 | 10.004 | 0.129 | 0.017 | −3 | −3.704 | 0.396 | 0.651 | −1 | −1.086 | 0.554 | 0.313 |
| CT-HMM | 10 | 9.999 | 0.130 | 0.017 | −3 | −3.058 | 0.395 | 0.159 | −1 | −1.079 | 0.550 | 0.309 |
| PH-HMM | 10 | 10.010 | 0.128 | 0.017 | −3 | −2.952 | 0.366 | 0.136 | −1 | −1.054 | 0.541 | 0.295 | |
| 1.3 | PMM | 10 | 9.997 | 0.128 | 0.016 | −3 | −2.609 | 0.178 | 0.184 | −1 | −0.772 | 0.241 | 0.110 |
| logistic | DT-HMM | 10 | 9.996 | 0.122 | 0.015 | −3 | −3.007 | 0.218 | 0.048 | −1 | −1.029 | 0.320 | 0.103 |
| CT-HMM | 10 | 9.996 | 0.122 | 0.015 | −3 | −3.072 | 0.208 | 0.048 | −1 | −0.970 | 0.298 | 0.089 | |
| PH-HMM | 10 | 9.996 | 0.122 | 0.015 | −3 | −3.072 | 0.208 | 0.048 | −1 | −0.970 | 0.298 | 0.089 | |
| 2.1 | PMM | 10 | 9.995 | 0.128 | 0.016 | −2 | −1.991 | 0.127 | 0.016 | −5 | −3.757 | 0.475 | 1.769 |
| survival | DT-HMM | 10 | 9.992 | 0.126 | 0.016 | −2 | −0.134 | 0.170 | 3.509 | −5 | −5.616 | 0.561 | 0.693 |
| CT-HMM | 10 | 9.835 | 0.145 | 0.048 | −2 | −2.250 | 0.104 | 0.073 | −5 | −3.348 | 0.364 | 2.860 |
| PH-HMM | 10 | 9.960 | 0.128 | 0.018 | −2 | −2.123 | 0.107 | 0.027 | −5 | −4.356 | 0.360 | 0.544 | |
| 2.2 | PMM | 10 | 9.973 | 0.322 | 0.104 | −2 | −1.512 | 0.271 | 0.312 | −5 | −2.785 | 0.481 | 5.137 |
| survival | DT-HMM | 10 | 10.000 | 0.127 | 0.016 | −2 | −2.663 | 0.452 | 0.644 | −5 | −5.608 | 1.509 | 2.643 |
| CT-HMM | 10 | 9.989 | 0.129 | 0.017 | −2 | −2.120 | 0.402 | 0.176 | −5 | −5.425 | 1.083 | 1.350 |
| PH-HMM | 10 | 10.006 | 0.127 | 0.016 | −2 | −1.924 | 0.341 | 0.122 | −5 | −4.802 | 0.911 | 0.868 | |
| 2.3 | PMM | 10 | 9.994 | 0.132 | 0.017 | −2 | −2.014 | 0.148 | 0.022 | −5 | −2.628 | 0.248 | 5.689 |
| logistic | DT-HMM | 10 | 9.990 | 0.129 | 0.017 | −2 | −2.004 | 0.235 | 0.055 | −5 | −5.151 | 0.736 | 0.564 |
| CT-HMM | 10 | 9.988 | 0.129 | 0.017 | −2 | −2.339 | 0.163 | 0.141 | −5 | −3.434 | 0.314 | 2.552 | |
| PH-HMM | 10 | 9.990 | 0.129 | 0.017 | −2 | −2.327 | 0.162 | 0.133 | −5 | −3.392 | 0.309 | 2.680 | |
| 3.1 | PMM | 5 | 5.024 | 0.120 | 0.015 | −3 | −2.514 | 0.070 | 0.241 | −1 | −0.571 | 0.099 | 0.194 |
| survival | DT-HMM | 5 | 5.028 | 0.107 | 0.012 | −3 | −1.309 | 0.148 | 2.880 | −1 | −1.104 | 0.213 | 0.056 |
| CT-HMM | 5 | 4.826 | 0.120 | 0.044 | −3 | −3.430 | 0.146 | 0.206 | −1 | −1.101 | 0.193 | 0.047 |
| PHHMM | 5 | 4.641 | 0.145 | 0.150 | −3 | −4.122 | 0.251 | 1.322 | −1 | −1.421 | 0.317 | 0.277 | |
| 3.2 | PMM | 5 | 4.994 | 0.243 | 0.059 | −3 | −0.689 | 0.194 | 5.378 | −1 | −0.749 | 0.210 | 0.107 |
