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Christian Ritz Concentration- Semi-parametric and response setup non-parametric approaches to Parametric models concentration-response modelling Non- parametric approach Semi- parametric Christian Ritz models Results University of Copenhagen, Denmark Concluding remarks Leuven, September 25 2008 Christian Ritz (U. Copenhagen) NCS 2008 1 / 15

Concentration-response setup Parameter of interest: effect concentration (such as EC50) Christian Ritz Concentration- Concentration-response setting: response setup biological response y i to stimulus x i Parametric models (stimulus applied for a range of concentrations) Non- parametric approach Semi- Response types: parametric models continuous (length, weight) Results Concluding counts (number of fronds, juveniles, offspring, roots) remarks quantal (number of organisms responding out of a total) (active/inactive, dead/alive, immobile/mobile) Christian Ritz (U. Copenhagen) NCS 2008 2 / 15

Concentration-response setup Parameter of interest: effect concentration (such as EC50) Christian Ritz Concentration- Concentration-response setting: response setup biological response y i to stimulus x i Parametric models (stimulus applied for a range of concentrations) Non- parametric approach Semi- Response types: parametric models continuous (length, weight) Results Concluding counts (number of fronds, juveniles, offspring, roots) remarks quantal (number of organisms responding out of a total) (active/inactive, dead/alive, immobile/mobile) Christian Ritz (U. Copenhagen) NCS 2008 2 / 15

Concentration-response setup Parameter of interest: effect concentration (such as EC50) Christian Ritz Concentration- Concentration-response setting: response setup biological response y i to stimulus x i Parametric models (stimulus applied for a range of concentrations) Non- parametric approach Semi- Response types: parametric models continuous (length, weight) Results Concluding counts (number of fronds, juveniles, offspring, roots) remarks quantal (number of organisms responding out of a total) (active/inactive, dead/alive, immobile/mobile) Christian Ritz (U. Copenhagen) NCS 2008 2 / 15

Parametric models General conditional mean structure: Christian Ritz Concentration- E ( y i | x i ) = f P ( x i , β ) response setup Details: Parametric models f P nonlinear mean function in β Non- parametric ◮ monotonous: log-logistic, Weibull, . . . approach ◮ non-monotonous: polynomials, biphasic models Semi- parametric models β unknown parameter to be estimated Results Concluding Methods of estimation: remarks least squares maximum likelihood quasi-likelihood Christian Ritz (U. Copenhagen) NCS 2008 3 / 15

Limitations Rough figures obtained from ECVAM: Christian Ritz 50% fitted nicely by common parametric models Concentration- response 20% borderline fits setup Parametric 30% no acceptable fit achievable models Non- parametric approach Problem: Empirically based models Semi- parametric models Consequences: Results Concluding Inadequate summary of the data structure remarks Risk of bias in estimates of EC values and other parameters of interest Christian Ritz (U. Copenhagen) NCS 2008 4 / 15

Limitations Rough figures obtained from ECVAM: Christian Ritz 50% fitted nicely by common parametric models Concentration- response 20% borderline fits setup Parametric 30% no acceptable fit achievable models Non- parametric approach Problem: Empirically based models Semi- parametric models Consequences: Results Concluding Inadequate summary of the data structure remarks Risk of bias in estimates of EC values and other parameters of interest Christian Ritz (U. Copenhagen) NCS 2008 4 / 15

Limitations Rough figures obtained from ECVAM: Christian Ritz 50% fitted nicely by common parametric models Concentration- response 20% borderline fits setup Parametric 30% no acceptable fit achievable models Non- parametric approach Problem: Empirically based models Semi- parametric models Consequences: Results Concluding Inadequate summary of the data structure remarks Risk of bias in estimates of EC values and other parameters of interest Christian Ritz (U. Copenhagen) NCS 2008 4 / 15

Non-parametric models Complete unspecified conditional mean: Christian Ritz E ( y i | x i ) = f NP ( x i ) Concentration- response setup Parametric Estimation by local linear regression: models Non- parametric approach choose a bandwidth h ( x ) 1 Semi- � � x i ′ − x parametric calculate weights w i ′ ( x ) = W 2 h ( x ) models (only using x i s in the interval ] x − h ( x ) , x + h ( x )[ ) Results Concluding fit weighted linear regression of y i ′ versus x i ′ with 3 remarks weights w i ′ ( x ) define ˆ f NP ( x ) to be the estimated intercept 4 Christian Ritz (U. Copenhagen) NCS 2008 5 / 15

