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.

Method

AlgorithmGauss-Newton
Max iterations200
Tolerance0.00001

Starting Values for Parameters

ParameterValue
b11
b2-0.1
b30.005
b4-0.000001
b5-0.005
b60.001
b7-0.0000001

Equation

Expansion = (1.07764 - 0.122693 * Kelvin + 0.00408638 * Kelvin ** 2 - 1.42627E-06 * Kelvin **
     3) / (1 - 0.00576099 * Kelvin + 0.000240537 * Kelvin ** 2 - 1.23144E-07 * Kelvin ** 3)

Parameter Estimates

ParameterEstimateSE Estimate
b11.077640.170702
b2-0.122690.012000
b30.004090.000225
b4-0.000000.000000
b5-0.005760.000247
b60.000240.000010
b7-0.000000.000000
Expansion = (b1 + b2 * Kelvin + b3 * Kelvin ** 2 + b4 * Kelvin ** 3) / (1 + b5 * Kelvin + b6
     * Kelvin ** 2 + b7 * Kelvin ** 3)

Lack of Fit

SourceDFSSMSFP
Error2291.532440.0066919   
  Lack of Fit2281.525830.00669221.010.679
  Pure Error10.006610.0066125   

Summary

Iterations15
Final SSE1.53244
DFE229
MSE0.0066919
S0.0818039
* WARNING * Some parameter estimates are highly correlated. Consider simplifying the
expectation function or transforming predictors or parameters to reduce collinearities.