SlideShare a Scribd company logo

Presentation

T
Teja Turk

The document discusses approaches for calculating confidence and prediction intervals in nonlinear mixed-effects models. It defines the nonlinear mixed-effects model and describes confidence interval approaches including bootstrap methods and non-bootstrap Wald and adjusted Wald intervals. It also evaluates the coverage rates of different interval approaches across a variety of nonlinear mixed-effects models. Prediction interval approaches for observed and unobserved groups are discussed as well.

1 of 47
Download to read offline
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
Comparison of Confidence and Prediction Interval Approaches in Nonlinear Mixed-Effects Models Teja Turk Adviser: Dr. Lukas Meier ETH Zurich Department of Mathematics Seminar for Statistics January–June 2015
Motivating example Nonlinear regression Concentration measurements on the individual level Time since drug administration (hours) Concentration(mg/L) 0 100 200 300 0 5 10 15 1 0 5 10 15 2 0 5 10 15 3 0 5 10 15 4 0 100 200 300 5 6 7 8 0 100 200 300 9 10 11 12 Teja Turk ETH Zurich (D-MATH, SfS) 1/14
Motivating example Nonlinear regression Concentration measurements on the individual level - pooled fit Time since drug administration (hours) Concentration(mg/L) 0 100 200 300 0 5 10 15 1 0 5 10 15 2 0 5 10 15 3 0 5 10 15 4 0 100 200 300 5 6 7 8 0 100 200 300 9 10 11 12 Teja Turk ETH Zurich (D-MATH, SfS) 1/14
Motivating example Nonlinear regression Concentration measurements on the individual level - separate fits Time since drug administration (hours) Concentration(mg/L) 0 100 200 300 0 5 10 15 1 0 5 10 15 2 0 5 10 15 3 0 5 10 15 4 0 100 200 300 5 6 7 8 0 100 200 300 9 10 11 12 Teja Turk ETH Zurich (D-MATH, SfS) 1/14
Motivating example Nonlinear mixed-effects model Concentration measurements on the individual level Time since drug administration (hours) Concentration(mg/L) 0 100 200 300 0 5 10 15 1 0 5 10 15 2 0 5 10 15 3 0 5 10 15 4 0 100 200 300 5 6 7 8 0 100 200 300 9 10 11 12 Teja Turk ETH Zurich (D-MATH, SfS) 2/14
Motivating example Nonlinear mixed-effects model Concentration measurements on the individual level - population curve Time since drug administration (hours) Concentration(mg/L) 0 100 200 300 0 5 10 15 1 0 5 10 15 2 0 5 10 15 3 0 5 10 15 4 0 100 200 300 5 6 7 8 0 100 200 300 9 10 11 12 Teja Turk ETH Zurich (D-MATH, SfS) 2/14
Motivating example Nonlinear mixed-effects model Concentration measurements on the individual level - individual curves Time since drug administration (hours) Concentration(mg/L) 0 100 200 300 0 5 10 15 1 0 5 10 15 2 0 5 10 15 3 0 5 10 15 4 0 100 200 300 5 6 7 8 0 100 200 300 9 10 11 12 Teja Turk ETH Zurich (D-MATH, SfS) 2/14
Nonlinear mixed-effects model Definition (Single level NLME model) yij = f (xij; β, bi) + εij, i ∈ {1, . . . , M} , j ∈ {1, . . . , ni} bi ∼ N (0, Ψ) εij ∼ N 0, σ2    independent Teja Turk ETH Zurich (D-MATH, SfS) 3/14
Nonlinear mixed-effects model Definition (Single level NLME model) yij = f (xij; β, bi) + εij, i ∈ {1, . . . , M} , j ∈ {1, . . . , ni} bi ∼ N (0, Ψ) εij ∼ N 0, σ2    independent • β: fixed effects Teja Turk ETH Zurich (D-MATH, SfS) 3/14
Nonlinear mixed-effects model Definition (Single level NLME model) yij = f (xij; β, bi) + εij, i ∈ {1, . . . , M} , j ∈ {1, . . . , ni} bi ∼ N (0, Ψ) εij ∼ N 0, σ2    independent • β: fixed effects • bi: random effects Teja Turk ETH Zurich (D-MATH, SfS) 3/14
Nonlinear mixed-effects model Definition (Single level NLME model) yij = f (xij; β, bi) + εij, i ∈ {1, . . . , M} , j ∈ {1, . . . , ni} bi ∼ N (0, Ψ) εij ∼ N 0, σ2    independent • β: fixed effects • bi: random effects • Ψ: variance-covariance matrix of random effects Teja Turk ETH Zurich (D-MATH, SfS) 3/14
Nonlinear mixed-effects model Definition (Single level NLME model) yij = f (xij; β, bi) + εij, i ∈ {1, . . . , M} , j ∈ {1, . . . , ni} bi ∼ N (0, Ψ) εij ∼ N 0, σ2    independent • β: fixed effects • bi: random effects • Ψ: variance-covariance matrix of random effects • σ2: within-group variance Teja Turk ETH Zurich (D-MATH, SfS) 3/14
Confidence and prediction intervals • linear approximation • iterative approach Teja Turk ETH Zurich (D-MATH, SfS) 4/14
Confidence and prediction intervals • linear approximation • iterative approach • Confidence intervals: the distributions of β, Ψ, σ ? Teja Turk ETH Zurich (D-MATH, SfS) 4/14
Confidence and prediction intervals • linear approximation • iterative approach • Confidence intervals: the distributions of β, Ψ, σ ? • Prediction intervals: the distribution of yk0 :=    f xk0; β, 0 if k unobserved f xk0; β, bk if k observed ? Teja Turk ETH Zurich (D-MATH, SfS) 4/14
Bootstrap Teja Turk ETH Zurich (D-MATH, SfS) 5/14
Bootstrap Parametric bi εij Random effects: Errors: bi ∼N 0,Ψ εij ∼N 0,σ2 y∗ ij = f xij ; β, b∗ i + ε∗ ij Teja Turk ETH Zurich (D-MATH, SfS) 5/14
Bootstrap Parametric bi εij Random effects: Errors: bi ∼N 0,Ψ εij ∼N 0,σ2 y∗ ij = f xij ; β, b∗ i + ε∗ ij Nonparametric bi rij BLUPs: Residuals: · · · · · · ... ... i y∗ ij = f xij ; β, b∗ i + r∗ ij Teja Turk ETH Zurich (D-MATH, SfS) 5/14
Bootstrap Parametric bi εij Random effects: Errors: bi ∼N 0,Ψ εij ∼N 0,σ2 y∗ ij = f xij ; β, b∗ i + ε∗ ij Nonparametric bi rij BLUPs: Residuals: · · · · · · ... ... i y∗ ij = f xij ; β, b∗ i + r∗ ij Case fi rij Fitted values: Residuals: y∗ ij = f ∗ ij + r∗ ij Teja Turk ETH Zurich (D-MATH, SfS) 5/14