| survival | DTHMM | 5 | 5.012 | 0.094 | 0.009 | −3 | −3.669 | 0.482 | 0.679 | −1 | −1.135 | 0.618 | 0.400 |
| CTHMM | 5 | 4.999 | 0.095 | 0.009 | −3 | −3.161 | 0.480 | 0.256 | −1 | −1.133 | 0.612 | 0.392 |
| PH−HMM | 5 | 5.067 | 0.100 | 0.015 | −3 | −2.300 | 0.396 | 0.647 | −1 | −0.895 | 0.676 | 0.467 | |
| 3.3 | PMM | 5 | 5.001 | 0.157 | 0.024 | −3 | −1.454 | 0.113 | 2.404 | −1 | −0.382 | 0.120 | 0.396 |
| logistic | DTHMM | 5 | 5.004 | 0.093 | 0.009 | −3 | −3.011 | 0.253 | 0.064 | −1 | −1.054 | 0.332 | 0.113 |
| CTHMM | 5 | 5.001 | 0.093 | 0.009 | −3 | −3.100 | 0.239 | 0.067 | −1 | −1.005 | 0.306 | 0.094 | |
| PHHMM | 5 | 5.001 | 0.093 | 0.009 | −3 | −3.099 | 0.239 | 0.067 | −1 | −1.000 | 0.306 | 0.093 | |
| 4.1 | PMM | 5 | 5.002 | 0.153 | 0.023 | −2 | −1.858 | 0.096 | 0.029 | −5 | −2.078 | 0.185 | 8.572 |
| survival | DTHMM | 5 | 5.004 | 0.090 | 0.008 | −2 | −0.135 | 0.221 | 3.525 | −5 | −5.821 | 0.909 | 1.500 |
| CT−HMM | 5 | 4.295 | 0.207 | 0.539 | −2 | −3.219 | 0.228 | 1.539 | −5 | −1.710 | 0.136 | 10.845 |
| PH-HMM | 5 | 4.850 | 0.129 | 0.039 | −2 | −2.952 | 0.135 | 0.924 | −5 | −2.798 | 0.226 | 4.900 | |
| 4.2 | PMM | 5 | 5.003 | 0.135 | 0.018 | −2 | −0.487 | 0.111 | 2.302 | −5 | −1.234 | 0.137 | 14.205 |
| survival | DT-HMM | 5 | 5.006 | 0.094 | 0.009 | −2 | −2.532 | 0.448 | 0.484 | −5 | −5.597 | 1.840 | 3.737 |
| CT-HMM | 5 | 4.986 | 0.095 | 0.009 | −2 | −2.268 | 0.488 | 0.310 | −5 | −5.040 | 1.129 | 1.273 |
| PH-HMM | 5 | 5.080 | 0.103 | 0.017 | −2 | −1.243 | 0.236 | 0.628 | −5 | −2.493 | 0.428 | 6.470 | |
| 4.3 | PMM | 5 | 5.002 | 0.162 | 0.026 | −2 | −1.157 | 0.081 | 0.716 | −5 | −1.200 | 0.100 | 14.451 |
| logistic | DT-HMM | 5 | 5.008 | 0.094 | 0.009 | −2 | −1.967 | 0.331 | 0.111 | −5 | −5.195 | 1.091 | 1.225 |
| CT-HMM | 5 | 5.007 | 0.093 | 0.009 | −2 | −2.389 | 0.185 | 0.185 | −5 | −3.246 | 0.303 | 3.167 | |
| PH-HMM | 5 | 5.016 | 0.093 | 0.009 | −2 | −2.328 | 0.177 | 0.139 | −5 | −3.010 | 0.283 | 4.038 |
Note: Mean parameter estimates (Est.), empirical standard error (SE) and mean squared error (MSE) based on 500 replicates. Cases 1.1–1.3 looked at low incidence rate and large distributional difference; Cases 2.1–2.3 looked at high incidence rate and large distributional difference; Cases 3.1–3.3 looked at low incidence rate and small distributional difference; and Cases 4.1–4.3 looked at high incidence rate and small distributional difference.
