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NIST StRD Misra1a: nonlinear least squares y = b1 (1 − exp(−b2 x)) from both certified starts

ToolRegression and Curve Fitting
SourceNational Institute of Standards and Technology (2012). Statistical Reference Datasets: Nonlinear Regression. NIST, Gaithersburg, MD. Misra1a dataset (lower difficulty), certified values and starting values (copy in References/Regression-StRD/Misra1a.dat).
LevelGraduate

Problem

Misra's 14 dental-research observations of y against x follow y = b1 (1 − exp(−b2 x)). Fit the two parameters by nonlinear least squares from NIST's two starting points, b = (500, 1 × 10⁻⁴) and (250, 5 × 10⁻⁴), and report the estimates, their standard deviations and the residual sum of squares.

Inputs to Enter

  • Model: user-written f(x) = b1*(1-exp(-b2*x)), parameters b1, b2, starting values 500, 0.0001 (then 250, 0.0005)
  • Data (x first): 77.6 10.07; 114.9 14.73; 141.1 17.94; 190.8 23.93; 239.9 29.61; 289.0 35.18; 332.8 40.02; 378.4 44.82; 434.8 50.76; 477.3 55.05; 536.8 61.01; 593.1 66.40; 689.1 75.47; 760.0 81.78
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Answers

  • b1 = 238.94212918 (SD 2.7070075241), b2 = 5.5015643181 × 10⁻⁴ (SD 7.2668688436 × 10⁻⁶)
  • Residual SS = 0.12455138894 on 12 df, residual SD = 0.10187876330
  • Both starting points converge to the same certified values

Notes

Choose the user model; the formula, parameter and starting-value fields already hold the Misra1a model and NIST start 1 as their defaults, so only the data need pasting. Parameters reproduced to 1e-8 and standard deviations (central-difference Jacobian) to 1e-6 relative (test_regression.php).

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