Population Modeling Examples¶

Examples demonstrating inter-individual variability (IIV), inter-occasion variability (IOV), and covariate modeling.

Examples¶

Example	Description	Directory
Basic IIV	Log-normal inter-individual variability	01_basic_iiv
IOV	Inter-occasion variability	02_iov
Static Covariates	Weight and age effects	03_static_covariates
Time-Varying Covariates	Creatinine clearance over time	04_time_varying_covariates
Complex Model	Multiple covariates, IIV, correlations	05_complex_covariate_model

Concepts¶

Inter-Individual Variability (IIV)¶

IIV describes differences between subjects:

θᵢ = θ_pop × exp(ηᵢ)

Where: - θᵢ = individual parameter - θ_pop = population typical value - ηᵢ ~ N(0, ω²)

Inter-Occasion Variability (IOV)¶

IOV describes within-subject variability across occasions:

θᵢⱼ = θ_pop × exp(ηᵢ + κⱼ)

Where: - κⱼ ~ N(0, π²) is the occasion-specific random effect

Covariate Models¶

Covariates explain part of the variability:

Power model (weight on CL):

CL = CL_pop × (WT/70)^0.75 × exp(η_CL)

Linear model (age on V):

V = V_pop × (1 + θ_age × (AGE - 40)) × exp(η_V)

Categorical (sex on CL):

CL = CL_pop × θ_sex^SEX × exp(η_CL)

File Structure¶

Each example contains:

01_example/
├── README.md       # Detailed explanation
├── julia.jl        # Julia implementation
├── python.py       # Python implementation
└── cli.json        # CLI specification (if applicable)

Running Examples¶

# Julia
julia --project=packages/core docs/examples/population/01_basic_iiv/julia.jl

# Python
python docs/examples/population/01_basic_iiv/python.py

Key Parameters¶

Omega (ω) Interpretation¶

ω²	ω (SD)	CV%	Interpretation
0.01	0.1	~10%	Low variability
0.09	0.3	~30%	Moderate variability
0.25	0.5	~50%	High variability
0.49	0.7	~70%	Very high variability

Note: For log-normal, CV ≈ ω for ω < 0.5

Common Covariate Exponents¶

Covariate	Parameter	Exponent	Rationale
Weight	CL	0.75	Allometric scaling
Weight	V	1.0	Linear scaling
Age	CL	-0.01 to -0.02	Linear decline
CrCL	CL	0.5-1.0	Renal function
Sex	CL	0.8-1.2	Categorical