Table 2
Simulation: estimates, standard errors and mean square errors for μ1 and β1.
| Case | Model | μ1 | Est. | SE | MSE | β10 | Est. | SE | MSE | β11 | Est. | SE | MSE |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 | PMM | 10 | 10.000 | 0.134 | 0.018 | −3 | −2.913 | 0.093 | 0.016 | −1 | −0.919 | 0.126 | 0.022 |
| survival | DT-HMM | 10 | 9.996 | 0.131 | 0.017 | −3 | −1.290 | 0.107 | 2.935 | −1 | −1.087 | 0.161 | 0.034 |
| CT-HMM | 10 | 9.902 | 0.138 | 0.029 | −3 | −3.072 | 0.098 | 0.015 | −1 | −1.028 | 0.141 | 0.021 |
| PH-HMM | 10 | 9.884 | 0.140 | 0.033 | −3 | −3.138 | 0.104 | 0.030 | −1 | −1.071 | 0.149 | 0.027 | |
| 1.2 | PMM | 10 | 10.009 | 0.135 | 0.018 | −3 | −2.109 | 0.250 | 0.856 | −1 | −1.162 | 0.517 | 0.293 |
| survival | DT-HMM | 10 | 10.004 | 0.129 | 0.017 | −3 | −3.704 | 0.396 | 0.651 | −1 | −1.086 | 0.554 | 0.313 |
| CT-HMM | 10 | 9.999 | 0.130 | 0.017 | −3 | −3.058 | 0.395 | 0.159 | −1 | −1.079 | 0.550 | 0.309 |
| PH-HMM | 10 | 10.010 | 0.128 | 0.017 | −3 | −2.952 | 0.366 | 0.136 | −1 | −1.054 | 0.541 | 0.295 | |
| 1.3 | PMM | 10 | 9.997 | 0.128 | 0.016 | −3 | −2.609 | 0.178 | 0.184 | −1 | −0.772 | 0.241 | 0.110 |
| logistic | DT-HMM | 10 | 9.996 | 0.122 | 0.015 | −3 | −3.007 | 0.218 | 0.048 | −1 | −1.029 | 0.320 | 0.103 |
| CT-HMM | 10 | 9.996 | 0.122 | 0.015 | −3 | −3.072 | 0.208 | 0.048 | −1 | −0.970 | 0.298 | 0.089 | |
| PH-HMM | 10 | 9.996 | 0.122 | 0.015 | −3 | −3.072 | 0.208 | 0.048 | −1 | −0.970 | 0.298 | 0.089 | |
| 2.1 | PMM | 10 | 9.995 | 0.128 | 0.016 | −2 | −1.991 | 0.127 | 0.016 | −5 | −3.757 | 0.475 | 1.769 |
| survival | DT-HMM | 10 | 9.992 | 0.126 | 0.016 | −2 | −0.134 | 0.170 | 3.509 | −5 | −5.616 | 0.561 | 0.693 |
| CT-HMM | 10 | 9.835 | 0.145 | 0.048 | −2 | −2.250 | 0.104 | 0.073 | −5 | −3.348 | 0.364 | 2.860 |
| PH-HMM | 10 | 9.960 | 0.128 | 0.018 | −2 | −2.123 | 0.107 | 0.027 | −5 | −4.356 | 0.360 | 0.544 | |
| 2.2 | PMM | 10 | 9.973 | 0.322 | 0.104 | −2 | −1.512 | 0.271 | 0.312 | −5 | −2.785 | 0.481 | 5.137 |
| survival | DT-HMM | 10 | 10.000 | 0.127 | 0.016 | −2 | −2.663 | 0.452 | 0.644 | −5 | −5.608 | 1.509 | 2.643 |