Confidence interval approaches • Bootstrap: basic 2λ − q∗ 1− α 2 , 2λ − q∗ α 2 percentile q∗ α 2 , q∗ 1− α 2 normal λ − e∗ + σ∗ zα 2 , λ − e∗ + σ∗ z1− α 2 Teja Turk ETH Zurich (D-MATH, SfS) 6/14
Confidence interval approaches • Bootstrap: basic 2λ − q∗ 1− α 2 , 2λ − q∗ α 2 percentile q∗ α 2 , q∗ 1− α 2 normal λ − e∗ + σ∗ zα 2 , λ − e∗ + σ∗ z1− α 2 • Non-bootstrap: intervals Teja Turk ETH Zurich (D-MATH, SfS) 6/14
Confidence interval approaches • Bootstrap: basic 2λ − q∗ 1− α 2 , 2λ − q∗ α 2 percentile q∗ α 2 , q∗ 1− α 2 normal λ − e∗ + σ∗ zα 2 , λ − e∗ + σ∗ z1− α 2 • Non-bootstrap: intervals Wald: λ − λ se λ ∼ tdf Teja Turk ETH Zurich (D-MATH, SfS) 6/14
Confidence interval approaches • Bootstrap: basic 2λ − q∗ 1− α 2 , 2λ − q∗ α 2 percentile q∗ α 2 , q∗ 1− α 2 normal λ − e∗ + σ∗ zα 2 , λ − e∗ + σ∗ z1− α 2 • Non-bootstrap: intervals Wald: λ − λ se λ ∼ tdf adjusted Wald: λ − λ se λ ∼ m + s · tdf Teja Turk ETH Zurich (D-MATH, SfS) 6/14
SSweibull 1 random effect 0.5 0.6 0.7 0.8 0.9 1.0 CR A D lrc pwr ΨA,A σ 2 uncorrelated random effects A D lrc pwr ΨA,A Ψpwr,pwr σ 2 correlated random effects A D lrc pwr ΨA,A ΨA,pwr Ψpwr,pwr σ intervals Wald adjusted Wald 1 random effect 0.5 0.6 0.7 0.8 0.9 1.0 CR A D lrc pwr ΨA,A σ 2 uncorrelated random effects A D lrc pwr ΨA,A Ψpwr,pwr σ 2 correlated random effects A D lrc pwr ΨA,A ΨA,pwr Ψpwr,pwr σ parametric: basic parametric: perc parametric: norm case: basic case: perc case: norm nonparametric: basic nonparametric: perc nonparametric: norm Teja Turk ETH Zurich (D-MATH, SfS) 7/14
Coverage rate averaged over parameters and models (95%-CI) CR averaged over random intervals Wald adjusted Wald parametric: basic parametric: perc parametric: norm case: basic case: perc case: norm nonparametric: basic nonparametric: perc nonparametric: norm dissProf logPKcomp sigEmax SSasymp SSasympOff SSasympOrig SSbiexp SSfol SSfpl SSgompertz SSlogis SSmicmen SSweibull fixed varcov sigma Teja Turk ETH Zurich (D-MATH, SfS) 8/14
Coverage rate averaged over parameters and models (95%-CI) CR averaged over random intervals Wald adjusted Wald parametric: basic parametric: perc parametric: norm case: basic case: perc case: norm nonparametric: basic nonparametric: perc nonparametric: norm dissProf logPKcomp sigEmax SSasymp SSasympOff SSasympOrig SSbiexp SSfol SSfpl SSgompertz SSlogis SSmicmen SSweibull 0.4 0.5 0.6 0.7 0.8 0.9 1.0 CR CR aggregated over the models fixed varcov sigma Teja Turk ETH Zurich (D-MATH, SfS) 8/14
Coverage rate by method and parameter type (CI) 0.2 0.4 0.6 0.8 1.0 CR intervals 0.2 0.4 0.6 0.8 1.0 CR Wald 0.50 0.80 0.95 0.2 0.4 0.6 0.8 1.0 1 − α CR adjusted Wald parametric: basic parametric: perc 0.50 0.80 0.95 1 − α parametric: norm case: basic case: perc 0.50 0.80 0.95 1 − α case: norm nonparametric: basic nonparametric: perc 0.50 0.80 0.95 1 − α nonparametric: norm fixed varcov sigma Teja Turk ETH Zurich (D-MATH, SfS) 9/14
Prediction interval approaches Unobserved group Var fk0 β, 0 + Var[fk0(β, bk)] Teja Turk ETH Zurich (D-MATH, SfS) 10/14
Prediction interval approaches Unobserved group Var fk0 β, 0 + Var[fk0(β, bk)] Non-bootstrap: • linear approximation: fk0 β, 0 , fk0(β, bk ) at (β, 0) Teja Turk ETH Zurich (D-MATH, SfS) 10/14
Prediction interval approaches Unobserved group Var fk0 β, 0 + Var[fk0(β, bk)] Non-bootstrap: • linear approximation: fk0 β, 0 , fk0(β, bk ) at (β, 0) • distribution-free approach: empirical quantiles of the population level residuals Teja Turk ETH Zurich (D-MATH, SfS) 10/14
Prediction interval approaches Unobserved group Var fk0 β, 0 + Var[fk0(β, bk)] Non-bootstrap: • linear approximation: fk0 β, 0 , fk0(β, bk ) at (β, 0) • distribution-free approach: empirical quantiles of the population level residuals Bootstrap: • linear approximation Teja Turk ETH Zurich (D-MATH, SfS) 10/14
Prediction interval approaches Unobserved group Var fk0 β, 0 + Var[fk0(β, bk)] Non-bootstrap: • linear approximation: fk0 β, 0 , fk0(β, bk ) at (β, 0) • distribution-free approach: empirical quantiles of the population level residuals Bootstrap: • linear approximation • prediction error sample: yk0−yk0−e∗ s∗ ∼ z Teja Turk ETH Zurich (D-MATH, SfS) 10/14
Prediction interval approaches Unobserved group Var fk0 β, 0 + Var[fk0(β, bk)] Non-bootstrap: • linear approximation: fk0 β, 0 , fk0(β, bk ) at (β, 0) • distribution-free approach: empirical quantiles of the population level residuals Bootstrap: • linear approximation • prediction error sample: yk0−yk0−e∗ s∗ ∼ z Observed group Var fk0 β, bk − fk0(β, bk) Teja Turk ETH Zurich (D-MATH, SfS) 10/14
Prediction interval approaches Unobserved group Var fk0 β, 0 + Var[fk0(β, bk)] Non-bootstrap: • linear approximation: fk0 β, 0 , fk0(β, bk ) at (β, 0) • distribution-free approach: empirical quantiles of the population level residuals Bootstrap: • linear approximation • prediction error sample: yk0−yk0−e∗ s∗ ∼ z Observed group Var fk0 β, bk − fk0(β, bk) Non-bootstrap: • the Delta method: fk0 β, bk − fk0(β, bk ) • the FOCE method: fk0(β, bk ) at β, bk • the FO method: fk0 β, bk , fk0(β, bk ) at (β, 0) Teja Turk ETH Zurich (D-MATH, SfS) 10/14
Prediction interval approaches Unobserved group Var fk0 β, 0 + Var[fk0(β, bk)] Non-bootstrap: • linear approximation: fk0 β, 0 , fk0(β, bk ) at (β, 0) • distribution-free approach: empirical quantiles of the population level residuals Bootstrap: • linear approximation • prediction error sample: yk0−yk0−e∗ s∗ ∼ z Observed group Var fk0 β, bk − fk0(β, bk) Non-bootstrap: • the Delta method: fk0 β, bk − fk0(β, bk ) • the FOCE method: fk0(β, bk ) at β, bk • the FO method: fk0 β, bk , fk0(β, bk ) at (β, 0) Bootstrap: • the Delta method • the FOCE method • the FO method Teja Turk ETH Zurich (D-MATH, SfS) 10/14
