Shrinkage and Interpreting Variability

Understand shrinkage and learn how to interpret variability estimates cautiously.
Tip

Big picture: Population models estimate individual behavior, but not all individual estimates are equally reliable.

Learning Objectives

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

  • explain shrinkage conceptually
  • understand why shrinkage occurs
  • interpret ETA estimates cautiously
  • distinguish population and individual interpretation
  • explain why data quantity affects variability estimation

Key Ideas

  • Individual estimates are uncertain.
  • Sparse data increases shrinkage.
  • Population estimates are usually more stable.
  • Shrinkage affects interpretation.

Setup

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

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

Fit 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)

  })

}

fit <-
  nlmixr2(
    one_comp_model,
    theo_sd,
    est = "focei",
    control = list(
      print = 0
    )
  )

Why Shrinkage Exists

Suppose one subject has:

Many Samples

and another has:

Few Samples

Should both subjects receive equally strong individual estimates?

Usually no.

Population models balance:

Individual Data
+
Population Information

This balancing creates shrinkage.

Worked Example 1: Conceptualize Shrinkage

Conceptually:

Rich Individual Data
↓
Estimate Moves Toward Subject

versus:

Sparse Data
↓
Estimate Moves Toward Population

Shrinkage is not failure.

Shrinkage is stabilization.

Worked Example 2: Locate Shrinkage in the Fit Object

Inspect:

print(fit)
── nlmixr² FOCEi (outer: nlminb) ──

          OBJF      AIC      BIC Log-likelihood Condition#(Cov) Condition#(Cor)
FOCEi 116.8039 373.4036 393.5832      -179.7018        68.64196        9.387133

── Time (sec $time): ──

           setup optimize covariance table    other
elapsed 0.001817 0.145883   0.145884  0.02 3.683416

── Population Parameters ($parFixed or $parFixedDf): ──

         Est.     SE %RSE Back-transformed(95%CI) BSV(CV%) Shrink(SD)%
tka     0.463  0.195 42.1       1.59 (1.08, 2.33)     70.5      1.86% 
tcl      1.01 0.0751 7.42       2.75 (2.37, 3.19)     26.8      3.98% 
tv       3.46 0.0436 1.26       31.8 (29.2, 34.6)     13.9      10.4% 
add.err 0.694                               0.694                     
 
  Covariance Type ($covMethod): r,s
  No correlations in between subject variability (BSV) matrix
  Full BSV covariance ($omega) or correlation ($omegaR; diagonals=SDs) 
  Distribution stats (mean/skewness/kurtosis/p-value) available in $shrink 
  Information about run found ($runInfo):
   • gradient problems with initial estimate and covariance; see $scaleInfo 
   • ETAs were reset to zero during optimization; (Can control by foceiControl(resetEtaP=.)) 
   • initial ETAs were nudged; (can control by foceiControl(etaNudge=., etaNudge2=)) 
  Censoring ($censInformation): No censoring
  Minimization message ($message):  
    relative convergence (4) 

── Fit Data (object is a modified tibble): ──
# A tibble: 132 × 22
  ID     TIME    DV  PRED    RES   WRES IPRED   IRES  IWRES CPRED   CRES  CWRES
  <fct> <dbl> <dbl> <dbl>  <dbl>  <dbl> <dbl>  <dbl>  <dbl> <dbl>  <dbl>  <dbl>
1 1      0     0.74  0     0.74   1.07   0     0.74   1.07   0     0.74   1.07 
2 1      0.25  2.84  3.26 -0.422 -0.225  3.85 -1.01  -1.45   3.22 -0.378 -0.177
3 1      0.57  6.57  5.83  0.740  0.297  6.78 -0.215 -0.310  5.77  0.796  0.287
# ℹ 129 more rows
# ℹ 10 more variables: eta.ka <dbl>, eta.cl <dbl>, eta.v <dbl>, depot <dbl>,
#   central <dbl>, ka <dbl>, cl <dbl>, v <dbl>, tad <dbl>, dosenum <dbl>

Look for:

Shrink(SD)%

Example output:

eta.cl

Shrink(SD)%

Interpretation:

Shrinkage summarizes how strongly individual estimates rely on population information.

Worked Example 3: Connect ETA and Shrinkage

Recall:

Typical Parameter
+
ETA
↓
Individual Parameter

Shrinkage influences:

ETA
↓
How Individualized

Higher shrinkage:

Individual Estimates

closer to

Population Mean

Lower shrinkage:

More Subject Information

Worked Example 4: Why Shrinkage Happens

Shrinkage may increase because of:

  • sparse sampling
  • weak information
  • large residual variability
  • limited subject data

Conceptually:

Less Information
↓
More Borrowing

Population models estimate subjects jointly.

Worked Example 5: Interpret Variability Carefully

Suppose:

ETA(CL)

appears small.

Question:

True Similarity?

or

Shrinkage?

Interpretation:

Observed variability and estimated variability are not identical.

Interpret variability together with model context.

Shrinkage Is Not Model Failure

High shrinkage does not automatically mean:

Bad Model

Low shrinkage does not automatically mean:

Good Model

Interpret shrinkage in context.

Population Thinking

Population models prioritize:

Stable Population Estimates

over:

Perfect Individual Estimates

Population modeling and individual forecasting are related but not identical goals.

Connect to Covariates

Shrinkage matters because later we ask:

Can variability be explained?

Strong shrinkage may affect how easily variability relationships appear.

Looking Ahead

So far we built:

Typical Parameters
↓
Random Effects
↓
Residual Error
↓
Variability Structure
↓
Shrinkage

Next we ask:

Can variability be explained?

The next module introduces covariate modeling.

Strategies

  • Interpret shrinkage cautiously.
  • Think population first.
  • Consider information content.

Common Mistakes

  • Overinterpreting individual estimates.
  • Assuming low shrinkage means correctness.
  • Ignoring uncertainty.

Practice Problems

  1. What is shrinkage?

  2. Why does shrinkage occur?

  3. Compare:

Rich Data

vs

Sparse Data

  1. Why does shrinkage affect ETA interpretation?

  2. Explain:

Individual Data + Population Information

Problem 1

Shrinkage reflects borrowing from the population.

Problem 2

Individual information may be limited.

Problem 3

Rich data: more individualized estimates

Sparse data: more population influence

Problem 4

Shrinkage changes how strongly ETAs reflect subjects.

Problem 5

Population models estimate subjects jointly.

Summary

  • Shrinkage stabilizes estimation.
  • Sparse data increases borrowing.
  • Population estimates remain central.
  • Variability should be interpreted carefully.
  • Shrinkage ≠ failure
  • ETA ≠ truth
  • Population first