Calibration lines passing through the origin with errors in both axes

Abstract 

Linear regression of calibration lines passing through the origin was investigated for three models of y-direction random errors: normally distributed errors with an invariable standard deviation (SD) and log normally and normally distributed errors with an invariable relative standard deviation (RSD). The weighted (weighting factor is x ² _i ), geometric and arithmetic means of the ratios y _i /x _i estimate the calibration slope for these models, respectively. Regression of the calibration lines with errors in both directions was also studied. The x-direction errors were assumed to be normally distributed random errors with either an invariable SD or invariable RSD, both combined with a constant relative systematic error. The random errors disperse the true, unknown x-values about the plotted, demanded x-values, which are shifted by the constant relative systematic error. The systematic error biases the slope estimate while the random errors do not. They only increase automatically the slope estimate uncertainty, in which the uncertainty component reflecting the range of the possible values of the systematic error must be additionally included.