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Parameter Estimation Examples

Examples demonstrating non-linear mixed effects (NLME) parameter estimation.

Estimation Methods

Method Description Best For
FOCE-I First-Order Conditional Estimation with Interaction Most PK/PD models, standard choice
SAEM Stochastic Approximation Expectation Maximization Complex models, multi-modal posteriors
Laplacian Laplace approximation Sparse data, categorical outcomes

Examples

Example Description Directory
FOCE-I Basic One-compartment estimation 01_foce_onecomp
SAEM Two-Comp Two-compartment with SAEM 02_saem_twocomp
Laplacian Sparse pediatric data 03_laplacian_sparse
Diagnostics GOF, CWRES, ETAs 04_estimation_diagnostics
Model Comparison AIC/BIC selection 05_model_comparison

Estimation Workflow

1. Data Preparation
   ├── Format observed data (CDISC or custom)
   └── Define dosing events

2. Model Specification
   ├── Choose structural model
   ├── Define IIV structure
   └── Specify residual error model

3. Initial Values
   ├── Set θ₀ (fixed effects)
   ├── Set Ω₀ (random effects variance)
   └── Set σ₀ (residual error)

4. Run Estimation
   ├── Choose method (FOCE-I, SAEM, Laplacian)
   └── Set convergence criteria

5. Evaluate Results
   ├── Convergence diagnostics
   ├── Parameter estimates ± SE
   ├── Goodness-of-fit plots
   └── Model selection (AIC/BIC)

Key Parameters

Estimation Configuration

config = EstimationConfig(
    method = FOCEI(),
    # Fixed effects
    theta_init = [5.0, 50.0, 1.5],
    theta_lower = [0.1, 1.0, 0.1],
    theta_upper = [100.0, 500.0, 10.0],
    # Random effects
    omega_init = [0.09, 0.04, 0.16],
    omega_fixed = [false, false, false],
    # Residual error
    sigma_init = [0.01],
    # Algorithm settings
    maxiter = 500,
    tol = 1e-6
)

Error Models

Model Formula Use Case
Additive Y = F + ε Low concentrations
Proportional Y = F × (1 + ε) Wide concentration range
Combined Y = F × (1 + ε₁) + ε₂ Best of both

Output Interpretation

Parameter Estimates

Parameter   Estimate    SE        RSE%      95% CI
CL          5.23        0.31      5.9       [4.62, 5.84]
V           48.7        3.2       6.6       [42.4, 55.0]
Ka          1.42        0.18      12.7      [1.07, 1.77]
ω_CL        0.32        0.04      12.5      [0.24, 0.40]
ω_V         0.21        0.03      14.3      [0.15, 0.27]
σ_prop      0.12        0.01      8.3       [0.10, 0.14]

Objective Function

  • OFV (Objective Function Value): -2 × log-likelihood
  • Lower OFV = better fit
  • ΔOFV > 3.84 (df=1, α=0.05) = significant improvement

Common Issues

Issue Symptom Solution
Non-convergence Max iterations reached Better initial values, simpler model
Boundary estimates ω → 0 or → ∞ Fix parameter, check data
High RSE SE > 50% More data, simpler model
Poor GOF Systematic bias Check structural model

See Also