Fitting models to biological data using linear and nonlinear regression. A practical guide to curve fitting.

*(English)*Zbl 1081.62100
Oxford: Oxford University Press (ISBN 0-19-517180-2/pbk; 0-19-517179-9/hbk). 351 p. (2004).

The present book is designed to provide a useful presentation of ten detailed, well written parts of regression analysis techniques, especially nonlinear analysis, a useful tool in the analysis of biological (and other) data. In particular the aim of the book is to give a consistent and self-contained overview on fitting models to biological data using linear and nonlinear regression including new examples and a number of more recent developments in order to help biologists in the analysis of data and in the interpretation of the results. The book is written as a companion to the computer program GraphPadPrism, available for both Windows and Macintoch, which combines scientific graphics, basic biostatistics and nonlinear regression.

The more analytically Part A presents a complete example of nonlinear regression in order to introduce the readers to the problem of fitting curves to data, including the preparation of data and the choices of the best-fit values of the parameters in the nonlinear regression model, regarding the questioning in the interpretation and reviewing of the results such as confidence and prediction bands, sum of squares, residuals, etc. To this direction five examples of bad fits are presented explaining the wrong situation in each one and how this can be salvaged. Since nonlinear regression is a special case of linear regression and any nonlinear regression program can be used to fit a linear model to data, Part B is devoted to fitting data by using \(k\)-near regression and to the interpretation of the results.

It is known that linear and nonlinear regression fit a mathematical model to the data and determine the best-fit values of the parameters of the model in order to describe and analyze processes and their mechanisms via this model as good as best. Following this idea, Part C presents an introduction to the choice of models via regression giving the suitable terminology, tips on choosing the suitable model for the respective process as well as the notions of global models defining a family of curves rather than just a simple curve, and of compartmental models defining a model with a differential equation. Part D presents the iterative approach required for the nonlinear regression to work. In particular this part, except for describing the modeling experimental error and the unequal weighting of data points, mostly explains the method of steepest descent and the method of Gauss-Newton used by nonlinear regression programs.

Part E presents the asymptotic standard errors and the confidence intervals containing the true or false values of the parameters in the model. In particular, this part points out the generation of confidence intervals via the asymptotic method, via Monte Carlo simulations or via mode comparison giving examples for comparing the three abovementioned methods for creating confidence intervals. Part F is devoted to the comparison of suitable regression models fitting biological data by using various methods, such as the extra sum-of-squares F-test and the Akaike information criterion through useful examples.

Part G is an extension of the previous part in the cases of comparing one or more parameters of the same model to different data sets, such as the use of global fitting to test a treatmen effect in one experiment, the use of two-way ANOVA to compare curves, etc.

Part H is devoted to fitting radioligand and enzyme kinetics data using nonlinear regression models.

Part I is devoted to fitting dose-response curves for making plots of the results of many kinds of experiments.

The book concludes with the study of fitting linear or nonlinear regression models to data by using a special computer program, the GraphPadPrism, providing nonlinear regression statistical techniques.

The book is of great interest to biologists, research workers and advanced students, and to those using the nonlinear regression statistical method in data analysis and modeling.

The more analytically Part A presents a complete example of nonlinear regression in order to introduce the readers to the problem of fitting curves to data, including the preparation of data and the choices of the best-fit values of the parameters in the nonlinear regression model, regarding the questioning in the interpretation and reviewing of the results such as confidence and prediction bands, sum of squares, residuals, etc. To this direction five examples of bad fits are presented explaining the wrong situation in each one and how this can be salvaged. Since nonlinear regression is a special case of linear regression and any nonlinear regression program can be used to fit a linear model to data, Part B is devoted to fitting data by using \(k\)-near regression and to the interpretation of the results.

It is known that linear and nonlinear regression fit a mathematical model to the data and determine the best-fit values of the parameters of the model in order to describe and analyze processes and their mechanisms via this model as good as best. Following this idea, Part C presents an introduction to the choice of models via regression giving the suitable terminology, tips on choosing the suitable model for the respective process as well as the notions of global models defining a family of curves rather than just a simple curve, and of compartmental models defining a model with a differential equation. Part D presents the iterative approach required for the nonlinear regression to work. In particular this part, except for describing the modeling experimental error and the unequal weighting of data points, mostly explains the method of steepest descent and the method of Gauss-Newton used by nonlinear regression programs.

Part E presents the asymptotic standard errors and the confidence intervals containing the true or false values of the parameters in the model. In particular, this part points out the generation of confidence intervals via the asymptotic method, via Monte Carlo simulations or via mode comparison giving examples for comparing the three abovementioned methods for creating confidence intervals. Part F is devoted to the comparison of suitable regression models fitting biological data by using various methods, such as the extra sum-of-squares F-test and the Akaike information criterion through useful examples.

Part G is an extension of the previous part in the cases of comparing one or more parameters of the same model to different data sets, such as the use of global fitting to test a treatmen effect in one experiment, the use of two-way ANOVA to compare curves, etc.

Part H is devoted to fitting radioligand and enzyme kinetics data using nonlinear regression models.

Part I is devoted to fitting dose-response curves for making plots of the results of many kinds of experiments.

The book concludes with the study of fitting linear or nonlinear regression models to data by using a special computer program, the GraphPadPrism, providing nonlinear regression statistical techniques.

The book is of great interest to biologists, research workers and advanced students, and to those using the nonlinear regression statistical method in data analysis and modeling.

Reviewer: Cryssoula Ganatsiou (Larissa)

##### MSC:

62P10 | Applications of statistics to biology and medical sciences; meta analysis |

62J02 | General nonlinear regression |

62-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to statistics |

62J05 | Linear regression; mixed models |