Residuals and Goodness-of-Fit

Understand residuals, goodness-of-fit plots, and how they reveal problems in pharmacometric models.
Tip

What you’ll build today: the ability to interpret residuals and goodness-of-fit plots to diagnose model problems.

Learning Objectives

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

  • Define residuals in pharmacometric models
  • Interpret common goodness-of-fit plots
  • Recognize patterns that indicate model misspecification
  • Connect diagnostics to model assumptions

Key Ideas

After fitting a model, we compare:

  • observed data
  • model predictions

The difference is called a residual:

\[ \text{Residual} = \text{Observed} - \text{Predicted} \]

Residuals help answer:

How well does the model explain the data?

Types of Residuals

Raw Residuals

Raw residuals measure the direct difference between observation and prediction.

\[ RES = DV - PRED \]

Where:

  • \(DV\) = observed value
  • \(PRED\) = model prediction

Interpretation:

  • positive residual → model underpredicts
  • negative residual → model overpredicts

Weighted Residuals (WRES)

Weighted residuals scale residuals by expected variability.

Conceptually:

\[ WRES \approx \frac{DV-PRED} {SD} \]

Where:

  • \(SD\) = standard deviation (the expected spread of observations)

This makes residuals:

  • easier to compare across observations
  • more interpretable when variability changes

Interpretation:

  • values near zero → predictions align with expectations
  • large positive or negative values → unusual prediction errors

Conditional Weighted Residuals (CWRES)

CWRES extend weighted residuals by accounting for:

  • individual random effects (\(\eta\))
  • population uncertainty

Conceptually:

\[ CWRES \approx \frac{DV-PRED} {\text{Conditional SD}} \]

Where:

  • \(DV\) = observed value
  • \(PRED\) = model prediction
  • Conditional SD = the variability expected under the model for that specific observation

Unlike WRES, CWRES account for both:

  • expected observation variability
  • uncertainty introduced by individual and population effects

CWRES are commonly used in:

  • population PK models
  • residual diagnostics
  • goodness-of-fit evaluation

Insight: CWRES ask whether observations are unusual given the model and expected variability.

Worked Example: Residual Concept

A residual is the vertical difference between an observed value and the model prediction.

Each residual measures:

\[ \text{Residual} = \text{Observed} - \text{Predicted} \]

Positive residuals mean observations are above predictions.

Negative residuals mean observations are below predictions.

Goodness-of-Fit (GOF) Plots

Common plots include:

Observed vs Predicted

  • Should lie along identity line

Residuals vs Time

  • Should show no pattern

Residuals vs Predictions

  • Should be randomly scattered

Expanding the Idea: What Patterns Mean

Residual plots are useful because different patterns suggest different problems.

Interpretation:

  • Random scatter → residual behavior looks acceptable
  • Trend → model may be missing time-dependent structure
  • Funnel shape → residual variability changes with magnitude

The goal is not to make residuals zero.

The goal is to remove systematic structure.

Insight

Residuals should look like random noise—patterns indicate problems.

Note

A useful model does not produce perfect residuals.

It produces residuals with no meaningful systematic structure.

Why This Matters

Residual diagnostics help detect:

  • model misspecification
  • incorrect variability assumptions
  • poor predictive performance

Strategies

  • Always inspect multiple GOF plots
  • Look for patterns, not individual points
  • Combine visual and statistical checks

Common Mistakes

  • Ignoring residual patterns
  • Overreacting to single outliers
  • Assuming fit is good without diagnostics

Practice Problems

  1. What is a residual?
  2. What should residual plots look like?
  3. What does a trend indicate?
  1. Observed minus predicted
  2. Random scatter with no pattern
  3. Model misspecification

Summary

Residuals:

  • measure model fit
  • reveal patterns
  • diagnose problems
  • Residuals should look random
  • Patterns = problems
  • Use multiple plots
  • Focus on trends, not noise