An Introduction to Nonlinear
Mixed Effects Models and PK/PD
Analysis
Marie Davidian
Department of Statistics
North Carolina State University
http://www.stat.ncsu.edu/∼davidian
1
Outline
1. Introduction
2. Pharmacokinetics and pharmacodynamics
3. Model formulation
4. Model interpretation and inferential objectives
5. Inferential approaches
6. Applications
7. Extensions
8. Discussion
2
Some references
Material in this webinar is drawn from:
Davidian, M. and Giltinan, D.M. (1995). Nonlinear Models for Repeated
Measurement Data. Chapman & Hall/CRC Press.
Davidian, M. and Giltinan, D.M. (2003). Nonlinear models for repeated
measurement data: An overview and update. Journal of Agricultural,
Biological, and Environmental Statistics 8, 387–419.
Davidian, M. (2009). Non-linear mixed-effects models. In Longitudinal Data
Analysis, G. Fitzmaurice, M. Davidian, G. Verbeke, and G. Molenberghs
(eds). Chapman & Hall/CRC Press, ch. 5, 107–141.
Shameless promotion:
3
Introduction
Common situation in the biosciences:
• A continuous outcome evolves over time (or other condition) within
individuals from a population of interest
• Scientific interest focuses on features or mechanisms that underlie
individual time trajectories of the outcome and how these vary
across the population
• A theoretical or empirical model for such individual profiles, typically
nonlinear in parameters that may be interpreted as representing such
features or mechanisms, is available
• Repeated measurements over time are available on each individual in
a sample drawn from the population
• Inference on the scientific questions of interest is to be made in the
context of the model and its parameters
4
Introduction
Nonlinear mixed effects model:
• Also known as the hierarchical nonlinear model
• A formal statistical framework for this situation
• Much statistical methodological research in the early 1990s
• Now widely accepted and used, with applications routinely reported
and commercial and free software available
• Extensions and methodological innovations are still ongoing
Objectives:
• Provide an introduction to the formulation, utility, and
implementation of nonlinear mixed models
• For definiteness, focus on pharmacokinetics and pharmacodynamics
as major application area
5
Pharmacokinetics and pharmacodynamics
Premise: Understanding what goes on between dose (administration)
and response can yield information on
• How best to choose doses at which to evaluate a drug
• Suitable dosing regimens to recommend to the population,
subpopulations of patients, and individual patients
• Labeling
Key concepts:
• Pharmacokinetics (PK ) – “what the body does to the drug”
• Pharmacodynamics (PD ) – “what the drug does to the body”
An outstanding overview: “Pharmacokinetics and
pharmacodynamics ,” by D.M. Giltinan, in Encyclopedia of Biostatistics,
2nd edition
6
Pharmacokinetics and pharmacodynamics
dose
....
PK
concentration
-
response
PD
-
....
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7
Pharmacokinetics and pharmacodynamics
Dosing regimen: Achieve therapeutic objective while minimizing
toxicity and difficulty of administration
• How much ? How often ? To whom ? Under what conditions ?
Information on this: Pharmacokinetics
• Study of how the drug moves through the body and the processes
that govern this movement
(Elimination = metabolism and excretion )
8
Pharmacokinetics and pharmacodynamics
Basic assumptions and principles:
• There is an “effect site ” where drug will have its effect
• Magnitudes of response and toxicity depend on drug concentration
at the effect site
• Drug cannot be placed directly at effect site, must move there
• Concentrations at the effect site are determined by ADME
• Concentrations must be kept high enough to produce a desirable
response, but low enough to avoid toxicity
=⇒ “Therapeutic window ”
• (Usually ) cannot measure concentration at effect site directly, but
can measure in blood/plasma/serum; reflect those at site
9
Pharmacokinetics and pharmacodynamics
Pharmacokinetics (PK): First part of the story
• Broad goal of PK analysis : Understand and characterize
intra-subject ADME processes of drug absorption , distribution ,
metabolism and excretion (elimination ) governing achieved drug
concentrations
• . . . and how these processes vary across subjects (inter-subject
variation )
10
Pharmacokinetics and pharmacodynamics
PK studies in humans: “Intensive studies ”
• Small number of subjects (often healthy volunteers )
• Frequent samples over time, often following single dose
• Usually early in drug development
• Useful for gaining initial information on “typical ” PK behavior in
humans and for identifying an appropriate PK model. . .
• Preclinical PK studies in animals are generally intensive studies
11
Pharmacokinetics and pharmacodynamics
PK studies in humans: “Population studies ”
• Large number of subjects (heterogeneous patients)
• Often in later stages of drug development or after a drug is in
routine use
• Haphazard , sparse sampling over time, multiple dosing intervals
• Extensive demographic and physiologic characteristics
• Useful for understanding associations between patient characteristics
and PK behavior =⇒ tailored dosing recommendations
12
Pharmacokinetics and pharmacodynamics
10
8
6
4
2
0
Theophylline Concentration (mg/L)
12
Theophylline study: 12 subjects, same oral dose (mg/kg)
0
5
10
15
Time (hours)
13
20
25
Pharmacokinetics and pharmacodynamics
Features:
• Intensive study
• Similarly shaped concentration-time profiles across subjects
• . . . but peak, rise, decay vary
• Attributable to inter-subject variation in underlying PK behavior
(absorption, distribution, elimination)
Standard: Represent the body by a simple system of compartments
• Gross simplification but extraordinarily useful. . .
14
Pharmacokinetics and pharmacodynamics
One-compartment model with first-order absorption, elimination:
-
oral dose D
ka
dA(t)
dt
dAa (t)
dt
-
A(t)
ke
= ka Aa (t) − ke A(t), A(0) = 0
= −ka Aa (t),
Aa (0) = D
F = bioavailability, Aa (t) = amount at absorption site
ka DF
A(t)
=
{exp(−ke t) − exp(−ka t)},
Concentration at t : m(t) =
V
V (ka − ke )
ke = Cl/V, V = “volume ” of compartment, Cl = clearance
15
Pharmacokinetics and pharmacodynamics
One-compartment model for theophylline:
• Single “blood compartment ” with fractional rates of absorption ka
and elimination ke
• Deterministic mathematical model
• Individual PK behavior characterized by PK parameters
θ = (ka , V, Cl)′
By-product:
• The PK model assumes PK processes are dose-independent
• =⇒ Knowledge of the values of θ = (ka , V, Cl)′ allows
determination of concentrations achieved at any time t under
different doses
• Can be used to develop dosing regimens
16
Pharmacokinetics and pharmacodynamics
Objectives of analysis:
• Estimate “typical ” values of θ = (ka , V, Cl)′ and how they vary in
the population of subjects based on the longitudinal concentration
data from the sample of 12 subjects
• =⇒ Must incorporate the (theoretical ) PK model in an appropriate
statistical model (somehow. . . )
17
Pharmacokinetics and pharmacodynamics
Quinidine population PK study: N = 136 patients undergoing
treatment with oral quinidine for atrial fibrillation or arrhythmia
• Demographic/physiologic characteristics : Age, weight, height,
ethnicity/race, smoking status, ethanol abuse, congestive heart
failure, creatinine clearance, α1 -acid glycoprotein concentration, . . .
• Samples taken over multiple dosing intervals =⇒
(dose time, amount) = (sℓ , Dℓ ) for the ℓth dose interval
• Standard assumption: “Principle of superposition ” =⇒ multiple
doses are “additive ”
• One compartment model gives expression for concentration
at time t. . .
