Approximate Methods for Maximum Likelihood Estimation of Multivariate Nonlinear Mixed-Effects Models
Abstract
:1. Introduction
2. Five Approximate ML Procedures
2.1. PNLS-MLME Procedure
2.2. Laplacian Procedure
2.3. Pseudo-ECM Algorithm
- E step:
- Evaluate the expected complete-data log-likelihood Function Equation (16) conditioning on the current estimates and the pseudo-responses , which linearize the regression function around the previous estimates of mixed effects and should be updated at each iteration. This gives rise to the so-called Q-function:
- CM step:
- Update the current estimates , , and by maximizing the Q-functionEquation (17). We obtain:
2.4. Monte Carlo EM Algorithm
2.5. Importance Sampling EM Algorithm
2.6. Expected Information Matrix
2.7. Initialization
- (i)
- A direct way of obtaining the initial value for β is to fit the NLMMs to each outcome variable separately by using the nlme R package [12].
- (ii)
- Using the fitting results of NLMMs for each outcome, we take the initial value as a (block) diagonal form with the diagonal entry being the variances (covariances) of random effects under the fitted NLMMs.
- (iii)
- For the initial value for ∑, we use the sample variance-covariance matrix of the data. That is, take , where and .
- (iv)
- The initial values for ϕ, depending on the structure, are simply chosen to give a condition of nearly uncorrelated errors.
3. Application: ACTG 315 Data
Parameter | PNLS-MLME | Laplacian | Pseudo-ECM | MCEM | ISEM |
---|---|---|---|---|---|
β1 | 12.0477 | 12.9800 | 12.0485 | 12.0784 | 12.114 |
(0.2513) | (0.2858) | (0.2530) | (0.2626) | (0.2652) | |
β2 | −2.6558 | −2.6476 | −2.6543 | −2.6198 | −2.6069 |
(0.1781) | (0.1970) | (0.1777) | (0.1950) | (0.1992) | |
β3 | 1.3039 | 1.3001 | 1.3039 | 1.3012 | 1.3000 |
(0.0274) | (0.0248) | (0.0273) | (0.0253) | (0.0249) | |
β4 | 16.8604 | 16.8577 | 16.8605 | 16.8875 | 16.9058 |
(0.3911) | (0.3340) | (0.3914) | (0.3863) | (0.3829) | |
β5 | −1.7324 | −1.7791 | −1.7312 | −1.7721 | −1.7643 |
(0.4936) | (0.4590) | (0.4930) | (0.4632) | (0.4585) | |
β6 | 1.3081 | 1.3514 | 1.3078 | 1.3604 | 1.3463 |
(0.3262) | (0.2899) | (0.3259) | (0.2972) | (0.2896) | |
0.0000 | 0.7457 | 0.0583 | 0.1183 | 0.1398 | |
(0.4665) | (0.5763) | (0.4753) | (0.4673) | (0.4612) | |
−0.0020 | −0.1400 | 0.0144 | −0.2386 | 0.0838 | |
(0.5414) | (0.5203) | (0.5479) | (0.5401) | (0.5295) | |
4.7425 | 3.8251 | 4.7585 | 5.4602 | 5.4894 | |
(1.3803) | (0.9953) | (1.3826) | (1.3561) | (1.3361) | |
σ11 | 0.4655 | 0.4267 | 0.4622 | 0.4379 | 0.4329 |
(0.0458) | (0.0411) | (0.0455) | (0.0420) | (0.0414) | |
σ21 | −0.2232 | −0.1738 | −0.2164 | −0.2185 | −0.2225 |
(0.0965) | (0.0747) | (0.0962) | (0.0786) | (0.0754) | |
σ22 | 5.7063 | 3.5558 | 5.6929 | 3.8956 | 3.6033 |
(0.5991) | (0.3520) | (0.5980) | (0.3874) | (0.3541) | |
ϕ | 0.6824 | 0.5447 | 0.6818 | 0.5674 | 0.5343 |
(0.0311) | (0.0422) | (0.0312) | (0.0400) | (0.0425) |
PNLS-MLME | Laplacian | Pseudo-ECM | MCEM | ISEM | |
---|---|---|---|---|---|