| CT-HMM | 10 | 9.989 | 0.129 | 0.017 | −2 | −2.120 | 0.402 | 0.176 | −5 | −5.425 | 1.083 | 1.350 |
| PH-HMM | 10 | 10.006 | 0.127 | 0.016 | −2 | −1.924 | 0.341 | 0.122 | −5 | −4.802 | 0.911 | 0.868 | |
| 2.3 | PMM | 10 | 9.994 | 0.132 | 0.017 | −2 | −2.014 | 0.148 | 0.022 | −5 | −2.628 | 0.248 | 5.689 |
| logistic | DT-HMM | 10 | 9.990 | 0.129 | 0.017 | −2 | −2.004 | 0.235 | 0.055 | −5 | −5.151 | 0.736 | 0.564 |
| CT-HMM | 10 | 9.988 | 0.129 | 0.017 | −2 | −2.339 | 0.163 | 0.141 | −5 | −3.434 | 0.314 | 2.552 | |
| PH-HMM | 10 | 9.990 | 0.129 | 0.017 | −2 | −2.327 | 0.162 | 0.133 | −5 | −3.392 | 0.309 | 2.680 | |
| 3.1 | PMM | 5 | 5.024 | 0.120 | 0.015 | −3 | −2.514 | 0.070 | 0.241 | −1 | −0.571 | 0.099 | 0.194 |
| survival | DT-HMM | 5 | 5.028 | 0.107 | 0.012 | −3 | −1.309 | 0.148 | 2.880 | −1 | −1.104 | 0.213 | 0.056 |
| CT-HMM | 5 | 4.826 | 0.120 | 0.044 | −3 | −3.430 | 0.146 | 0.206 | −1 | −1.101 | 0.193 | 0.047 |
| PHHMM | 5 | 4.641 | 0.145 | 0.150 | −3 | −4.122 | 0.251 | 1.322 | −1 | −1.421 | 0.317 | 0.277 | |
| 3.2 | PMM | 5 | 4.994 | 0.243 | 0.059 | −3 | −0.689 | 0.194 | 5.378 | −1 | −0.749 | 0.210 | 0.107 |
| survival | DTHMM | 5 | 5.012 | 0.094 | 0.009 | −3 | −3.669 | 0.482 | 0.679 | −1 | −1.135 | 0.618 | 0.400 |
| CTHMM | 5 | 4.999 | 0.095 | 0.009 | −3 | −3.161 | 0.480 | 0.256 | −1 | −1.133 | 0.612 | 0.392 |
| PH−HMM | 5 | 5.067 | 0.100 | 0.015 | −3 | −2.300 | 0.396 | 0.647 | −1 | −0.895 | 0.676 | 0.467 | |
| 3.3 | PMM | 5 | 5.001 | 0.157 | 0.024 | −3 | −1.454 | 0.113 | 2.404 | −1 | −0.382 | 0.120 | 0.396 |
| logistic | DTHMM | 5 | 5.004 | 0.093 | 0.009 | −3 | −3.011 | 0.253 | 0.064 | −1 | −1.054 | 0.332 | 0.113 |
| CTHMM | 5 | 5.001 | 0.093 | 0.009 | −3 | −3.100 | 0.239 | 0.067 | −1 | −1.005 | 0.306 | 0.094 | |
| PHHMM | 5 | 5.001 | 0.093 | 0.009 | −3 | −3.099 | 0.239 | 0.067 | −1 | −1.000 | 0.306 | 0.093 | |
| 4.1 | PMM | 5 | 5.002 | 0.153 | 0.023 | −2 | −1.858 | 0.096 | 0.029 | −5 | −2.078 | 0.185 | 8.572 |
| survival | DTHMM | 5 | 5.004 | 0.090 | 0.008 | −2 | −0.135 | 0.221 | 3.525 | −5 | −5.821 | 0.909 | 1.500 |
| CT−HMM | 5 | 4.295 | 0.207 | 0.539 | −2 | −3.219 | 0.228 | 1.539 | −5 | −1.710 | 0.136 | 10.845 |
