Nonlinear Regression
Nonlinear regression is a regression in which the dependent or criterion variables are modeled as a non-linear function of model parameters and one or more independent variables. There are several common models, such as Asymptotic Regression/Growth Model, which is given by:
b1 + b2 * exp(b3 * x)
Logistic Population Growth Model, which is given by:
b1 / (1 + exp(b2 + b3 * x)), and
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Asymptotic Regression/Decay Model, which is given by:
b1 – (b2 * (b3 * x)) etc.
The reason that these models are called nonlinear regression is because the relationships between the dependent and independent parameters are not linear.
This test in SPSS is done by selecting “analyze” from the menu. Then, select “regression” from analyze. After this, select “linear from regression,” and then click on “perform nonlinear regression.”
There are certain terminologies in nonlinear regression which will help in understanding nonlinear regression in a much better manner. These terminologies are as follows:
Model Expression is the model used, the first task is to create a model. The selection of the model in is based on theory and past experience in the field. For example, in demographics, for the study of population growth, logistic nonlinear regression growth model is useful.
Parameters are those which are estimated. For example, in logistic nonlinear regression growth model, the parameters are b1, b2 and b3.
Segmented model is required for those models which have multiple different equations of different ranges, equations are then specified as a term in multiple conditional logic statements.
Loss function is a function which is required to be minimized. This is done by nonlinear regression.
Assumptions
The data level in must be quantitative, the categorical variables must be coded as binary variables.
The value of the coefficients can be correctly interpreted, only if the correct model has been fitted, therefore it is important to identify useful models.
A good choice of starting points can lead to a desirable output, a poor choice will make the output misleading.
*For assistance with conducting a nonlinear regression or other quantitative analyses click here.
Resources
Bates, D. M., & Watts, D. G. (1988). Nonlinear regression analysis and its applications. New York: John Wiley & Sons.
Crainiceanu, C. M., & Ruppert, D. (2004). Likelihood ratio tests for goodness-of-fit of a nonlinear regression model. Journal of Multivariate Analysis, 91(1), 35-52.
Fujii, T., & Konishi, S. (2006). Nonlinear regression modeling via regularized wavelets and smoothing parameter selection. Journal of Multivariate Analysis, 97(9), 2023-2033.
Gross, A. L., & Fleishman, L. E. (1987). The correction for restriction of range and nonlinear regressions: An analytic study. Applied Psychological Measurement, 11(2), 211-217.
Hanson, S. J. (1978). Confidence intervals for nonlinear regression: A BASIC program. Behavior Research Methods & Instrumentation, 10(3), 437-441.
Huet, S., Bouvier, A., Poursat, M. -A., & Jolivet, E. (2004). Statistical tools for nonlinear regression: A practical guide with S-PLUS and R examples (2nd ed.). New York: Springer.
McGwin, G., Jr., Jackson, G. R., & Owsley, C. (1999). Using nonlinear regression to estimate parameters of dark adaptation. Behavior Research Methods, Instruments & Computers, 31(4), 712-717.
Rao, B. L. S. P. (2004). Estimation of cusp in nonregular nonlinear regression models. Journal of Multivariate Analysis, 88(2), 243-251.
Seber, G. A. F., & Wild, C. J. (2003). Nonlinear regression. New York: John Wiley & Sons.
Sheu, C. -F., & Heathcote, A. (2001). A nonlinear regression approach to estimating signal detection models for rating data. Behavior Research Methods, Instruments & Computers, 33(2), 108-114.
Verboon, P. (1993). Robust nonlinear regression analysis. British Journal of Mathematical and Statistical Psychology, 46(1), 77-94.
Wang, J. (1995). Asymptotic normality of L-sub-1-estimators in nonlinear regression. Journal of Multivariate Analysis, 54(2), 227-238.
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