NONLINEAR MIXED EFFECTSMODELS

An Overview and Update

Marie DavidianDepartment of Statistics

North Carolina State University

http://www.stat.ncsu.edu/∼davidian

Based on: Davidian, M. and Giltinan, D.M. (2003), “Nonlinear Models

for Repeated Measurement Data: An Overview and Update,”

JABES 8, 387–419

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Outline

• Introduction

• The Setting

• The Model

• Inferential Objectives and Model Interpretation

• Implementation

• Extensions and Recent Developments

• Discussion

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Introduction

Common situation in agricultural, environmental, and biomedical

applications:

• A continuous response evolves over time (or other condition) within

individuals from a population of interest

• Inference focuses on features or mechanisms that underlie individual

profiles of repeated measurements of the response and how these vary

in the population

• A theoretical or empirical model for individual profiles with parameters

that may be interpreted as representing such features or mechanisms is

available

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Introduction

Nonlinear mixed effects model: aka hierarchical nonlinear model

• A formal statistical framework for this situation

• A “hot ” methodological research area in the early 1990s

• Now widely accepted as a suitable approach to inference, with

applications routinely reported and commercial software available

• Many recent extensions, innovations

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Introduction

Nonlinear mixed effects model: aka hierarchical nonlinear model

• A formal statistical framework for this situation

• A “hot ” methodological research area in the early 1990s

• Now widely accepted as a suitable approach to inference, with

applications routinely reported and commercial software available

• Many recent extensions, innovations

Objective of this talk: An updated review of the model and survey of

recent advances

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The Setting

Example 1: Pharmacokinetics

• Broad goal : Understand intra-subject processes of drug absorption,

distribution, and elimination governing achieved concentrations

• . . . and how these vary across subjects

• Critical for developing dosing strategies and guidelines

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The Setting

Theophylline study: 12 subjects, same oral dose (mg/kg)

Time (hr)

The

ophy

lline

Con

c. (

mg/

L)

0 5 10 15 20 25

02

46

810

12

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The Setting

Example 1: Pharmacokinetics (PK)

• Similarly-shaped concentration-time profiles across subjects

• . . . but peak, rise, decay vary considerably

• Attributable to inter-subject variation in underlying PK processes

(absorption, etc)

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The Setting

Example 1: Pharmacokinetics (PK)

• Standard: approximate representation of the body by simple

compartment models (differential equations)

• One-compartment model for theophylline following oral dose D at

time t = 0 leads to description of concentration C(t) at time t ≥ 0

C(t) =Dka

V (ka − Cl/V )

{exp(−kat)− exp

(−Cl

Vt

)}

ka fractional rate of absorption (1/time)

Cl clearance rate (volume/time)

V volume of distribution

• (ka, Cl, V ) summarize PK processes underlying observed

concentration profiles for a given subject

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The Setting

Example 1: Pharmacokinetics (PK)

• Goal, more precisely stated : Determine mean/median values of

(ka, Cl, V ) and how they vary in the population of subjects

• Elucidate whether some of this variation is associated with subject

characteristics (e.g. weight, age, renal function)

• Develop dosing strategies for subpopulations with certain

characteristics (e.g. the elderly)

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The Setting

Example 2: HIV Dynamics

• Monitoring of “viral load ” (concentration of virus) is now routine for

HIV-infected patients

• Broad goal : Characterize mechanisms underlying the interaction

between HIV virus and the immune system governing decay (and

rebound) of virus levels following treatment with Highly Active

AntiRetroviral Therapy (HAART)

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The Setting

ACTG 315: (log) Viral load profiles for 10 subjects following HAART

0 20 40 60 80

12

34

56

7

Days

log1

0 P

lasm

a R

NA

(co

pies

/ml)

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The Setting

Example 2: HIV Dynamics

• Similarly-shaped profiles with different decay patterns

• Complication – Viral load assay has lower limit of quantification

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The Setting

Example 2: HIV Dynamics

• Represent body by system of ordinary differential equations, e.g.

dX

dt= (1− ε)kV T − δX

dV

dt= pX − cV

X, T size of infected, uninfected immune cell populations

V size of viral population, c viral clearance

δ infected cell death rate, p viral production rate

k probability of infection, ε treatment efficacy

• Parameters characterize intra-subject mechanisms related to

interaction between virus and immune system

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The Setting

Example 2: HIV Dynamics

• Complication – Expression for V (viral load) may not be available in a

closed form

• Further complication – All states of the system of ODEs may not have

been measured

• Goal, more precisely stated : Elucidate “typical ” parameter values

(mean/median), variation across subjects, associations with measures

of pre-treatment disease status

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The Setting

Example 3: Forestry

• Interest in impact of silvicultural treatments and soil types on features

of profiles of forest growth yield

• Individual-tree growth model, e.g. Richards model for dominant height

H(t) at stand age t

H(t) = A{1− exp(−bt)}c

A asymptotic value of dominant height

b rate parameter

c shape parameter

• Goal: Determine “typical ” values, whether variation in parameters is

associated with factors such as treatments and soil types

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The Setting

Further applications:

