nonlinear mixed effectsdavidian/nlmmtalk.pdfnonlinear mixed efiects model: aka hierarchical...
TRANSCRIPT
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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
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