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Bayesian analysis for heavy-tailed nonlinear mixed effects models Cibele M. Russo in collaboration with Danilo A. Silva Universidade de S ao Paulo, Brazil May 2013 Bayes 2013, Rotterdam Cibele Russo (USP, Brazil) Bayesian analysis for

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Bayesian analysis for heavy-tailed nonlinear mixed effects models

Cibele M. Russo

in collaboration with Danilo A. Silva

Universidade de S˜ ao Paulo, Brazil May 2013 Bayes 2013, Rotterdam

Cibele Russo (USP, Brazil) Bayesian analysis for nonlinear models May 2013 1 / 22

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Outline

Introduction Motivating example Nonlinear mixed effects models Heavy-tailed nonlinear mixed effects models Application Discussion and remarks Work in progress Bibliography

Cibele Russo (USP, Brazil) Bayesian analysis for nonlinear models May 2013 2 / 22

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Introduction

Nonlinear mixed effects models are suitable for longitudinal data or growth curves, specially in pharmacokinetics.

Cibele Russo (USP, Brazil) Bayesian analysis for nonlinear models May 2013 3 / 22

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Introduction

Nonlinear mixed effects models are suitable for longitudinal data or growth curves, specially in pharmacokinetics. Random effects and errors are usually assumed to be independent and normally distributed.

Cibele Russo (USP, Brazil) Bayesian analysis for nonlinear models May 2013 3 / 22

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Introduction

Nonlinear mixed effects models are suitable for longitudinal data or growth curves, specially in pharmacokinetics. Random effects and errors are usually assumed to be independent and normally distributed. We assume that the random effects and errors jointly follow heavy tailed distributions for the random effects and errors.

Cibele Russo (USP, Brazil) Bayesian analysis for nonlinear models May 2013 3 / 22

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Introduction

Nonlinear mixed effects models are suitable for longitudinal data or growth curves, specially in pharmacokinetics. Random effects and errors are usually assumed to be independent and normally distributed. We assume that the random effects and errors jointly follow heavy tailed distributions for the random effects and errors. This assumption may produce more robust estimates against outlying

  • r influential observations (see, for instance, Meza et al., 2011 and

Russo et al., 2009).

Cibele Russo (USP, Brazil) Bayesian analysis for nonlinear models May 2013 3 / 22

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Motivating example: Theophylline concentration

Kinetics of anti-asthmatic agent theophylline (Pinheiro & Bates 2000) Y : Theophylline Concentration (mg / L) (response variable) T: Time (h) (covariate) D: Applied dose (mg / kg)

Cibele Russo (USP, Brazil) Bayesian analysis for nonlinear models May 2013 4 / 22

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Motivating example: Theophylline concentration

Kinetics of anti-asthmatic agent theophylline (Pinheiro & Bates 2000) Y : Theophylline Concentration (mg / L) (response variable) T: Time (h) (covariate) D: Applied dose (mg / kg) E(Y ) = D exp(lKe + lKa − lCl)[exp(−elKeT) − exp(−elKaT)] elKa − elKe lKa: logarithm of the absorption rate, lKe: logarithm of the elimination rate lCl: the logarithm of clearance.

Cibele Russo (USP, Brazil) Bayesian analysis for nonlinear models May 2013 4 / 22

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Motivating example: Theophylline concentration

In an experiment, serum concentration (in mg/L) of theophylline was measured in eleven times (in h) after the administration of D dose (in mg/kg) in each of twelve patients.

