Example of Nonlinear Regression

Researchers for the NIST (National Institute of Standards and Technology) want to understand the relationship between the coefficient of thermal expansion for copper and the temperature in degrees Kelvin.

Previous research indicates that a nonlinear model with 7 parameters provides an adequate fit. The researchers use nonlinear regression to estimate the parameters in the model.

  1. Open the sample data, CopperExpansion.MTW.
  2. Choose Stat > Regression > Nonlinear Regression.
  3. In Response, enter Expansion.
  4. In Edit directly, copy and paste, or type the following: (b1+b2*Kelvin+b3*Kelvin^2+b4*Kelvin^3)/(1+b5*Kelvin+b6*Kelvin^2+b7*Kelvin^3)
  5. Click Parameters.
  6. In Required starting values, enter these values:
    Parameter Values
    b1 1
    b2 -0.1
    b3 0.005
    b4 -1e-6
    b5 -0.005
    b6 0.001
    b7 -1e-7
  7. Click OK in each dialog box.

Interpret the results

The fitted line plot shows that the fitted line follows the observed values, which visually indicates that the model fits the data. The p-value for the lack-of-fit test is 0.679, which provides no evidence that the model fits the data poorly.

The warning about highly correlated parameters indicates that at least one pair of parameters has a correlation greater than an absolute value of 0.99. However, because previous studies indicate that a nonlinear model with 7 parameters provides an adequate fit to the data, the researchers do not change the model.

Nonlinear Regression: Expansion = (b1 + b2 * Kelvin + b3 * Kelvin ** 2 + ...

Method Algorithm Gauss-Newton Max iterations 200 Tolerance 0.00001
Starting Values for Parameters Parameter Value b1 1 b2 -0.1 b3 0.005 b4 -0.000001 b5 -0.005 b6 0.001 b7 -0.0000001
Equation Expansion = (1.07764 - 0.122693 * Kelvin + 0.00408638 * Kelvin ** 2 - 1.42627e-006 * Kelvin ** 3) / (1 - 0.00576099 * Kelvin + 0.000240537 * Kelvin ** 2 - 1.23144e-007 * Kelvin ** 3)
Parameter Estimates Parameter Estimate SE Estimate b1 1.07764 0.170702 b2 -0.12269 0.012000 b3 0.00409 0.000225 b4 -0.00000 0.000000 b5 -0.00576 0.000247 b6 0.00024 0.000010 b7 -0.00000 0.000000 Expansion = (b1 + b2 * Kelvin + b3 * Kelvin ** 2 + b4 * Kelvin ** 3) / (1 + b5 * Kelvin + b6 * Kelvin ** 2 + b7 * Kelvin ** 3)
Lack of Fit Source DF SS MS F P Error 229 1.53244 0.0066919 Lack of Fit 228 1.52583 0.0066922 1.01 0.679 Pure Error 1 0.00661 0.0066125
Summary Iterations 15 Final SSE 1.53244 DFE 229 MSE 0.0066919 S 0.0818039 * WARNING * Some parameter estimates are highly correlated. Consider simplifying the expectation function or transforming predictors or parameters to reduce collinearities.
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