18
Pharmacokinetics and pharmacodynamics
For a subject not yet at a steady state:
Aa (sℓ ) = Aa (sℓ−1 ) exp{−ka (sℓ − sℓ−1 )} + Dℓ ,
ka
m(sℓ ) = m(sℓ−1 ) exp{−ke (sℓ − sℓ−1 )} + Aa (sℓ−1 )
V (ka − ke )
h
i
× exp{−ke (sℓ − sℓ−1 )} − exp{−ka (sℓ − sℓ−1 )} .
m(t) =
ka
m(sℓ ) exp{−ke (t − sℓ )} + Aa (sℓ )
V (ka − ke )
h
i
× exp{−ke (t − sℓ )} − exp{−ka (t − sℓ )} , sℓ < t < sℓ+1
ke = Cl/V,
θ = (ka , V, Cl)′
Objective of analysis: Characterize typical values of and variation in
θ = (ka , V, Cl)′ across the population and elucidate systematic
associations between θ and patient characteristics
19
Pharmacokinetics and pharmacodynamics
Data for a representative subject:
time
(hours)
conc.
(mg/L)
dose
(mg)
age
(years)
weight
(kg)
creat.
(ml/min)
glyco.
(mg/dl)
0.00
6.00
11.00
17.00
23.00
27.67
29.00
35.00
41.00
47.00
53.00
65.00
71.00
77.00
161.00
168.75
–
–
–
–
–
0.7
–
–
–
–
–
–
–
0.4
–
0.6
166
166
166
166
166
–
166
166
166
166
166
166
166
–
166
–
75
75
75
75
75
75
75
75
75
75
75
75
75
75
75
75
108
108
108
108
108
108
108
108
108
108
108
108
108
108
108
108
> 50
> 50
> 50
> 50
> 50
> 50
> 50
> 50
> 50
> 50
> 50
> 50
> 50
> 50
> 50
> 50
69
69
69
69
69
69
94
94
94
94
94
94
94
94
88
88
height=72 inches, Caucasian, smoker, no ethanol abuse, no CHF
20
Pharmacokinetics and pharmacodynamics
Pharmacodynamics (PD): Second part of the story
• What is a “good ” drug concentration?
• What is the “therapeutic window ?” Is it wide or narrow ? Is it the
same for everyone ?
• Relationship of response to drug concentration
• PK/PD study : Collect both concentration and response data from
each subject
21
Pharmacokinetics and pharmacodynamics
Argatroban PK/PD study: Anticoagulant
• N = 37 subjects assigned to different constant infusion rates
(doses) of 1 to 5 µ/kg/min of argatroban
• Administered by intravenous infusion for 4 hours (240 min)
• PK (blood samples) at
(30,60,90,115,160,200,240,245,250,260,275,295,320) min
• PD : additional samples at 5–9 time points, measured activated
partial thromboplastin time (aPTT, the response)
Effect site: The blood
22
Pharmacokinetics and pharmacodynamics
1000
800
600
0
200
400
Argatroban Concentration (ng/ml)
800
600
400
200
0
Argatroban Concentration (ng/ml)
1000
1200
Infustion rate 4.5 µg/kg/min
1200
Infusion rate 1.0 µg/kg/min
0
100
200
300
0
Time (min)
100
200
Time (min)
23
300
Pharmacokinetics and pharmacodynamics
Argatroban PK model: One-compartment model with constant
intravenous infusion rate D (µg/kg/min) for duration tinf = 240 min
D
Cl
Cl
mP K (t) =
exp − (t − tinf )+ − exp − t , θ = (Cl, V )′
Cl
V
V
x+ = 0 if x ≤ 0 and x+ = x if x > 0
• PK parameters : θ P K = (Cl, V )′
• Estimate “typical ” values of θ P K = (Cl, V )′ and how they vary in
the population of subjects
• Understand relationship between achieved concentrations and
response (pharmacodynamics . . . )
24
0
0
400
600
800
1000
0
100
200
300
Time (minutes)
25
200
400
600
800
1000
Argatroban Concentration (ng/ml)
200
Argatroban Concentration (ng/ml)
1200
1200
Pharmacokinetics and pharmacodynamics
0
100
200
Time (minutes)
300
Pharmacokinetics and pharmacodynamics
Response-concentration for 4 subjects:
80
60
0
20
40
60
40
0
20
aPTT (seconds)
80
100
Subject 19
100
Subject 15
0
500
1000
1500
2000
2500
0
500
2000
2500
2000
2500
100
80
0
20
40
60
80
60
40
20
0
aPTT (seconds)
1500
Subject 33
100
Subject 28
1000
0
500
1000
1500
2000
2500
0
Predicted argatroban conc. (ng/ml)
500
1000
1500
Predicted argatroban conc. (ng/ml)
26
Pharmacokinetics and pharmacodynamics
Argatroban PD model: “Emax model ”
PD
m
Emax − E0
(t) = E0 +
1 + EC50 /mP K (t)
• Response at time t depends on concentration at effect site at time t
(same as concentration in blood here)
• PD parameters : θ P D = (E0 , Emax , EC50 )′
• Also depends on PK parameters
• Estimate “typical ” values of θ P D = (E0 , Emax , EC50 )′ and how
they vary in the population of subjects
27
Pharmacokinetics and pharmacodynamics
Ultimate objective: Put PK and PD together to
• Characterize the “therapeutic window ” and how it varies across
subjects
• Develop dosing regimens targeting achieved concentrations leading
to therapeutic response
• For population, subpopulations, individuals
• Decide on dose(s) to carry forward to future studies
28
Pharmacokinetics and pharmacodynamics
Summary: Common themes
• An outcome (or outcomes) evolves over time; e.g., concentration in
PK, response in PD
• Interest focuses on underlying mechanisms/processes taking place
within an individual leading to outcome trajectories and how these
vary across the population
• A (usually deterministic ) model is available representing
mechanisms explicitly by scientifically meaningful model parameters
• Mechanisms cannot be observed directly
• =⇒ Inference on mechanisms must be based on repeated
measurements of the outcome(s) over time on each of a sample of
N individuals from the population
29
Pharmacokinetics and pharmacodynamics
Other application areas:
• Toxicokinetics (Physiologically-based pharmacokinetic – PBPK –
models)
• HIV dynamics
• Stability testing
• Agriculture
• Forestry
• Dairy science
• Cancer dynamics
• More . . .
30
Model formulation
Nonlinear mixed effects model: Embed the (deterministic ) model
describing individual trajectories in a statistical model
• Formalizes knowledge and assumptions about variation in outcomes
and mechanisms within and among individuals
• Provides a framework for inference based on repeated measurement
data from N individuals
• For simplicity : Focus on univariate outcome (= drug concentration
in PK); multivariate (PK/PD ) outcome later
Basic set-up: N individuals from a population of interest, i = 1, . . . , N
• For individual i, observe ni measurements of the outcome
Yi1 , Yi2 , . . . , Yini
at times
ti1 , ti2 , . . . , tini
• I.e., for individual i, Yij at time tij , j = 1, . . . , ni
31
Model formulation
Within-individual conditions of observation: For individual i, U i
• Theophylline : U i = Di = oral dose for i at time 0 (mg/kg)
• Argatroban : U i = (Di , tinf ) = infusion rate and duration for i
• Quinidine : For subject i observed over di dosing intervals, U i has
elements (siℓ , Diℓ )′ , ℓ = 1, . . . , di
• U i are “within-individual covariates ” – needed to describe
outcome-time relationship at the individual level
32
Model formulation
Individual characteristics: For individual i, Ai
• Age, weight, ethnicity, smoking status, renal function, etc. . .
• For now : Elements of Ai do not change over observation period
• Ai are “among-individual covariates ” – relevant only to how
individuals differ but are not needed to describe outcome-time
relationship at the individual level
Observed data: (Y ′i , X ′i )′ , i = 1, . . . , N , assumed independent across i
• Y i = (Yi1 , . . . , Yini )′
• X i = (U ′i , A′i )′ = combined within- and among-individual
covariates (for brevity later)
Basic model: A two-stage hierarchy
33
Model formulation
Stage 1 – Individual-level model:
Yij = m(tij , U i , θi ) + eij , j = 1, . . . , ni , θ i (r × 1)
• E.g., for theophylline (F ≡ 1)
kai Di
{exp(−Cli t/Vi ) − exp(−kai t)}
m(t, U i , θi ) =
Vi (kai − Cli /Vi )
θ i = (kai , Vi , Cli )′ = (θi1 , θi2 , θi3 )′ , r = 3, U i = Di
• Assume eij = Yij − m(tij , U i , θ i ) satisfy E(eij | U i , θ i ) = 0
=⇒
E(Yij | U i , θ i ) = m(tij , U i , θ i ) for each j
• Standard assumption : eij and hence Yij are conditionally normally
distributed (on U i , θ i )
• More shortly. . .
