Abstract
This chapter expands on linear models through the introduction of random effects, where the independent variables in the model include those variables that are fixed and those that vary across subjects. The general linear mixed effects model (LMEM) is introduced as methods for parameter estimation – maximum likelihood and restricted maximum likelihood. Model selection and goodness of fit in the context of LMEMs are discussed. The concept of empirical Bayes estimates and how shrinkage affects these estimates are also discussed. Building upon LMEMs, partial LMEMs are introduced, which use penalized spline regression to obtain a nonparametric-type smooth to the data, and how these might be used for covariate selection when knowing the exact nature of the effect of time is not needed. Three examples of LMEMs are provided: results from a food effect study, modeling tumor growth in a mouse xenograft model, and a detailed analysis of QT prolongation in clinical studies.
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Notes
- 1.
The word “subject” will be used generically referring to an experimental unit, not necessarily to “subject” in the human sense.
- 2.
Note that the R-matrix in linear mixed effects models should not be confused with the R-matrix in NONMEM, which is the Hessian of the variance–covariance matrix. The two matrices are different.
- 3.
A symmetric matrix is positive semidefinite if all its eigenvalues are nonnegative. A positive definite matrix is one where all its eigenvalues are strictly positive and nonzero.
- 4.
This was the first published paper studying the relationship between concentration and QT interval prolongation.
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Recommended Reading
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Appendices
Appendix 1: SAS Code Used to Model Data in Fig. 6 Using Mixed Model Framework (Quadratic Spline Basis Function)
It should be noted that sometimes the additional statement
Random intercept/subject = subject;
is added to the model to further account for between-subject variability. See Maringwa et al. ( 2009 ) for an example using a 4-period crossover study in dogs with QT intervals as the biomarker under consideration.
Appendix 2: Tumor Volume Data
Control group | |||||||||
Group | Day | Weight | Volume | Weight | Volume | Weight | Volume | Weight | Volume |
Mouse 1 | Mouse 1 | Mouse 2 | Mouse 2 | Mouse 3 | Mouse 3 | Mouse 4 | Mouse 4 | ||
1 | 1 | 25.7 | 93.3 | 28.4 | 107.9 | 25.5 | 160.1 | 23.6 | 226.5 |
1 | 5 | 25.8 | 124.5 | 29.6 | 193.6 | 28.6 | 209.1 | 23.7 | 287.6 |
1 | 8 | 24.2 | 155.6 | 28.6 | 220.3 | 28.7 | 358.4 | 23.8 | 364.1 |
1 | 12 | 24.6 | 188.1 | 29.0 | 277.2 | 29.5 | 491.8 | 24.2 | 479.2 |
1 | 15 | 26.7 | 204.2 | 28.3 | 366.5 | 28.5 | 552.5 | 24.4 | 566.8 |
1 | 20 | 25.6 | 230.0 | 20.0 | 435.2 | 29.2 | 670.3 | 24.3 | 585.8 |
1 | 22 | 25.4 | 229.1 | 29.2 | 442.8 | 29.7 | 698.3 | 24.8 | 635.1 |
1 | 26 | 26.8 | 264.7 | 30.7 | 579.5 | 30.1 | 973.2 | 25.9 | 855.4 |
