Digital Public Health Program Modeling Life Science - Dynamical Processes
Nonlinear Mixed-Effects Modeling (population approach)
Fabien Crauste Chargé de Recherches CNRS Institut de Mathématiques de Bordeaux UMR 5251 Université de Bordeaux
1. Motivation
2. Linear mixed-effects models
3. Nonlinear mixed-effects models
4. Identifiability
Motivation
MotivationsMain objective: reproducing data using a mathematical model
Math ModelData
MotivationsMain objective: reproducing data using a mathematical model
Math ModelData
What data?
?
?
MotivationsMain objective: reproducing data using a mathematical model
Math ModelData
What data?
?
?
What model?
?
?
MotivationsMain objective: reproducing data using a mathematical model
Math ModelData
What data?
?
?
What model?
?
?
How to compare them?
Collinson-Pautz et al. (2016) Crauste et al. (2015)C
D8+
T c
ell c
ount
s(lo
g sc
ale)
Days post-infection0 5 10 15 20 25 30 35 40 45 50
101
102
103
104
105
106
Time (days)
CD
8 T
cell
counts
Lm
Flu
VV
MotivationsLongitudinal data: mean +/- standard deviation
2 examples: Tumor growth (left) and immune cell counts (right), in various conditions ; mean and standard deviations
MotivationsLongitudinal data: individual dynamics
CD
8+ T
cel
ls(lo
g sc
ale)
Days after infection
Number of CD8+ T cells in Balbc mice
mean +/- s.d. and individual dynamics
Time (hours) after injection
War
farin
Con
cent
ratio
n (m
g/L)
Concentration of warfarin in 32 healthy patients
(1.5 mg/kg for each patient)
Bazzoli et al. (2014)
From Lavielle and Bleakley (2015)
1. Viral load of Hepatitis C (4 patients)
MotivationsLongitudinal data: individual dynamics
From Lavielle and Bleakley (2015)
2. Weight curves (4 rats)
1. Viral load of Hepatitis C (4 patients)
MotivationsLongitudinal data: individual dynamics
From Lavielle and Bleakley (2015)
2. Weight curves (4 rats)
1. Viral load of Hepatitis C (4 patients)
MotivationsLongitudinal data: individual dynamics
MotivationsHow to account for inter-individual variability?
Option 1: Use a model to characterize each individual’s behavior-> not enough data to estimate all parameter values -> identifiability issue -> impossibility to characterize individuals’ behaviors
MotivationsHow to account for inter-individual variability?
Option 1: Use a model to characterize each individual’s behavior-> not enough data to estimate all parameter values -> identifiability issue -> impossibility to characterize individuals’ behaviors
Option 2: Use a Population Approach and Mixed-Effects Models-> consider individuals belong to the same population -> use statistical model -> validated in PKPD works
MotivationsHow to account for inter-individual variability?
Option 1: Use a model to characterize each individual’s behavior-> not enough data to estimate all parameter values -> identifiability issue -> impossibility to characterize individuals’ behaviors
Option 2: Use a Population Approach and Mixed-Effects Models-> consider individuals belong to the same population -> use statistical model -> validated in PKPD works
MotivationsHow to account for inter-individual variability?
(Non)Linear (mixed-effects) models
(Non)Linear (mixed-effects) modelsLinear models
Assume a linear relationship between observations and explanatory variables
Methodology (simple): - pick up explanatory variables - estimate parameters using least-square method
(Non)Linear (mixed-effects) modelsLinear models
Assume a linear relationship between observations and explanatory variables
Methodology (simple): - pick up explanatory variables - estimate parameters using least-square method
But: a linear relationship is not always appropriate
(Non)Linear (mixed-effects) modelsLinear models
Assume a linear relationship between observations and explanatory variables
Methodology (simple): - pick up explanatory variables - estimate parameters using least-square method
But: a linear relationship is not always appropriate
d = 1
d = 3 d = 2
(Non)Linear (mixed-effects) modelsLinear models
Assume a linear relationship between observations and explanatory variables
Methodology (simple): - pick up explanatory variables - estimate parameters using least-square method
But: a linear relationship is not always appropriate
d = 1
d = 3 d = 2
Question: is weight a polynomial function of time?