34
Model formulation
Stage 2 – Population model:
θ i = d(Ai , β, bi ), i = 1, . . . , N, (r × 1)
• d is r-dimensional function describing relationship between θ i and
Ai in terms of . . .
• β (p × 1) fixed parameter (“fixed effects ”)
• bi (q × 1) “random effects ”
• Characterizes how elements of θ i vary across individual due to
– Systematic associations with Ai (modeled via β)
– “Unexplained variation ” in the population (represented by bi )
• Usual assumptions :
E(bi | Ai ) = E(bi ) = 0 and Cov(bi | Ai ) = Cov(bi ) = G, bi ∼ N (0, G)
35
Model formulation
Stage 2 – Population model:
θ i = d(Ai , β, bi ), i = 1, . . . , N
Example: Quinidine, θ i = (kai , Vi , Cli )′ (r = 3)
• Ai = (wi , δi , ai )′ , wi = weight, , ai = age,
δi = I(creatinine clearance > 50 ml/min)
• bi = (bi1 , bi2 , bi3 )′ (q = 3), β = (β1 , . . . , β7 )′ (p = 7)
kai
Vi
Cli
= θi1 = d1 (Ai , β, bi ) = exp(β1 + bi1 ),
= θi2 = d2 (Ai , β, bi ) = exp(β2 + β4 wi + bi2 ),
= θi3 = d3 (Ai , β, bi ) = exp(β3 + β5 wi + β6 δi + β7 ai + bi3 ),
• Positivity of kai , Vi , Cli enforced
• If bi ∼ N (0, G), kai , Vi , Cli are each lognormally distributed in the
population
36
Model formulation
Stage 2 – Population model:
θ i = d(Ai , β, bi ), i = 1, . . . , N
Example: Quinidine, continued, θ i = (kai , Vi , Cli )′ (r = 3)
• “Are elements of θ i fixed or random effects ?”
• “Unexplained variation ” in one component of θ i “small ” relative to
others – no associated random effect, e.g., r = 3, q = 2
kai
Vi
Cli
= exp(β1 + bi1 )
= exp(β2 + β4 wi ) (all population variation due to weight)
= exp(β3 + β5 wi + β6 δi + β7 ai + bi3 )
• An approximation – usually biologically implausible ; used for
parsimony, numerical stability
37
Model formulation
Stage 2 – Population model:
θ i = d(Ai , β, bi ), i = 1, . . . , N
• Allows nonlinear (in β and bi ) specifications for elements of θ i
• May be more appropriate than linear specifications (positivity
requirements, skewed distributions)
Some accounts: Restrict to linear specification
θ i = Ai β + Bi bi
• Ai (r × p) “design matrix ” depending on elements of Ai
• Bi (r × q) typically 0s and 1s (identity matrix if r = q)
• Mainly in the statistical literature
38
Model formulation
Stage 2 – Linear population model:
θ i = Ai β + Bi bi
Example: Quinidine, continued
∗
∗
= log(kai ),
, Vi∗ , Cli∗ )′ , kai
• Reparameterize in terms of θ i = (kai
Vi∗ = log(Vi ), and Cli∗ = log(Cli ) (r = 3)
∗
kai
= β1 + bi1 ,
Vi∗
= β2 + β4 wi + bi2 ,
Cli∗
= β3 + β5 wi + β6 δi + β7 ai + bi3



1 0 0 0
0 0 0
1





Ai =  0 1 0 wi 0 0 0  ,
Bi = 
 0
0 0 1 0 wi δi ai
0
39
0 0


1 0 

0 1
Model formulation
Stage 2 – Population model:
θ i = d(Ai , β, bi ), i = 1, . . . , N
Simplest example: Argatroban PK - No among-individual covariates
K
• θP
= (Cli , Vi )′ (r = 2), take
i
Cli
= exp(β1 + bi1 )
Vi
= exp(β2 + bi2 )
K
• Or reparameterize with θ P
= (Cli∗ , Vi∗ )′ , Cli∗ = log(Cli ),
i
Vi∗ = log(Vi )
Cli∗
= β1 + bi1
Vi∗
= β2 + bi2
so Ai = Bi = (2 × 2) identity matrix
40
Model formulation
Within-individual considerations: Complete the Stage 1
individual-level model
• Assumptions on the distribution of Y i given U i and θ i
• Focus on a single individual i observed under conditions U i
• Yij at times tij viewed as intermittent observations on a
stochastic process
Yi (t, U i ) = m(t, U i , θi ) + ei (t, U i )
E{ei (t, U i ) | U i , θ i } = 0, E{Yi (t, U i ) | U i , θ i } = m(t, U i , θ i ) for all t
• Yij = Yi (tij , U i ), eij = ei (tij , U i )
• “Deviation ” process ei (t, U i ) represents all sources of variation
acting within an individual causing a realization of Yi (t, U i ) to
deviate from the “smooth ” trajectory m(t, U i , θ i )
41
Model formulation
800
600
400
200
0
Concentration
1000
Conceptualization:
0
50
100
150
200
Time
42
250
300
350
Model formulation
Conceptual interpretation:
• Solid line : m(t, U i , θi ) represents “inherent tendency ” for i’s
outcome to evolve over time; depends on i’s “inherent
characteristics ” θ i
• Dashed line : Actual realization of the outcome – fluctuates about
solid line because m(t, U i , θi ) is a simplification of complex truth
• Symbols : Actual, intermittent measurements of the dashed line –
deviate from the dashed line due to measurement error
Result: Two sources of intra-individual variation
• “Realization deviation ”
• Measurement error variation
• m(t, U i , θ i ) is the average of all possible realizations of measured
outcome trajectory that could be observed on i
43
Model formulation
To formalize: ei (t, U i ) = eR,i (t, U i ) + eM,i (t, U i )
• Within-individual stochastic process
Yi (t, U i ) = m(t, U i , θ i ) + eR,i (t, U i ) + eM,i (t, U i )
E{eR,i (t, U i ) | U i , θi } = E{eM,i (t, U i ) | U i , θ i } = 0
• =⇒ Yij = Yi (tij , U i ), eR,i (tij , U i ) = eR,ij , eM,i (tij , U i ) = eM,ij
Yij = m(tij , U i , θi ) + eR,ij + eM,ij
|
{z
}
eij
eR,i = (eR,i1 , . . . , eR,ini )′ , eM,i = (eM,i1 , . . . , eM,ini )′
• eR,i (t, U i ) = “realization deviation process ”
• eM,i (t, U i ) = “measurement error deviation process ”
• Assumptions on eR,i (t, U i ) and eM,i (t, U i ) lead to a model for
Cov(ei | U i , θ i ) and hence Cov(Y i | U i , θ i )
44
Model formulation
800
600
400
200
0
Concentration
1000
Conceptualization:
0
50
100
150
200
Time
45
250
300
350
Model formulation
Realization deviation process:
• Natural to expect eR,i (t, U i ) and eR,i (s, U i ) at times t and s to be
positively correlated , e.g.,
corr{eR,i (t, U i ), eR,i (s, U i ) | U i , θ i } = exp(−ρ|t − s|), ρ ≥ 0
• Assume variation of realizations about m(t, U i , θi ) are of similar
magnitude over time and individuals, e.g.,
2
Var{eR,i (t, U i ) | U i , θ i } = σR
≥ 0 (constant for all t)
• Or assume variation depends on m(t, U i , θ i ), e.g.,
2
Var{eR,i (t, U i ) | U i , θ i } = σR
{m(t, U i , θ i )}2η , η > 0
• Result : Assumptions imply a covariance model (ni × ni )
2
2
Cov(eR,i | U )i , θ i ) = VR,i (U i , θ i , αR ), αR = (σR
, ρ)′ or αR = (σR
, ρ, η)′
46
Model formulation
800
600
400
200
0
Concentration
1000
Conceptualization:
0
50
100
150
200
Time
47
250
300
350
Model formulation
Measurement error deviation process:
• Measuring devices commit haphazard errors =⇒
corr{eM,i (t, U i ), eM,i (s, U i ) | U i , θi } = 0 for all t > s
• Assume magnitude of errors is similar regardless of level, e.g.,
2
Var{eM,i (t, U i ) | U i , θ i } = σM
≥ 0 (constant for all t)
• Or assume magnitude changes with level; often approximated under
assumption Var{eR,i (t, U i ) | U i , θ i } << Var{eM,i (t, U i ) | U i , θ i }
2
Var{eM,i (t, U i ) | U i , θ i } = σM
{m(t, U i , θ i )}2ζ , ζ > 0
• Result : Assumptions imply a covariance model (ni × ni )
(diagonal matrix )
2
2
Cov(eM,i | U )i , θ i ) = VM,i (U i , θ i , αR ), αM = σM
or αM = (σM
, ζ)′
48
Model formulation
Combining:
• Standard assumption : eR,i (t, U i ) and eM,i (t, U i ) are independent
Cov(ei | U i , θ i ) = Cov(eR,i | U i , θ i ) + Cov(eM,i | U i , θ i )
= VR,i (U i , θ i , αR ) + VM,i (U i , θ i , αM )
= Vi (U i , θ i , α)
α = (α′R , α′M )′
• This assumption may or may not be realistic
Practical considerations: Quite complex intra-individual covariance
models can result from faithful consideration of the situation. . .
