A New Linearized Formula for the Law of Total Effective Temperature and the Evaluation of Line-Fitting Methods with Both Variables Subject to Error

Abstract

In the quantitative analysis of experimental data regarding temperature-dependent development, the so-called law of total effective temperature is sometimes expressed in the linearized equation 1: 1/D = −(t/k) + (1/k)T.D indicates the duration of development; T, temperature; t, the estimated developmental zero temperature; and k, the effective cumulative temperature. The method of fitting usually involves the regression of y = 1/D on x = T. Although the degree of fitting of equation 1 to data within optimum temperature ranges is fairly satisfactory, we have in the current study addressed three problems regarding the use of equation 1 and methods of fitting involving the regression of y on x. First, we found that the detection of optimum temperature ranges is frequently difficult with equation 1. Second, in applying the method of regression of y on x with equation 1, the weights of the data points are disproportionate between those in the upper and lower parts of the line and they are not homogeneous along the temperature axis. The lower the temperature, the more disproportionate weight is burdened and the less weight is loaded. Third, in most of the data, errors in the x-variable are ignored. The second and third problems would in most cases result in a reduction in the slope of the line, a smaller t, and a larger k. Therefore, we proposed a new linearized formula: (DT) = k + tD. We further propose the use of the reduced major axis, obtained as the solution of the functional model among bivariate errors-in-variables models, in the method of fitting to data. We demonstrated that the majority of the problems raised above could be unraveled under this new approach based on statistical analysis.