1 | 29 | 26.5 | 274.6 | 30.1 | 691.6 | 29.9 | 1,026.4 | 25.3 | 878.4 |
1 | 33 | 26.4 | 273.8 | 29.1 | 765.1 | 30.6 | 1078.7 | 25.9 | 925.9 |
Group | Day | Weight | Volume | Weight | Volume | Weight | Volume | ||
Mouse 5 | Mouse 5 | Mouse 6 | Mouse 6 | Mouse 7 | Mouse 7 | ||||
1 | 1 | 25.5 | 73.9 | 24.6 | 183.8 | 29.0 | 64.8 | ||
1 | 5 | 25.2 | 123.9 | 25.5 | 222.9 | 28.6 | 141.5 | ||
1 | 8 | 25.0 | 222.0 | 25.1 | 282.9 | 28.6 | 327.7 | ||
1 | 12 | 25.5 | 226.6 | 25.0 | 302.7 | 29.6 | 421.6 | ||
1 | 15 | 25.0 | 271.8 | 25.5 | 366.1 | 29.1 | 426.3 | ||
1 | 20 | 26.4 | 310.3 | 26.4 | 418.2 | 28.8 | 454.7 | ||
1 | 22 | 26.1 | 313.3 | 26.0 | 435.9 | 29.1 | 469.2 | ||
1 | 26 | 27.3 | 337.7 | 27.2 | 382.9 | 29.4 | 650.1 | ||
1 | 29 | 27.0 | 404.6 | 26.8 | 409.1 | 30.0 | 627.3 | ||
1 | 33 | 26.1 | 463.4 | 26.7 | 415.9 | 29.8 | 862.0 | ||
Drug X (100 mg/kg) PO | |||||||||
Group | Day | Weight | Volume | Weight | Volume | Weight | Volume | Weight | Volume |
Mouse 8 | Mouse 8 | Mouse 9 | Mouse 9 | Mouse 10 | Mouse 10 | Mouse 11 | Mouse 11 | ||
2 | 1 | 28.3 | 135.8 | 27.0 | 127.1 | 24.9 | 61.4 | 23.9 | 84.6 |
2 | 5 | 30.1 | 149.5 | 26.0 | 116.5 | 25.3 | 62.2 | 25.5 | 106.4 |
2 | 8 | 30.4 | 204.2 | 26.1 | 167.9 | 25.0 | 116.5 | 25.8 | 105.3 |
2 | 12 | 30.1 | 258.5 | 26.3 | 237.7 | 25.5 | 155.7 | 25.8 | 141.0 |
2 | 15 | 29.4 | 311.9 | 25.9 | 302.7 | 24.8 | 176.4 | 25.3 | 154.8 |
2 | 20 | 30.0 | 317.7 | 26.2 | 404.0 | 25.3 | 303.1 | 25.9 | 154.8 |
2 | 22 | 30.2 | 525.3 | 26.3 | 422.3 | 25.2 | 291.2 | 26.0 | 184.3 |
2 | 26 | 31.4 | 682.3 | 27.2 | 455.5 | 26.4 | 622.4 | 27.0 | 240.1 |
2 | 29 | 31.2 | 734.8 | 27.8 | 804.8 | 26.0 | 489.8 | 25.1 | 266.5 |
2 | 33 | 30.4 | 1145.2 | 27.0 | 909.4 | 26.0 | 685.4 | 25.9 | 306.1 |
Group | Day | Weight | Volume | Weight | Volume | Weight | Volume | ||
Mouse 12 | Mouse 12 | Mouse 13 | Mouse 13 | Mouse 14 | Mouse 14 | ||||
2 | 1 | 28.7 | 201.1 | 27.0 | 136.4 | 25.2 | 162.7 | ||
2 | 5 | 27.1 | 244.5 | 28.0 | 167.4 | 24.3 | 199.1 | ||
2 | 8 | 27.2 | 380.9 | 27.8 | 222.2 | 23.7 | 240.3 | ||
2 | 12 | 27.4 | 320.6 | 28.5 | 259.6 | 23.8 | 278.7 | ||
2 | 15 | 27.1 | 472.4 | 28.4 | 282.3 | 23.5 | 318.2 | ||
2 | 20 | 26.1 | 530.0 | 28.3 | 343.7 | 22.6 | 369.8 | ||
2 | 22 | 26.8 | 484.4 | 29.4 | 320.6 | 22.8 | 349.3 | ||
2 | 26 | 28.3 | 499.5 | 29.4 | 444.3 | 24.1 | 383.3 | ||
2 | 29 | 27.9 | 726.7 | 29.9 | 426.9 | 24.7 | 323.8 | ||
2 | 33 | 27.5 | 820.2 | 29.9 | 517.6 | 25.0 | 450.6 | ||
Note: Weight in grams; volume in cubic millimeter | |||||||||
Drug X (10 mg/kg) PO | |||||||||
Group | Day | Weight | Volume | Weight | Volume | Weight | Volume | Weight | Volume |
Mouse 15 | Mouse 15 | Mouse 16 | Mouse 16 | Mouse 17 | Mouse 17 | Mouse 18 | Mouse 18 | ||
3 | 1 | 27.1 | 99.1 | 25.6 | 129.0 | 28.5 | 162.7 | 27.6 | 117.0 |
3 | 5 | 23.4 | 183.8 | 25.1 | 142.2 | 29.2 | 184.0 | 28.3 | 202.3 |
3 | 8 | 20.1 | 237.1 | 26.0 | 165.3 | 29.6 | 240.3 | 24.3 | 210.7 |
3 | 12 | 24.0 | 334.1 | 26.2 | 173.2 | 30.1 | 153.8 | 28.4 | 255.9 |
3 | 15 | 23.9 | 399.4 | 25.9 | 304.2 | 29.7 | 230.0 | 27.9 | 290.5 |