| PH-HMM | 5 | 4.850 | 0.129 | 0.039 | −2 | −2.952 | 0.135 | 0.924 | −5 | −2.798 | 0.226 | 4.900 | |
| 4.2 | PMM | 5 | 5.003 | 0.135 | 0.018 | −2 | −0.487 | 0.111 | 2.302 | −5 | −1.234 | 0.137 | 14.205 |
| survival | DT-HMM | 5 | 5.006 | 0.094 | 0.009 | −2 | −2.532 | 0.448 | 0.484 | −5 | −5.597 | 1.840 | 3.737 |
| CT-HMM | 5 | 4.986 | 0.095 | 0.009 | −2 | −2.268 | 0.488 | 0.310 | −5 | −5.040 | 1.129 | 1.273 |
| PH-HMM | 5 | 5.080 | 0.103 | 0.017 | −2 | −1.243 | 0.236 | 0.628 | −5 | −2.493 | 0.428 | 6.470 | |
| 4.3 | PMM | 5 | 5.002 | 0.162 | 0.026 | −2 | −1.157 | 0.081 | 0.716 | −5 | −1.200 | 0.100 | 14.451 |
| logistic | DT-HMM | 5 | 5.008 | 0.094 | 0.009 | −2 | −1.967 | 0.331 | 0.111 | −5 | −5.195 | 1.091 | 1.225 |
| CT-HMM | 5 | 5.007 | 0.093 | 0.009 | −2 | −2.389 | 0.185 | 0.185 | −5 | −3.246 | 0.303 | 3.167 | |
| PH-HMM | 5 | 5.016 | 0.093 | 0.009 | −2 | −2.328 | 0.177 | 0.139 | −5 | −3.010 | 0.283 | 4.038 |
| Case | Model | μ1 | Est. | SE | MSE | β10 | Est. | SE | MSE | β11 | Est. | SE | MSE |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 | PMM | 10 | 10.000 | 0.134 | 0.018 | −3 | −2.913 | 0.093 | 0.016 | −1 | −0.919 | 0.126 | 0.022 |
| survival | DT-HMM | 10 | 9.996 | 0.131 | 0.017 | −3 | −1.290 | 0.107 | 2.935 | −1 | −1.087 | 0.161 | 0.034 |
| CT-HMM | 10 | 9.902 | 0.138 | 0.029 | −3 | −3.072 | 0.098 | 0.015 | −1 | −1.028 | 0.141 | 0.021 |
| PH-HMM | 10 | 9.884 | 0.140 | 0.033 | −3 | −3.138 | 0.104 | 0.030 | −1 | −1.071 | 0.149 | 0.027 | |
| 1.2 | PMM | 10 | 10.009 | 0.135 | 0.018 | −3 | −2.109 | 0.250 | 0.856 | −1 | −1.162 | 0.517 | 0.293 |
| survival | DT-HMM | 10 | 10.004 | 0.129 | 0.017 | −3 | −3.704 | 0.396 | 0.651 | −1 | −1.086 | 0.554 | 0.313 |
| CT-HMM | 10 | 9.999 | 0.130 | 0.017 | −3 | −3.058 | 0.395 | 0.159 | −1 | −1.079 | 0.550 | 0.309 |
| PH-HMM | 10 | 10.010 | 0.128 | 0.017 | −3 | −2.952 | 0.366 | 0.136 | −1 | −1.054 | 0.541 | 0.295 | |
| 1.3 | PMM | 10 | 9.997 | 0.128 | 0.016 | −3 | −2.609 | 0.178 | 0.184 | −1 | −0.772 | 0.241 | 0.110 |
| logistic | DT-HMM | 10 | 9.996 | 0.122 | 0.015 | −3 | −3.007 | 0.218 | 0.048 | −1 | −1.029 | 0.320 | 0.103 |
| CT-HMM | 10 | 9.996 | 0.122 | 0.015 | −3 | −3.072 | 0.208 | 0.048 | −1 | −0.970 | 0.298 | 0.089 | |