Approximate | −974.360 | −986.794 | −974.592 | −966.763 | −1010.370 |
Exact | −1063.338 | −991.754 | −978.269 | −981.384 | −978.758 |
AD | 88.978 | 4.96 | 3.677 | 14.621 | 31.612 |
4. Simulation Study
4.1. Bivariate Linear Case
N | ρ | PNLS-MLME | Laplacian | Pseudo-ECM | MCEM | ISEM | |
---|---|---|---|---|---|---|---|
25 | 0 | Time | 4.077 | 25.954 | 1.970 | 8789.093 | 5862.499 |
Iter | 2.150 | 12.140 | 9.800 | 138.440 | 58.390 | ||
−576.769 | −610.274 | −577.121 | −556.914 | −642.139 | |||
RB | 0.008 | −0.033 | 0.008 | 0.045 | −0.100 | ||
RMSE | 2.229 | 2.441 | 2.169 | 2.176 | 2.177 | ||
0.5 | Time | 4.370 | 30.803 | 2.045 | 2403.145 | 1680.319 | |
Iter | 2.120 | 11.430 | 9.930 | 35.650 | 15.750 | ||
−559.366 | −582.622 | −559.907 | −536.608 | −625.736 | |||
RB | 0.009 | −0.022 | 0.008 | 0.052 | −0.103 | ||
RMSE | 0.580 | 0.672 | 0.561 | 0.601 | 0.602 | ||
0.9 | Time | 3.646 | 25.006 | 1.749 | 1252.625 | 1158.028 | |
Iter | 2.000 | 8.940 | 8.570 | 18.330 | 10.760 | ||
−468.270 | −474.786 | −468.909 | −423.555 | −535.591 | |||
RB | 0.011 | −0.003 | 0.009 | 0.118 | −0.120 | ||
RMSE | 0.470 | 0.484 | 0.450 | 0.486 | 0.477 | ||
50 | 0 | Time | 8.365 | 41.545 | 8.927 | 6825.341 | 3967.824 |
Iter | 2.240 | 10.050 | 9.260 | 56.240 | 20.170 | ||
−1159.337 | −1177.863 | −1159.675 | −1120.721 | −1292.848 | |||
RB | 0.004 | −0.010 | 0.004 | 0.039 | −0.094 | ||
RMSE | 1.688 | 1.747 | 1.685 | 1.692 | 1.689 | ||
0.5 | Time | 9.776 | 56.560 | 10.210 | 2112.857 | 1706.392 | |
Iter | 2.140 | 9.760 | 9.530 | 11.800 | 9.690 | ||
−1124.354 | −1140.195 | −1124.911 | −1079.401 | −1258.644 | |||
RB | 0.004 | −0.009 | 0.004 | 0.046 | −0.098 | ||
RMSE | 0.277 | 0.324 | 0.270 | 0.313 | 0.315 | ||
0.9 | Time | 8.185 | 34.382 | 6.666 | 1512.85 | 1091.661 | |
Iter | 2.000 | 6.070 | 6.210 | 7.320 | 6.850 | ||
−933.662 | −943.973 | −934.566 | −843.025 | −1069.55 | |||
RB | 0.005 | −0.006 | 0.004 | 0.113 | −0.116 | ||
RMSE | 0.226 | 0.229 | 0.226 | 0.237 | 0.234 |
4.2. Bivariate Nonlinear Case
Sample Size N | Methods | Comparison Criteria | ||||
---|---|---|---|---|---|---|
Time | Iter | RB | RMSE | |||
25 | PNLS-MLME | 5.071 | 3.533 | −847.968 | 0.009 | 1.671 |
Laplacian | 21.199 | 7.133 | −860.383 | −0.012 | 2.000 | |
Pseudo-ECM | 2.709 | 12.000 | −847.994 | 0.009 | 1.967 | |
MCEM | 9062.743 | 380.000 | −847.217 | 0.010 | 2.099 | |
MCEM | 9569.619 | 213.733 | −847.346 | 0.010 | 2.072 | |
MCEM | 11,375.297 | 131.400 | −847.896 | 0.009 | 2.029 | |
ISEM | 17,008.449 | 333.733 | −887.996 | −0.028 | 1.999 | |
ISEM | 4635.601 | 93.400 | −881.169 | −0.018 | 1.882 | |
ISEM | 1086.651 | 22.200 | −862.842 | −0.020 | 2.077 | |
50 | PNLS-MLME | 14.149 | 3.940 | −1710.123 | 0.007 | 1.119 |
Laplacian | 53.066 | 7.690 | −1763.046 | −0.010 | 1.134 | |
Pseudo-ECM | 11.331 | 13.070 | −1710.216 | 0.007 | 1.110 | |
MCEM | 15,860.866 | 392.595 | −1713.939 | 0.005 | 1.184 | |
MCEM | 24,077.335 | 238.470 | −1714.151 | 0.005 | 1.157 | |
MCEM | 26,328.930 | 134.750 | −1714.447 | 0.004 | 1.151 | |