Prediction interval approaches Unobserved group Var fk0 β, 0 + Var[fk0(β, bk)] Non-bootstrap: • linear approximation: fk0 β, 0 , fk0(β, bk ) at (β, 0) • distribution-free approach: empirical quantiles of the population level residuals Bootstrap: • linear approximation • prediction error sample: yk0−yk0−e∗ s∗ ∼ z Observed group Var fk0 β, bk − fk0(β, bk) Non-bootstrap: • the Delta method: fk0 β, bk − fk0(β, bk ) • the FOCE method: fk0(β, bk ) at β, bk • the FO method: fk0 β, bk , fk0(β, bk ) at (β, 0) Bootstrap: • the Delta method • the FOCE method • the FO method • prediction error sample Teja Turk ETH Zurich (D-MATH, SfS) 10/14
SSasympOff: diag_2 0.80 0.85 0.90 0.95 1.00 1.05 1.10 CR unobserved: nonboot Lin. approximation Distr.-free unobserved: boot Lin. approximation Pred. error sample 16.67 50 66.67 105 120 150 0.80 0.85 0.90 0.95 1.00 1.05 1.10 CR observed: nonboot FOCE method FO method Delta method 16.67 50 66.67 105 120 150 observed: boot Pred. error sample FOCE method FO method Delta method Teja Turk ETH Zurich (D-MATH, SfS) 11/14
Coverage rate averaged over parameters and models (95%-PI) CR averaged over random Unobs.: nonBLA Unobs.: nonBDF Unobs.: BLA Unobs.: BPES Obs.: nonBFOCE Obs.: nonBFO Obs.: nonB∆ Obs.: BPES Obs.: BFOCE Obs.: BFO Obs.: B∆ dissProf logPKcomp sigEmax SSasymp SSasympOff SSasympOrig SSbiexp SSfol SSfpl SSgompertz SSlogis SSmicmen SSweibull interpolation extrapolation Teja Turk ETH Zurich (D-MATH, SfS) 12/14
Coverage rate averaged over parameters and models (95%-PI) CR averaged over random Unobs.: nonBLA Unobs.: nonBDF Unobs.: BLA Unobs.: BPES Obs.: nonBFOCE Obs.: nonBFO Obs.: nonB∆ Obs.: BPES Obs.: BFOCE Obs.: BFO Obs.: B∆ dissProf logPKcomp sigEmax SSasymp SSasympOff SSasympOrig SSbiexp SSfol SSfpl SSgompertz SSlogis SSmicmen SSweibull 0.75 0.80 0.85 0.90 0.95 1.00 CR CR aggregated over the models interpolation extrapolation Teja Turk ETH Zurich (D-MATH, SfS) 12/14
Coverage rate by method and parameter type (PI) 0.4 0.6 0.8 1.0 CR Unobs.: nonBLA Unobs.: nonBDF Unobs.: BLA Unobs.: BPES 0.4 0.6 0.8 1.0 CR Obs.: nonBFOCE Obs.: nonBFO Obs.: nonB∆ 0.50 0.80 0.95 1 − α Obs.: BPES 0.50 0.80 0.95 0.4 0.6 0.8 1.0 1 − α CR Obs.: BFOCE 0.50 0.80 0.95 1 − α Obs.: BFO 0.50 0.80 0.95 1 − α Obs.: B∆ interpolation extrapolation Teja Turk ETH Zurich (D-MATH, SfS) 13/14
Conclusions • differences in performance of CIs between the fixed effects, the variance-covariance components and σ Teja Turk ETH Zurich (D-MATH, SfS) 14/14
Conclusions • differences in performance of CIs between the fixed effects, the variance-covariance components and σ • no significant differences in performance of PIs between the interpolation and extrapolation covariate points Teja Turk ETH Zurich (D-MATH, SfS) 14/14
Conclusions • differences in performance of CIs between the fixed effects, the variance-covariance components and σ • no significant differences in performance of PIs between the interpolation and extrapolation covariate points • parametric bootstrap successfully used for both CIs and PIs Teja Turk ETH Zurich (D-MATH, SfS) 14/14
Conclusions • differences in performance of CIs between the fixed effects, the variance-covariance components and σ • no significant differences in performance of PIs between the interpolation and extrapolation covariate points • parametric bootstrap successfully used for both CIs and PIs • CIs tend to be too narrow or shifted Teja Turk ETH Zurich (D-MATH, SfS) 14/14
Conclusions • differences in performance of CIs between the fixed effects, the variance-covariance components and σ • no significant differences in performance of PIs between the interpolation and extrapolation covariate points • parametric bootstrap successfully used for both CIs and PIs • CIs tend to be too narrow or shifted • PIs tend to be too wide Teja Turk ETH Zurich (D-MATH, SfS) 14/14
Coverage rate averaged over parameters (95%-CI) diag_1 diag_1 diag_1 diag_1 diag_1 diag_1 diag_1 diag_1 diag_1 diag_1 diag_1 diag_1 diag_1 diag_2 diag_2 diag_2 diag_2 diag_2 diag_2 diag_2 diag_2 diag_2 diag_2 diag_2 diag_2 diag_2 symm_2 symm_2 symm_2 symm_2 symm_2 symm_2 symm_2 symm_2 symm_2 symm_2 symm_2 symm_2 symm_2 dissProf logPKcomp sigEmax SSasymp SSasympOff SSasympOrig SSbiexp SSfol SSfpl SSgompertz SSlogis SSmicmen SSweibull intervals Wald adjusted Wald parametric: basic parametric: perc parametric: norm case: basic case: perc case: norm nonparametric: basic nonparametric: perc nonparametric: norm fixed varcov sigma Teja Turk ETH Zurich (D-MATH, SfS)
Coverage rate averaged over subjects (95%-PI) diag_1 diag_1 diag_1 diag_1 diag_1 diag_1 diag_1 diag_1 diag_1 diag_1 diag_1 diag_1 diag_1 diag_2 diag_2 diag_2 diag_2 diag_2 diag_2 diag_2 diag_2 diag_2 diag_2 diag_2 diag_2 diag_2 symm_2 symm_2 symm_2 symm_2 symm_2 symm_2 symm_2 symm_2 symm_2 symm_2 symm_2 symm_2 symm_2 dissProf logPKcomp sigEmax SSasymp SSasympOff SSasympOrig SSbiexp SSfol SSfpl SSgompertz SSlogis SSmicmen SSweibull Unobs.: nonBLA Unobs.: nonBDF Unobs.: BLA Unobs.: BPES Obs.: nonBFOCE Obs.: nonBFO Obs.: nonB∆ Obs.: BPES Obs.: BFOCE Obs.: BFO Obs.: B∆ interpolation extrapolation Teja Turk ETH Zurich (D-MATH, SfS)