| PH-HMM | 10 | 9.996 | 0.122 | 0.015 | −3 | −3.072 | 0.208 | 0.048 | −1 | −0.970 | 0.298 | 0.089 | |
| 2.1 | PMM | 10 | 9.995 | 0.128 | 0.016 | −2 | −1.991 | 0.127 | 0.016 | −5 | −3.757 | 0.475 | 1.769 |
| survival | DT-HMM | 10 | 9.992 | 0.126 | 0.016 | −2 | −0.134 | 0.170 | 3.509 | −5 | −5.616 | 0.561 | 0.693 |
| CT-HMM | 10 | 9.835 | 0.145 | 0.048 | −2 | −2.250 | 0.104 | 0.073 | −5 | −3.348 | 0.364 | 2.860 |
| PH-HMM | 10 | 9.960 | 0.128 | 0.018 | −2 | −2.123 | 0.107 | 0.027 | −5 | −4.356 | 0.360 | 0.544 | |
| 2.2 | PMM | 10 | 9.973 | 0.322 | 0.104 | −2 | −1.512 | 0.271 | 0.312 | −5 | −2.785 | 0.481 | 5.137 |
| survival | DT-HMM | 10 | 10.000 | 0.127 | 0.016 | −2 | −2.663 | 0.452 | 0.644 | −5 | −5.608 | 1.509 | 2.643 |
| CT-HMM | 10 | 9.989 | 0.129 | 0.017 | −2 | −2.120 | 0.402 | 0.176 | −5 | −5.425 | 1.083 | 1.350 |
| PH-HMM | 10 | 10.006 | 0.127 | 0.016 | −2 | −1.924 | 0.341 | 0.122 | −5 | −4.802 | 0.911 | 0.868 | |
| 2.3 | PMM | 10 | 9.994 | 0.132 | 0.017 | −2 | −2.014 | 0.148 | 0.022 | −5 | −2.628 | 0.248 | 5.689 |
| logistic | DT-HMM | 10 | 9.990 | 0.129 | 0.017 | −2 | −2.004 | 0.235 | 0.055 | −5 | −5.151 | 0.736 | 0.564 |
| CT-HMM | 10 | 9.988 | 0.129 | 0.017 | −2 | −2.339 | 0.163 | 0.141 | −5 | −3.434 | 0.314 | 2.552 | |
| PH-HMM | 10 | 9.990 | 0.129 | 0.017 | −2 | −2.327 | 0.162 | 0.133 | −5 | −3.392 | 0.309 | 2.680 | |
| 3.1 | PMM | 5 | 5.024 | 0.120 | 0.015 | −3 | −2.514 | 0.070 | 0.241 | −1 | −0.571 | 0.099 | 0.194 |
| survival | DT-HMM | 5 | 5.028 | 0.107 | 0.012 | −3 | −1.309 | 0.148 | 2.880 | −1 | −1.104 | 0.213 | 0.056 |
| CT-HMM | 5 | 4.826 | 0.120 | 0.044 | −3 | −3.430 | 0.146 | 0.206 | −1 | −1.101 | 0.193 | 0.047 |
| PHHMM | 5 | 4.641 | 0.145 | 0.150 | −3 | −4.122 | 0.251 | 1.322 | −1 | −1.421 | 0.317 | 0.277 | |
| 3.2 | PMM | 5 | 4.994 | 0.243 | 0.059 | −3 | −0.689 | 0.194 | 5.378 | −1 | −0.749 | 0.210 | 0.107 |
| survival | DTHMM | 5 | 5.012 | 0.094 | 0.009 | −3 | −3.669 | 0.482 | 0.679 | −1 | −1.135 | 0.618 | 0.400 |
| CTHMM | 5 | 4.999 | 0.095 | 0.009 | −3 | −3.161 | 0.480 | 0.256 | −1 | −1.133 | 0.612 | 0.392 |