• . . . But may be difficult to implement
49
Model formulation
Standard model simplifications: One or more might be adopted
• Negligible measurement error =⇒
Vi (U i , θ i , α) = VR,i (U i , θ i , αR )
• The tij may be at widely spaced intervals =⇒ autocorrelation
among eR,ij negligible =⇒ Vi (U i , θ i , α) is diagonal
• Var{eR,i (t, U i ) | U i , θ i } << Var{eM,i (t, U i ) | U i , θ i } =⇒
measurement error is dominant source
• Simplifications should be justifiable in the context at hand
Note: All of these considerations apply to any mixed effects model
formulation, not just nonlinear ones!
50
Model formulation
Routine assumption: Vi (U i , θ i , α) = σe2 Ini α = σe2
• Often made by “default ” with little consideration of the
assumptions it implies !
• Assumes autocorrelation among eR,ij negligible
2
• Assumes constant variances , i.e., Var{eR,i (t, U i ) | U i , θ i } = σR
2
2
2
and Var{eM,i (t, U i ) | U i , θ i } = σM
=⇒ σe2 = σR
+ σM
2
• If measurement error is negligible =⇒ σe2 = σR
• If Var{eR,i (t, U i ) | U i , θi } << Var{eM,i (t, U i ) | U i , θ i }
2
=⇒ σe2 ≈ σM
51
Model formulation
Standard assumptions in PK:
• Sampling times are sufficiently far apart that autocorrelation among
eR,ij negligible (not always justifiable!)
• Measurement error dominates realization error so that
Var(eR,ij | U i , θi ) << Var(eM,ij | U i , θ i )
(often reasonable )
• Measurement error variance depends on level, approximated by
2
Var(eM,ij | U i , θ i ) = σM
{m(tij , U i , θ i )}2ζ
so that Vi (U i , θ i , α) = VM,i (U i , θi , αM ) is diagonal with these
elements (almost always the case)
• Often, ζ ≈ 1
52
Model formulation
Distributional assumption:
• Specification for E(Y i | U i , θ i ) = mi (U i , θ i ),
mi (U i , θ i ) = {m(ti1 , U i , θi ), . . . , m(tini , U i , θ i )}′ (ni × 1)
• Specification for Cov(Y i |U i , θ i ) = Vi (U i , θ i , α)
• Standard assumption : Distribution of Y i given U i and θ i is
multivariate normal with these moments
• Alternatively, model on the log scale =⇒ Yij are conditionally (on
U i and θ i ) lognormal
• In what follows : Yij denotes the outcome on the original or
transformed scale as appropriate
53
Model formulation
Summary of the two-stage model: Recall X i = (U ′i , A′i )′
• Substitute population model for θ i in individual-level model
• Stage 1 – Individual-level model :
E(Y i | X i , bi ) =
Cov(Y i | X i , bi ) =
E(Y i | U i , θ i ) = mi (U i , θi ) = mi (X i , β, bi ),
Cov(Y i | U i , θi ) = Vi (U i , θi , α) = Vi (X i , β, bi , α)
• Stage 2 – Population model :
θ i = d(Ai , β, bi ),
• Standard assumptions :
bi ∼ (0, G)
– Y i given X i and bi multivariate normal (perhaps transformed )
– bi ∼ N (0, G)
– All of these can be relaxed
54
Model interpretation and inferential objectives
“Subject-specific” model: Individual behavior modeled explicitly at
Stage 1, depending on individual-specific parameters θ i that have
scientifically meaningful interpretation
• Models for E(Y i | U i , θ i ) and θ i , and hence E(Y i | X i , bi ),
specified . . .
• . . . in contrast to a “population-averaged ”model, where a model for
E(Y i | X i ) is specified directly
• This is consistent with the inferential objectives
• Interest is in “typical ” values of θ i and how they vary in the
population. . .
• . . . not in the “typical ” response pattern described by a
“population-averaged ”model for E(Y i | X i )
R
• =⇒ E(Y i | X i ) = E(Y i | X i , bi ) p(bi ; G) dbi
55
Model interpretation and inferential objectives
Main inferential objectives: May be formalized in terms of the model
• For a specific population model d, the fixed effect β characterizes
the mean or median (“typical ”) value of θ i in the population
(perhaps for individuals with given value of Ai )
• =⇒ Determining an appropriate population model d(Ai , β, bi ) and
inference on elements of β in it is of central interest
• Variation of θ i across individuals beyond that attributable to
systematic associations with among-individual covariates Ai is
described by G (“unexplained variation ”)
• =⇒ Inference on G is of interest (in particular, diagonal elements )
56
Model interpretation and inferential objectives
Additional inferential objectives: In some contexts
• Inference on θ i and/or m(t0 , U i , θ i ) at some specific time t0 for
i = 1, . . . , N or for future individuals is of interest
• Example : “Individualized ” dosing in PK
• The model is a natural framework for “borrowing strength ” across
similar individuals (more later)
57
Inferential approaches
Reminder – summary of the two-stage model: X i = (U ′i , A′i )′
• Stage 1 – Individual-level model :
E(Y i | X i , bi ) =
Cov(Y i | X i , bi ) =
E(Y i | U i , θ i ) = mi (U i , θi ) = mi (X i , β, bi ),
Cov(Y i | U i , θi ) = Vi (U i , θi , α) = Vi (X i , β, bi , α)
• Stage 2 – Population model :
θ i = d(Ai , β, bi ),
bi ∼ (0, G)
• Standard assumptions :
– Y i given X i and bi multivariate normal (perhaps transformed )
=⇒ probability density function fi (y i | xi , bi ; β, α)
– bi ∼ N (0, G) =⇒ density f (bi ; G)
• Observed data : {(Y i , X i ), i = 1, . . . , N } = (Y , X),
(Y i , X i ) assumed independent across i
58
Inferential approaches
Natural basis for inference on β, G: Maximum likelihood
• Joint density of Y given X (by independence )
f (y | x; γ, G) =
N
Y
i=1
fi (y i | xi ; γ, G), γ = (β ′ , α′ )′
• fi (y i , bi | xi ; γ, G) = fi (y i | xi , bi ; γ)f (bi ; G)
• Log-likelihood for (γ, G)
ℓ(γ, G) = log
(N
Y
i=1
= log
fi (y i | xi ; γ, G)
(N Z
Y
i=1
)
fi (y i | xi , bi ; γ) f (bi ; G) dbi
• Involves N q−dimensional integrals
59
)
Inferential approaches
ℓ(γ, G) = log
(N Z
Y
i=1
fi (y i | xi , bi ; γ) f (bi ; G) dbi
)
Major practical issue: These integrals are analytically intractable in
general and may be high-dimensional
• Some means of approximation of the integrals required
• Analytical approximation (the approach used historically , first by
pharmacokineticists) – will discuss first
• Numerical approximation (more recent, as computational resources
have improved)
60
Inferential approaches
Inference based on individual estimates: If ni ≥ r, can (in principle)
bi
obtain individual regression estimates θ
• E.g., if Vi (U i , θ i , α) = σe2 Ini can use ordinary least squares
for each i
• For fancier Vi (U i , θ i , α) can use generalized (weighted ) least
squares for each i with an estimate of α substituted
• α can be estimated by “pooling ” residuals across all N individuals
• Realistically : Require ni >> r
• Described in Chapter 5 of Davidian and Giltinan (1995)
bi , i = 1, . . . , N , as “data ” to estimate β and G. . .
