Biased standard error estimations in transport model calibration due to heteroscedasticity arising from the variability of linear data projection

Reliable transport models calibrated from accurate traffic data are crucial for predicating transportation system performance and ensuring better traffic planning. However, due to the impracticability of collecting data from an entire population, methods of data inference such as the linear data projection are commonly adopted. A recent study has shown that systematic bias may be embedded in the parameters calibrated due to linearly projected data that do not account for scaling factor variability. Adjustment factors for reducing such biases in the calibrated parameters have been proposed for a generalized multivariate polynomial model. However, the effects of linear data projection on the dispersion of and confidence in the adjusted parameters have not been explored. Without appropriate statistics examining the statistical significance of the adjusted model, their validity in applications remains unknown and dubious. This study reveals that heteroscedasticity is inherently introduced by data projection with a varying scaling factor. Parameter standard errors that are estimated by linearly projected data without any appropriate treatments for non-homoscedasticity are definitely biased, and possibly above or below their true values. To ensure valid statistical tests of significance and prevent exposure to uninformed and unnecessary risk in applications, a generic analytical distribution-free (ADF) method and an equivalent scaling factor (ESF) method are proposed to adjust the parameter standard errors for a generalized multivariate polynomial model, based on the reported residual sum of squares. The ESF method transforms a transport model into a linear function of the scaling factor before calibration, which provides an alternative solution path for achieving unbiased parameter estimations. Simulation results demonstrate the robustness of the ESF method compared with the ADF method at high model nonlinearity. Case studies are conducted to illustrate the applicability of the ESF method for the parameter standard error estimations of six Macroscopic Bureau of Public Road functions, which are calibrated using real-world global positioning system data obtained from Hong Kong.