More Related Content

PDF
Estimation of the score vector and observed information matrix in intractable...
Pierre Jacob
 
PDF
Imprecision in learning: an overview
Sebastien Destercke
 
PDF
Estimation of the score vector and observed information matrix in intractable...
Pierre Jacob
 
PDF
asymptotics of ABC
Christian Robert
 
PPTX
5 8 к
sk1ll
 
PDF
Architecting Security across global networks
EQS Group
 
PDF
Coves delToll
Araceli Arias Roy
 
PPTX
5 клас урок 8
Olga Sokolik
 
Estimation of the score vector and observed information matrix in intractable...
Pierre Jacob
 
Imprecision in learning: an overview
Sebastien Destercke
 
Estimation of the score vector and observed information matrix in intractable...
Pierre Jacob
 
asymptotics of ABC
Christian Robert
 
5 8 к
sk1ll
 
Architecting Security across global networks
EQS Group
 
Coves delToll
Araceli Arias Roy
 
5 клас урок 8
Olga Sokolik
 

Viewers also liked (6)

PPTX
C1 Topic 1 Language and Communication
Nina's Cabbages
 
PDF
Geografija 7-klas-bojko
kreidaros1
 
PDF
BK 7210 Urban analysis and design principles – ir. Evelien Brandes
jornvorn
 
PDF
Geografija 6-klas-skuratovych
kreidaros1
 
PPTX
Introduction to Web Architecture
Chamnap Chhorn
 
PPTX
Principles of Teaching:Different Methods and Approaches
justindoliente
 
C1 Topic 1 Language and Communication
Nina's Cabbages
 
Geografija 7-klas-bojko
kreidaros1
 
BK 7210 Urban analysis and design principles – ir. Evelien Brandes
jornvorn
 
Geografija 6-klas-skuratovych
kreidaros1
 
Introduction to Web Architecture
Chamnap Chhorn
 
Principles of Teaching:Different Methods and Approaches
justindoliente
 
Ad

Similar to Presentation (20)

PDF
Formulas statistics
Prashi_Jain
 
PDF
Nonparametric testing for exogeneity with discrete regressors and instruments
GRAPE
 
PDF
Bayesian inference for mixed-effects models driven by SDEs and other stochast...
Umberto Picchini
 
PDF
the ABC of ABC
Christian Robert
 
PPTX
Stat2013
Sucheta Tripathy
 
PPT
comparison of two population means - chapter 8
und3rpant
 
PPT
comparison of two population means - chapter 8
und3rpant
 
PDF
Laplace's Demon: seminar #1
Christian Robert
 
PDF
Incremental and Multi-feature Tensor Subspace Learning applied for Background...
Andrews Cordolino Sobral
 