Idea: Use the θ
61
Inferential approaches
bi , i = 1, . . . , N , as “data ” to estimate β and G
Idea: Use the θ
• Consider linear population model θ i = Ai β + Bi bi
• Standard large-ni asymptotic theory =⇒
·
bi | U i , θ i ∼
N (θ i , Ci ),
θ
Ci depends on θ i , α
·
bi | U i , θ i ∼
bi , α
bi ) and
b =⇒ θ
N (θ i , C
• Estimate Ci by substituting θ
bi as fixed
treat C
• Write as
·
bi ≈ θ i + e∗ , e∗ | U i , θ i ∼
bi )
N (0, C
θ
i
i
bi
• =⇒ Approximate “linear mixed effects model ” for “outcome ” θ
bi ≈ Ai β + Bi bi + e∗ ,
θ
i
bi ∼ N (0, G),
·
bi )
e∗i | U i , θ i ∼ N (0, C
• Can be fitted (estimate β, G) using standard linear mixed model
bi as fixed )
methods (treating C
62
Inferential approaches
bi ≈ Ai β + Bi bi + e∗ ,
θ
i
bi ∼ N (0, G),
Fitting the “linear mixed model”:
·
bi )
e∗i | U i , θ i ∼ N (0, C
• “Global two-stage algorithm ” (GTS ): Fit using the EM algorithm ;
see Davidian and Giltinan (1995, Chapter 5)
• Use standard linear mixed model software such as SAS proc
mixed , R function lme – requires some tweaking to handle the fact
bi is regarded as known
that C
• Appeal to usual large-N asymptotic theory for the “linear mixed
b confidence
model ” to obtain standard errors for elements of β,
intervals for elements of β, etc (generally works well )
Common misconception: This method is often portrayed in the
literature as having no relationship to the nonlinear mixed effects model
63
Inferential approaches
How does this approximate the integrals? Not readily apparent
bi as approximate “sufficient statistics ” for the θ i
• May view the θ
• Change of variables in the integrals and replace fi (y i | xi , bi ; γ) by
bi | U i , θ i ; α) corresponding to the
the (normal) density f (θ
asymptotic approximation
Remarks:
• When all ni are sufficiently large to justify the asymptotic
approximation (e.g., intensive PK studies), I like this method!
• Easy to explain to collaborators
• Gives similar answers to other analytical approximation methods
(coming up)
• Drawback : No standard software (although see my website for
R/SAS code)
64
Inferential approaches
In many settings: “Rich ” individual data not available for all i
(e.g., population PK studies); i.e., ni “not large ” for some or all i
• Approximate the integrals more directly by approximating
fi (y i | xi ; γ, G)
Write model with normality assumptions at both stages:
1/2
Y i = mi (X i , β, bi ) + Vi
1/2
• Vi
(X i , β, bi , α) ǫi , bi ∼ N (0, G)
1/2
(ni × ni ) such that Vi
1/2 ′
(Vi
) = Vi
• ǫi | X i , bi ∼ N (0, Ini ) (ni × 1)
• First-order Taylor series about bi = b∗i “close” to bi , ignoring
cross-product (bi − b∗i )ǫi as negligible =⇒
1/2
Y i ≈ mi (X i , β, b∗i )−Zi (X i , β, b∗i )b∗i +Zi (X i , β, b∗i )bi +Vi
(X i , β, b∗i , α) ǫi
Zi (X i , β, b∗i ) = ∂/∂bi {mi (X i , β, bi )}|bi = b∗i
65
Inferential approaches
1/2
Y i ≈ mi (X i , β, b∗i )−Zi (X i , β, b∗i )b∗i +Zi (X i , β, b∗i )bi +Vi
(X i , β, b∗i , α) ǫi
“First-order” method: Take b∗i = 0 (mean of bi )
• =⇒ Distribution of Y i given X i approximately normal with
E(Y i | X i ) ≈ mi (X i , β, 0),
Cov(Y i | X i ) ≈ Zi (X i , β, 0) G Zi′ (X i , β, 0) + Vi (X i , β, 0, α)
• =⇒ Approximate fi (y i | xi ; γ, G) by a normal density with these
moments, so that ℓ(γ, G) is in a closed form
• =⇒ Estimate (β, α, G) by maximum likelihood – because integrals
are eliminated, is a direct optimization (but still very messy. . . )
• First proposed by Beal and Sheiner in early 1980s in the context of
population PK
66
Inferential approaches
“First-order” method: Software
• fo method in the Fortran package nonmem (widely used by PKists)
• SAS proc nlmixed using the method=firo option
Alternative implementation: View as an approximate
“population-averaged ” model for mean and covariance
E(Y i | X i ) ≈ mi (X i , β, 0),
Cov(Y i | X i ) ≈ Zi (X i , β, 0) G Zi′ (X i , β, 0) + Vi (X i , β, 0, α)
• =⇒ Estimate (β, α, G) by solving a set of generalized estimating
equations (GEEs; specifically, “GEE-1 ”)
• Is a different method from maximum likelihood (“GEE-2 ”)
• Software : SAS macro nlinmix with expand=zero
67
Inferential approaches
Problem: These approximate moments are clearly poor approximations
to the true moments
• In particular, poor approximation to E(Y i | X i ) =⇒ biased
estimators for β
“First-order conditional methods”: Use a “better ” approximation
• Take b∗i “closer ”to bi
• Natural choice: b
bi = mode of the posterior density
fi (y i | xi , bi ; γ) f (bi ; G)
f (bi | y i , xi ; γ, G) =
fi (y i | xi ; γ, G)
• =⇒ Approximate moments
E(Y i | X i ) ≈ mi (X i , β, b
bi ) − Zi (X i , β, b
bi )b
bi
Cov(Y i | X i ) ≈ Zi (X i , β, b
bi ) G Z ′ (X i , β, b
bi ) + Vi (X i , β, b
bi , α)
i
68
Inferential approaches
Fitting algorithms: Iterate between
(i) Update b
bi , i = 1, . . . , N , by maximizing the posterior density (or
b substituted and held fixed
b and G
approximation to it) with γ
(ii) Hold the b
bi fixed and update estimation of γ and G by either
(a) Maximizing the approximate normal log-likelihood based on
treating Y i given X i as normal with these moments, OR
(b) Solving a corresponding set of GEEs
• Usually “converges ” (although no guarantee )
Software:
• nonmem with foce option implements (ii)(a)
• R function nlme, SAS macro nlinmix with expand=blup option
implement (ii)(b)
69
Inferential approaches
Standard errors, etc: For both “first-order ” approximations
• Pretend that the approximate moments are exact and use the usual
large-N asymptotic theory for maximum likelihood or GEEs
• Provides reliable inferences in problems where N is reasonably large
and the magnitude of among-individual variation is not huge
My experience:
• Even without the integration, these are nasty computational
problems , and good starting values for the parameters are required
(may have to try several sets of starting values).
