Bayesian analysis for heavy-tailed nonlinear mixed effects models - PowerPoint PPT Presentation

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

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

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

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

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

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 or 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

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

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( lK e + lK a − lC l )[exp( − e lK e T ) − exp( − e lK a T )] e lK a − e lK e lK a : logarithm of the absorption rate, lK e : logarithm of the elimination rate lC l : the logarithm of clearance. Cibele Russo (USP, Brazil) Bayesian analysis for nonlinear models May 2013 4 / 22

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

Motivating example: Theophylline data 0 5 10 15 20 25 0 5 10 15 20 25 12 10 1 5 ● ● 10 ● ●● ● ● ● ● ● ● ● 8 ● ● ● ●● ●● ● ●● ● ● 6 ● ● ● ● ● ● ● 4 ● Theophylline concentrarion ( mg L) ● ● ● ● 2 ● ● ● ● ● ● 0 ● ● 3 2 4 9 10 ● ● ● ● ● ● ● 8 ●● ●● ● ● ● ● ● ●● ●●● 6 ●● ●● ● ● ● ● ●● 4 ● ● ● 2 ● ● ● ● ● ● ● ● ● ● 0 6 7 8 11 10 8 ● ● ● ●● ● ● ● ● ●● ● 6 ● ● ● ● ● ● ●● ● ● ● 4 ● ● ● ● ● ● ● ● ● ● ● 2 ● ● ● ● ● ● 0 ● ● ● ● 0 5 10 15 20 25 0 5 10 15 20 25 Time (h) Theophylline concentration versus time Cibele Russo (USP, Brazil) Bayesian analysis for nonlinear models May 2013 6 / 22

Nonlinear model with random effects For the j th measurement taken in the i th subject, Y ij , in the time T ij , a possible mixed effects nonlinear model would be Cibele Russo (USP, Brazil) Bayesian analysis for nonlinear models May 2013 7 / 22

Nonlinear model with random effects For the j th measurement taken in the i th subject, Y ij , in the time T ij , a possible mixed effects nonlinear model would be Y ij = D exp( φ 1 i + φ 2 i − φ 3 i )[exp( − e φ 1 i T ij ) − exp( − e φ 2 i T ij )] + ǫ ij e φ 2 i − e φ 1 i Cibele Russo (USP, Brazil) Bayesian analysis for nonlinear models May 2013 7 / 22

Nonlinear model with random effects For the j th measurement taken in the i th subject, Y ij , in the time T ij , a possible mixed effects nonlinear model would be Y ij = D exp( φ 1 i + φ 2 i − φ 3 i )[exp( − e φ 1 i T ij ) − exp( − e φ 2 i T ij )] + ǫ ij e φ 2 i − e φ 1 i where φ 1 i = lK e + b 1 i φ 2 i = lK a + b 2 i φ 3 i = lC l + b 3 i lK e , lK a and lC l are fixed effects b 1 i , b 2 i and b 3 i are random effects ǫ ij are random errors. Cibele Russo (USP, Brazil) Bayesian analysis for nonlinear models May 2013 7 / 22

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

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

Interpretation of random effects in the nonlinear model Inclusion of b 1 Inclusion of b 2 Inclusion of b 3 20 20 20 Theophylline concentration (mg/L) Theophylline concentration (mg/L) Theophylline concentration (mg/L) b 3 = − 0.9 b 2 = − 0.18 b 3 = − 0.9 b 3 = − 0.6 b 2 = − 0.12 b 3 = − 0.6 b 3 = − 0.3 b 2 = − 0.06 b 3 = − 0.3 15 b 3 =0 15 b 2 =0 15 b 3 =0 b 3 =0.3 b 2 =0.06 b 3 =0.3 b 3 =0.6 b 2 =0.12 b 3 =0.6 b 3 =0.9 b 2 =0.18 b 3 =0.9 10 10 10 5 5 5 0 0 0 0 5 10 15 20 25 30 0 5 10 15 20 25 30 0 5 10 15 20 25 30 Time since drug administration (h) Time since drug administration (h) Time since drug administration (h) Interpretation of random effects in the nonlinear model Cibele Russo (USP, Brazil) Bayesian analysis for nonlinear models May 2013 9 / 22

Nonlinear mixed-effects models n ) ⊤ is a vector of observed continuous Suppose that y = ( y ⊤ 1 , . . . , y ⊤ multivariate responses with y i a ( n i × 1) vector containing the observations for the experimental unit i , i = 1 , . . . , n , such that y i = g ( φ i , X i ) + ǫ i , i = 1 , . . . , n , (1) φ i = β + b i , with X i = ( X i 1 , . . . , X in i ) ⊤ a matrix of explanatory variables for the i -th unit, b i is a ( q × 1) vector of random effects, ǫ i is an ( n i × 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

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 / 2 Z , 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

Heavy-tailed nonlinear mixed-effects models Characterization of some SMN distributions. Distribution κ ( u ) U MN m ( µ , Σ ) 1 U = 1 � ν 2 , ν U ind. � 1 ∼ Gamma , MSt m ( µ , Σ , ν ) 2 u u > 0 , ν > 0 1 U ∼ Beta ( ν, 1) , MSl m ( µ , Σ , ν ) u 0 < u < 1 , ν > 0 1 h ( u , ν ) = ν I ( u = γ ) + (1 − ν ) I ( u =1) , MCN m ( µ , Σ , ν, γ ) u 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

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. 0.5 0.5 0.5 t 3 t 5 normal 0.4 0.4 0.4 0.3 0.3 0.3 density density density 0.2 0.2 0.2 0.1 0.1 0.1 0.0 0.0 0.0 −4 −4 −4 −2 −2 −2 0 0 0 2 2 2 4 4 4 x x x Normal and Student-t distributions Cibele Russo (USP, Brazil) Bayesian analysis for nonlinear models May 2013 13 / 22