PDF
Bayesian Hierarchical Models
Ammar Rashed
 
PDF
8.pdf
ChrisMartin260004
 
PPTX
Lecture 11 Paired t test.pptx
shakirRahman10
 
PPT
Presentation lab meeting NRE
farnaznojavan
 
PDF
Estimation of the score vector and observed information matrix in intractable...
Pierre Jacob
 
PPT
Ttestrrrrrrrrrrrrrr2dfsssssssssssss008.ppt
EndrisHEbrahim
 
PPTX
Resampling methods
Setia Pramana
 
PDF
MUMS Opening Workshop - UQ Data Fusion: An Introduction and Case Study - Robe...
The Statistical and Applied Mathematical Sciences Institute
 
PPTX
ders 3.3 Unit root testing section 3 .pptx
Ergin Akalpler
 
DOC
Ch 4 Slides.doc655444444444444445678888776
ohenebabismark508
 
PDF
Dr.Dinesh-BIOSTAT-Tests-of-significance-1-min.pdf
HassanMohyUdDin2
 
Formulas statistics
Prashi_Jain
 
Nonparametric testing for exogeneity with discrete regressors and instruments
GRAPE
 
Bayesian inference for mixed-effects models driven by SDEs and other stochast...
Umberto Picchini
 
the ABC of ABC
Christian Robert
 
Stat2013
Sucheta Tripathy
 
comparison of two population means - chapter 8
und3rpant
 
comparison of two population means - chapter 8
und3rpant
 
Laplace's Demon: seminar #1
Christian Robert
 
Incremental and Multi-feature Tensor Subspace Learning applied for Background...
Andrews Cordolino Sobral
 
Bayesian Hierarchical Models
Ammar Rashed
 
8.pdf
ChrisMartin260004
 
Lecture 11 Paired t test.pptx
shakirRahman10
 
Presentation lab meeting NRE
farnaznojavan
 
Estimation of the score vector and observed information matrix in intractable...
Pierre Jacob
 
Ttestrrrrrrrrrrrrrr2dfsssssssssssss008.ppt
EndrisHEbrahim
 
Resampling methods
Setia Pramana
 
MUMS Opening Workshop - UQ Data Fusion: An Introduction and Case Study - Robe...
The Statistical and Applied Mathematical Sciences Institute
 
