Residual Error Models

Understand how population models represent disagreement between predictions and observations.
Tip

Big picture: Population models describe not only subject differences but also observation-level variability.

Learning Objectives

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

  • explain residual error conceptually
  • distinguish residual error from between-subject variability
  • compare additive, proportional, and combined residual error models
  • recognize residual error syntax in nlmixr2
  • explain when different error models may be useful

Key Ideas

  • Predictions are never perfectly observed.
  • Residual error represents remaining disagreement.
  • Different residual error models imply different assumptions.
  • Residual error is not model failure.

Setup

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

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

Why Residual Error Exists

Suppose the model predicts:

Concentration = 10

Do we always observe:

Concentration = 10

No.

Observed values differ.

Sources include:

  • assay variability
  • sampling timing
  • biological fluctuations
  • model simplification

Conceptually:

Prediction
+
Residual Error
↓
Observation

Worked Example 1: Locate Residual Error

Inspect earlier model syntax.

linCmt() ~ add(add.err)

Interpretation:

Prediction
↓
Add Residual Error
↓
Observation

This line specifies the observation model.

Worked Example 2: Additive Error Model

One possibility:

\[ DV = IPRED + \epsilon \]

Interpretation:

Residual variability remains approximately constant.

Example:

Prediction = 10

Observed = 11
Observed = 9
Observed = 10

Same absolute spread.

In nlmixr2:

linCmt() ~ add(add.err)

Worked Example 3: Proportional Error Model

Another possibility:

\[ DV = IPRED(1+\epsilon) \]

Interpretation:

Residual variability increases as prediction increases.

Example:

Prediction = 10 ±10%

In nlmixr2:

linCmt() ~ prop(prop.err)

Larger concentrations show larger variability.

Worked Example 4: Combined Error Model

A common alternative:

\[ DV = IPRED(1+\epsilon_p)+\epsilon_a \]

Interpretation:

Combine:

  • fixed variability
  • relative variability

In nlmixr2:

linCmt() ~ add(add.err) + prop(prop.err)

This often provides flexibility across concentration ranges.

Worked Example 5: Compare Variability Sources

Do not confuse:

Source Meaning
ETA subject differences
Residual error observation disagreement

Conceptually:

Typical Parameter + ETA
↓
Prediction + Residual Error
↓
Observation

Population models contain both.

Additive vs Proportional Thinking

Conceptually:

Additive

Same Spread

versus

Proportional

Spread Grows

Choose based on scientific behavior.

Connect to Residuals

Earlier we used:

RES

IRES

Residual quantities measure disagreement.

Residual error models describe assumptions about that disagreement.

Residuals do not create the model.

Residual assumptions create the model.

Looking Ahead

We now understand:

Prediction
↓
Residual Error
↓
Observation

Next we ask:

Can variability components interact?

The next lesson introduces variability structures and correlations.

Strategies

  • Think biologically.
  • Compare assumptions.
  • Interpret spread.

Common Mistakes

  • Using one error model everywhere.
  • Treating residuals as error models.
  • Forgetting observation noise.

Practice Problems

  1. What does residual error represent?

  2. Compare additive and proportional error.

  3. Why might combined error be useful?

  4. Interpret:

linCmt() ~ prop(prop.err)
  1. Explain:
Prediction
↓
Residual Error
↓
Observation

Problem 1

Residual error represents remaining disagreement.

Problem 2

Additive:

constant spread

Proportional:

spread increases

Problem 3

Combined error allows both behaviors.

Problem 4

Residual variability scales with prediction size.

Problem 5

Observations differ from predictions because of residual variability.

Summary

  • Residual error models observation variability.
  • Additive and proportional assumptions differ.
  • Combined models offer flexibility.
  • Residual error is not model failure.
  • ETA ≠ residual error
  • Error ≠ failure
  • Residual assumptions matter