More on local linear regression How to balance bias-variance trade-off? Christian Ritz Concentration- How to choose the bandwidth? Variable bandwidth? response setup Parametric In practice used for both continuous and quantal data! models Non- Local likelihood approaches exist (Loader, 1999) parametric approach Semi- Implementations in R : parametric models Results ◮ loess() in stats (standard installation) Concluding ◮ locfit() in the locfit package remarks Christian Ritz (U. Copenhagen) NCS 2008 6 / 15

Semi-parametric models Maybe there exists a compromise: Christian Ritz imposing some basic concentration-response structure Concentration- response leaving enough flexibility for capturing non-standard setup Parametric patterns in the data models Non- parametric Model-robust approach (Nottingham & Birch, 2000): approach Semi- parametric f MR ( x ) = λ f NP ( x ) + ( 1 − λ ) f P ( x , β ) models Results Concluding λ ∈ [ 0 , 1 ] controls the mixing of components remarks Separate estimation of parametric and non-parametric components Christian Ritz (U. Copenhagen) NCS 2008 7 / 15

Semi-parametric models Maybe there exists a compromise: Christian Ritz imposing some basic concentration-response structure Concentration- response leaving enough flexibility for capturing non-standard setup Parametric patterns in the data models Non- parametric Model-robust approach (Nottingham & Birch, 2000): approach Semi- parametric f MR ( x ) = λ f NP ( x ) + ( 1 − λ ) f P ( x , β ) models Results Concluding λ ∈ [ 0 , 1 ] controls the mixing of components remarks Separate estimation of parametric and non-parametric components Christian Ritz (U. Copenhagen) NCS 2008 7 / 15

Combining model fits Optimal mixing parameter λ determined from: Christian Ritz Concentration- response n setup PRESS ∗ = � ˆ � f MR � g i − i ( x i ) , λ Parametric models i = 1 Non- parametric using leave-one-out predictions: ˆ f MR approach − i ( x i ) Semi- parametric models Least squares criterion (common choice): Results Concluding g i ( z , λ ) = w i ( y i − z ) 2 / g 0 ( λ ) remarks ( g 0 some weight function) Christian Ritz (U. Copenhagen) NCS 2008 8 / 15

Implementation R package: mrdrc Christian Ritz Concentration- also available as a GUI: response setup Parametric ◮ http://130.75.68.4:8080/deploy/doseresponse/ models Non- parametric approach Semi- parametric models Results Concluding remarks Christian Ritz (U. Copenhagen) NCS 2008 9 / 15

Quantal data ( ˆ λ = 0 . 65) Christian Ritz Concentration- response 1.0 setup Parametric 0.8 models ● ● Non- parametric Matured/total 0.6 approach ● Semi- parametric 0.4 models Results 0.2 Concluding ● remarks ● 0.0 0.0 0.2 0.4 0.6 0.8 Concentration Christian Ritz (U. Copenhagen) NCS 2008 10 / 15

Continuous data ( ˆ λ = 1) Christian Ritz Concentration- 0.8 response setup ● ● 0.7 ● Parametric ● ● ● ● ● ● models ● ● ● ● ● ● ● ● ● ● ● 0.6 Non- ● ● ● ● parametric ● ● ● approach ● Response 0.5 ● ● Semi- parametric ● ● 0.4 models ● ● ● ● Results 0.3 ● ● ● Concluding ● remarks ● 0.2 ● ● ● ● ● ● 10 20 50 100 200 500 1000 Concentration Christian Ritz (U. Copenhagen) NCS 2008 11 / 15

Simulation: continuous data - null Christian Ritz Model Method Replicates EC True Mean Width Coverage (%) Concentration- Log-logistic Parametric 1 10 1.46 1.53 2.73 95.3 response model 20 1.92 1.97 2.51 94.5 setup 7 concs 50 3.06 3.10 2.23 92.8 Parametric 2 10 1.46 1.49 1.17 95.3 models 20 1.92 1.95 1.11 95.2 50 3.06 3.09 1.06 94.1 Non- parametric 3 10 1.46 1.48 0.88 97.3 approach 20 1.92 1.94 0.84 97.1 50 3.06 3.07 0.82 94.4 Semi- parametric Semi 1 10 1.46 1.36 1.66 85.1 models -parametric 20 1.92 1.91 1.18 84.2 Results (0.23) 50 3.06 3.25 1.32 78.6 Concluding 2 10 1.46 1.39 0.93 76.2 remarks 20 1.92 1.88 0.67 76.5 (0.14) 50 3.06 3.08 0.72 83.5 3 10 1.46 1.40 0.68 77.6 20 1.92 1.89 0.57 79.0 (0.11) 50 3.06 3.07 0.60 85.5 Christian Ritz (U. Copenhagen) NCS 2008 12 / 15