Cibele Russo (USP, Brazil) Bayesian analysis for nonlinear models May 2013 5 / 22

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Motivating example: Theophylline data

Time (h) Theophylline concentrarion ( mg L)

2 4 6 8 10 5 10 15 20 25

  • ● ●
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  • 3
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2 4 6 8 10

  • 9

2 4 6 8 10

  • 12

5 10 15 20 25

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  • 1

5 10 15 20 25

  • 5

Theophylline concentration versus time

Cibele Russo (USP, Brazil) Bayesian analysis for nonlinear models May 2013 6 / 22

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Nonlinear model with random effects

For the jth measurement taken in the ith subject, Yij, in the time Tij, a possible mixed effects nonlinear model would be

Cibele Russo (USP, Brazil) Bayesian analysis for nonlinear models May 2013 7 / 22

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Nonlinear model with random effects

For the jth measurement taken in the ith subject, Yij, in the time Tij, a possible mixed effects nonlinear model would be Yij = D exp(φ1i + φ2i − φ3i)[exp(−eφ1iTij) − exp(−eφ2iTij)] eφ2i − eφ1i + ǫij

Cibele Russo (USP, Brazil) Bayesian analysis for nonlinear models May 2013 7 / 22

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Nonlinear model with random effects

For the jth measurement taken in the ith subject, Yij, in the time Tij, a possible mixed effects nonlinear model would be Yij = D exp(φ1i + φ2i − φ3i)[exp(−eφ1iTij) − exp(−eφ2iTij)] eφ2i − eφ1i + ǫij where φ1i = lKe + b1i φ2i = lKa + b2i φ3i = lCl + b3i lKe, lKa and lCl are fixed effects b1i, b2i and b3i are random effects ǫij are random errors.

Cibele Russo (USP, Brazil) Bayesian analysis for nonlinear models May 2013 7 / 22

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Nonlinear model with random effects

It is usual to assume that the random effects and errors are independent and normally distributed.

Cibele Russo (USP, Brazil) Bayesian analysis for nonlinear models May 2013 8 / 22

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Nonlinear model with random effects

It is usual to assume that the random effects and errors are independent and normally distributed. Here we assume the random effects and errors to jointly follow a multivariate scale mixture of normal distributions.

Cibele Russo (USP, Brazil) Bayesian analysis for nonlinear models May 2013 8 / 22

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Interpretation of random effects in the nonlinear model

5 10 15 20 25 30 5 10 15 20

Inclusion of b1

Time since drug administration (h) Theophylline concentration (mg/L)

b3=−0.9 b3=−0.6 b3=−0.3 b3=0 b3=0.3 b3=0.6 b3=0.9

5 10 15 20 25 30 5 10 15 20

Inclusion of b2

Time since drug administration (h) Theophylline concentration (mg/L)

b2=−0.18 b2=−0.12 b2=−0.06 b2=0 b2=0.06 b2=0.12 b2=0.18

5 10 15 20 25 30 5 10 15 20

Inclusion of b3

Time since drug administration (h) Theophylline concentration (mg/L)

b3=−0.9 b3=−0.6 b3=−0.3 b3=0 b3=0.3 b3=0.6 b3=0.9

Interpretation of random effects in the nonlinear model

Cibele Russo (USP, Brazil) Bayesian analysis for nonlinear models May 2013 9 / 22

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Nonlinear mixed-effects models

Suppose that y = (y⊤

1 , . . . , y⊤ n )⊤ is a vector of observed continuous

multivariate responses with yi a (ni × 1) vector containing the

  • bservations for the experimental unit i, i = 1, . . . , n, such that

yi = g(φi, Xi) + ǫi, i = 1, . . . , n, φi = β + bi, (1) with Xi = (Xi1, . . . , Xini)⊤ a matrix of explanatory variables for the i-th unit, bi is a (q × 1) vector of random effects, ǫi is an (ni × 1) vector of random errors values for i = 1, . . . , n, β is a (p × 1) location vector.

Cibele Russo (USP, Brazil) Bayesian analysis for nonlinear models May 2013 10 / 22

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Heavy-tailed nonlinear mixed-effects models

Let a m−dimensional random vector W follows a scale mixture of normal distribution (SMN) in its stochastic form, that is W = µ + κ(U)1/2Z, where µ is the location vector, U is a positive random variable with cumulative distribution function (cdf) H(u, ν) and probability density function (pdf) h(u, ν), ν is a scalar or vector parameter indexing the distribution of U, κ(U) is the weight function, Z ∼ N(0, Σ) Z and U independent.