• The “first-order ” approximation is too crude and should be avoided
in general (although can be a good way to get reasonable starting
values for other methods)
• The “first-order conditional ” methods often work well, are
numerically well-behaved , and yield reliable inferences
70
Inferential approaches
ℓ(γ, G) = log
(N Z
Y
i=1
fi (y i | xi , bi ; γ) f (bi ; G) dbi
)
Numerical approximation methods: Approximate the integrals using
deterministic or stochastic numerical integration techniques
(q-dimensional numerical integration ) and maximize the log-likelihood
• Issue : For each iteration of the likelihood optimization algorithm,
must approximate N q-dimensional integrals
• Infeasible until recently: Numerical integration embedded repeatedly
in an optimization routine is computationally intensive
• Gets worse with larger q (the “curse of dimensionality ”)
71
Inferential approaches
Deterministic techniques:
• Normality of bi =⇒ Gauss-Hermite quadrature
• Quadrature rule : Approximate an integral by a suitable weighted
average of the integrand evaluated at a q-dimensional grid of values
=⇒ accuracy increases with more grid points, but so does
computational burden
• Adaptive Gaussian quadrature : “Center ” and “scale ” the grid
about b
bi =⇒ can greatly reduce the number of grid points needed
Software: SAS proc nlmixed
• Adaptive Gaussian quadrature : The default
• Gaussian quadrature : method=gauss noad
72
Inferential approaches
ℓ(γ, G) = log
(N Z
Y
i=1
fi (y i | xi , bi ; γ) f (bi ; G) dbi
)
Stochastic techniques:
• “Brute force ” Monte Carlo integration: Represent integral for i by
B −1
B
X
b=1
fi (y i | xi , b(b) ; γ),
b(b) are draws from N (0, G) (at the current estimates of γ, G)
• Can require very large B for acceptable accuracy (inefficient )
• Importance sampling : Replace this by a suitably weighted version
that is more efficient
Software: SAS proc nlmixed implements importance sampling
(method=isamp)
73
Inferential approaches
My experience with SAS proc nlmixed:
• Good starting values are essential (may have to try many sets) –
starting values are required for all of β, G, α
• Could obtain starting values from an analytical approximation
method
• Practically speaking, quadrature is infeasible for q > 2 almost always
with the mechanism-based nonlinear models in PK and other
applications
74
Inferential approaches
Other methods: Maximize the log-likelihood via an EM algorithm
• For nonlinear mixed models , the conditional expectation in the
E-step is not available in a closed form
• Monte Carlo EM algorithm : Approximate the E-step by ordinary
Monte Carlo integration
• Stochastic approximation EM algorithm : Approximate the E-step by
Monte Carlo simulation and stochastic approximation
• Software : MONOLIX (http://www.monolix.org/)
75
Inferential approaches
Bayesian inference : Natural approach to hierarchical models
Big picture: In the Bayesian paradigm
• View β, α, G, and bi , i = 1, . . . , N , as random parameters (on
equal footing) with prior distributions (priors for bi , i = 1, . . . , N ,
are N (0, G))
• Bayesian inference on β and G is based on their posterior
distributions
• The posterior distributions involve high-dimensional integration and
cannot be derived analytically . . .
• . . . but samples from the posterior distributions can be obtained via
Markov chain Monte Carlo (MCMC)
76
Inferential approaches
Bayesian hierarchy:
• Stage 1 – Individual-level model : Assume normality
E(Y i | X i , bi ) =
Cov(Y i | X i , bi ) =
E(Y i | U i , θ i ) = mi (U i , θi ) = mi (X i , β, bi ),
Cov(Y i | U i , θi ) = Vi (U i , θi , α) = Vi (X i , β, bi , α)
• Stage 2 – Population model : θ i = d(Ai , β, bi ), bi ∼ N (0, G)
• Stage 3 – Hyperprior : (β, α, G) ∼ f (β, α, G) = f (β)f (α)g(G)
• Joint posterior density
f (γ, G, b | y, x) =
QN
i=1 fi (y i
| xi , bi ; γ) f (bi ; G)f (β, α, G)
;
f (y | x)
denominator is numerator integrated wrt (γ, G, bi , i = 1, . . . , N )
• E.g., posterior for β, f (β | y, x): Integrate out α, G, bi , i = 1, . . . , N
77
Inferential approaches
Estimator for β: Mode of posterior
• Uncertainty measured by spread of f (β | y, x)
• Similarly for α, G, and bi , i = 1, . . . , N
Implementation: By simulation via MCMC
• Samples from the full conditional distributions (eventually) behave
like samples from the posterior distributions
• The mode and measures of uncertainty may be calculated
empirically from these samples
• Issue : Sampling from some of the full conditionals is not entirely
straightforward because of nonlinearity of m in θ i and hence bi
• =⇒ “All-purpose ” software not available in general, but has been
implemented for popular m in add-ons to WinBUGS (e.g., PKBugs)
78
Inferential approaches
Experience:
• With weak hyperpriors and “good ” data, inferences are very similar
to those based on maximum likelihood and first-order conditional
methods
• Convergence of the chain must be monitored carefully; “false
convergence ” can happen
• Advantage of Bayesian framework : Natural mechanism to
incorporate known constraints and prior scientific knowledge
79
Inferential approaches
Inference on individuals: Follows naturally from a Bayesian perspective
• Goal : “Estimate ” bi or θ i for a randomly chosen individual i from
the population
• “Borrowing strength ”: Individuals sharing common characteristics
can enhance inference
• =⇒ Natural “estimator” is the mode of the posterior f (bi | y, x) or
f (θ i | y, x)
• Frequentist perspective : (γ, G) are fixed – relevant posterior is
f (bi | y i , xi ; γ, G) =
fi (y i | xi , bi ; γ) f (bi ; G)
fi (y i | xi ; γ, G)
=⇒ substitute estimates for (γ, G)
bi = d(Ai , β,
b b
• θ
bi )
• “Empirical Bayes ”
80
Inferential approaches
Selecting the population model d: Everything so far is predicated on
a fixed d(Ai , β, bi )
• A key objective in many analyses (e.g., population PK) is to identify
an appropriate d(Ai , β, bi )
• Must identify elements of Ai to include in each component of
d(Ai , β, bi ) and the functional form of each component
• Likelihood inference : Use nested hypothesis tests or information
criteria (AIC, BIC, etc)
• Challenging when Ai is high-dimensional. . .
• . . . Need a way of selecting among large number of variables and
functional forms in each component (still an open problem. . . )
81
Inferential approaches
Selecting the population model d: Continued
• Graphical methods : Based on Bayes or empirical Bayes “estimates”
– Fit an initial population model with no covariates (elements of
Ai and obtain B/EB estimates b
bi , i = 1 . . . , N
bi against elements of Ai , look for
– Plot components of b
relationships
– Postulate and fit an updated population model d incorporating
relationships and obtain updated B/EB estimates b
bi and re-plot
– If model is adequate, plots should show haphazard scatter ;
otherwise, repeat
– Issue 1 : “Shrinkage ” of B/EB estimates could obscure
relationships (especially if bi really aren’t normally distributed )
– Issue 2 : “One-at-a-time ” assessment of relationships could miss
important features
82
Inferential approaches
Normality of bi : The assumption bi ∼ N (0, G) is standard in mixed
effects model analysis; however
• Is it always realistic ?
• Unmeasured binary among-individual covariate systematically
associated with θ i =⇒ bi has bimodal distribution
• Or a normal distribution may just not be the best model!
(heavy tails , skewness. . . )
• Consequences ?
