Nonlinear Least Squares Regression

Nonlinear least squares regression fits a curve whose parameters enter nonlinearly, such as logistic, Gompertz or Michaelis–Menten models, by minimising the sum of squared residuals iteratively.

비선형 최소제곱 회귀는 로지스틱·곰페르츠·미카엘리스-멘텐처럼 모수가 비선형으로 들어가는 곡선을 잔차제곱합을 반복적으로 최소화해 맞추는 방법입니다.

My growth data level off at a ceiling; how do I fit a logistic curve and get standard errors for its parameters?

성장 자료가 어느 수준에서 포화되는데, 로지스틱 곡선을 맞추고 모수의 표준오차를 구하려면 어떻게 하나요?

Nonlinear least squares estimates curve parameters such as an asymptote, midpoint and scale by iteratively minimising squared residuals with the Levenberg–Marquardt algorithm. Standard errors come from the Jacobian at the solution, and AIC compares alternative curve shapes fitted to the same data.

When to use Nonlinear Least Squares Regression

• Theory specifies a curve form such as saturation, exponential growth or decay.
• Parameters with a substantive meaning (asymptote, rate, half-saturation constant) must be estimated.
• Several candidate curve shapes need to be compared by AIC.

When to use something else

• The curve shape is unknown and only a smooth fit is needed; use a natural spline or a generalized additive model.
• The outcome is binary or a count; use a generalized linear model.

Assumptions and requirements

• Errors are independent with constant variance (or known weights).
• The chosen function correctly describes the mean.
• Starting values lie in the basin of the least-squares solution.

What the analysis reports

• Parameter estimates with Jacobian-based standard errors, t tests and Wald confidence intervals
• Residual standard error, R-squared, RMSE
• Log-likelihood, AIC and BIC; AIC of other built-in curves
• Residual normality, spread and runs diagnostics

How to report it

A [model] curve fitted by nonlinear least squares gave [parameter] = [value] (SE = [value]) and [parameter] = [value] (SE = [value]); residual SE = [value] on [df] df.

Related methods

• Polynomial Regression
• Natural Cubic Spline Regression
• Generalized Additive Models (GAM)

Run Nonlinear Least Squares Regression on Quorant