| PH−HMM | 5 | 5.067 | 0.100 | 0.015 | −3 | −2.300 | 0.396 | 0.647 | −1 | −0.895 | 0.676 | 0.467 | |
| 3.3 | PMM | 5 | 5.001 | 0.157 | 0.024 | −3 | −1.454 | 0.113 | 2.404 | −1 | −0.382 | 0.120 | 0.396 |
| logistic | DTHMM | 5 | 5.004 | 0.093 | 0.009 | −3 | −3.011 | 0.253 | 0.064 | −1 | −1.054 | 0.332 | 0.113 |
| CTHMM | 5 | 5.001 | 0.093 | 0.009 | −3 | −3.100 | 0.239 | 0.067 | −1 | −1.005 | 0.306 | 0.094 | |
| PHHMM | 5 | 5.001 | 0.093 | 0.009 | −3 | −3.099 | 0.239 | 0.067 | −1 | −1.000 | 0.306 | 0.093 | |
| 4.1 | PMM | 5 | 5.002 | 0.153 | 0.023 | −2 | −1.858 | 0.096 | 0.029 | −5 | −2.078 | 0.185 | 8.572 |
| survival | DTHMM | 5 | 5.004 | 0.090 | 0.008 | −2 | −0.135 | 0.221 | 3.525 | −5 | −5.821 | 0.909 | 1.500 |
| CT−HMM | 5 | 4.295 | 0.207 | 0.539 | −2 | −3.219 | 0.228 | 1.539 | −5 | −1.710 | 0.136 | 10.845 |
| PH-HMM | 5 | 4.850 | 0.129 | 0.039 | −2 | −2.952 | 0.135 | 0.924 | −5 | −2.798 | 0.226 | 4.900 | |
| 4.2 | PMM | 5 | 5.003 | 0.135 | 0.018 | −2 | −0.487 | 0.111 | 2.302 | −5 | −1.234 | 0.137 | 14.205 |
| survival | DT-HMM | 5 | 5.006 | 0.094 | 0.009 | −2 | −2.532 | 0.448 | 0.484 | −5 | −5.597 | 1.840 | 3.737 |
| CT-HMM | 5 | 4.986 | 0.095 | 0.009 | −2 | −2.268 | 0.488 | 0.310 | −5 | −5.040 | 1.129 | 1.273 |
| PH-HMM | 5 | 5.080 | 0.103 | 0.017 | −2 | −1.243 | 0.236 | 0.628 | −5 | −2.493 | 0.428 | 6.470 | |
| 4.3 | PMM | 5 | 5.002 | 0.162 | 0.026 | −2 | −1.157 | 0.081 | 0.716 | −5 | −1.200 | 0.100 | 14.451 |
| logistic | DT-HMM | 5 | 5.008 | 0.094 | 0.009 | −2 | −1.967 | 0.331 | 0.111 | −5 | −5.195 | 1.091 | 1.225 |
| CT-HMM | 5 | 5.007 | 0.093 | 0.009 | −2 | −2.389 | 0.185 | 0.185 | −5 | −3.246 | 0.303 | 3.167 | |
| PH-HMM | 5 | 5.016 | 0.093 | 0.009 | −2 | −2.328 | 0.177 | 0.139 | −5 | −3.010 | 0.283 | 4.038 |
Note: Mean parameter estimates (Est.), empirical standard error (SE) and mean squared error (MSE) based on 500 replicates. Cases 1.1–1.3 looked at low incidence rate and large distributional difference; Cases 2.1–2.3 looked at high incidence rate and large distributional difference; Cases 3.1–3.3 looked at low incidence rate and small distributional difference; and Cases 4.1–4.3 looked at high incidence rate and small distributional difference.