ders 3.3 Unit root testing section 3 .pptx
Ergin Akalpler
 
Ch 4 Slides.doc655444444444444445678888776
ohenebabismark508
 
Dr.Dinesh-BIOSTAT-Tests-of-significance-1-min.pdf
HassanMohyUdDin2
 
Ad

Presentation

  • 1. Comparison of Confidence and Prediction Interval Approaches in Nonlinear Mixed-Effects Models Teja Turk Adviser: Dr. Lukas Meier ETH Zurich Department of Mathematics Seminar for Statistics January–June 2015
  • 2. Motivating example Nonlinear regression Concentration measurements on the individual level Time since drug administration (hours) Concentration(mg/L) 0 100 200 300 0 5 10 15 1 0 5 10 15 2 0 5 10 15 3 0 5 10 15 4 0 100 200 300 5 6 7 8 0 100 200 300 9 10 11 12 Teja Turk ETH Zurich (D-MATH, SfS) 1/14
  • 3. Motivating example Nonlinear regression Concentration measurements on the individual level - pooled fit Time since drug administration (hours) Concentration(mg/L) 0 100 200 300 0 5 10 15 1 0 5 10 15 2 0 5 10 15 3 0 5 10 15 4 0 100 200 300 5 6 7 8 0 100 200 300 9 10 11 12 Teja Turk ETH Zurich (D-MATH, SfS) 1/14
  • 4. Motivating example Nonlinear regression Concentration measurements on the individual level - separate fits Time since drug administration (hours) Concentration(mg/L) 0 100 200 300 0 5 10 15 1 0 5 10 15 2 0 5 10 15 3 0 5 10 15 4 0 100 200 300 5 6 7 8 0 100 200 300 9 10 11 12 Teja Turk ETH Zurich (D-MATH, SfS) 1/14
  • 5. Motivating example Nonlinear mixed-effects model Concentration measurements on the individual level Time since drug administration (hours) Concentration(mg/L) 0 100 200 300 0 5 10 15 1 0 5 10 15 2 0 5 10 15 3 0 5 10 15 4 0 100 200 300 5 6 7 8 0 100 200 300 9 10 11 12 Teja Turk ETH Zurich (D-MATH, SfS) 2/14
  • 6. Motivating example Nonlinear mixed-effects model Concentration measurements on the individual level - population curve Time since drug administration (hours) Concentration(mg/L) 0 100 200 300 0 5 10 15 1 0 5 10 15 2 0 5 10 15 3 0 5 10 15 4 0 100 200 300 5 6 7 8 0 100 200 300 9 10 11 12 Teja Turk ETH Zurich (D-MATH, SfS) 2/14
  • 7. Motivating example Nonlinear mixed-effects model Concentration measurements on the individual level - individual curves Time since drug administration (hours) Concentration(mg/L) 0 100 200 300 0 5 10 15 1 0 5 10 15 2 0 5 10 15 3 0 5 10 15 4 0 100 200 300 5 6 7 8 0 100 200 300 9 10 11 12 Teja Turk ETH Zurich (D-MATH, SfS) 2/14
  • 8. Nonlinear mixed-effects model Definition (Single level NLME model) yij = f (xij; β, bi) + εij, i ∈ {1, . . . , M} , j ∈ {1, . . . , ni} bi ∼ N (0, Ψ) εij ∼ N 0, σ2    independent Teja Turk ETH Zurich (D-MATH, SfS) 3/14
  • 9. Nonlinear mixed-effects model Definition (Single level NLME model) yij = f (xij; β, bi) + εij, i ∈ {1, . . . , M} , j ∈ {1, . . . , ni} bi ∼ N (0, Ψ) εij ∼ N 0, σ2    independent • β: fixed effects Teja Turk ETH Zurich (D-MATH, SfS) 3/14
  • 10. Nonlinear mixed-effects model Definition (Single level NLME model) yij = f (xij; β, bi) + εij, i ∈ {1, . . . , M} , j ∈ {1, . . . , ni} bi ∼ N (0, Ψ) εij ∼ N 0, σ2    independent • β: fixed effects • bi: random effects Teja Turk ETH Zurich (D-MATH, SfS) 3/14
  • 11. Nonlinear mixed-effects model Definition (Single level NLME model) yij = f (xij; β, bi) + εij, i ∈ {1, . . . , M} , j ∈ {1, . . . , ni} bi ∼ N (0, Ψ) εij ∼ N 0, σ2    independent • β: fixed effects • bi: random effects • Ψ: variance-covariance matrix of random effects Teja Turk ETH Zurich (D-MATH, SfS) 3/14
  • 12. Nonlinear mixed-effects model Definition (Single level NLME model) yij = f (xij; β, bi) + εij, i ∈ {1, . . . , M} , j ∈ {1, . . . , ni} bi ∼ N (0, Ψ) εij ∼ N 0, σ2    independent • β: fixed effects • bi: random effects • Ψ: variance-covariance matrix of random effects • σ2: within-group variance Teja Turk ETH Zurich (D-MATH, SfS) 3/14
  • 13. Confidence and prediction intervals • linear approximation • iterative approach Teja Turk ETH Zurich (D-MATH, SfS) 4/14
  • 14. Confidence and prediction intervals • linear approximation • iterative approach • Confidence intervals: the distributions of β, Ψ, σ ? Teja Turk ETH Zurich (D-MATH, SfS) 4/14
  • 15. Confidence and prediction intervals • linear approximation • iterative approach • Confidence intervals: the distributions of β, Ψ, σ ? • Prediction intervals: the distribution of yk0 :=    f xk0; β, 0 if k unobserved f xk0; β, bk if k observed ? Teja Turk ETH Zurich (D-MATH, SfS) 4/14
  • 16. Bootstrap Teja Turk ETH Zurich (D-MATH, SfS) 5/14
  • 17. Bootstrap Parametric bi εij Random effects: Errors: bi ∼N 0,Ψ εij ∼N 0,σ2 y∗ ij = f xij ; β, b∗ i + ε∗ ij Teja Turk ETH Zurich (D-MATH, SfS) 5/14
  • 18. Bootstrap Parametric bi εij Random effects: Errors: bi ∼N 0,Ψ εij ∼N 0,σ2 y∗ ij = f xij ; β, b∗ i + ε∗ ij Nonparametric bi rij BLUPs: Residuals: · · · · · · ... ... i y∗ ij = f xij ; β, b∗ i + r∗ ij Teja Turk ETH Zurich (D-MATH, SfS) 5/14
  • 19. Bootstrap Parametric bi εij Random effects: Errors: bi ∼N 0,Ψ εij ∼N 0,σ2 y∗ ij = f xij ; β, b∗ i + ε∗ ij Nonparametric bi rij BLUPs: Residuals: · · · · · · ... ... i y∗ ij = f xij ; β, b∗ i + r∗ ij Case fi rij Fitted values: Residuals: y∗ ij = f ∗ ij + r∗ ij Teja Turk ETH Zurich (D-MATH, SfS) 5/14
  • 20. Confidence interval approaches • Bootstrap: basic 2λ − q∗ 1− α 2 , 2λ − q∗ α 2 percentile q∗ α 2 , q∗ 1− α 2 normal λ − e∗ + σ∗ zα 2 , λ − e∗ + σ∗ z1− α 2 Teja Turk ETH Zurich (D-MATH, SfS) 6/14
