Can you trust the parametric standard errors in nonlinear least squares? Yes, with provisos.

BACKGROUND

Questions about the reliability of parametric standard errors (SEs) from nonlinear least squares (LS) algorithms have led to a general mistrust of these precision estimators that is often unwarranted.

METHODS

The importance of non-Gaussian parameter distributions is illustrated by converting linear models to nonlinear by substituting e, ln A, and 1/A for a linear parameter a. Monte Carlo (MC) simulations characterize parameter distributions in more complex cases, including when data have varying uncertainty and should be weighted, but weights are neglected. This situation leads to loss of precision and erroneous parametric SEs, as is illustrated for the Lineweaver-Burk analysis of enzyme kinetics data and the analysis of isothermal titration calorimetry data.

RESULTS

Non-Gaussian parameter distributions are generally asymmetric and biased. However, when the parametric SE is <10% of the magnitude of the parameter, both the bias and the asymmetry can usually be ignored. Sometimes nonlinear estimators can be redefined to give more normal distributions and better convergence properties.

CONCLUSION

Variable data uncertainty, or heteroscedasticity, can sometimes be handled by data transforms but more generally requires weighted LS, which in turn require knowledge of the data variance.

GENERAL SIGNIFICANCE

Parametric SEs are rigorously correct in linear LS under the usual assumptions, and are a trustworthy approximation in nonlinear LS provided they are sufficiently small - a condition favored by the abundant, precise data routinely collected in many modern instrumental methods.