Variability Structures and Correlations

Understand how population models represent independent and correlated variability.
Tip

Big picture: Subjects may differ independently or their differences may move together.

Learning Objectives

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

  • explain variability structures
  • distinguish variance and covariance
  • interpret correlated ETAs conceptually
  • recognize covariance syntax in nlmixr2
  • understand why variability assumptions matter

Key Ideas

  • Random effects can be independent or correlated.
  • Covariance measures joint variability.
  • Correlated ETAs may reflect biology or data structure.
  • More parameters increase complexity.

Setup

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

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

Why Variability Structure Matters

Suppose two subjects differ.

Question:

Does higher CL occur with larger V?

Sometimes:

No
↓
Independent

Sometimes:

Yes
↓
Correlated

Population models can represent both situations.

Worked Example 1: Independent Variability

The simplest assumption:

ETA(CL)

independent

ETA(V)

Conceptually:

Higher CL

does not imply

Higher V

This is often a good starting point.

Worked Example 2: Variance

Variance describes spread.

Conceptually:

\[ Var(\eta) \]

Interpretation:

Small Variance

Subjects Similar

versus

Large Variance

Subjects More Different

Variance describes magnitude.

Not direction.

Worked Example 3: Covariance

Covariance describes joint behavior.

Conceptually:

\[ Cov( \eta_1, \eta_2 ) \]

Interpretation:

Positive:

Higher CL
↓
Higher V

Negative:

Higher CL
↓
Lower V

Near zero:

No Relationship

Worked Example 4: Covariance in nlmixr2

Independent ETAs:

ini({

    eta.cl ~ 0.1

    eta.v ~ 0.1

})

Correlated ETAs:

ini({

    eta.cl + eta.v ~ c(0.1,
                       0.02, 0.1)

})

Interpretation:

Variance
Covariance Variance

The off-diagonal element represents covariance.

You do not need to memorize syntax.

Focus on interpretation.

Worked Example 5: Inspect Variability Output

Fit summaries often report:

BSV(CV%)

and:

fit$omega

Inspect:

fit$omega

Interpretation:

Diagonal:

Variance

Off-diagonal:

Covariance

At this stage:

recognize structure.

Do not evaluate model quality.

Covariance vs Correlation

Conceptually:

Covariance:

Joint Scale

Correlation:

Standardized Relationship

Correlation ranges:

−1 to +1

Examples:

0

No Linear Relationship
+1

Strong Positive
−1

Strong Negative

Should We Always Estimate Correlations?

No.

Adding covariance increases complexity.

Conceptually:

More Parameters
↓
More Flexibility
↓
More Uncertainty

Start simple.

Add complexity only when justified.

Connect to Covariates

Correlations may suggest:

Shared Biology

or:

Missing Covariates

Later modules ask:

Can covariates explain variability?

Looking Ahead

So far:

Typical Parameters
↓
ETA
↓
Prediction
↓
Residual Error

Next we ask:

How reliable are ETA estimates?

The next lesson introduces shrinkage.

Strategies

  • Start independent.
  • Add structure cautiously.
  • Interpret biologically.

Common Mistakes

  • Assuming covariance proves mechanism.
  • Overfitting variability.
  • Forgetting parameter count.

Practice Problems

  1. What does variance represent?

  2. What does covariance represent?

  3. Interpret:

eta.cl + eta.v ~ c(0.1,
                   0.02, 0.1)

  1. Why not estimate every correlation?

  2. Explain:

Variance

vs

Covariance

Problem 1

Variance describes spread.

Problem 2

Covariance describes joint movement.

Problem 3

Two ETAs share covariance.

Problem 4

More parameters increase uncertainty.

Problem 5

Variance: individual spread

Covariance: joint spread

Summary

  • Variance measures spread.
  • Covariance measures joint variability.
  • Correlated ETAs may reflect biology.
  • Simpler variability structures are often preferred.
  • Variance ≠ covariance
  • Correlation ≠ causation
  • Start simple