Nonlinear Mixed Effects Models: an Overview and Update

NONLINEAR MIXED EFFECTS MODELS An Overview and Update

Marie Davidian Department 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

IBC2004 1 Outline

Introduction • The Setting • The Model • Inferential Objectives and Model Interpretation • Implementation • Extensions and Recent Developments • Discussion •

IBC2004 2 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

IBC2004 3 Introduction

Nonlinear mixed eﬀects model: aka hierarchical nonlinear model Aformalstatistical 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 •

IBC2004 4 Introduction

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

IBC2004 4 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 •

IBC2004 5 The Setting

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

Theophylline Conc. (mg/L) Theophylline Conc. 024681012

0 5 10 15 20 25

Time (hr)

IBC2004 6 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)

IBC2004 7 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 =0leads to description of concentration C(t) at time t 0 ≥ Dk Cl C(t)= a exp( k t) exp t V (k Cl/V ) ½ − a − µ− V ¶¾ a − ka fractional rate of absorption (1/time) Cl clearance rate (volume/time) V volume of distribution (k ,Cl,V) summarize PK processes underlying observed • a concentration proﬁles for a given subject

IBC2004 8 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)

IBC2004 9 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)

IBC2004 10 The Setting

ACTG 315: (log) Viral load proﬁles for 10 subjects following HAART log10 Plasma RNA (copies/ml) 1234567

0 20406080

Days

IBC2004 11 The Setting

Example 2: HIV Dynamics Similarly-shaped profiles with different decay patterns • Complication – Viral load assay has lower limit of quantification •

IBC2004 12 The Setting

Example 2: HIV Dynamics Represent body by system of ordinary diﬀerential equations, e.g. • dX =(1 ²)kV T δX dt − − dV = pX cV dt −

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, ² treatment eﬃcacy Parameters characterize intra-subject mechanisms related to • interaction between virus and immune system

IBC2004 13 The Setting

Example 2: HIV Dynamics Complication – Expression for V (viral load) may not be available in a • closed form Further complication –Allstates 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

IBC2004 14 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

IBC2004 15 The Setting

Further applications: Dairy science • Wildlife science • Fisheries science • Biomedical science •

IBC2004 16 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

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

(yi, ui, ai) are independent across i Example: Theophylline pharmacokinetics y is drug concentration for subject i at time t post-dose • ij ij u = D is dose given to subject i at time zero • i i a contains subject characteristics such as weight, age, renal function, • i smoking status, etc.

IBC2004 17 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 β parameters of f speciﬁc to individual i (p 1) i × e satisfy E(e u , β )=0 ij ij| i i Example: Theophylline pharmacokinetics f is the one-compartment model with dose u = D • i i β =(k ,V ,Cl )T =(β ,β ,β )T , where k , V , and Cl are • i ai i i 1i 2i 3i ai i i absorption rate, volume, and clearance for subject i D k Cl f(t, u , β )= i ai exp( k t) exp i t i i V (k Cl /V ) ½ − ai − µ− V ¶¾ i ai − i i i

IBC2004 18 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) × b random effects (k 1) i × Characterizes how elements of βi vary across individuals due to Systematic association with a (modeled via β) • i Unexplained variation in the population (represented by b ) • i Usual assumption E(b a )=E(b )=0,var(b a )=var(b )=D • i| i i i| i i (can be relaxed )

IBC2004 19 The Model

Basic model: Stage 2 – Population model

βi = d(ai, β, bi),i=1,...,m

Example: Theophylline pharmacokinetics

E.g. a =(c ,w )T , c = I( creatinine clearance > 50 ml/min ), • i i i i wi =weight(kg) b =(b ,b ,b )T (p = k =3), β =(β ,...,β )T (r =7) • i 1i 2i 3i 1 7

kai =exp(β1 + b1i)

Vi =exp(β2 + β3wi + b2i)