  • 21. Confidence interval approaches • Bootstrap: basic 2λ − q∗ 1− α 2 , 2λ − q∗ α 2 percentile q∗ α 2 , q∗ 1− α 2 normal λ − e∗ + σ∗ zα 2 , λ − e∗ + σ∗ z1− α 2 • Non-bootstrap: intervals Teja Turk ETH Zurich (D-MATH, SfS) 6/14
  • 22. Confidence interval approaches • Bootstrap: basic 2λ − q∗ 1− α 2 , 2λ − q∗ α 2 percentile q∗ α 2 , q∗ 1− α 2 normal λ − e∗ + σ∗ zα 2 , λ − e∗ + σ∗ z1− α 2 • Non-bootstrap: intervals Wald: λ − λ se λ ∼ tdf Teja Turk ETH Zurich (D-MATH, SfS) 6/14
  • 23. Confidence interval approaches • Bootstrap: basic 2λ − q∗ 1− α 2 , 2λ − q∗ α 2 percentile q∗ α 2 , q∗ 1− α 2 normal λ − e∗ + σ∗ zα 2 , λ − e∗ + σ∗ z1− α 2 • Non-bootstrap: intervals Wald: λ − λ se λ ∼ tdf adjusted Wald: λ − λ se λ ∼ m + s · tdf Teja Turk ETH Zurich (D-MATH, SfS) 6/14
  • 24. SSweibull 1 random effect 0.5 0.6 0.7 0.8 0.9 1.0 CR A D lrc pwr ΨA,A σ 2 uncorrelated random effects A D lrc pwr ΨA,A Ψpwr,pwr σ 2 correlated random effects A D lrc pwr ΨA,A ΨA,pwr Ψpwr,pwr σ intervals Wald adjusted Wald 1 random effect 0.5 0.6 0.7 0.8 0.9 1.0 CR A D lrc pwr ΨA,A σ 2 uncorrelated random effects A D lrc pwr ΨA,A Ψpwr,pwr σ 2 correlated random effects A D lrc pwr ΨA,A ΨA,pwr Ψpwr,pwr σ parametric: basic parametric: perc parametric: norm case: basic case: perc case: norm nonparametric: basic nonparametric: perc nonparametric: norm Teja Turk ETH Zurich (D-MATH, SfS) 7/14
  • 25. Coverage rate averaged over parameters and models (95%-CI) CR averaged over random intervals Wald adjusted Wald parametric: basic parametric: perc parametric: norm case: basic case: perc case: norm nonparametric: basic nonparametric: perc nonparametric: norm dissProf logPKcomp sigEmax SSasymp SSasympOff SSasympOrig SSbiexp SSfol SSfpl SSgompertz SSlogis SSmicmen SSweibull fixed varcov sigma Teja Turk ETH Zurich (D-MATH, SfS) 8/14
  • 26. Coverage rate averaged over parameters and models (95%-CI) CR averaged over random intervals Wald adjusted Wald parametric: basic parametric: perc parametric: norm case: basic case: perc case: norm nonparametric: basic nonparametric: perc nonparametric: norm dissProf logPKcomp sigEmax SSasymp SSasympOff SSasympOrig SSbiexp SSfol SSfpl SSgompertz SSlogis SSmicmen SSweibull 0.4 0.5 0.6 0.7 0.8 0.9 1.0 CR CR aggregated over the models fixed varcov sigma Teja Turk ETH Zurich (D-MATH, SfS) 8/14
  • 27. Coverage rate by method and parameter type (CI) 0.2 0.4 0.6 0.8 1.0 CR intervals 0.2 0.4 0.6 0.8 1.0 CR Wald 0.50 0.80 0.95 0.2 0.4 0.6 0.8 1.0 1 − α CR adjusted Wald parametric: basic parametric: perc 0.50 0.80 0.95 1 − α parametric: norm case: basic case: perc 0.50 0.80 0.95 1 − α case: norm nonparametric: basic nonparametric: perc 0.50 0.80 0.95 1 − α nonparametric: norm fixed varcov sigma Teja Turk ETH Zurich (D-MATH, SfS) 9/14
  • 28. Prediction interval approaches Unobserved group Var fk0 β, 0 + Var[fk0(β, bk)] Teja Turk ETH Zurich (D-MATH, SfS) 10/14
  • 29. Prediction interval approaches Unobserved group Var fk0 β, 0 + Var[fk0(β, bk)] Non-bootstrap: • linear approximation: fk0 β, 0 , fk0(β, bk ) at (β, 0) Teja Turk ETH Zurich (D-MATH, SfS) 10/14
  • 30. Prediction interval approaches Unobserved group Var fk0 β, 0 + Var[fk0(β, bk)] Non-bootstrap: • linear approximation: fk0 β, 0 , fk0(β, bk ) at (β, 0) • distribution-free approach: empirical quantiles of the population level residuals Teja Turk ETH Zurich (D-MATH, SfS) 10/14
  • 31. Prediction interval approaches Unobserved group Var fk0 β, 0 + Var[fk0(β, bk)] Non-bootstrap: • linear approximation: fk0 β, 0 , fk0(β, bk ) at (β, 0) • distribution-free approach: empirical quantiles of the population level residuals Bootstrap: • linear approximation Teja Turk ETH Zurich (D-MATH, SfS) 10/14
  • 32. Prediction interval approaches Unobserved group Var fk0 β, 0 + Var[fk0(β, bk)] Non-bootstrap: • linear approximation: fk0 β, 0 , fk0(β, bk ) at (β, 0) • distribution-free approach: empirical quantiles of the population level residuals Bootstrap: • linear approximation • prediction error sample: yk0−yk0−e∗ s∗ ∼ z Teja Turk ETH Zurich (D-MATH, SfS) 10/14
  • 33. Prediction interval approaches Unobserved group Var fk0 β, 0 + Var[fk0(β, bk)] Non-bootstrap: • linear approximation: fk0 β, 0 , fk0(β, bk ) at (β, 0) • distribution-free approach: empirical quantiles of the population level residuals Bootstrap: • linear approximation • prediction error sample: yk0−yk0−e∗ s∗ ∼ z Observed group Var fk0 β, bk − fk0(β, bk) Teja Turk ETH Zurich (D-MATH, SfS) 10/14
  • 34. Prediction interval approaches Unobserved group Var fk0 β, 0 + Var[fk0(β, bk)] Non-bootstrap: • linear approximation: fk0 β, 0 , fk0(β, bk ) at (β, 0) • distribution-free approach: empirical quantiles of the population level residuals Bootstrap: • linear approximation • prediction error sample: yk0−yk0−e∗ s∗ ∼ z Observed group Var fk0 β, bk − fk0(β, bk) Non-bootstrap: • the Delta method: fk0 β, bk − fk0(β, bk ) • the FOCE method: fk0(β, bk ) at β, bk • the FO method: fk0 β, bk , fk0(β, bk ) at (β, 0) Teja Turk ETH Zurich (D-MATH, SfS) 10/14
  • 35. Prediction interval approaches Unobserved group Var fk0 β, 0 + Var[fk0(β, bk)] Non-bootstrap: • linear approximation: fk0 β, 0 , fk0(β, bk ) at (β, 0) • distribution-free approach: empirical quantiles of the population level residuals Bootstrap: • linear approximation • prediction error sample: yk0−yk0−e∗ s∗ ∼ z Observed group Var fk0 β, bk − fk0(β, bk) Non-bootstrap: • the Delta method: fk0 β, bk − fk0(β, bk ) • the FOCE method: fk0(β, bk ) at β, bk • the FO method: fk0 β, bk , fk0(β, bk ) at (β, 0) Bootstrap: • the Delta method • the FOCE method • the FO method Teja Turk ETH Zurich (D-MATH, SfS) 10/14