Relaxing the normality assumption: Represent the density of bi by a
flexible form
• Estimate the density along with the model parameters
• =⇒ Insight into possible omitted covariates
83
Applications
Example 1: A basic analysis – argatroban PK
• N = 37, different infusion rates Di for tinf = 240 min
• YijP K = concentrations at tij = 30,60,90,115,160,200,240,
245,250,260,275,295,320,360 min (ni = 14),
• One compartment model
Cl∗
Cl∗ i
e
Di
e i
PK
m (t, U i , θi ) = Cl∗ exp − V ∗ (t − tinf )+ − exp − V ∗ t
e i
e i
e i
K
θP
= (Cli∗ , Vi∗ )′ , U i = (Di , tinf )
i
x+ = 0 if x ≤ 0 and x+ = x if x > 0
• Cli∗ = log(Cli ), Vi∗ = log(Vi )
• No among-individual covariates Ai
84
Applications
Profiles for subjects receiving 1.0 and 4.5 µg/kg-min:
1000
800
600
0
200
400
Argatroban Concentration (ng/ml)
800
600
400
200
0
Argatroban Concentration (ng/ml)
1000
1200
Infustion rate 4.5 µg/kg/min
1200
Infusion rate 1.0 µg/kg/min
0
100
200
300
0
Time (min)
100
200
Time (min)
85
300
Applications
Nonlinear mixed model:
• Stage 1 – Individual-level model : Yij normal with
E(Yij | U i , θi ) = mP K (tij , U i , θ i )
Cov(Y i | U i , Ai ) = Vi (U i , θ i , α) = σe2 diag{. . . , mP K (tij , U i , θi )2ζ , . . .}
=⇒ negligible autocorrelation , measurement error dominates
• Stage 2 – Population model
θ i = β + bi ,
β = (β1 , β2 )′ ,
bi ∼ N (0, G)
=⇒ β1 , β2 represent population means of log clearance, volume;
equivalently, exp(β1 ), exp(β2 ) are population medians
√
√
=⇒ G11 , G22 ≈ coefficients of variation of clearance, volume
86
Applications
Implementation: Using
bi found using “pooled ” generalized least
• Individual estimates θ
squares including estimation of ζ (customized R code) followed by
fitting the “linear mixed model ” (SAS proc mixed)
• First-order method via version 8.01 of SAS macro nlinmix with
expand=zero – fix ζ = 0.22 (estimate from above)
• First-order conditional method via version 8.01 of SAS macro
nlinmix with expand=eblup – fix ζ = 0.22
• First-order conditional method via R function nlme (estimate ζ)
• Maximum likelihood via SAS proc nlmixed with adaptive Gaussian
quadrature (estimate ζ)
Code: Can be found at
http://www.stat.ncsu.edu/people/davidian/courses/st762/
87
Applications
Method
β1
β2
σe
ζ
G11
Indiv. est.
−5.433
(0.062)
−1.927
(0.026)
23.47
0.22
0.137
6.06
6.17
First-order
nlinmix
−5.490
(0.066)
−1.828
(0.034)
26.45
–
0.158
−3.08
16.76
First-order cond.
nlinmix
−5.432
(0.062)
−1.926
(0.026)
23.43
–
0.138
5.67
4.76
First-order cond.
nlme
−5.433
(0.063)
−1.918
(0.025)
20.42
0.24
0.138
6.73
4.56
ML
nlmixed
−5.423
(0.064)
−1.897
(0.038)
11.15
0.34
0.150
13.71
11.26
Values for G12 , G22 are multiplied by 103
88
G12
G22
Applications
Interpretation: Based on the individual estimates results
• Concentrations measured in ng/ml = 1000 µg/ml
• Median argatroban clearance ≈ 4.4 µg/ml/kg
(≈ exp(−5.43) × 1000)
• Median argatroban volume ≈ 145.1 ml/kg =⇒ ≈ 10 liters for
a 70 kg subject
• Assuming Cli , Vi approximately lognormal
√
– G11 ≈ 0.14× 100 ≈ 37% coefficient of variation for clearance
√
– G22 ≈ 0.00617× 100 ≈ 8% CV for volume
89
Applications
1200
1000
800
600
400
0
200
Argatroban Concentration (ng/ml)
1000
800
600
400
200
0
Argatroban Concentration (ng/ml)
1200
Individual inference: Individual estimate (dashed) and empirical Bayes
estimate (solid)
0
100
200
300
0
Time (minutes)
100
200
Time (minutes)
90
300
Applications
Example 2: A simple population PK study analysis: phenobarbital
• World-famous example
• N = 59 preterm infants treated with phenobarbital for seizures
• ni = 1 to 6 concentration measurements per infant, total of 155
• Among-infant covariates (Ai ): Birth weight wi (kg), 5-minute
Apgar score δi = I[Apgar < 5]
• Multiple intravenous doses : U i = (siℓ , Diℓ ), ℓ = 1, . . . , di
• One-compartment model (principle of superposition )
X Diℓ
Cli
m(t, U i , θ i ) =
exp −
(t − siℓ )
Vi
Vi
ℓ:siℓ <t
• Objectives : Characterize PK and its variation – Mean/median Cli ,
Vi ? Systematic associations with among-infant covariates ? Extent
of unexplained variation ?
91
Applications
0
Phenobarbital conc. (mcg/ml)
20
40
60
Dosing history and concentrations for one infant:
0
50
100
150
200
Time (hours)
92
250
300
Applications
Nonlinear mixed model:
• Stage 1 – Individual-level model
Cov(Y i | U i , Ai ) = Vi (U i , θi , α) = σe2 Ini
E(Yij | U i , θ i ) = m(tij , U i , θ i ),
=⇒ negligible autocorrelation , measurement error dominates and
has constant variance
• Stage 2 – Population model
– Without among-infant covariates Ai
log Cli = β1 + bi1 ,
log Vi = β2 + bi2
– With among-infant covariates Ai
log Cli = β1 + β3 wi + bi1 ,
93
log Vi = β2 + β4 wi + β5 δi + bi2
Applications
.
.
.
.
.
.
.
.
.
1.0
1.5 2.0 2.5
Birth weight
3.0
3.5
.
.
..
. . ... .
. .. ..
. . . . .. . .
. .
.
.
. . .. .
.. ..
.
.. .
0.5
1.0
.
.
1.5 2.0 2.5
Birth weight
.
3.0
Volume random effect
-0.5 0.0 0.5 1.0
Clearance random effect
-0.5 0.0 0.5 1.0 1.5
0.5
. . .. . .. . .
.
. . .. .. . .. . .
.
.. .. ... .. .
.
.
.
... .
.
Volume random effect
-0.5 0.0 0.5 1.0
Clearance random effect
-0.5 0.0 0.5 1.0 1.5
Empirical Bayes estimates vs. covariates: Fit without
Apgar<5
Apgar>=5
Apgar<5
Apgar score
Apgar>=5
Apgar score
94
.
3.5
Applications
.
.. ..
. . . ...
. . . .. .. .. .. ... . .. . .
. . ... . . . . .
.
.
.
..
.
.
.
1.0
1.5 2.0 2.5
Birth weight
3.0
3.5
.
.
.. .
.
.
.
. . .. .
.
..
. . .. . . .
. .. .
.
. .. ..
..
. . . .. .
.
0.5
.
1.0
.
.
.
.
.
.
1.5 2.0 2.5
Birth weight
3.0
Volume random effect
-0.2
0.0 0.1 0.2 0.3
Clearance random effect
-0.5 0.0 0.5
0.5
.
Volume random effect
-0.2
0.0 0.1 0.2 0.3
Clearance random effect
-0.5 0.0 0.5
Empirical Bayes estimates vs. covariates: Fit with
Apgar<5
Apgar>=5
Apgar<5
Apgar score
Apgar>=5
Apgar score
95
3.5
Applications
Relaxing the normality assumption on bi : Represent the density of bi
by a flexible form , fit by maximum likelihood
(b)
0 2
4
h
6
h
0 1 2 3 4 5 6
8 10
(a)
0 .4
0.6
0 .2
Vo 0
lum -0
e .2
0.5
-0
.4
-0.5
0 ce
n
ara
0.4
0.2
Vo 0
lum-0
e .2
0.5
-0
Cle
0.2
Volume
-0.6 -0.4 -0.2 0.0
Volume
0.5
-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6
0.4
0.6
(d)
.
. ...
..
... . .. .
. ............. .
.
.
. . ..... .. ..
.. .
