##### Document Text Contents

Page 1

Springer Series in Statistics

Advisors:

P. Bickel, P. Diggle, S. Fienberg, K. Krickeberg,

I. Olkin, N. Wermuth, S. Zeger

Springer

New York

Berlin

Heidelberg

Hong Kong

London

Milan

Paris

Tokyo

Page 2

S. Huet

A. Bouvier

M.-A. Poursat

E. Jolivet

Statistical Tools for

Nonlinear Regression

A Practical Guide With S-PLUS and

R Examples

Second Edition

Page 120

4.2 Diagnostics of Model Misspecifications with Graphics 109

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��x� b��

Figure 4.13. Simulated example: Graph of sensitivity functions calculated in θ̂

versus x

where the εi, i = 1, . . . 10, are independent variables with the same distribu-

tion as ε.

x 0.1 0.6 1.1 1.6 2.1

Y 0.1913 0.3436 0.6782 0.8954 0.7809

x 2.6 3.1 3.6 4.1 4.6

Y 0.9220 0.9457 0.9902 0.9530 1.0215

Model The regression function is

f(x, θ) = θ1(1 − exp(−θ2x)),

and the variances are homogeneous: Var(εi) = σ2.

Method The parameters are estimated by minimizing the sum of squares,

C(θ); see Equation (1.10).

Results

Parameters Estimated Values Estimated Standard Errors

θ1 1.1006 0.04

θ2 0.9512 0.14

As expected, the estimated standard errors are smaller than they were

in the first data set. Comparing Figures 4.14 and 4.12 clearly shows that the

experimental design in the second data set allows us to estimate the asymptote

θ1 better.

Page 121

110 4 Diagnostics of Model Misspecification

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Figure 4.14. Simulated example: Second data set, graph of simulated data and

fitted value of the response

4.3 Diagnostics of Model Misspecifications with Tests

4.3.1 RIA of Cortisol: Comparison of Nested Models

In some situations it is easy to find possible alternatives to the regression

function or the variance function. For example, if we choose a symmetric

sigmoidally shaped curve for the cortisol data, it is reasonable to wonder if an

asymmetric curve would not be more appropriate.

Let us consider the models defined in Equations (4.4) and (4.5). If we

suspect an asymmetry of the curve, we can test hypothesis H: θ5 = 1 against

the alternative A: θ5 �= 1. Let us apply a likelihood ratio test. Let θ̂H be

the estimator of θ under hypothesis H and σ̂2Hf

2(xi, θ̂H) be the estimator

of Var(εij). In the same way, let us define σ̂2Af

2(xi, θ̂A) as the estimator of

Var(εij) under the alternative A. The test statistic

SL =

k∑

i=1

log σ̂2Hf

2(xi, θ̂H) −

k∑

i=1

log σ̂2Af

2(xi, θ̂A)

equals 39, which must be compared to the 0.95 quantile of a χ2 with one degree

of freedom. The hypothesis of a symmetric regression function is rejected.

The following section presents a likelihood ratio test based on the com-

parison between the considered model and a bigger model defined when the

response is observed with replications.

4.3.2 Tests Using Replications

Let us return to the general nonlinear regression model with replications. We

assume that for each value of x, xi, i = 1, . . . k, the number ni of observed

Page 240

Statistical Tools for Nonlinear Regression, (Second Edition), presents

methods for analyzing data using parametric nonlinear regression models.

The new edition has been expanded to include binomial, multinomial and

Poisson non-linear models. Using examples from experiments in agronomy

and biochemistry, it shows how to apply these methods. It concentrates on

presenting the methods in an intuitive way rather than developping the the-

oretical backgrounds.

The examples are analyzed with the free software nls2 updated to deal

with the new models included in the second edition. The nls2 package is

implemented in S-Plus and R. Its main advantages are to make the model

building, estimation and validation tasks, easy to do. More precisely,

• complex models can be easily described using a symbolic syntax. The re-

gression function as well as the variance function can be defined explicitly

as functions of independent variables and of unknown parameters or they

can be defined as the solution to a system of differential equations. More-

over, constraints on the parameters can easily be added to the model. It is

thus possible to test nested hypotheses and to compare several data sets.

• several additional tools are included in the package for calculating con-

fidence regions for functions of parameters or calibration intervals, using

classical methodology or bootstrap. Moreover, some graphical tools are

proposed for visualizing the fitted curves, the residuals, the confidence

regions, and the numerical estimation procedure.

This book is aimed at scientists who are not familiar with statistical theory,

but have a basic knowledge of statistical concepts. It includes methods based

on classical nonlinear regression theory and more modern methods, such as

bootstrap, which have proved effective in practice. The additional chapters of

the second edition assume some practical experience in data analysis using

generalized linear models. The book will be of interest both for practitioners

as a guide and reference book, and for students, as a tutorial book.

Sylvie Huet and Emmanuel Jolivet are senior researchers and Annie Bou-

vier is computing engineer at INRA, National Institute of Agronomical Re-

search, France; Marie-Anne Poursat is Associate professor of statistics at the

University Paris XI.