• Dairy science

• Wildlife science

• Fisheries science

• Biomedical science

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The Model

Basic model: The data are repeated measurements on each of m subjects

yij response at jth “time” tij for subject i

ui vector of additional conditions under which i is observed

ai vector of characteristics for subject i

i = 1, . . . , m, j = 1, . . . , ni, yi = (yi1, . . . , yini)T

(yi, ui, ai) are independent across i

Example: Theophylline pharmacokinetics

• yij is drug concentration for subject i at time tij post-dose

• ui = Di is dose given to subject i at time zero

• ai contains subject characteristics such as weight, age, renal function,

smoking status, etc.

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The Model

Basic model: Stage 1 – Individual-level model

yij = f(tij , ui, βi) + eij , i = 1, . . . , m, j = 1, . . . , ni

f function governing within-individual behavior

βi parameters of f specific to individual i (p× 1)

eij satisfy E(eij |ui, βi) = 0

Example: Theophylline pharmacokinetics

• f is the one-compartment model with dose ui = Di

• βi = (kai, Vi, Cli)T = (β1i, β2i, β3i)T , where kai, Vi, and Cli are

absorption rate, volume, and clearance for subject i

f(t, ui, βi) =Dikai

Vi(kai − Cli/Vi)

{exp(−kait)− exp

(−Cli

Vit

)}

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The Model

Basic model: Stage 2 – Population model

βi = d(ai, β, bi), i = 1, . . . , m

d p-dimensional function

β fixed effects (r × 1)

bi random effects (k × 1)

Characterizes how elements of βi vary across individuals due to

• Systematic association with ai (modeled via β)

• Unexplained variation in the population (represented by bi)

• Usual assumption E(bi|ai) = E(bi) = 0, var(bi|ai) = var(bi) = D

(can be relaxed )

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The Model

Basic model: Stage 2 – Population model

βi = d(ai, β, bi), i = 1, . . . , m

Example: Theophylline pharmacokinetics

• E.g. ai = (ci, wi)T , ci = I( creatinine clearance > 50 ml/min ),

wi = weight (kg)

• bi = (b1i, b2i, b3i)T (p = k = 3), β = (β1, . . . , β7)T (r = 7)

kai = exp(β1 + b1i)

Vi = exp(β2 + β3wi + b2i)

Cli = exp(β4 + β5wi + β6ci + β7wici + b3i)

• If bi ∼ N (0, D), kai, Vi, Cli are lognormal

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The Model

Basic model: Stage 2 – Population model

βi = d(ai, β, bi), i = 1, . . . , m

Example: Theophylline pharmacokinetics, continued

• “Are elements of βi fixed or random effects ?”

• “Unexplained variation ” in one component of βi “small” relative to

others – no associated random effect, e.g.

kai = exp(β1 + b1i)

Vi = exp(β2 + β3wi) (all population variation due to weight)

Cli = exp(β4 + β5wi + β6ci + β7wici + b3i)

• An approximation – usually biologically implausible ; used for

parsimony, numerical stability

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The Model

Basic model: Stage 2 – Population model

βi = d(ai, β, bi), i = 1, . . . , m

Example: Theophylline pharmacokinetics, continued

• Alternative parameterization – reparameterize f in terms of

(k∗a, V ∗, Cl∗)T = (log ka, log V, log Cl)T , βi = (k∗ai, V∗i , Cl∗i )T ,

k∗ai = β1 + b1i

V ∗i = β2 + β3wi + b2i

Cl∗i = β4 + β5wi + β6ci + β7wici + b3i

• Common special case – linear population model

βi = Aiβ + Bibi

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The Model

Within-individual variation: Often misunderstood

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The Model

Within-individual variation: Often misunderstood

t

C(t

)

0 5 10 15 20

02

46

810

12

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The Model

Within-individual variation: Conceptual perspective

• E(yij |ui, βi) = f(tij , ui, βi) =⇒ f represents i’s “on-average ”

profile (smooth curve)

• f may not capture all within-individual processes perfectly, “local

fluctuations ” =⇒ actual realized profile (jittery line)