0.0
-0.5
-0
.6
(c)
-0.5
.4
0 ce
n
ara
Cle
.. .
. ............. ..
.. . .
. . ..........
.. .
-0.5
Clearance
0.0
Clearance
96
.
. . .. .
.
.
0.5
Extensions
Multivariate outcome: More than one type of outcome measured
longitudinally on each individual
• Objectives: Understand the relationships between the outcome
trajectories and the processes underlying them
• Key example : Pharmacokinetics/pharmacodynamics (PK/PD) as in
argatroban
97
Extensions
Required: A joint model for PK and PD
• Data :
K
– YijP K at times tP
(PK concentrations)
ij
D
(PD aPTT responses)
– YijP D at times tP
ij
K
• PK model with θ P
= (Cli∗ , Vi∗ )′ , U i = (Di , tinf )
i
Cl∗ Cl∗
i
e i
D
e
i
PK
PK
− exp − V ∗ t
m (t, U i , θ i ) = Cl∗ exp
∗ (t − tinf )+
V
i
i
e
e i
e
D
∗
∗
∗
• PD model with θ P
= (E0i
, Emax,i
, EC50,i
)′
i
∗
D
mP D (conc, θ P
)=e
i
∗
E0i
98
eEmax,i − eE0i
+
∗
EC50,i
1+e
/conc
∗
Extensions
Concentration-PD response relationship:
80
60
0
20
40
60
40
0
20
aPTT (seconds)
80
100
Subject 19
100
Subject 15
0
500
1000
1500
2000
2500
0
500
2000
2500
2000
2500
100
80
0
20
40
60
80
60
40
20
0
aPTT (seconds)
1500
Subject 33
100
Subject 28
1000
0
500
1000
1500
2000
2500
Predicted argatroban conc. (ng/ml)
0
500
1000
1500
Predicted argatroban conc. (ng/ml)
99
Extensions
Result: Assuming measurement error dominates realization variation, so
K
)
“true ” PK concentration for i at t ≈ m(t, U i , θ P
i
• Stage 1 – Individual-level model
YijP K
PK
K
K
= mP K (tP
) + eP
ij , U i , θ i
ij
YijP D
PK
D
D
D
= mP D {mP K (tP
), θ P
} + eP
ij , U i , θ i
i
ij
PD
K
mutually independent (primarily measurement error )
• eP
ij , eij
100
Extensions
K′
D′ ′
Full model: Combined outcomes Y i = (Y P
,Y P
)
i
i
K′
P D′ ′
∗
∗
∗
∗
∗ ′
θ i = (θ P
,
θ
)
=
(Cl
,
V
,
E
,
E
,
EC
i
i
i
i
0i
max,i
50i )
• Stage 1 – Individual-level model
PK
K
E(YijP K |U i , θ i ) = mP K (tP
,
U
,
θ
)
i
ij
i
PK
PD
D
E(YijP D |U i , θ i ) = mP D {mP K (tP
,
U
,
θ
),
θ
}
i
ij
i
i
Cov(Y i |U i , θ i ) = block diag{ViP K (U i , θ i , αP K ), ViP D (U i , θ i , αP D )}
PK
P K 2ζ
2
PK PK
ViP K (U i , θ i , αP K ) = σe,P
diag{.
.
.
,
m
(t
,
U
,
θ
)
, . . .}
i
K
ij
i
h
i
PD
P
K
P
D
2
PD
D
ViP D (U i , θ i , αP D ) = σe,P
{mP K (tP
), θ i }2ζ , . . .
D diag . . . , m
ij , U i , θ i
• Stage 2 – Population model
θ i = β + bi ,
β = (β1 , . . . , β5 )′ ,
101
bi ∼ N (0, G)
Extensions
Implementation:
PK
b
• Sequentially : Fit PK model =⇒ individual estimates θ i and
bP K ) at the PD sampling
predicted concentrations mP K (tP D , U i , θ
ij
i
times
• Sequentially : Substitute in PD model and treat as known , fit PD
model
• Jointly : Fit full model directly
102
Extensions
Fits: Using nlme
β1
β2
β3
β4
β5
(σeP K , ζ P K )
(σeP D , ζ P D )
G11
G22
G33
G44
G55
Sequential
Joint
−5.433 (0.062)
−1.918 (0.025)
3.361 (0.018)
4.410 (0.045)
6.295 (0.150)
(20.42,0.24)
(0.164,0.80)
0.138
4.56
8.16
0.025
0.377
−5.439 (0.061)
−1.894 (0.024)
3.360 (0.019)
4.427 (0.045)
6.362 (0.155)
(16.66,0.28)
(0.154,0.82)
0.133
5.51
8.51
0.026
0.425
Values for G22 , G33 are multiplied by 103
103
Extensions
Time-dependent among-individual covariates: Among-individual
covariates change over time within an individual
• In principle , one could write θ ij for each tij ; however . . .
• Key issue : Does this make scientific sense ?
• PK : Do pharmacokinetic processes vary within an individual?
Example: Quinidine study
• Creatinine clearance, α1 -acid glycoprotein concentration, etc,
change over dosing intervals
• How to incorporate dependence of Cli , Vi on α1 -acid glycoprotein
concentration?
104
Extensions
Data for a representative subject:
time
(hours)
conc.
(mg/L)
dose
(mg)
age
(years)
weight
(kg)
creat.
(ml/min)
glyco.
(mg/dl)
0.00
6.00
11.00
17.00
23.00
27.67
29.00
35.00
41.00
47.00
53.00
65.00
71.00
77.00
161.00
168.75
–
–
–
–
–
0.7
–
–
–
–
–
–
–
0.4
–
0.6
166
166
166
166
166
–
166
166
166
166
166
166
166
–
166
–
75
75
75
75
75
75
75
75
75
75
75
75
75
75
75
75
108
108
108
108
108
108
108
108
108
108
108
108
108
108
108
108
> 50
> 50
> 50
> 50
> 50
> 50
> 50
> 50
> 50
> 50
> 50
> 50
> 50
> 50
> 50
> 50
69
69
69
69
69
69
94
94
94
94
94
94
94
94
88
88
height=72 inches, Caucasian, smoker, no ethanol abuse, no CHF
105
Extensions
Population model: Standard approach in PK
• For subject i: α1 -acid glycoprotein concentration likely measured
intermittently at times 0, 29, 161 hours and assumed constant over
the intervals (0,29), (29,77), (161,·) hours
• For intervals Ik , k = 1, . . . , a (a = 3 here), Aik = among-individual
covariates for tij ∈ Ik =⇒ e.g., linear model
θ ij = Aik β + bi
• This population model assumes “within subject inter-interval
variation ” entirely “explained ” by changes in covariate values
• Alternatively : Nested random effects
θ ij = Aik β + bi + bik , bi , bik independent
106
Extensions
Multi-level models: More generally
• Nesting : E.g., outcomes Yikj , j = 1, . . . , nik , on several trees
(k = 1, . . . , vi ) within each of several plots (i = 1, . . . , N )
θ ik = Aik β + bi + bik , bi , bik independent
Missing/mismeasured covariates: Ai , U i , tij
Censored outcome: E.g., due to an assay quantification limit
Semiparametric models: Allow m(t, U i , θ i ) to depend on an
unspecified function g(t, θ i )
• Flexibility , model misspecification
Clinical trial simulation: “Virtual ” subjects simulated from a nonlinear
mixed effects model for PK/PD/disease progression linked to a clinical
end-point
107
Discussion
Summary:
• The nonlinear mixed effects model is now a standard statistical
framework in many areas of application
• Is appropriate when scientific interest focuses on within-individual
mechanisms/processes that can be represented by parameters in a
nonlinear (often theoretical ) model for individual time course
• Free and commercial software is available, but implementation is
still complicated
• Specification of models and assumptions, particularly the population
model , is somewhat an art-form
• Current challenge : High-dimensional Ai (e.g., genomic information)
• Still plenty of methodological research to do
108
Discussion
See the references on slide 3 for an extensive bibliography
109