Instead of assuming a linear relationship between observations and explanatory variables
(Non)Linear (mixed-effects) modelsNonlinear models
assume a nonlinear relationship between observations and explanatory variables:
where f is a structural model, defined by a set (vector) of parameter values and an explanatory variable .
Instead of assuming a linear relationship between observations and explanatory variables
(Non)Linear (mixed-effects) modelsNonlinear models
assume a nonlinear relationship between observations and explanatory variables:
where f is a structural model, defined by a set (vector) of parameter values and an explanatory variable .
In the case of rats weight, one can for instance assume
Instead of assuming a linear relationship between observations and explanatory variables
(Non)Linear (mixed-effects) modelsNonlinear models
assume a nonlinear relationship between observations and explanatory variables:
where f is a structural model, defined by a set (vector) of parameter values and an explanatory variable .
In the case of rats weight, one can for instance assume
Instead of assuming a linear relationship between observations and explanatory variables
(Non)Linear (mixed-effects) modelsNonlinear models
assume a nonlinear relationship between observations and explanatory variables:
where f is a structural model, defined by a set (vector) of parameter values and an explanatory variable .
In the case of rats weight, one can for instance assume
Assume a linear relationship between observations and explanatory variables for each individual:
(Non)Linear (mixed-effects) modelsLinear mixed-effects models
with
Assume a linear relationship between observations and explanatory variables for each individual:
(Non)Linear (mixed-effects) modelsLinear mixed-effects models
with
fixed effect
Assume a linear relationship between observations and explanatory variables for each individual:
(Non)Linear (mixed-effects) modelsLinear mixed-effects models
with
fixed effect random effect
Assume a linear relationship between observations and explanatory variables for each individual:
(Non)Linear (mixed-effects) modelsLinear mixed-effects models
with
fixed effect random effect
Random effects and residuals are assumed to be random variables, with mean = 0 and a variance to be estimated. Assuming they are normally distributed, then:
Then one needs relevant and appropriate methods to estimate parameter values… (I’ll come back to that)
Assume a linear relationship between observations and explanatory variables for each individual:
(Non)Linear (mixed-effects) modelsLinear mixed-effects models
with
fixed effect random effect
Random effects and residuals are assumed to be random variables, with mean = 0 and a variance to be estimated. Assuming they are normally distributed, then:
Then one needs relevant and appropriate methods to estimate parameter values… (I’ll come back to that)
Nonlinear mixed-effects models
- observed data of individual i
- explanatory variables
- vector of parameters, with
- the structural model, here a function of explanatory variables and a vector of parameter values
- the error model,
Nonlinear mixed-effects modelsStructural and error models
From the previous considerations, we can then assume
where
[constant]
[proportional]
[combined]
- observed data of individual i
- explanatory variables
- vector of parameters, with
- the structural model, here a function of explanatory variables and a vector of parameter values
- the error model,
Nonlinear mixed-effects modelsStructural and error models
From the previous considerations, we can then assume
where
[constant]
[proportional]
[combined]
- observed data of individual i
- explanatory variables
- vector of parameters, with
- the structural model, here a function of explanatory variables and a vector of parameter values
- the error model,
Nonlinear mixed-effects modelsStructural and error models
From the previous considerations, we can then assume
where
[constant]
[proportional]
[combined]
- observed data of individual i
- explanatory variables
- vector of parameters, with
- the structural model, here a function of explanatory variables and a vector of parameter values
- the error model,
Nonlinear mixed-effects modelsStructural and error models
From the previous considerations, we can then assume
where
[constant]
[proportional]
[combined]
- observed data of individual i
- explanatory variables
- vector of parameters, with
- the structural model, here a function of explanatory variables and a vector of parameter values
- the error model,
Nonlinear mixed-effects modelsStructural and error models
From the previous considerations, we can then assume
where
[constant]
[proportional]
[combined]
- observed data of individual i
- explanatory variables
- vector of parameters, with
- the structural model, here a function of explanatory variables and a vector of parameter values