• f(t, ui, βi) is average over all possible realizations =⇒ “inherent

tendency ” for i’s profile evolution

• =⇒ βi is “inherent characteristic ” of i

• =⇒ Interest focuses on inherent properties of individuals rather than

actual response realizations

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The Model

Within-individual variation: Conceptual perspective

• Within-individual stochastic process

yi(t, ui) = f(t, ui, βi) + eR,i(t, ui) + eM,i(t, ui)

E{eR,i(t, ui)|ui, βi} = E{eM,i(t, ui)|ui, βi} = 0

• Thus yij = yi(tij , ui), eR,i(tij , ui) = eR,ij , eM,i(tij , ui) = eM,ij

yij = f(tij , ui, βi) + eR,ij + eM,ij︸ ︷︷ ︸eij

eR,i = (eR,i1, . . . , eR,ini)T , eM,i = (eM,i1, . . . , eM,ini

)T

• eR,i(t, ui) = “realization deviation process ”

• eM,i(t, ui) = “measurement error process ”

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The Model

Within-individual variation: Conceptual perspective

• Model for eR,i(t, ui) and hence eR,i based on assumptions about

actual realization variance, correlation

var(eR,i|ui, βi) = T1/2i (ui, βi, δ)Γi(ρ)T 1/2

i (ui, βi, δ), (ni × ni)

• Model for eM,i(t, ui) and hence eM,i based on assumptions about

measurement error variance

var(eM,i|ui, βi) = Λi(ui, βi, θ), (ni × ni) diagonal matrix

• Common assumption – realization, measurement error processes

independent =⇒var(yi|ui, βi) = var(eR,i|ui, βi) + var(eM,i|ui, βi) = Ri(ui, βi, ξ)

ξ = (δT , ρT , θT )T

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The Model

var(yi|ui, βi) = var(eR,i|ui, βi) + var(eM,i|ui, βi)

= T1/2i (ui, βi, δ)Γi(ρ)T 1/2

i (ui, βi, δ) + Λi(ui, βi, θ)

= Ri(ui, βi, ξ)

Example: Theophylline pharmacokinetics

• Usual assumption – tij are sufficiently far apart that correlation among

eR,ij is negligible (Γi(ρ) = I)

• Usual assumption – Local fluctuations are negligible, measurement

error dominates realization error

• Ri(ui, βi, ξ) = Λi(ui, βi, θ) diagonal with diagonal elements

var(eij |ui, βi) = var(eM,ij |ui, βi) = σ2Mf2θ(tij , ui, βi)

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The Model

Summary: f i(ui, βi) = {f(xi1, βi), . . . , f(xini, βi)}T , zi = (uT

i , aTi )T

• Stage 1 – Individual-level model

E(yi|zi, bi) = f i(ui, βi) = f i(zi, β, bi)

var(yi|zi, bi) = Ri(ui, βi, ξ) = Ri(zi, β, bi, ξ)

• Stage 2 – Population model

βi = d(ai, β, bi), bi ∼ (0, D)

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The Model

“Within-individual correlation”

• Implies marginal moments

E(yi|zi) =∫

f i(zi, β, bi) dFb(bi)

var(yi|zi) = E{Ri(zi, β, bi, ξ)|zi}+ var{f i(zi, β, bi)|zi}

• E{Ri(zi, β, bi, ξ)|zi} = average of realization/measurement variation

over population =⇒ diagonal only if correlation of within-individual

realizations negligible

• var{f i(zi, β, bi)|zi} = population variation in “inherent trajectories ”

=⇒ non-diagonal in general

• Result – Overall pattern of marginal correlation is the aggregate of

correlation due to both sources

• Prefer “aggregate correlation ” to “within-individual correlation ”

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Inferential Objectives and Model Interpretation

Main goal:

• Elements of βi represent underlying features

• “Typical ” values of underlying features, variation in these, and

association with individual characteristics =⇒ inference on β and D

• =⇒ Deduce an appropriate d

Additional goal: “Individual-level prediction

• Inference on βi, f(t0, ui, βi)

• “Borrow strength ” across similar subjects

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Inferential Objectives and Model Interpretation

Subject-specific model:

• Not the same as the population averaged approach of modeling

E(yi|zi), var(yi|zi) directly

• Explicitly acknowledges individual behavior

• Interest in the “typical value,” variation of underlying features βi, not

in the “typical response profile” and overall variation about it

• Incorporates scientific assumptions embedded in the model f for

individual behavior

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Implementation

Likelihood: With distributional assumptions on (yi|zi, bi) and bi

(almost always normal )

L(β, ξ, D) =m∏

i=1

∫p(yi, bi|zi, ; β, ξ, D) dbi =

m∏i=1

∫p(yi|zi, bi; β, ξ)p(bi; D) dbi

• Maximize jointly in (β, ξ, D)