Cibele Russo (USP, Brazil) Bayesian analysis for nonlinear models May 2013 11 / 22

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Heavy-tailed nonlinear mixed-effects models

Characterization of some SMN distributions.

Distribution κ(u) U MNm(µ, Σ) 1 U = 1 MStm(µ, Σ, ν) 1 u U ind. ∼ Gamma ν 2, ν 2

  • ,

u > 0, ν > 0 MSlm(µ, Σ, ν) 1 u U ∼ Beta(ν, 1), 0 < u < 1, ν > 0 MCNm(µ, Σ, ν, γ) 1 u h(u, ν) = νI(u=γ) + (1 − ν)I(u=1), u = γ, 1; 0 ≤ ν ≤ 1, 0 ≤ γ ≤ 1,

with d = (y − µ)⊤Σ−1(y − µ)

Cibele Russo (USP, Brazil) Bayesian analysis for nonlinear models May 2013 12 / 22

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Heavy-tailed nonlinear mixed-effects models

Scale mixture of normal distributions include symmetrical distributions with heavier tailed than the Gaussian distributions, including the normal itself.

−4 −2 2 4 0.0 0.1 0.2 0.3 0.4 0.5 x density −4 −2 2 4 0.0 0.1 0.2 0.3 0.4 0.5 x density −4 −2 2 4 0.0 0.1 0.2 0.3 0.4 0.5 x density t3 t5 normal

Normal and Student-t distributions

Cibele Russo (USP, Brazil) Bayesian analysis for nonlinear models May 2013 13 / 22

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Heavy-tailed nonlinear mixed-effects models

In the nonlinear model yi = g(φi, Xi) + ǫi, i = 1, . . . , n, φi = β + bi, we assume that

  • ǫi

bi

  • ind.

∼ SMNni+q

  • ,
  • Σi

D

  • ; H
  • ,

(2) where D and Σi are positive-definite dispersion matrices. For simplicity, we assume that D = diag(τ) and Σi = σ2Ini for i = 1, . . . , n and σ > 0 a scalar.

Cibele Russo (USP, Brazil) Bayesian analysis for nonlinear models May 2013 14 / 22

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Application

In the nonlinear model for the theophylline data: θ = (β, σ2, τ)⊤ = (lKe, lKa, lCl, σ2, τ1, τ2, τ3)⊤

Cibele Russo (USP, Brazil) Bayesian analysis for nonlinear models May 2013 15 / 22

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Application

In the nonlinear model for the theophylline data: θ = (β, σ2, τ)⊤ = (lKe, lKa, lCl, σ2, τ1, τ2, τ3)⊤ Prior distributions: βi ∼ N(0, v), τi ∼ Gamma(ai, bi), σ2 ∼ Gamma(c, d).

Cibele Russo (USP, Brazil) Bayesian analysis for nonlinear models May 2013 15 / 22

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Application

In the nonlinear model for the theophylline data: θ = (β, σ2, τ)⊤ = (lKe, lKa, lCl, σ2, τ1, τ2, τ3)⊤ Prior distributions: βi ∼ N(0, v), τi ∼ Gamma(ai, bi), σ2 ∼ Gamma(c, d). The Monte Carlo estimates using OpenBUGS were obtained generating chains of size 30000 spaced by 50 (burn-in of 5000). The posterior means and the corresponding standard deviations of the posterior distributions are presented in the following slides.