Page 241

BAYESIAN NONPARAMETRICS

JAYANTA K. GHOSH and R.V. RAMAMOORTHI

Bayesian nonparametrics has grown tremen-

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time series data, the book also presents an up-to-

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els, including ARCH/GARCH models and

threshold models. A compact view on linear

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www.springer-ny.com

Springer Series in Statistics

Advisors:

P. Bickel, P. Diggle, S. Fienberg, K. Krickeberg,

I. Olkin, N. Wermuth, S. Zeger

Springer

New York

Berlin

Heidelberg

Hong Kong

London

Milan

Paris

Tokyo

Page 2

S. Huet

A. Bouvier

M.-A. Poursat

E. Jolivet

Statistical Tools for

Nonlinear Regression

A Practical Guide With S-PLUS and

R Examples

Second Edition

Page 120

4.2 Diagnostics of Model Misspecifications with Graphics 109

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Figure 4.13. Simulated example: Graph of sensitivity functions calculated in θ̂

versus x

where the εi, i = 1, . . . 10, are independent variables with the same distribu-

tion as ε.

x 0.1 0.6 1.1 1.6 2.1

Y 0.1913 0.3436 0.6782 0.8954 0.7809

x 2.6 3.1 3.6 4.1 4.6

Y 0.9220 0.9457 0.9902 0.9530 1.0215

Model The regression function is

f(x, θ) = θ1(1 − exp(−θ2x)),

and the variances are homogeneous: Var(εi) = σ2.

Method The parameters are estimated by minimizing the sum of squares,

C(θ); see Equation (1.10).

Results

Parameters Estimated Values Estimated Standard Errors

θ1 1.1006 0.04

θ2 0.9512 0.14

As expected, the estimated standard errors are smaller than they were

in the first data set. Comparing Figures 4.14 and 4.12 clearly shows that the

experimental design in the second data set allows us to estimate the asymptote

θ1 better.

Page 121

110 4 Diagnostics of Model Misspecification

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Figure 4.14. Simulated example: Second data set, graph of simulated data and

fitted value of the response

4.3 Diagnostics of Model Misspecifications with Tests

4.3.1 RIA of Cortisol: Comparison of Nested Models

In some situations it is easy to find possible alternatives to the regression

function or the variance function. For example, if we choose a symmetric

sigmoidally shaped curve for the cortisol data, it is reasonable to wonder if an

asymmetric curve would not be more appropriate.

Let us consider the models defined in Equations (4.4) and (4.5). If we

suspect an asymmetry of the curve, we can test hypothesis H: θ5 = 1 against

the alternative A: θ5 �= 1. Let us apply a likelihood ratio test. Let θ̂H be

the estimator of θ under hypothesis H and σ̂2Hf

2(xi, θ̂H) be the estimator

of Var(εij). In the same way, let us define σ̂2Af

2(xi, θ̂A) as the estimator of

Var(εij) under the alternative A. The test statistic

SL =

k∑

i=1

log σ̂2Hf

2(xi, θ̂H) −

k∑

i=1

log σ̂2Af

2(xi, θ̂A)

equals 39, which must be compared to the 0.95 quantile of a χ2 with one degree

of freedom. The hypothesis of a symmetric regression function is rejected.

The following section presents a likelihood ratio test based on the com-

parison between the considered model and a bigger model defined when the

response is observed with replications.

4.3.2 Tests Using Replications

Let us return to the general nonlinear regression model with replications. We

assume that for each value of x, xi, i = 1, . . . k, the number ni of observed

Page 240

Statistical Tools for Nonlinear Regression, (Second Edition), presents

methods for analyzing data using parametric nonlinear regression models.

The new edition has been expanded to include binomial, multinomial and

Poisson non-linear models. Using examples from experiments in agronomy

and biochemistry, it shows how to apply these methods. It concentrates on

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oretical backgrounds.

The examples are analyzed with the free software nls2 updated to deal

with the new models included in the second edition. The nls2 package is

implemented in S-Plus and R. Its main advantages are to make the model

building, estimation and validation tasks, easy to do. More precisely,

• complex models can be easily described using a symbolic syntax. The re-

gression function as well as the variance function can be defined explicitly

as functions of independent variables and of unknown parameters or they

can be defined as the solution to a system of differential equations. More-

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• several additional tools are included in the package for calculating con-

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This book is aimed at scientists who are not familiar with statistical theory,

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on classical nonlinear regression theory and more modern methods, such as

bootstrap, which have proved effective in practice. The additional chapters of

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generalized linear models. The book will be of interest both for practitioners

as a guide and reference book, and for students, as a tutorial book.

Sylvie Huet and Emmanuel Jolivet are senior researchers and Annie Bou-

vier is computing engineer at INRA, National Institute of Agronomical Re-

search, France; Marie-Anne Poursat is Associate professor of statistics at the

University Paris XI.

Page 241

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