Introduction to
Mixed-Effects modelling
Eugenio Cinquemani
INSA, December 14, 2016
Cell-to-cell variability
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Example: gene response to osmotic shocks in yeast
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Similar but quantitatively different expression pattern
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How to model and recover individual cell “identities”?
Mixed-effects modelling: Origins
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Pharmacokinetics
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Drug absorption
in human body
Davidian, Giltinan, J Agric Biol Env Stat, 2013
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Immunology
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HIV dynamics
Davidian, Giltinan, J Agric Biol Env Stat, 2013
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Dairy science
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Forestry
…
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Microbiology
Scenario
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Responses similar across individuals of a same
population
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Inter-individual variability
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Few observations per individual
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Measurement error
Objectives
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Modelling framework accounting for:
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Characters of the population (typical response, variability of the
response across individuals)
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Specific features of the observed individuals
Inference of the characteristics of a population from
observations of a few individuals
Prediction of the response of new individuals from the same
population
Hierarchical modelling
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Population model: Typical response of an individual
y=f(φ,x)+ε
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x “predictor” value
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φ individual parameters
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ε intra-individual variability (modelling/observation “error”)
Individual model: Law underlying parameter values of the
different individuals i
φi ~ p ( θ i , Θ )
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θi individual “covariates”
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Θ population parameters
Mixed-Effects (ME) models (linear)
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φi composed of fixed effects and random effects
φi = Ai · β + Bi · bi ,
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β fixed effects (population feature)
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bi random effects (individual outcome)
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Ψ inter-individual variability (population feature)
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Ai , Bi individual covariates
Different observations j from different individuals i
yi,j = f ( φi , xi,j ) + εi,j ,
–
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bi ~ N ( 0 , Ψ )
i.i.d. εi,j ~ N ( 0 , σ2 )
σ2 error variance
Nonlinear mixed-effects models: Various generalizations
Example
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Drug absorption in human body
Aim: Infer a Mixed-Effects model to predict optimal repeated drug
delivery to (new) patients (as a function of weight, age, … )
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Nonlinear ME model:
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Linear ME model (approximation):
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Individual covariates:
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Weight wi
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Renal efficiency ci
Inference
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Aim: Given data ( xi,j , yi,j )V i,j estimate:
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Population parameters β, Ψ (and σ2)
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Individual parameters φi (i.e. random effects bi)
Naive approach (case φi = β + bi):
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Separately for every i, estimate φi from ( xi,j , yi,j )V j
^φi =
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arg minφ ∑j [ yi,j – f ( φ , xi,j ) ]2
Compute ^β=mean( ^φ• ), ^Ψ=Var( ^φ• )
Ignores the fact that random effects come from a common
distribution
ME (empirical Bayes) approach
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Exploits the fact that random effects come from a
common distribution
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Use all data to estimate population parameters first
^β, ^Ψ, ^σ
= arg max β, Ψ, σ p(y•,• | x•,• , β, Ψ, σ2 )
where the marginal likelihood can be expanded as
∏i,j ∫ dφi p(yi,j | xi,j , φi , σ2) p(φi | β ,Ψ ; Ai , Bi )
population model
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individual model
Then, if of interest, estimate individual effects
^φi =
arg max p(φi | ^β ,^Ψ ; Ai , Bi )
Software
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Monolix (dedicated, free for academic use)
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R commands
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Matlab commands
nlmefit, nlmefitsa
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...