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:
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Davidian, M. and Giltinan, D.M. (2003), “Nonlinear Models
for Repeated Measurement Data: An Overview and Update,”
JABES 8, 387–419
1
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
8
6
4
0
2
Theophylline Conc. (mg/L)
10
12
Theophylline study: 12 subjects, same oral dose (mg/kg)
0
5
10
15
Time (hr)
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20
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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
½
µ
¶¾
Cl
Dka
exp(−ka t) − exp − t
C(t) =
V (ka − Cl/V )
V
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
5
4
3
1
2
log10 Plasma RNA (copies/ml)
6
7
ACTG 315: (log) Viral load profiles for 10 subjects following HAART
0
20
40
Days
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60
80
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
dV
dt
= (1 − ²)kV T − δX
= 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,
²
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 , y i = (yi1 , . . . , yini )T
(y i , 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 , β i ) + eij ,
f
i = 1, . . . , m, j = 1, . . . , ni
function governing within-individual behavior
βi
parameters of f specific to individual i (p × 1)
eij
satisfy E(eij |ui , β i ) = 0
Example: Theophylline pharmacokinetics
• f is the one-compartment model with dose ui = Di
• β i = (kai , Vi , Cli )T = (β1i , β2i , β3i )T , where kai , Vi , and Cli are
absorption rate, volume, and clearance for subject i
½
µ
¶¾
Di kai
Cli
exp(−kai t) − exp −
t
f (t, ui , β i ) =
Vi (kai − Cli /Vi )
Vi
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The Model
Basic model: Stage 2 – Population model
β i = d(ai , β, bi ),
d
p-dimensional function
β
fixed effects (r × 1)
bi
random effects (k × 1)
i = 1, . . . , m
Characterizes how elements of β i 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
β i = 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 + β3 wi + b2i )
Cli
=
exp(β4 + β5 wi + β6 ci + β7 wi ci + b3i )
• If bi ∼ N (0, D), kai , Vi , Cli are lognormal
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The Model
Basic model: Stage 2 – Population model
β i = d(ai , β, bi ),
i = 1, . . . , m
Example: Theophylline pharmacokinetics, continued
• “Are elements of β i fixed or random effects ?”
• “Unexplained variation ” in one component of β i “small” relative to
others – no associated random effect, e.g.
kai
=
exp(β1 + b1i )
Vi
=
exp(β2 + β3 wi ) (all population variation due to weight)
Cli
=
exp(β4 + β5 wi + β6 ci + β7 wi ci + b3i )
• An approximation – usually biologically implausible ; used for
parsimony, numerical stability
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The Model
Basic model: Stage 2 – Population model
β i = d(ai , β, bi ),
i = 1, . . . , m
Example: Theophylline pharmacokinetics, continued
• Alternative parameterization – reparameterize f in terms of
∗
, Vi∗ , Cli∗ )T ,
(ka∗ , V ∗ , Cl∗ )T = (log ka , log V, log Cl)T , β i = (kai
∗
kai
=
β1 + b1i
Vi∗
=
β2 + β3 wi + b2i
Cli∗
=
β4 + β5 wi + β6 ci + β7 wi ci + b3i
• Common special case – linear population model
β i = Ai β + B i bi
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The Model
Within-individual variation: Often misunderstood
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The Model
6
0
2
4
C(t)
8
10
12
Within-individual variation: Often misunderstood
0
5
10
15
t
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The Model
Within-individual variation: Conceptual perspective
• E(yij |ui , β i ) = f (tij , ui , β i ) =⇒ 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 , β i ) is average over all possible realizations =⇒ “inherent
tendency ” for i’s profile evolution
• =⇒ β i 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 , β i ) + eR,i (t, ui ) + eM,i (t, ui )
E{eR,i (t, ui )|ui , β i } = E{eM,i (t, ui )|ui , β i } = 0
• Thus yij = yi (tij , ui ), eR,i (tij , ui ) = eR,ij , eM,i (tij , ui ) = eM,ij
yij = f (tij , ui , β i ) + eR,ij + eM,ij
|
{z
}
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
1/2
1/2