ISEM | 31,224.663 | 386.120 | −1789.168 | −0.021 | 1.255 | |
ISEM | 7065.363 | 106.350 | −1780.396 | −0.015 | 1.138 | |
ISEM | 2805.677 | 26.870 | −1779.298 | −0.018 | 1.153 |
Sample Size N | Methods | Parameter | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
25 | PNLS-MLME | 0.046 | 0.960 | 0.046 | 0.041 | 18.505 | 11.559 | 21.698 | 87.774 | 4.565 | 0.998 | 20.003 | 0.909 |
Laplacian | 0.045 | 0.965 | 0.046 | 0.033 | 18.600 | 11.766 | 20.875 | 120.340 | 4.609 | 1.412 | 20.013 | 1.325 | |
Pseudo-ECM | 0.043 | 0.964 | 0.046 | 0.026 | 18.501 | 11.570 | 20.066 | 118.786 | 4.759 | 0.988 | 20.010 | 0.909 | |
MCEM | 0.046 | 0.956 | 0.045 | 0.039 | 18.736 | 11.781 | 20.926 | 130.477 | 4.549 | 1.389 | 19.682 | 1.299 | |
MCEM | 0.047 | 0.969 | 0.045 | 0.038 | 18.589 | 11.668 | 21.160 | 127.668 | 4.797 | 1.383 | 19.559 | 1.314 | |
MCEM | 0.047 | 0.970 | 0.046 | 0.036 | 18.602 | 11.669 | 20.402 | 123.817 | 4.606 | 1.404 | 19.935 | 1.315 | |
ISEM | 0.046 | 0.960 | 0.046 | 0.028 | 18.590 | 11.666 | 20.510 | 120.740 | 4.609 | 1.400 | 20.013 | 1.315 | |
ISEM | 0.047 | 0.969 | 0.046 | 0.040 | 18.420 | 11.476 | 19.930 | 110.919 | 4.000 | 1.463 | 19.619 | 1.271 | |
ISEM | 0.043 | 0.993 | 0.045 | 0.021 | 18.785 | 11.899 | 26.451 | 122.587 | 4.407 | 1.631 | 19.474 | 1.377 | |
50 | PNLS-MLME | 0.053 | 0.433 | 0.019 | 0.083 | 8.038 | 6.437 | 9.609 | 62.998 | 3.126 | 0.355 | 20.445 | 0.290 |
Laplacian | 0.054 | 0.433 | 0.019 | 0.040 | 8.056 | 6.501 | 10.172 | 63.165 | 3.121 | 0.787 | 20.025 | 1.051 | |
Pseudo-ECM | 0.053 | 0.432 | 0.019 | 0.043 | 8.055 | 6.452 | 8.858 | 62.921 | 3.013 | 0.355 | 20.493 | 0.289 | |
MCEM | 0.052 | 0.420 | 0.019 | 0.087 | 8.117 | 6.505 | 10.334 | 67.836 | 3.055 | 0.875 | 20.054 | 1.033 | |
MCEM | 0.054 | 0.420 | 0.019 | 0.085 | 8.149 | 6.508 | 10.099 | 65.263 | 3.113 | 0.881 | 20.003 | 1.063 | |
MCEM | 0.054 | 0.418 | 0.019 | 0.075 | 8.120 | 6.494 | 10.185 | 64.350 | 3.120 | 0.892 | 20.274 | 1.070 | |
ISEM | 0.059 | 0.415 | 0.019 | 0.080 | 8.034 | 6.429 | 18.313 | 67.054 | 3.142 | 0.924 | 20.045 | 1.011 | |
ISEM | 0.055 | 0.429 | 0.019 | 0.053 | 8.131 | 6.508 | 9.040 | 64.614 | 3.143 | 0.861 | 19.819 | 1.080 | |
ISEM | 0.052 | 0.431 | 0.019 | 0.035 | 8.194 | 6.554 | 10.182 | 64.165 | 3.011 | 0.987 | 20.542 | 1.147 |
5. Discussion and Conclusions
Acknowledgments
Conflicts of Interest
Appendix
A. Score Vector and Hessian Matrix
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Wang, W.-L. Approximate Methods for Maximum Likelihood Estimation of Multivariate Nonlinear Mixed-Effects Models. Entropy 2015, 17, 5353-5381. https://doi.org/10.3390/e17085353
Wang W-L. Approximate Methods for Maximum Likelihood Estimation of Multivariate Nonlinear Mixed-Effects Models. Entropy. 2015; 17(8):5353-5381. https://doi.org/10.3390/e17085353
Chicago/Turabian StyleWang, Wan-Lun. 2015. "Approximate Methods for Maximum Likelihood Estimation of Multivariate Nonlinear Mixed-Effects Models" Entropy 17, no. 8: 5353-5381. https://doi.org/10.3390/e17085353