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

If b (0, D), k ,V ,Cl are lognormal • i ∼N ai i i

IBC2004 20 The Model

Basic model: Stage 2 – Population model

βi = d(ai, β, bi),i=1,...,m

Example: Theophylline pharmacokinetics, continued “Are elements of β fixed or random effects ?” • i “Unexplained variation ” in one component of β “small” relative to • i 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 ;usedfor • parsimony, numerical stability

IBC2004 21 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 • T T T (ka∗,V∗,Cl∗) = (log ka, log V,log Cl) , βi =(kai∗ ,Vi∗,Cli∗) ,

kai∗ = β1 + b1i

Vi∗ = β2 + β3wi + b2i

Cli∗ = β4 + β5wi + β6ci + β7wici + b3i

Common special case – linear population model •

βi = Aiβ + Bibi

IBC2004 22 The Model

Within-individual variation: Often misunderstood

IBC2004 23 The Model

Within-individual variation: Often misunderstood C(t) 024681012

0 5 10 15 20

IBC2004 23 The Model

Within-individual variation: Conceptual perspective

E(y u , β )=f(t , u , β ) = f represents i’s “on-average ” • ij| i i ij i i ⇒ profile (smooth curve) f may not capture all within-individual processes perfectly,“local • fluctuations ” = actual realized profile (jittery line) ⇒ f(t, u , β ) is average over all possible realizations = “inherent • i i ⇒ tendency ”fori’s profile evolution

= β is “inherent characteristic ”ofi • ⇒ i = Interest focuses on inherent properties of individuals rather than • ⇒ actual response realizations

IBC2004 24 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 e (t, u ) u , β = E e (t, u ) u , β =0 { R,i i | i i} { M,i i | i i} Thus y = y (t , u ), e (t , u )=e , e (t , u )=e • ij i ij i R,i ij i R,ij M,i ij i M,ij

yij = f(tij, ui, βi)+eR,ij + eM,ij

eij | {z } T T eR,i =(eR,i1,...,eR,ini ) , eM,i =(eM,i1,...,eM,ini ) e (t, u ) =“realization deviation process ” • R,i i e (t, u ) =“measurement error process ” • M,i i

IBC2004 25 The Model

Within-individual variation: Conceptual perspective Model for e (t, u ) and hence e based on assumptions about • R,i i R,i actual realization variance, correlation

var(e u , β )=T 1/2(u , β , δ)Γ (ρ)T 1/2(u , β , δ), (n n ) R,i| i i i i i i i i i i × i Model for e (t, u ) and hence e based on assumptions about • M,i i M,i measurement error variance

var(e u , β )=Λ (u , β , θ), (n n ) diagonal matrix M,i| i i i i i i × i Common assumption – realization, measurement error processes • independent = ⇒ var(y u , β )=var(e u , β )+var(e u , β )=R (u , β , ξ) i| i i R,i| i i M,i| i i i i i ξ =(δT , ρT , θT )T

IBC2004 26 The Model

var(y u , β )=var(e u , β )+var(e u , β ) i| i i R,i| i i M,i| i i 1/2 1/2 = T i (ui, βi, δ)Γi(ρ)T i (ui, βi, δ)+Λi(ui, βi, θ)

= Ri(ui, βi, ξ)

Example: Theophylline pharmacokinetics Usual assumption – t are suﬃciently far apart that correlation among • ij eR,ij is negligible (Γi(ρ)=I) Usual assumption – Local ﬂuctuations are negligible, measurement • error dominates realization error

R (u , β , ξ)=Λ (u , β , θ) diagonal with diagonal elements • i i i i i i var(e u , β )=var(e u , β )=σ2 f 2θ(t , u , β ) ij| i i M,ij| i i M ij i i

IBC2004 27 The Model

Summary: f (u , β )= f(x , β ),...,f(x , β ) T , z =(uT , aT )T i i i { i1 i ini i } i i i Stage 1 – Individual-level model • E(y z , b )=f (u , β )=f (z , β, b ) i| i i i i i i i i var(y z , b )=R (u , β , ξ)=R (z , β, b , ξ) i| i i i i i i i i Stage 2 – Population model • β = d(a , β, b ), b (0, D) i i i i ∼