  • 36. Prediction interval approaches Unobserved group Var fk0 β, 0 + Var[fk0(β, bk)] Non-bootstrap: • linear approximation: fk0 β, 0 , fk0(β, bk ) at (β, 0) • distribution-free approach: empirical quantiles of the population level residuals Bootstrap: • linear approximation • prediction error sample: yk0−yk0−e∗ s∗ ∼ z Observed group Var fk0 β, bk − fk0(β, bk) Non-bootstrap: • the Delta method: fk0 β, bk − fk0(β, bk ) • the FOCE method: fk0(β, bk ) at β, bk • the FO method: fk0 β, bk , fk0(β, bk ) at (β, 0) Bootstrap: • the Delta method • the FOCE method • the FO method • prediction error sample Teja Turk ETH Zurich (D-MATH, SfS) 10/14
  • 37. SSasympOff: diag_2 0.80 0.85 0.90 0.95 1.00 1.05 1.10 CR unobserved: nonboot Lin. approximation Distr.-free unobserved: boot Lin. approximation Pred. error sample 16.67 50 66.67 105 120 150 0.80 0.85 0.90 0.95 1.00 1.05 1.10 CR observed: nonboot FOCE method FO method Delta method 16.67 50 66.67 105 120 150 observed: boot Pred. error sample FOCE method FO method Delta method Teja Turk ETH Zurich (D-MATH, SfS) 11/14
  • 38. Coverage rate averaged over parameters and models (95%-PI) CR averaged over random Unobs.: nonBLA Unobs.: nonBDF Unobs.: BLA Unobs.: BPES Obs.: nonBFOCE Obs.: nonBFO Obs.: nonB∆ Obs.: BPES Obs.: BFOCE Obs.: BFO Obs.: B∆ dissProf logPKcomp sigEmax SSasymp SSasympOff SSasympOrig SSbiexp SSfol SSfpl SSgompertz SSlogis SSmicmen SSweibull interpolation extrapolation Teja Turk ETH Zurich (D-MATH, SfS) 12/14
  • 39. Coverage rate averaged over parameters and models (95%-PI) CR averaged over random Unobs.: nonBLA Unobs.: nonBDF Unobs.: BLA Unobs.: BPES Obs.: nonBFOCE Obs.: nonBFO Obs.: nonB∆ Obs.: BPES Obs.: BFOCE Obs.: BFO Obs.: B∆ dissProf logPKcomp sigEmax SSasymp SSasympOff SSasympOrig SSbiexp SSfol SSfpl SSgompertz SSlogis SSmicmen SSweibull 0.75 0.80 0.85 0.90 0.95 1.00 CR CR aggregated over the models interpolation extrapolation Teja Turk ETH Zurich (D-MATH, SfS) 12/14
  • 40. Coverage rate by method and parameter type (PI) 0.4 0.6 0.8 1.0 CR Unobs.: nonBLA Unobs.: nonBDF Unobs.: BLA Unobs.: BPES 0.4 0.6 0.8 1.0 CR Obs.: nonBFOCE Obs.: nonBFO Obs.: nonB∆ 0.50 0.80 0.95 1 − α Obs.: BPES 0.50 0.80 0.95 0.4 0.6 0.8 1.0 1 − α CR Obs.: BFOCE 0.50 0.80 0.95 1 − α Obs.: BFO 0.50 0.80 0.95 1 − α Obs.: B∆ interpolation extrapolation Teja Turk ETH Zurich (D-MATH, SfS) 13/14
  • 41. Conclusions • differences in performance of CIs between the fixed effects, the variance-covariance components and σ Teja Turk ETH Zurich (D-MATH, SfS) 14/14
  • 42. Conclusions • differences in performance of CIs between the fixed effects, the variance-covariance components and σ • no significant differences in performance of PIs between the interpolation and extrapolation covariate points Teja Turk ETH Zurich (D-MATH, SfS) 14/14
  • 43. Conclusions • differences in performance of CIs between the fixed effects, the variance-covariance components and σ • no significant differences in performance of PIs between the interpolation and extrapolation covariate points • parametric bootstrap successfully used for both CIs and PIs Teja Turk ETH Zurich (D-MATH, SfS) 14/14
  • 44. Conclusions • differences in performance of CIs between the fixed effects, the variance-covariance components and σ • no significant differences in performance of PIs between the interpolation and extrapolation covariate points • parametric bootstrap successfully used for both CIs and PIs • CIs tend to be too narrow or shifted Teja Turk ETH Zurich (D-MATH, SfS) 14/14
  • 45. Conclusions • differences in performance of CIs between the fixed effects, the variance-covariance components and σ • no significant differences in performance of PIs between the interpolation and extrapolation covariate points • parametric bootstrap successfully used for both CIs and PIs • CIs tend to be too narrow or shifted • PIs tend to be too wide Teja Turk ETH Zurich (D-MATH, SfS) 14/14
  • 46. Coverage rate averaged over parameters (95%-CI) diag_1 diag_1 diag_1 diag_1 diag_1 diag_1 diag_1 diag_1 diag_1 diag_1 diag_1 diag_1 diag_1 diag_2 diag_2 diag_2 diag_2 diag_2 diag_2 diag_2 diag_2 diag_2 diag_2 diag_2 diag_2 diag_2 symm_2 symm_2 symm_2 symm_2 symm_2 symm_2 symm_2 symm_2 symm_2 symm_2 symm_2 symm_2 symm_2 dissProf logPKcomp sigEmax SSasymp SSasympOff SSasympOrig SSbiexp SSfol SSfpl SSgompertz SSlogis SSmicmen SSweibull intervals Wald adjusted Wald parametric: basic parametric: perc parametric: norm case: basic case: perc case: norm nonparametric: basic nonparametric: perc nonparametric: norm fixed varcov sigma Teja Turk ETH Zurich (D-MATH, SfS)
  • 47. Coverage rate averaged over subjects (95%-PI) diag_1 diag_1 diag_1 diag_1 diag_1 diag_1 diag_1 diag_1 diag_1 diag_1 diag_1 diag_1 diag_1 diag_2 diag_2 diag_2 diag_2 diag_2 diag_2 diag_2 diag_2 diag_2 diag_2 diag_2 diag_2 diag_2 symm_2 symm_2 symm_2 symm_2 symm_2 symm_2 symm_2 symm_2 symm_2 symm_2 symm_2 symm_2 symm_2 dissProf logPKcomp sigEmax SSasymp SSasympOff SSasympOrig SSbiexp SSfol SSfpl SSgompertz SSlogis SSmicmen SSweibull Unobs.: nonBLA Unobs.: nonBDF Unobs.: BLA Unobs.: BPES Obs.: nonBFOCE Obs.: nonBFO Obs.: nonB∆ Obs.: BPES Obs.: BFOCE Obs.: BFO Obs.: B∆ interpolation extrapolation Teja Turk ETH Zurich (D-MATH, SfS)