Cibele Russo (USP, Brazil) Bayesian analysis for nonlinear models May 2013 15 / 22

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Results: Theophylline data

Normal Student-t4 Slash4 mean (sd) mean (sd) mean (sd) lKe

  • 2.465

(0.102)

  • 2.452

(0.099)

  • 2.450

(0.099) lKa 0.462 (0.142) 0.460 (0.146) 0.459 (0.147) lCl

  • 3.232

(0.095)

  • 3.221

(0.097)

  • 3.220

(0.094) (σ2)−1 0.636 (0.104) 0.552 (0.145) 0.123 (0.035) (τ1)−1 5.580 (1.691) 5.053 (1.492) 5.052 (1.505) (τ2)−1 13.060 (3.458) 13.150 (3.451) 13.210 (3.451) (τ3)−1 11.830 (3.087) 11.740 (3.082) 11.760 (3.085) DIC 323.5 303.6 298.3

Cibele Russo (USP, Brazil) Bayesian analysis for nonlinear models May 2013 16 / 22

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Results: Theophylline data

Fitted curves

Time (h) (b) Theophylline concentrarion ( mg L)

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Fitted curves for theophylline problem under Slash4 model.

Cibele Russo (USP, Brazil) Bayesian analysis for nonlinear models May 2013 17 / 22

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Convergence

Convergence of chains in Slash4 case

Cibele Russo (USP, Brazil) Bayesian analysis for nonlinear models May 2013 18 / 22

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Implementation

Easy implementation in OpenBUGS,

Cibele Russo (USP, Brazil) Bayesian analysis for nonlinear models May 2013 19 / 22

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Implementation

Easy implementation in OpenBUGS, Computationally expensive,

Cibele Russo (USP, Brazil) Bayesian analysis for nonlinear models May 2013 19 / 22

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Implementation

Easy implementation in OpenBUGS, Computationally expensive, For nonlinear mixed effects models, convergence may be difficult to reach depending on the initial values.

Cibele Russo (USP, Brazil) Bayesian analysis for nonlinear models May 2013 19 / 22

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Discussion and remarks

Nonlinear mixed effects models plays an important role in nonlinear problems with correlated data.

Cibele Russo (USP, Brazil) Bayesian analysis for nonlinear models May 2013 20 / 22

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Discussion and remarks

Nonlinear mixed effects models plays an important role in nonlinear problems with correlated data. Considering a stochastic formulation in a bayesian approach, we propose the use of heavy-tailed distributions in a bayesian context to provide alternatives to the gaussian model.

Cibele Russo (USP, Brazil) Bayesian analysis for nonlinear models May 2013 20 / 22

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Discussion and remarks

Nonlinear mixed effects models plays an important role in nonlinear problems with correlated data. Considering a stochastic formulation in a bayesian approach, we propose the use of heavy-tailed distributions in a bayesian context to provide alternatives to the gaussian model. Easy implementation in Bayesian approach.

Cibele Russo (USP, Brazil) Bayesian analysis for nonlinear models May 2013 20 / 22

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Discussion and remarks

Nonlinear mixed effects models plays an important role in nonlinear problems with correlated data. Considering a stochastic formulation in a bayesian approach, we propose the use of heavy-tailed distributions in a bayesian context to provide alternatives to the gaussian model. Easy implementation in Bayesian approach. Estimation of individual profiles.

Cibele Russo (USP, Brazil) Bayesian analysis for nonlinear models May 2013 20 / 22

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Work in progress

Model diagnostics Robustness assessment Other nonlinear data sets.

Cibele Russo (USP, Brazil) Bayesian analysis for nonlinear models May 2013 21 / 22

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References

Meza, C., Osorio, F. and De la Cruz, R. (2012) Estimation in nonlinear mixed-effects models using heavy-tailed distributions. Statistics and Computing 22, 121-139. Pinheiro, J. C. and Bates, D. M. (2000) Mixed-Effects Models in S and S-Plus, Springer, New York. Russo, C. M., and Paula, G. A. and Aoki, R. (2009). Influence diagnostics in nonlinear mixed-effects elliptical models. Computational Statistics & Data Analysis 53, 4143–4156. Thank you!

Cibele Russo (USP, Brazil) Bayesian analysis for nonlinear models May 2013 22 / 22