Understanding Parameter Estimates

Interpret fixed effects, uncertainty, variability estimates, and residual error from the first population PK model.

Tip

Big picture: Estimation produces numbers, but interpretation turns those numbers into scientific understanding.

Learning Objectives

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

• Interpret fixed effects estimates.
• Understand parameter uncertainty conceptually.
• Recognize variability outputs without fully interpreting them.
• Recognize residual error estimates.
• Connect estimates back to PK meaning.

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Key Ideas

• Parameters have biological meaning.
• Population estimates represent typical values.
• Estimates always contain uncertainty.
• Variability and residual error will be developed further later.

---

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 Interpretation Matters

Model fitting does not end with convergence.

After estimation we ask:

• Are values plausible?
• Are estimates stable?
• Does biology make sense?

Interpretation transforms output into knowledge.

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Worked Example 1: Fixed Effects

Extract estimates.

coef(fit)$fixed %>%
  enframe(
    name = "parameter",
    value = "estimate"
  )

# A tibble: 4 × 2
  parameter estimate
  <chr>        <dbl>
1 tka          0.463
2 tcl          1.01 
3 tv           3.46 
4 add.err      0.694

Typical fixed effects include:

Parameter	Interpretation
tka	typical absorption
tcl	typical clearance
tv	typical volume

These represent the typical subject.

Not any individual subject.

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Worked Example 2: Translate Back to PK Parameters

Our estimates were modeled on the log scale.

Convert to original units.

coef(fit)$fixed %>%
  exp() %>%
  enframe(
    name = "parameter",
    value = "estimate_original_scale"
  )

# A tibble: 4 × 2
  parameter estimate_original_scale
  <chr>                       <dbl>
1 tka                          1.59
2 tcl                          2.75
3 tv                          31.8 
4 add.err                      2.00

Interpretation:

• larger CL → faster elimination
• larger V → lower concentrations
• larger ka → faster absorption

Interpret parameters in PK language.

---

Worked Example 3: Understand Parameter Uncertainty

Inspect fit output.

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 0.001713 0.142397   0.142397 0.019 3.656493

── 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>

Focus on:

Est.
SE
%RSE

Interpretation:

• Est. → estimated value
• SE → standard error
• %RSE → relative uncertainty

Conceptually:

Estimate

↓

Uncertainty

Smaller uncertainty suggests more stable estimates.

Do not apply strict thresholds yet.

---

Worked Example 4: Recognize Variability Outputs

Inspect:

VarCorr(fit)

         Variance    StdDev
eta.ka 0.40363397 0.6353219
eta.cl 0.06927252 0.2631967
eta.v  0.01925920 0.1387775

This output describes variability estimates.

Conceptually:

Typical Subject

↓

Individual Differences

At this stage:

recognize where variability appears.

The next module explains:

• ETA models
• variability assumptions
• random effects structures

Do not interpret exact values yet.

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Worked Example 5: Recognize Residual Error

Inspect the model.

fit

\[\begin{align*} {ka} & = \exp\left({tka}+{eta.ka}\right) \\ {cl} & = \exp\left({tcl}+{eta.cl}\right) \\ {v} & = \exp\left({tv}+{eta.v}\right) \\ linCmt() & \sim add({add.err}) \end{align*}\]

Locate:

add.err

Residual error represents variation not explained by the model.

Examples:

• assay variability
• measurement uncertainty
• model simplification
• unexplained biology

Residual error does not necessarily indicate model failure.

Residual error models come later.

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Population Thinking

Suppose estimation returns:

Typical CL = 3

Population interpretation:

Subject A → around 3

Subject B → around 3

Subject C → around 3

Population modeling estimates:

Typical Values + Variability + Residual Error

Not one universal profile.

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Parameter Language

As terminology evolves, you may hear:

Term	Meaning
THETA	fixed effects
ETA	variability
SIGMA	residual error

You do not need to memorize these today.

They become more useful later.

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Interpreting Estimates Systematically

Recommended order:

Fixed Effects

↓

Uncertainty

↓

Variability

↓

Residual Error

↓

Diagnostics

Interpret estimates before evaluating diagnostics.

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Strategies

• Interpret on original scale.
• Think biologically.
• Separate estimate categories.

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Common Mistakes

• Treating estimates as exact truth.
• Overinterpreting variability.
• Jumping to diagnostics.

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Practice Problems

1. What do fixed effects represent?

2. Why transform estimates back to original units?

3. What does %RSE represent?

4. Extract estimates.

coef(fit)$fixed %>%
  exp() %>%
  enframe(
    name = "parameter",
    value = "estimate_original_scale"
  )

Interpret one parameter biologically.

5. Run:

VarCorr(fit)

What type of information does this provide?

Do not interpret exact values.

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TipStep-by-Step Solutions

Problem 1

Fixed effects describe typical parameter values.

Examples:

• typical clearance
• typical volume
• typical absorption

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Problem 2

Parameters are often estimated on transformed scales.

Interpretation becomes easier after returning to PK units.

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Problem 3

%RSE describes relative uncertainty.

Conceptually:

Estimate

↓

Uncertainty

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Problem 4

Interpret on original units.

Examples:

Higher CL

↓

Faster elimination

Higher V

↓

Lower concentration

Higher ka

↓

Faster absorption

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Problem 5

VarCorr() reports variability estimates.

This output becomes more important in the next module.

For now:

recognize where variability appears.

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Summary

• Fixed effects describe typical values.
• Estimates contain uncertainty.
• Variability and residual error appear in model output.
• Interpretation converts estimates into understanding.

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TipQuick Tips

• Interpret biologically.
• Convert scales.
• Uncertainty is expected.
• Variability comes next.