17.1 Introduction to Non-Linear Regression

Non-linear regression models relationships between variables that aren't straight lines. It's used when linear models fall short, like for population growth or drug concentration over time. These models can capture curves, asymptotes, and changing rates.

Fitting non-linear models can be tricky. You need to choose the right function, find good starting values, and deal with multiple local optima. Interpreting results and getting enough data can also be challenging. But when done right, they offer powerful insights.

Non-linear Regression

Definition and Characteristics

• Non-linear regression is a statistical technique used to model the relationship between a dependent variable and one or more independent variables when the relationship is not linear
• In non-linear regression, the model parameters are estimated by minimizing a loss function, typically the sum of squared residuals, using iterative optimization algorithms
  • Common optimization algorithms include Gauss-Newton or Levenberg-Marquardt methods
• Non-linear regression models can take various forms, depending on the nature of the relationship between the variables
  • Examples of non-linear functions: exponential, logarithmic, power, or sigmoidal functions
• The choice of the non-linear function is based on domain knowledge, theoretical considerations, or empirical evidence suggesting a specific type of non-linear relationship

Appropriate Situations for Non-linear Regression

• Non-linear regression is appropriate when the relationship between the dependent and independent variables cannot be adequately described by a straight line
• Situations where the rate of change in the dependent variable varies with the level of the independent variable(s) often require non-linear regression
  • For example, the growth rate of a population may slow down as the population size approaches the carrying capacity of the environment
• Non-linear regression is suitable for modeling phenomena that exhibit saturation, exponential growth or decay, or asymptotic behavior
• Non-linear regression is commonly applied in various fields:
  • Population growth (logistic growth models)
  • Pharmacokinetics (drug concentration over time)
  • Enzyme kinetics (Michaelis-Menten equation)
  • Economic models of diminishing returns (Cobb-Douglas production function)

Linear Regression vs Non-linear Relationships

Limitations of Linear Regression

• Linear regression assumes a constant rate of change in the dependent variable for a unit change in the independent variable(s), which may not hold for non-linear relationships
  • For instance, the effect of fertilizer on crop yield may diminish at higher application rates
• Fitting a linear model to non-linear data can lead to biased and inconsistent parameter estimates, as well as poor model fit and predictive performance
• Linear regression may fail to capture important features of non-linear relationships
  • Curvature
  • Asymptotes
  • Inflection points
• Extrapolating predictions beyond the range of the observed data using a linear model fitted to non-linear data can result in unrealistic or nonsensical predictions
  • Negative values for strictly positive quantities (population size, drug concentration)

Challenges of Non-linear Modeling

Model Specification and Parameter Estimation

• Non-linear regression often requires specifying the functional form of the relationship a priori, which may not be known with certainty and can lead to model misspecification
  • Choosing between exponential, power, or logarithmic functions
• The choice of starting values for the model parameters can influence the convergence and final estimates of the optimization algorithm, potentially leading to suboptimal or unstable solutions
• Non-linear models may have multiple local optima in the parameter space, making it difficult to find the global optimum and obtain reliable parameter estimates

Interpretation and Sample Size Requirements

• The interpretation of the model parameters in non-linear regression can be more complex than in linear regression, as the effects of the independent variables on the dependent variable may vary across the range of the data
  • The slope of a logistic growth curve changes over time
• Non-linear models often require larger sample sizes to obtain precise parameter estimates and achieve adequate statistical power compared to linear models
  • More data points are needed to capture the curvature and asymptotic behavior of non-linear relationships