IntroductionNonlinear Models

Sigmoid CurvesAssess the Approximation

Numerical Case StudyConclusions

Sigmoid curves and a case for close-to-linearnonlinear models

Charles Y. TanMBSW

May 19th, 2009

All models are wrong, some are more useful than others.

Charles Y. Tan MBSW May 19th, 2009 Sigmoid curves and a case for close-to-linear nonlinear models

IntroductionNonlinear Models

Sigmoid CurvesAssess the Approximation

Numerical Case StudyConclusions

Outline

Introduction

Nonlinear Models

Sigmoid Curves

Assess the Approximation

Numerical Case Study

Conclusions

Charles Y. Tan MBSW May 19th, 2009 Sigmoid curves and a case for close-to-linear nonlinear models

IntroductionNonlinear Models

Sigmoid CurvesAssess the Approximation

Numerical Case StudyConclusions

Applications and ContextStatistical Models

Sigmoid curves are common in biological sciences

I Quantitative bioanalytical methodsI ImmunoassaysI BioassaysI Hill equation (1910)

I PharmacologyI Concentration-effect or dose-response curvesI Emax model (1964)

I Growth curvesI (Population or organ) size as function of timeI Mechanistic and empiricalI Autocatalytic model (1838, 1908)

Charles Y. Tan MBSW May 19th, 2009 Sigmoid curves and a case for close-to-linear nonlinear models

IntroductionNonlinear Models

Sigmoid CurvesAssess the Approximation

Numerical Case StudyConclusions

Applications and ContextStatistical Models

Statistics: old favorite and new question

I Classic models: (four-parameter) logistic modelsI Hill equation, Emax model, and autocatalytic model are the

same models: logistic models.I Theyre symmetric.

I New question: what model to use when data are asymmetric

I Answer from some quarters: five-parameter logistic (5PL)(Richards model)

I Ratkowsky (1983, 1990): significant intrinsic curvature, aparticularly unfortunate model, abuse of Occams Razor

I Seber and Wild (1989): Bad ill-conditioning and convergenceproblems

Charles Y. Tan MBSW May 19th, 2009 Sigmoid curves and a case for close-to-linear nonlinear models

IntroductionNonlinear Models

Sigmoid CurvesAssess the Approximation

Numerical Case StudyConclusions

Applications and ContextStatistical Models

Statistics: old favorite and new question

I Classic models: (four-parameter) logistic modelsI Hill equation, Emax model, and autocatalytic model are the

same models: logistic models.I Theyre symmetric.

I New question: what model to use when data are asymmetricI Answer from some quarters: five-parameter logistic (5PL)

(Richards model)

I Ratkowsky (1983, 1990): significant intrinsic curvature, aparticularly unfortunate model, abuse of Occams Razor

I Seber and Wild (1989): Bad ill-conditioning and convergenceproblems

Charles Y. Tan MBSW May 19th, 2009 Sigmoid curves and a case for close-to-linear nonlinear models

IntroductionNonlinear Models

Sigmoid CurvesAssess the Approximation

Numerical Case StudyConclusions

Applications and ContextStatistical Models

Statistics: old favorite and new question

I Classic models: (four-parameter) logistic modelsI Hill equation, Emax model, and autocatalytic model are the

same models: logistic models.I Theyre symmetric.

I New question: what model to use when data are asymmetricI Answer from some quarters: five-parameter logistic (5PL)

(Richards model)I Ratkowsky (1983, 1990): significant intrinsic curvature, a

particularly unfortunate model, abuse of Occams RazorI Seber and Wild (1989): Bad ill-conditioning and convergence

problems

Charles Y. Tan MBSW May 19th, 2009 Sigmoid curves and a case for close-to-linear nonlinear models

IntroductionNonlinear Models

Sigmoid CurvesAssess the Approximation

Numerical Case StudyConclusions

Standard ApproachRelative CurvatureClose-to-Linear

Nonlinear regression

yi = f (xi ;) + i , i = 1, 2, . . . , n,

I Nonlinearity of f with respect to : defining characteristics

I Nonlinearity of f with respect to x : incidental

I Homogeneous variance: i s are i.i.d. N(0, 2)

Maximum Likelihood = Least SquaresObjective function:

S() = (y f()) (y f())

Charles Y. Tan MBSW May 19th, 2009 Sigmoid curves and a case for close-to-linear nonlinear models

IntroductionNonlinear Models

Sigmoid CurvesAssess the Approximation

Numerical Case StudyConclusions

Standard ApproachRelative CurvatureClose-to-Linear

Nonlinear regression

yi = f (xi ;) + i , i = 1, 2, . . . , n,

I Nonlinearity of f with respect to : defining characteristics

I Nonlinearity of f with respect to x : incidental

I Homogeneous variance: i s are i.i.d. N(0, 2)