- the error model,
Nonlinear mixed-effects modelsStructural and error models
From the previous considerations, we can then assume
where
[constant]
[proportional]
[combined]
- observed data of individual i
- explanatory variables
- vector of parameters, with
- the structural model, here a function of explanatory variables and a vector of parameter values
- the error model,
Nonlinear mixed-effects modelsStructural and error models
From the previous considerations, we can then assume
where
[constant]
[proportional]
[combined]
- observed data of individual i
- explanatory variables
- vector of parameters, with
- the structural model, here a function of explanatory variables and a vector of parameter values
- the error model,
Nonlinear mixed-effects modelsStructural and error models
From the previous considerations, we can then assume
where
[constant]
[proportional]
[combined]
- observed data of individual i
- explanatory variables
- vector of parameters, with
- the structural model, here a function of explanatory variables and a vector of parameter values
- the error model,
Nonlinear mixed-effects modelsStructural and error models
From the previous considerations, we can then assume
where
[constant]
[proportional]
[combined]
- observed data of individual i
- explanatory variables
- vector of parameters, with
- the structural model, here a function of explanatory variables and a vector of parameter values
- the error model,
Nonlinear mixed-effects modelsStructural and error models
From the previous considerations, we can then assume
where
[constant]
[proportional]
[combined]
- observed data of individual i
- explanatory variables
- vector of parameters, with
- the structural model, here a function of explanatory variables and a vector of parameter values
- the error model,
Nonlinear mixed-effects modelsStructural and error models
From the previous considerations, we can then assume
where
[constant]
[proportional]
[combined]
Nonlinear mixed-effects modelsStructural and error models: comments
Comments
Nonlinear mixed-effects modelsStructural and error models: comments
Comments
- observed data of individual i-> the number of observations may vary from one individual to another
Nonlinear mixed-effects modelsStructural and error models: comments
Comments
- observed data of individual i-> the number of observations may vary from one individual to another
- explanatory variables-> time is often an explanatory variable-> solutions of a dynamical system may also be explanatory variables
Nonlinear mixed-effects modelsStructural and error models: comments
Comments
- observed data of individual i-> the number of observations may vary from one individual to another
- explanatory variables-> time is often an explanatory variable-> solutions of a dynamical system may also be explanatory variables
- vector of parameters, with -> the law of random effects has to be chosen-> it may depend on one’s knowledge or on classical laws
Nonlinear mixed-effects modelsStructural and error models: comments
Comments
- observed data of individual i-> the number of observations may vary from one individual to another
- explanatory variables-> time is often an explanatory variable-> solutions of a dynamical system may also be explanatory variables
- vector of parameters, with -> the law of random effects has to be chosen-> it may depend on one’s knowledge or on classical laws
- the error model,
-> often, a constant error is used, -> then
How does the drug circulate in the body?
Warfarin injections in 32 patients
Nonlinear mixed-effects modelsAn example in PharmacoKinetics-PharmacoDynamics (PK-PD)
Time (hours) after injectionW
arfa
rin C
once
ntra
tion
(mg/
L)
How does it affect the patient?
PCA
(%)
Time (hours) after injectionBazzoli et al. (2014)
Consider a simple model (1 compartment)
Drug Main Compartment
dose F
absorption ka clearance kevolume V
Warfarin injections in 32 patients PK model
Concentration: , where
Nonlinear mixed-effects modelsAn example in PharmacoKinetics-PharmacoDynamics (PK-PD)
Bazzoli et al. (2014)
Consider a classical model: direct response, with inhibitory effect
Effect(Y )PCA synthesis
Rindegradation
kout
function(C(t))
Warfarin injections in 32 patients PD model
Effect of the drug on PCA
Nonlinear mixed-effects modelsAn example in PharmacoKinetics-PharmacoDynamics (PK-PD)
Bazzoli et al. (2014)
Warfarin injections in 32 patients PK-PD model
Effect of the drug on PCA:
Drug concentration:
2 quantities, 9 parameters, and non-identifiability (see for instance F and V)
Nonlinear mixed-effects modelsAn example in PharmacoKinetics-PharmacoDynamics (PK-PD)
Bazzoli et al. (2014)
Warfarin injections in 32 patients PK-PD model
The observation model is
Nonlinear mixed-effects modelsAn example in PharmacoKinetics-PharmacoDynamics (PK-PD)
Bazzoli et al. (2014)
The observation model is
2 types of data (PK and PD)
