Between-Subject Variability and ETA Models

Understand how population models represent individual differences using ETA random effects.
Tip

Big picture: Population models estimate a typical subject and then allow individuals to differ through random effects.

Learning Objectives

By the end of this lesson, you will be able to:

  • explain between-subject variability
  • interpret ETA terms conceptually
  • distinguish fixed and random effects
  • compare additive, proportional, and exponential variability models
  • explain why exponential models are common in PK

Key Ideas

  • Individuals differ from the typical subject.
  • ETA terms represent subject-level deviations.
  • Different variability models imply different assumptions.
  • Exponential variability is common in PK.

Setup

library(tidyverse)
library(nlmixr2)
library(nlmixr2data)

data(
  "theo_sd",
  package = "nlmixr2data"
)

Why ETA Models Exist

Suppose we estimate:

Typical CL = 3

Should every subject have:

CL = 3

No.

Population models allow subjects to vary.

Conceptually:

Typical Parameter
+
Random Effect
↓
Individual Parameter

The random effect is commonly written:

η
(ETA)

Worked Example 1: Locate ETA Terms

Inspect the model.

one_comp_model <- function(){

    ini({

        tka <- log(1)

        tcl <- log(3)

        tv <- log(30)

        eta.ka ~ 0.1
        eta.cl ~ 0.1
        eta.v ~ 0.1

        add.err <- 0.1

    })

    model({

        ka <- exp(tka + eta.ka)

        cl <- exp(tcl + eta.cl)

        v <- exp(tv + eta.v)

        linCmt() ~ add(add.err)

    })

}

one_comp_model
function () 
{
    ini({
        tka <- log(1)
        tcl <- log(3)
        tv <- log(30)
        eta.ka ~ 0.1
        eta.cl ~ 0.1
        eta.v ~ 0.1
        add.err <- 0.1
    })
    model({
        ka <- exp(tka + eta.ka)
        cl <- exp(tcl + eta.cl)
        v <- exp(tv + eta.v)
        linCmt() ~ add(add.err)
    })
}

Interpretation:

tcl
↓
Typical Clearance
eta.cl
↓
Subject Difference

Worked Example 2: Additive Random Effects

One possibility:

\[ P_i = TVP + \eta_i \]

Interpretation:

Every subject differs by a fixed amount.

Example:

Typical CL = 3

Subject A = 2

Subject B = 4

This model assumes constant variability.

Worked Example 3: Proportional Random Effects

Another possibility:

\[ P_i = TVP(1+\eta_i) \]

Interpretation:

Variability scales with parameter size.

Example:

Typical CL = 3

10% variation

Larger parameters produce larger differences.

This form is less common in PK.

Worked Example 4: Exponential Random Effects

The most common PK formulation:

\[ P_i = TVP \times e^{\eta_i} \]

Interpretation:

Random effects become multiplicative.

Advantages:

  • parameters remain positive
  • variability scales naturally
  • convenient interpretation

Example:

Typical CL = 3

ETA = +0.2
↓
Higher CL

This is why earlier models used:

cl <- exp(tcl + eta.cl)

Worked Example 5: Connect ETA to Predictions

Population prediction:

PRED

Typical Subject

Individual prediction:

IPRED

Typical Subject
+
ETA

ETA contributes to:

Individual Parameter
↓
Individual Prediction

Why Exponential Models Dominate PK

PK parameters are positive.

Examples:

CL > 0

V > 0

ka > 0

Exponential random effects preserve positivity.

Negative clearance is impossible.

This makes exponential models attractive.

Fixed Effects vs Random Effects

Component Interpretation
Fixed effect typical value
ETA individual difference
Individual parameter resulting subject parameter

Conceptually:

Typical
+
ETA
↓
Individual

Looking Ahead

Today we introduced:

Individual Differences

Next we ask:

How do observations differ from predictions?

The next lesson introduces residual error models.

Strategies

  • Think biologically.
  • Interpret variability conceptually.
  • Start with exponential models.

Common Mistakes

  • Treating ETA as measurement noise.
  • Forgetting parameter scale.
  • Assuming all variability models behave similarly.

Practice Problems

  1. What does ETA represent?

  2. Why do population models use random effects?

  3. Compare:

  • additive variability
  • proportional variability
  • exponential variability

  1. Why are exponential models common in PK?

  2. Explain:

Typical Parameter
+
ETA
↓
Individual Parameter

Problem 1

ETA represents subject-to-subject differences.

Problem 2

Random effects allow individuals to differ.

Problem 3

Additive: constant changes

Proportional: relative changes

Exponential: multiplicative changes

Problem 4

Exponential models preserve positivity.

Problem 5

Population estimates define the typical subject.

ETA modifies those values for individuals.

Summary

  • ETA models represent subject differences.
  • Population models combine fixed and random effects.
  • Exponential variability is common in PK.
  • ETA influences individual predictions.
  • ETA ≠ residual error
  • Exponential is common
  • Fixed + ETA = individual