Maximum Likelihood = Least SquaresObjective function:

S() = (y f()) (y f())

Charles Y. Tan MBSW May 19th, 2009 Sigmoid curves and a case for close-to-linear nonlinear models

IntroductionNonlinear Models

Sigmoid CurvesAssess the Approximation

Numerical Case StudyConclusions

Standard ApproachRelative CurvatureClose-to-Linear

1st order approximation of the model (w.r.t. parameter)

f() f() + F( ),

where

F = F() =

(f (xi ;)

j

=

)nk

Plug it in the definition of S(), we have a partial 2nd orderexpansion of S() near :

S() 2F( ) + ( )FF( )

Charles Y. Tan MBSW May 19th, 2009 Sigmoid curves and a case for close-to-linear nonlinear models

IntroductionNonlinear Models

Sigmoid CurvesAssess the Approximation

Numerical Case StudyConclusions

Standard ApproachRelative CurvatureClose-to-Linear

Common framework for inference

S() S() ( )FF( ) F(FF)1F

S() (I F(FF)1F)

Since F(FF)1F is idempotent

Local inference:

( )FF( )S()

kn k

Fk,nk

Global inference:

S() S()S()

kn k

Fk,nk

Charles Y. Tan MBSW May 19th, 2009 Sigmoid curves and a case for close-to-linear nonlinear models

IntroductionNonlinear Models

Sigmoid CurvesAssess the Approximation

Numerical Case StudyConclusions

Standard ApproachRelative CurvatureClose-to-Linear

Common framework for inference

S() S() ( )FF( ) F(FF)1F

S() (I F(FF)1F)Since F(FF)

1F is idempotent

Local inference:

( )FF( )S()

kn k

Fk,nk

Global inference:

S() S()S()

kn k

Fk,nk

Charles Y. Tan MBSW May 19th, 2009 Sigmoid curves and a case for close-to-linear nonlinear models

IntroductionNonlinear Models

Sigmoid CurvesAssess the Approximation

Numerical Case StudyConclusions

Standard ApproachRelative CurvatureClose-to-Linear

Common framework for inference

S() S() ( )FF( ) F(FF)1F

S() (I F(FF)1F)Since F(FF)

1F is idempotentLocal inference:

( )FF( )S()

kn k

Fk,nk

Global inference:

S() S()S()

kn k

Fk,nk

Charles Y. Tan MBSW May 19th, 2009 Sigmoid curves and a case for close-to-linear nonlinear models

IntroductionNonlinear Models

Sigmoid CurvesAssess the Approximation

Numerical Case StudyConclusions

Standard ApproachRelative CurvatureClose-to-Linear

Implications of Local and Global Inferences

I Local: always produces ellipsoids in the parameter spaceI Implicitly assumes S() is an elliptic paraboloid near

I Both the shape and cutoff depend on the approximationI Is S() always elliptic paraboloid like near ?

I Global (S() c): shape varies case by caseI Follows the Likelihood Principle (Exact)I The cutoff c depends on the approximationI What if S() has multiple minimums and the difference in

their S values is small?

Charles Y. Tan MBSW May 19th, 2009 Sigmoid curves and a case for close-to-linear nonlinear models

IntroductionNonlinear Models

Sigmoid CurvesAssess the Approximation

Numerical Case StudyConclusions

Standard ApproachRelative CurvatureClose-to-Linear

Intrinsic and parameter-effect curvatures

Expectation surface or solution locus: f() RnIts approximation:

f() f() + F( )

I Planar assumption

I The expectation surface is close to its tangent plane.I Intrinsic curvature: deviation at f().

I Uniform-coordinate assumption

I Straight parallel equispaced lines in the parameter space Rkmap into straight parallel equispaced lines in the expectationsurface (as they do in the tangent plane).

I Parameter-effect curvature: deviation at f().

Charles Y. Tan MBSW May 19th, 2009 Sigmoid curves and a case for close-to-linear nonlinear models

IntroductionNonlinear Models

Sigmoid CurvesAssess the Approximation

Numerical Case StudyConclusions

Standard ApproachRelative CurvatureClose-to-Linear

Intrinsic and parameter-effect curvatures

Expectation surface or solution locus: f() RnIts approximation:

f() f() + F( )

I Planar assumptionI The expectation surface is close to its tangent plane.I Intrinsic curvature: deviation at f().

I Uniform-coordinate assumption

I Straight parallel equispaced lines in the parameter space Rkmap into straight parallel equispaced lines in the expectationsurface (as they do in the tangent plane).

I Parameter-effect curvature: deviation at f().

Charles Y. Tan MBSW May 19th, 2009 Sigmoid curves and a case for close-to-linear nonlinear models

IntroductionNonlinear Models

Sigmoid CurvesAssess the Approximation

Numerical Case StudyConclusions

Stan