Nonlinear mixed-effects modelsAn example in PharmacoKinetics-PharmacoDynamics (PK-PD)
Bazzoli et al. (2014)
Warfarin injections in 32 patients PK-PD model
The observation model is
structural models
Nonlinear mixed-effects modelsAn example in PharmacoKinetics-PharmacoDynamics (PK-PD)
Bazzoli et al. (2014)
Warfarin injections in 32 patients PK-PD model
The observation model is
explanatory variable (time)
Nonlinear mixed-effects modelsAn example in PharmacoKinetics-PharmacoDynamics (PK-PD)
Bazzoli et al. (2014)
Warfarin injections in 32 patients PK-PD model
The observation model is
individual-dependent parameters
Nonlinear mixed-effects modelsAn example in PharmacoKinetics-PharmacoDynamics (PK-PD)
Bazzoli et al. (2014)
Warfarin injections in 32 patients PK-PD model
The observation model is
One parameter fixed within the population
Nonlinear mixed-effects modelsAn example in PharmacoKinetics-PharmacoDynamics (PK-PD)
Bazzoli et al. (2014)
Warfarin injections in 32 patients PK-PD model
The observation model is
combined error model
Nonlinear mixed-effects modelsAn example in PharmacoKinetics-PharmacoDynamics (PK-PD)
Bazzoli et al. (2014)
Warfarin injections in 32 patients PK-PD model
The observation model is
constant error model
Nonlinear mixed-effects modelsAn example in PharmacoKinetics-PharmacoDynamics (PK-PD)
Bazzoli et al. (2014)
Warfarin injections in 32 patients PK-PD model
Nonlinear mixed-effects modelsCovariates
Covariates (age, weight, group…) allow to account for specificities in the population that could explain, at least partially, the data:
Covariates are a priori individual-dependent.
Nonlinear mixed-effects modelsCovariates
Covariates (age, weight, group…) allow to account for specificities in the population that could explain, at least partially, the data:
Covariates are a priori individual-dependent.
Example: weight curves of rats
Nonlinear mixed-effects modelsCovariates
Covariates (age, weight, group…) allow to account for specificities in the population that could explain, at least partially, the data:
Covariates are a priori individual-dependent.
Example: weight curves of rats
males
Nonlinear mixed-effects modelsCovariates
Covariates (age, weight, group…) allow to account for specificities in the population that could explain, at least partially, the data:
Covariates are a priori individual-dependent.
Example: weight curves of rats
males females
Nonlinear mixed-effects modelsCovariates on the PK-PD example
Covariates (age, weight, group…) allow to account for specificities in the population that could explain, at least partially, the data:
Covariates are a priori individual-dependent.
covariate on all parameters?
Bazzoli et al. (2014)
Categorical covariates (age, weight, group…) allow to account for specificities in the population that could explain, at least partially, the data:
Covariates are a priori individual-dependent.
covariate on all parameters?(Wald test)
log(individual weights)
log(
k a)
log(
V)
log(
k e)
gender
Nonlinear mixed-effects modelsCovariates on the PK-PD example
Bazzoli et al. (2014)
Categorical covariates (age, weight, group…) allow to account for specificities in the population that could explain, at least partially, the data:
Covariates are a priori individual-dependent.
covariate on all parameters?(Wald test)
log(individual weights)
log(
k a)
log(
V)
log(
k e)
gender
Nonlinear mixed-effects modelsCovariates on the PK-PD example
Bazzoli et al. (2014)
Categorical covariates (age, weight, group…) allow to account for specificities in the population that could explain, at least partially, the data:
Covariates are a priori individual-dependent.
covariate on all parameters?(Wald test)
log(individual weights)
log(
k a)
log(
V)
log(
k e)
gender
Nonlinear mixed-effects modelsCovariates on the PK-PD example
Bazzoli et al. (2014)
Categorical covariates (age, weight, group…) allow to account for specificities in the population that could explain, at least partially, the data:
Covariates are a priori individual-dependent.
covariate on all parameters?Wald test
Nonlinear mixed-effects modelsCovariates on the PK-PD example
Bazzoli et al. (2014)
Nonlinear mixed-effects modelsEstimating parameter values
This approach aims at modeling the behavior of a population of individuals, hence the objective is to characterize the population behavior. This is done by estimating population parameters, using data of individual dynamics.
Statistical methods used to estimate parameter values are based on specific probability distributions: we denote by the set of population parameters (fixed effects, error parameters…) and by the likelihood of observations y with respect to
Nonlinear mixed-effects modelsEstimating parameter values
This approach aims at modeling the behavior of a population of individuals, hence the objective is to characterize the population behavior. This is done by estimating population parameters, using data of individual dynamics.
