We can use nonlinear regression to describe complicated, nonlinear relationships between a response variable and one or more predictor variables. Chemists, engineers, scientists and others who want to model growth, decay, or other complex functions often need to use nonlinear regression. Minitab Statistical Software can help you harness this powerful statistical technique to model such relationships and make sound predictions.
Responses that can be modeled using nonlinear regression include:
Nonlinear regression often provides a better analysis for these types of responses than linear regression. For example, suppose we want to model the relationship between the ages and weights of adolescent boys so we can predict their weight based on age. Using linear regression, we arrive at a model that despite its large R-squared value, does not provide a good fit.
The fitted line plot above shows that this model underestimates weight around the ages of 3 and 4 and in the later teenage years, and overestimates weight for boys aged 6 through 10. In addition, this model violates the constant variance assumption, which can be evaluated using Minitab’s residual plots, further supporting the conclusion that this model does not provide a good fit. Attempting to use a quadratic model provides an equally poor fit.
If we wish to be able to accurately predict a child's weight with an equation based on these data, we need to use nonlinear regression. The nonlinear model better fits our data, and does not underestimate or overestimate weight at different ages as the linear model did. The fitted line plot below reveals how closely the nonlinear regression model mirrors our data.
Let’s look at the similarities and differences between linear and nonlinear regression. Both analyses:
The fundamental difference between linear and nonlinear regression is the form of the model. When statisticians talk about “linear” and “nonlinear” regression models, they are referring to the parameters in the model, not whether the model describes a curved or straight line. For example, cubic and quadratic regression can yield graphs that show curvature, but these are considered to be linear regression models. In a linear regression model, each predictor is additive and contains only one parameter that multiplies it. Therefore, a linear regression model always looks something like this:
response = constant + parameter * predictor + ... + parameter * predictor
or
y = β_{o} + β_{1}x_{1} + β_{2}x_{2} + ... + β_{k}x_{k}
In contrast, nonlinear regression equations have many different forms. Potential nonlinear shapes include concave, convex, exponential growth or decay, and sigmoidal (S) curves.
When performing nonlinear regression in Minitab, you select a function (or shape) that you expect describes the relationship between the predictor(s) and response variables, based on prior knowledge about the response curve's shape or the behavior of physical and chemical properties in the system you are investigating. You also need to supply estimated starting points for each parameter in the model. Once again, researchers frequently draw on prior knowledge of the system they are studying to provide these starting values.
Finding the function that provides the optimal fit for your data can require considerable effort. You need to specify a function that satisfies both the requirements of your prior knowledge and the nonlinear regression assumptions, and additional research, subject area knowledge, and trial and error analyses may be required. However, to make function selection easier, Minitab provides sample graphs that you can use to help choose the best function.
You can select a function that is supplied by Minitab or enter a custom function. If you use a custom function, Minitab automatically adds that function to its catalog so you can conveniently access it again in the future.
The flexibility nonlinear regression offers is very powerful, and Minitab 16’s Nonlinear Regression tool makes the benefits of this statistical technique easier for everyone.
If you’re not already using the power of Minitab to get the maximum value from your data, download a free, fully-functional 30-day trial of Minitab Statistical Software today.
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