• Intractable integrations in general

• Potentially high-dimensional, computationally expensive

• =⇒ Approximate L(β, ξ, D) by analytical approximation to

p(yi|zi; β, ξ, D) =∫

p(yi|zi, bi; β, ξ)p(bi; D) dbi

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Implementation

First-order methods: Combine both stages as

yi = f i(zi, β, bi) + R1/2i (zi, β, bi, ξ)εi, εi|zi, bi ∼ (0, Ini

)

• Taylor series about bi = 0 to linear terms

yi ≈ f i(zi, β,0) + Zi(zi, β,0)bi + R1/2i (zi, β,0, ξ)εi

Zi(zi, β, b∗) = ∂/∂bi{f i(zi, β, bi)}|bi=b∗

• Implies E(yi|zi) ≈ f i(zi, β,0)

var(yi|zi) ≈ Zi(zi, β,0)DZTi (zi, β,0) + Ri(zi, β,0, ξ)

• Estimate (β, ξ, D) by fitting this approximate marginal model

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Implementation

First-order methods: Software

• SAS macro nlinmix with expand=zero – Solve a set of generalized

estimating equations (“GEE-1 ”) based on these marginal moments

• nonmem fo method, SAS proc nlmixed with method=firo –

Maximize normal likelihood with these marginal moments (“GEE-2 ”)

• proc nlmixed cannot handle dependence of Ri on βi, β

• Obvious potential for bias

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Implementation

First-order conditional methods: More “refined ” approximation for

“ni large ” (several variations)

E(yi|zi) ≈ f i(zi, β, b̂i)−Zi(zi, β, b̂i)b̂i

var(yi|zi) ≈ Zi(zi, β, b̂i)DZTi (zi, β, b̂i) + Ri(zi, β, b̂i, ξ)

b̂i = DZTi (zi, β, b̂i)Ri(zi, β, b̂i, ξ){yi − f i(zi, β, b̂i)}

• May be derived by Taylor series argument or invoking Laplace’s

approximation

• Suggests iterative scheme – alternate between update of b̂i and fitting

the approximate marginal model

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Implementation

First-order conditional methods: Software

• nonmem foce – Based on normal likelihood (“GEE-2 ”)

• SAS macro nlinmix with expand=eblup and R/Splus function

nlme( ) – Solve a set of generalized estimating equations (“GEE-1 ”)

based on these marginal moments

Performance: Work well even for ni not large as long as within-individual

variation is not large

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Implementation

“Exact likelihood” methods: Maximize likelihood “directly ” using

deterministic or stochastic approximation to the integrals

• Deterministic approximation – Quadrature, Adaptive Gaussian

quadrature

• Stochastic approximation – Importance sampling, brute-force Monte

Carlo integration

“Exact likelihood” methods: Software

• proc nlmixed – quadrature methods, importance sampling when

bi ∼ N (0, D)

• Other non-commercial software

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Implementation

Bayesian formulation: Stage 3 – Hyperprior

(β, ξ, D) ∼ p(β, ξ, D)

• Markov chain Monte Carlo (MCMC) techniques to simulate samples

from posterior distributions for β, ξ, D

• Not possible in general in WinBUGS because nonlinearity of f may

require tailored approach

• PKBugs has tailored implementation for compartment models for f

used in PK

• Attractive feature – natural way to incorporate constraints and

subject-matter information

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Extensions and Recent Developments

Multi-level models: In many applications

• Nesting – response profiles (yihj , j = 1, . . . , nih) on several trees

(h = 1, ..., pi) within each of several plots (i = 1, . . . , m), e.g.,

βih = Aihβ + bi + bih, bi, bih independent

Multivariate response: More than one type of response profile

(` = 1, . . . , q) on each individual

• yij` = f`(tij`, ui, βi`) + eij`

• Pharmacokinetics (concentration-time) and pharmacodynamics

(response-concentration)

yij,PK = fPK(tij,PK , ui, βi,PK) + eij,PK

yij,PD = fPD{ fPK(tij,PK , ui, βi,PK), βi,PD }+ eij,PD

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Extensions and Recent Developments

Missing/mismeasured covariates: ai, ui, and tij

Censored response: E.g., due to quantification limit

Semiparametric models: Model misspecification, flexibility

• f depends on unspecified function g(t, βi)

Clinical trial simulation: Hypothetical subjects simulated from nonlinear

mixed models for population PK/PD, linked to clinical endpoint

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Discussion

• The nonlinear mixed model is now a standard inferential tool used

routinely in many applications

• For extensive references and more details see

Davidian, M. and Giltinan, D.M. (2003), “Nonlinear Models

for Repeated Measurement Data: An Overview and Update,”

JABES 8, 387–419

IBC2004 41