IBC2004 28 The Model

“Within-individual correlation” Implies marginal moments •

E(y zi)= f (zi, β, bi) dFb(bi) i| Z i var(y z )=E R (z , β, b , ξ) z + var f (z , β, b ) z i| i { i i i | i} { i i i | i} E R (z , β, b , ξ) z = average of realization/measurement variation • { i i i | i} over population = diagonal only if correlation of within-individual ⇒ realizations negligible var f (z , β, b ) z = population variation in “inherent trajectories ” • { i i i | i} = 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 ” •

IBC2004 29 Inferential Objectives and Model Interpretation

Main goal:

Elements of β represent underlying features • i “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 β , f(t , u , β ) • i 0 i i “Borrow strength ” across similar subjects •

IBC2004 30 Inferential Objectives and Model Interpretation

Subject-specific model: Not the same as the population averaged approach of modeling • E(y z ),var(y z ) directly i| i i| i Explicitly acknowledges individual behavior • Interest in the “typical value,” variation of underlying features β , not • i in the “typical response profile” and overall variation about it Incorporates scientific assumptions embedded in the model f for • individual behavior

IBC2004 31 Implementation

Likelihood: With distributional assumptions on (y z , b ) and b i| i i i (almost always normal )

m m L(β, ξ, D)= p(y , bi zi, ; β, ξ, D) dbi = p(y zi, bi; β, ξ)p(bi; D) dbi Z i | Z i| iY=1 iY=1 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(y zi; β, ξ, D)= p(y zi, bi; β, ξ)p(bi; D) dbi i| Z i|

IBC2004 32 Implementation

First-order methods: Combine both stages as

y = f (z , β, b )+R1/2(z , β, b , ξ)² , ² z , b (0, I ) i i i i i i i i i| i i ∼ ni Taylor series about b = 0 to linear terms • i y f (z , β, 0)+Z (z , β, 0)b + R1/2(z , β, 0, ξ)² i ≈ i i i i i i i i

Z (z , β, b∗)=∂/∂b f (z , β, b ) i i i{ i i i }|bi=b∗ Implies E(y z ) f (z , β, 0) • i| i ≈ i i var(y z ) Z (z , β, 0)DZT (z , β, 0)+R (z , β, 0, ξ) i| i ≈ i i i i i i Estimate (β, ξ, D) by ﬁtting this approximate marginal model •

IBC2004 33 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 R on β , β • i i Obvious potential for bias •

IBC2004 34 Implementation

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

“ni large ” (several variations)

E(y z ) f (z , β, b ) Z (z , β, b )b i| i ≈ i i i − i i i i T var(y zi) Zi(zi, β,bbi)DZ (zi, β,bbi)+b Ri(zi, β, bi, ξ) i| ≈ i b = DZT (z , β, b )Rb(z , β, b , ξ) yb f (z , β, b )b i i i i i i i { i − i i i } May beb derived by Taylorb series argumentb or invoking Laplace’sb • approximation

Suggests iterative scheme – alternate between update of b and ﬁtting • i the approximate marginal model b

IBC2004 35 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

IBC2004 36 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 • b (0, D) i ∼N Other non-commercial software •

IBC2004 37 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

IBC2004 38 Extensions and Recent Developments

Multi-level models: In many applications Nesting – response proﬁles (y , j =1,...,n ) on several trees • ihj ih (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 y = f (t , u , β )+e • ij` ` ij` i i` ij` Pharmacokinetics (concentration-time) and pharmacodynamics • (response-concentration)

yij,P K = fPK(tij,P K , ui, βi,P K )+eij,P K y = f f (t , u , β ), β + e ij,P D PD{ PK ij,P K i i,P K i,P D } ij,P D

IBC2004 39 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, β ) • i

Clinical trial simulation: Hypothetical subjects simulated from nonlinear mixed models for population PK/PD, linked to clinical endpoint

IBC2004 40 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