Statistical methods used to estimate parameter values are based on specific probability distributions: we denote by the set of population parameters (fixed effects, error parameters…) and by the likelihood of observations y with respect to
Frequentist approach: maximize the likelihood of observations with respect to parameters
Nonlinear mixed-effects modelsEstimating parameter values
This approach aims at modeling the behavior of a population of individuals, hence the objective is to characterize the population behavior. This is done by estimating population parameters, using data of individual dynamics.
Statistical methods used to estimate parameter values are based on specific probability distributions: we denote by the set of population parameters (fixed effects, error parameters…) and by the likelihood of observations y with respect to
Frequentist approach: maximize the likelihood of observations with respect to parameters
Bayesian approach: maximize the conditional probability of parameters with respect to observations
Nonlinear mixed-effects modelsEstimating parameter values
This approach aims at modeling the behavior of a population of individuals, hence the objective is to characterize the population behavior. This is done by estimating population parameters, using data of individual dynamics.
Statistical methods used to estimate parameter values are based on specific probability distributions: we denote by the set of population parameters (fixed effects, error parameters…) and by the likelihood of observations y with respect to
Frequentist approach: maximize the likelihood of observations with respect to parameters
Bayesian approach: maximize the conditional probability of parameters with respect to observations
Nonlinear mixed-effects modelsEstimating parameter values: likelihood maximization
- Model linearization: use an order 1 development of f, compute the integral, and obtain a ‘linearized’ likelihood
- Numerical integration: use a numerical integration of the integral, computationally very expensive
- Stochastic integration: use Monte-Carlo simulations instead of numerical integration of the integral, computationally very expensive
The first question is: how to compute the likelihood?
Nonlinear mixed-effects modelsEstimating parameter values: likelihood maximization
The second question is: how to estimate population parameters?
- Standard minimization algorithms: like the Newton-Ralphson algorithm…
- Specific algorithms: SAEM (stochastic approximation expectation maximization), a modified version of the EM algorithm, is adapted to NMEM, it:
๏ first computes the conditional expectancy of the likelihood, using a stochastic approximation,
๏ then maximizes this quantity with respect to the parameters.
Once population parameters have been estimated, individual parameters are estimated by computing the conditional distribution of
Nonlinear mixed-effects modelsEstimating parameter values: individual parameters
In the frequentist approach, one can focus on the maximum a posteriori (MAP) estimator, given by
and the conditional mean of is the mean of the conditional distribution of
Consequence: the more information the better for the estimation, so individuals with few information may tend to select the population parameter as an individual parameter…
Nonlinear mixed-effects modelsEstimating parameter values: individual parameters & shrinkage
We made the assumption
and when estimating parameter values we obtained values that should follow this law ; hence, we can compute the
Bazzoli et al. (2014)
Nonlinear mixed-effects modelsEstimating parameter values: individual parameters & shrinkage
We made the assumption
and when estimating parameter values we obtained values that should follow this law ; hence, we can compute the
Bazzoli et al. (2014)
PKPD(warfarin)example
75% 10%
Nonlinear mixed-effects modelsEstimating parameter values: individual parameters & shrinkage
We made the assumption
and when estimating parameter values we obtained values that should follow this law ; hence, we can compute the
Are probability laws correct?
Are data informative
enough?
Bazzoli et al. (2014)
PKPD(warfarin)example
75% 10%
Nonlinear mixed-effects modelsValidating model & assumptions
Nonlinear mixed-effects modelsValidating model & assumptions
The whole method is based on a set of assumptions/hypotheses:
Nonlinear mixed-effects modelsValidating model & assumptions
The whole method is based on a set of assumptions/hypotheses: The whole method is based on a set of assumptions/hypotheses:
- Structural model: which formalism? which interactions? how many parameters?
Nonlinear mixed-effects modelsValidating model & assumptions
The whole method is based on a set of assumptions/hypotheses: The whole method is based on a set of assumptions/hypotheses:
- Structural model: which formalism? which interactions? how many parameters?
The whole method is based on a set of assumptions/hypotheses:
- Structural model: which formalism? which interactions? how many parameters?
- Parameters: fixed effects? random effects? which probability law for random effects?
Nonlinear mixed-effects modelsValidating model & assumptions
The whole method is based on a set of assumptions/hypotheses: The whole method is based on a set of assumptions/hypotheses:
- Structural model: which formalism? which interactions? how many parameters?