var(eR,i |ui , β i ) = T i (ui , β i , δ)Γi (ρ)T i (ui , β i , δ),
(ni × ni )
• Model for eM,i (t, ui ) and hence eM,i based on assumptions about
measurement error variance
var(eM,i |ui , β i ) = Λi (ui , β i , θ),
(ni × ni ) diagonal matrix
• Common assumption – realization, measurement error processes
independent =⇒
var(y i |ui , β i ) = var(eR,i |ui , β i ) + var(eM,i |ui , β i ) = Ri (ui , β i , ξ)
ξ = (δ T , ρT , θ T )T
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The Model
var(y i |ui , β i ) = var(eR,i |ui , β i ) + var(eM,i |ui , β i )
=
1/2
1/2
T i (ui , β i , δ)Γi (ρ)T i (ui , β i , δ) + Λi (ui , β i , θ)
= Ri (ui , β i , 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 , β i , ξ) = Λi (ui , β i , θ) diagonal with diagonal elements
2
f 2θ (tij , ui , β i )
var(eij |ui , β i ) = var(eM,ij |ui , β i ) = σM
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The Model
Summary: f i (ui , β i ) = {f (xi1 , β i ), . . . , f (xini , β i )}T , z i = (uTi , aTi )T
• Stage 1 – Individual-level model
E(y i |z i , bi ) = f i (ui , β i ) = f i (z i , β, bi )
var(y i |z i , bi ) = Ri (ui , β i , ξ) = Ri (z i , β, bi , ξ)
• Stage 2 – Population model
β i = d(ai , β, bi ), bi ∼ (0, D)
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The Model
“Within-individual correlation”
• Implies marginal moments
Z
f i (z i , β, bi ) dFb (bi )
E(y i |z i ) =
var(y i |z i )
= E{Ri (z i , β, bi , ξ)|z i } + var{f i (z i , β, bi )|z i }
• E{Ri (z i , β, bi , ξ)|z i } = average of realization/measurement variation
over population =⇒ diagonal only if correlation of within-individual
realizations negligible
• var{f i (z i , β, bi )|z i } = 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 β i 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 β i , f (t0 , ui , β i )
• “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(y i |z i ), var(y i |z i ) directly
• Explicitly acknowledges individual behavior
• Interest in the “typical value,” variation of underlying features β i , 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 (y i |z i , bi ) and bi
(almost always normal )
m Z
m Z
Y
Y
L(β, ξ, D) =
p(y i , bi |z i , ; β, ξ, D) dbi =
p(y i |z i , bi ; β, ξ)p(bi ; D) dbi
i=1
i=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
Z
p(y i |z i ; β, ξ, D) = p(y i |z i , bi ; β, ξ)p(bi ; D) dbi
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Implementation
First-order methods: Combine both stages as
1/2
y i = f i (z i , β, bi ) + Ri (z i , β, bi , ξ)²i , ²i |z i , bi ∼ (0, I ni )
• Taylor series about bi = 0 to linear terms
1/2
y i ≈ f i (z i , β, 0) + Z i (z i , β, 0)bi + Ri (z i , β, 0, ξ)²i
Z i (z i , β, b∗ ) = ∂/∂bi {f i (z i , β, bi )}|bi =b∗
• Implies E(y i |z i )
var(y i |z i )
≈ f i (z i , β, 0)
≈ Z i (z i , β, 0)DZ Ti (z i , β, 0) + Ri (z i , 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 β i , β
• Obvious potential for bias
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Implementation
First-order conditional methods: More “refined ” approximation for
“ni large ” (several variations)
bi ) − Z i (z i , β, b
bi )b
bi
E(y i |z i ) ≈ f i (z i , β, b
var(y |z i ) ≈ Z i (z i , β, b
bi )DZ T (z i , β, b
bi ) + Ri (z i , β, b
bi , ξ)
i
i
b
bi )Ri (z i , β, b
bi , ξ){y i − f i (z i , β, b
bi )}
bi = DZ Ti (z i , β, b
• May be derived by Taylor series argument or invoking Laplace’s
approximation
• Suggests iterative scheme – alternate between update of b
bi 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.,
β ih = Aih β + bi + bih , bi , bih independent
Multivariate response: More than one type of response profile
(` = 1, . . . , q) on each individual
• yij` = f` (tij` , ui , β i` ) + eij`
• Pharmacokinetics (concentration-time) and pharmacodynamics
(response-concentration)
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yij,P K
=
fP K (tij,P K , ui , β i,P K ) + eij,P K
yij,P D
=
fP D { fP K (tij,P K , ui , β i,P K ), β i,P D } + eij,P D
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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, β i )
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
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