What happens when data are fitted to the wrong equation?

Abstract

In many problems of data analysis it is necessary to fit the data to a mathematical equation. Random errors of measurement will be responsible for deviations between the data and the equation, but superimposed on this there may be deviations that result from the equation being an inadequate description ofthe system from which the data were obtained. Plots of the residual (i.e. the difference between the experimental and calculated values of the dependent variable) against each of the experimental variables have been previously used to detect a misfit between the data and the equation. In the present paper, we show that the shape of the residual plots may be used as a guide in choosing a more appropriate equation. In addition, residual plots give useful information on the error structure of the data, and hence the weighting factors that should be used in the analysis. The purpose of the present paper is to show that, when data are fitted to the wrong equation, the shape of the residual plot contains valuable information that can be utilized to determine the way in which the equation should be modified to achieve a better description of the data.

Enantioselective modulation of GABAergic synaptic transmission by steroids and benz[e]indenes in hippocampal microcultures

“…Multi-exponential decays of synaptic currents were fit using programs in Axobasic, pCLAMP, or SigmaPlot (Jandel Scientific Inc., San Rafael, CA) and a Levenberg-Marquardt algorithm. The number of exponential terms in the decay was determined by inspection of residual plots and F-test (Ellis and Dubbleby, 1978).…”

Noninvasive Study of Neurovascular Coupling during Graded Neuronal Suppression

“…If the increase in the sum of squares associated with the reduced model is purely ascribed to statistical fluctuation, reduced model can be used to replace the original model because the original model overfits the data; if the increase in the sum of squares associated with the reduced model is due to lack of fitting (underfitting), the reduced model has to be rejected. Comparison between the variance due to lack of fit (s lf 2 ) in the reduced model and the variance due to pure error (s pe 2 ), which is assumed to be approximated by the variance of the fit by the original model, illustrates the relative fitting performances between the two models (Ellis and Duggleby, 1978). The value of F score is calculated from the ratio of these variances:…”

23Na‐NMR detects hypoxic injury in intact kidney: Increases in sodium inhibited by DMSO and DMTU

“…Briefly, a model function was fitted by a nonlinear regression program, DNRP53 (22), to the time-averaged data from each group of experiments to yield estimates and standard errors for each parameter, including rate constants, plus a residual sum of squares (RSS), which represents the weighted square of the difference between the fitted and experimental values, summed over all data points. Evaluation of the suitability of the model function itself, analyses of various individual data sets and various combinations of data sets were compared using the values of RSS in an F test (21,(23)(24)(25).…”