The whole method is based on a set of assumptions/hypotheses:
- Structural model: which formalism? which interactions? how many parameters?
- Parameters: fixed effects? random effects? which probability law for random effects?
The whole method is based on a set of assumptions/hypotheses:
- Structural model: which formalism? which interactions? how many parameters?
- Parameters: fixed effects? random effects? which probability law for random effects?
- Error model: constant? proportional? combined? something else?
Nonlinear mixed-effects modelsValidating model & assumptions
The whole method is based on a set of assumptions/hypotheses: The whole method is based on a set of assumptions/hypotheses:
- Structural model: which formalism? which interactions? how many parameters?
The whole method is based on a set of assumptions/hypotheses:
- Structural model: which formalism? which interactions? how many parameters?
- Parameters: fixed effects? random effects? which probability law for random effects?
The whole method is based on a set of assumptions/hypotheses:
- Structural model: which formalism? which interactions? how many parameters?
- Parameters: fixed effects? random effects? which probability law for random effects?
- Error model: constant? proportional? combined? something else?
The whole method is based on a set of assumptions/hypotheses:
- Structural model: which formalism? which interactions? how many parameters?
- Parameters: fixed effects? random effects? which probability law for random effects?
- Error model: constant? proportional? combined? something else?
- Covariates: meaningful? meaningless?
Nonlinear mixed-effects modelsValidating model & assumptions
The whole method is based on a set of assumptions/hypotheses: The whole method is based on a set of assumptions/hypotheses:
- Structural model: which formalism? which interactions? how many parameters?
The whole method is based on a set of assumptions/hypotheses:
- Structural model: which formalism? which interactions? how many parameters?
- Parameters: fixed effects? random effects? which probability law for random effects?
The whole method is based on a set of assumptions/hypotheses:
- Structural model: which formalism? which interactions? how many parameters?
- Parameters: fixed effects? random effects? which probability law for random effects?
- Error model: constant? proportional? combined? something else?
The whole method is based on a set of assumptions/hypotheses:
- Structural model: which formalism? which interactions? how many parameters?
- Parameters: fixed effects? random effects? which probability law for random effects?
- Error model: constant? proportional? combined? something else?
- Covariates: meaningful? meaningless?
Need for diagnosis tools
Nonlinear mixed-effects modelsValidating model & assumptions: Residuals
Estimated values versus Theoretical values
- k
- Other ‘shrinkage’ measure:
Individual Weighted Residuals (IWRES) should be normally distributed, with mean 0 and variance 1, so the should be close to 0.
If not, then this may indicate that data are scarce or that too many parameters are used
Nonlinear mixed-effects modelsValidating model & assumptions: Distribution errors
Estimated values versus Theoretical values
Normalized prediction distribution
errors (npde)
[normally distributed]
PK-PD(warfarin)example
npde
npde
time (hours) Predicted population
PK
PD
Bazzoli et al. (2014)
Nonlinear mixed-effects modelsValidating model & assumptions: Visual Predictive Check
Estimated values versus Theoretical values
Visual predictive check (VPC)
[percentiles, based on simulations]
Time (hours) after injection
War
farin
Con
cent
ratio
n (m
g/L)
PK
PCA
(%)
Time (hours) after injection
PD
PK-PD(warfarin)example
Bazzoli et al. (2014)
Nonlinear mixed-effects modelsValidating model & assumptions: Observations vs Predictions
Estimated values versus Theoretical values
Observations versus
Predictions
PD
Population predictions Individual predictions
Obs
erve
d PC
A (%
)
PK
Obs
erve
d C
once
ntra
tion
PK-PD(warfarin)example
Bazzoli et al. (2014)
Nonlinear mixed-effects modelsTo conclude: software
Software Algorithms for estimating parameter values
Monolix SAEM
R nlme, nlmixr, saemix (SAEM), lme (Laplacian approximation)
NONMEM several (Laplacian approximation, SAEM, first-order method…) [but PK modeling only]
SAS NLMIXED (adaptive gaussian quadrature, first-order method)
WinBugs Bayesian algorithm
Identifiability
IdentifiabiltyWhat is ‘identifiability’?
Main equation:
Initial condition equation:
Observation:
- given inputs - set of parameters (values to be determined) - solution at time t, with the parameters
,
IdentifiabiltyWhat is ‘identifiability’?