3 | 20 | 25.1 | 506.9 | 27.0 | 305.3 | 30.1 | 236.3 | 28.4 | 366.9 |
3 | 22 | 25.1 | 500.9 | 26.4 | 398.7 | 19.4 | 241.9 | 18.4 | 373.2 |
3 | 26 | 25.9 | 696.6 | 27.6 | 620.0 | 30.0 | 379.5 | 29.0 | 516.9 |
3 | 29 | 25.4 | 777.7 | 27.4 | 621.0 | 30.0 | 373.5 | 28.4 | 516.9 |
3 | 33 | 25.0 | 975.8 | 26.7 | 807.9 | 30.1 | 352.0 | 27.8 | 619.0 |
Group | Day | Weight | Volume | Weight | Volume | Weight | Volume | ||
Mouse 19 | Mouse 19 | Mouse 20 | Mouse 20 | Mouse 21 | Mouse 21 | ||||
3 | 1 | 27.1 | 65.7 | 28.8 | 142.9 | 25.4 | 190.4 | ||
3 | 5 | 27.1 | 60.1 | 29.8 | 213.2 | 25.2 | 328.1 | ||
3 | 8 | 27.3 | 124.5 | 29.9 | 223.0 | 25.7 | 308.4 | ||
3 | 12 | 27.2 | 172.1 | 28.5 | 304.2 | 25.6 | 337.9 | ||
3 | 15 | 26.6 | 262.8 | 29.0 | 354.3 | 25.3 | 740.9 | ||
3 | 20 | 27.5 | 404.6 | 30.0 | 435.7 | 24.0 | 656.7 | ||
3 | 22 | 17.6 | 393.8 | 29.5 | 482.4 | 24.4 | 773.3 | ||
3 | 26 | 29.1 | 438.9 | 30.1 | 479.2 | 26.9 | 1,125.0 | ||
3 | 29 | 29.1 | 573.9 | 29.7 | 423.4 | 26.9 | 1,130.1 | ||
3 | 33 | 29.3 | 786.5 | 29.7 | 562.2 | 27.4 | 1,105.4 | ||
Drug X (10 mg/kg) IP | |||||||||
Group | Day | Weight | Volume | Weight | Volume | Weight | Volume | Weight | Volume |
Mouse 22 | Mouse 22 | Mouse 23 | Mouse 23 | Mouse 24 | Mouse 24 | Mouse 25 | Mouse 25 | ||
4 | 1 | 26.5 | 151.6 | 30.1 | 129.5 | 24.1 | 84.1 | 26.8 | 164.8 |
4 | 5 | 25.6 | 216.6 | 28.5 | 139.2 | 27.2 | 102.8 | 27.8 | 206.7 |
4 | 8 | 26.3 | 202.2 | 28.7 | 157.6 | 26.7 | 125.4 | 27.2 | 222.9 |
4 | 12 | 26.7 | 218.5 | 29.5 | 184.5 | 27.3 | 174.6 | 27.8 | 239.7 |
4 | 15 | 25.4 | 240.9 | 28.3 | 234.5 | 26.6 | 189.8 | 26.8 | 239.8 |
4 | 20 | 27.2 | 275.2 | 28.7 | 289.9 | 28.2 | 249.9 | 28.0 | 276.8 |
4 | 22 | 27.6 | 288.0 | 29.8 | 265.9 | 29.1 | 245.1 | 28.9 | 277.7 |
4 | 26 | 29.1 | 387.2 | 31.0 | 317.6 | 29.8 | 296.5 | 28.9 | 269.8 |
4 | 29 | 29.1 | 487.4 | 30.2 | 358.7 | 28.7 | 315.0 | 28.6 | 338.7 |
4 | 33 | 28.3 | 488.4 | 29.4 | 516.6 | 28.2 | 386.9 | 28.5 | 379.3 |
Group | Day | Weight | Volume | Weight | Volume | Weight | Volume | ||
Mouse 26 | Mouse 26 | Mouse 27 | Mouse 27 | Mouse 28 | Mouse 28 | ||||
4 | 1 | 28.6 | 117.0 | 26.6 | 198.5 | 26.7 | 74.5 | ||
4 | 5 | 26.4 | 155.6 | 27.5 | 225.5 | 26.0 | 106.7 | ||
4 | 8 | 25.9 | 181.3 | 27.7 | 321.0 | 26.3 | 141.0 | ||
4 | 12 | 26.2 | 140.9 | 28.2 | 342.2 | 27.7 | 150.7 | ||
4 | 15 | 25.8 | 139.3 | 28.5 | 362.4 | 26.9 | 210.7 | ||
4 | 20 | 26.9 | 52.9 | 29.2 | 512.8 | 27.7 | 275.7 | ||
4 | 22 | 26.9 | 27.6 | 30.0 | 586.6 | 28.0 | 319.4 | ||
4 | 26 | 27.9 | 21.0 | 30.8 | 729.0 | 28.7 | 380.9 | ||
4 | 29 | 27.8 | 18.5 | 30.1 | 780.5 | 28.7 | 305.3 | ||
4 | 33 | 27.3 | 15.8 | 29.5 | 838.2 | 28.3 | 403.1 | ||
Note: Weight in grams; volume in cubic millimeter | |||||||||
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Bonate, P.L. (2011). Linear Mixed Effects Models. In: Pharmacokinetic-Pharmacodynamic Modeling and Simulation. Springer, Boston, MA. https://doi.org/10.1007/978-1-4419-9485-1_6
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