Approximation of nonlinear regression models by linear regression models

摘要

Nonlinear regression models are useful in many areas of application. Software for fitting nonlinear models is widely available in the comprehensive statistical packages often used by statisticians (e.g., SAS). However, many researchers do not have such software available to them or do not have the expertise to use such packages.;Using an appropriately-chosen transformation, the deterministic component of a nonlinear regression model can often be linearized. For example, a cell surviving fraction model in radiobiology has deterministic component $esp{beta X}.$ By log-transformation, it is linearized to $beta X$ on the log scale. The parameter of the model can then be estimated using standard methodology for fitting linear regression models; such methodology is widely available to non-statisticians. While such a transformation linearizes the deterministic component of the model, the effect on the assumed error structure is not always recognized or appreciated.;In order to investigate the consequences of such linearizing transformations, the least square estimators of the parameters of two models with nonlinear deterministic components $(esp{beta X}$ and $esp{alpha+beta X})$ are compared under two error structures (additive and multiplicative). The topics include: (1) justification of the assumption of log-normality of the response in the multiplicative error model, (2) justification of using the one-step Gauss-Newton estimator as an approximation to the final-step Gauss-Newton estimator, (3) analytical relationships between the parameter estimators and their variance estimators of the proposed nonlinear additive error models and those of the log-linearized multiplicative error models, (4) empirical comparisons of the estimators between the nonlinear models and the log-linearized models in terms of bias and precision.;The analytical relationships obtained can be used to approximate parameter estimates and their variances when nonlinear models are transformed to log-linear models. As one would expect, when the true error is additive, the nonlinear model performs somewhat better than the log-linearized model. Similarly, when the true error is multiplicative, the log-linearized model performs better. However, in nonlinear models with deterministic components of the form $esp{beta X}$ and $esp{alpha+beta X}$ the log-linearized model generally provides accurate parameter estimates.