Nonlinear Regression
Nonlinear regression models the dependent or criterion variables as a non-linear function of model parameters and one or more independent variables. In this case, the relationship between the variables is not a straight line, but instead follows a more complex curve, requiring specialized techniques to analyze. 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
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.
You estimate the parameters.. For example, in logistic nonlinear regression growth model, the parameters are b1, b2 and b3.
You require a segmented model for models with multiple different equations for different ranges. You specify the equations as terms 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 must be quantitative, and you must code the categorical variables as binary variables.
You can correctly interpret the value of the coefficients only if you have fitted the correct model. Therefore, identifying useful models is important.
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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