Main equation:
Initial condition equation:
Observation:
- given inputs - set of parameters (values to be determined) - solution at time t, with the parameters
,
Definition. Let be given. The model is identifiable in if for all parameters such that outputs are equal then The model is said to be identifiable if it is identifiable in every parameter
Methods
IdentifiabiltyTheoretical approaches
Linear models Nonlinear models
Laplace transform Taylor series expansion
Taylor series expansions (of the output) Algebro-Differential elimination
State isomorphism theorem
We will consider the case when the model is either an algebraic equation or a system of ODEs
IdentifiabiltyTheoretical approaches: an illustrative example
Hemodynamics model (C. Audebert’s PhD thesis)
Model
IdentifiabiltyTheoretical approaches: an illustrative example
Hemodynamics model (C. Audebert’s PhD thesis)
Model
Time (s)
Obs
erva
tion
(ml/s
)
Observation Qin
IdentifiabiltyTheoretical approaches: an illustrative example
Hemodynamics model (C. Audebert’s PhD thesis)
Model
Time (s)
Obs
erva
tion
(ml/s
)
Observation QinEstimate
IdentifiabiltyTheoretical approaches: an illustrative example
Model
, with
IdentifiabiltyTheoretical approaches: an illustrative example
using Laplace transform,
so
Hence parameters are identifiable if
IdentifiabiltyTheoretical approaches: an illustrative example
IdentifiabiltyTheoretical approaches: an illustrative example
IdentifiabiltyTheoretical approaches: an illustrative example
IdentifiabiltyTheoretical approaches: an illustrative example
IdentifiabiltyTheoretical approaches: an illustrative example
IdentifiabiltyTheoretical approaches: an illustrative example
IdentifiabiltyTheoretical approaches: an illustrative example
IdentifiabiltyTheoretical approaches: an illustrative example
IdentifiabiltyTheoretical approaches
Yet, most of the time it is not possible to prove identifiability
and data are not always ‘ideal’
IdentifiabiltyStatistical approaches
Raue et al. (2009)
‘Given a model that sufficiently describes the measured data, it is important to infer how well model parameters are
determined by the amount and quality of experimental data’ (Raue et al., 2009)
IdentifiabiltyStatistical approaches
Raue et al. (2009)
Estimate parameters using likelihood maximizationParameter estimation
IdentifiabiltyStatistical approaches
Raue et al. (2009)
Estimate parameters using likelihood maximizationParameter estimation
Define confidence intervals of the estimate of a parameter
where is the Hessian matrix, and is the covariance matrix of
the parameter estimates
Confidence intervals
IdentifiabiltyStatistical approaches
Raue et al. (2009)
One can also define likelihood-based confidence intervals, as a confidence region:
Estimate parameters using likelihood maximizationParameter estimation
Define confidence intervals of the estimate of a parameter
where is the Hessian matrix, and is the covariance matrix of
the parameter estimates
Confidence intervals
IdentifiabiltyStructural identifiability
Raue et al. (2009)
Identifiability
Structural Identifiability
IdentifiabiltyStructural identifiability
Raue et al. (2009)
Identifiability
Structural Identifiability
identifiable structurally non-identifiable
IdentifiabiltyStatistical approaches: an example
Raue et al. (2009)
IdentifiabiltyStatistical approaches: an example
Raue et al. (2009)
IdentifiabiltyStatistical approaches: an example
Raue et al. (2009)
Identifiable
IdentifiabiltyStatistical approaches: an example
Raue et al. (2009)
IdentifiabiltyStatistical approaches: an example
Raue et al. (2009)
Structurallynon-identifiable
IdentifiabiltyStatistical approaches: an example
Raue et al. (2009)
IdentifiabiltyStatistical approaches: an example
Raue et al. (2009)
????
IdentifiabiltyPractical identifiability
Raue et al. (2009)
identifiable practically non-identifiable
IdentifiabiltyConclusions
Identifiability.Some theoretical methods available, but most of them are limited. (There are also methods investigating ‘local’ identifiability)
Structural non-identifiability.The - structural - model is not well fitted to the data. The number of parameters must be reduced (e.g. using correlations).
Practical non-identifiability.Data being ‘non-ideal’, non-identifiability can come either from the model or from the data (amount, quality). Based on profile likelihood.
Top Related