Testing Linearity and Slope

Two slopes are compared in a similar manner as are two means the simplest case is obtained when both calibrations are carried out using identical calibration concentrations (as is usual when SOPs are followed) the average variance V u is used in a t-test  

Example 30 Two calibrations are carried out with the results b = 0.004144 0.000046 and b = 0.003986 0.000073 V is thus 0.000061 and t = 0.0000158/0.000061 = 2.6 since n = n = 5, f = 6 and l(6, 0.05) = 2.45 so that a significant difference in the slopes is found. The reader is reminded to round only final results if already rounded results had been used here, different conclusions would have been reached the use of five or four 

Eight combinations are possible with the true/false answers to the following three questions (1) is 5res, i = ires,2 , (2) is b = bjl, (3) is ymean.I = ymean.z A rigorous treatment is given in Ref. 34. First, question 1 must be answered if Hq is retained, question 2 must be answered. Only if this also leads to a positive result can question 3 be posed. 

There are several ways to test the linearity of a calibration line one can devise theory-based tests, or use common sense. The latter approach is suggested here because if only a few calibration points are available on which to rest one s judgement, a graph of the residuals will reveal a trend, if any is present, while numerical tests need to be adjusted to have the proper sensitivity. It is advisable to add two horizontal lines offset by the measure of repeatability accepted for the method unless the apparent curvature is such that points near the middle, respectively the end of the x-range are clearly outside this reproducibility band, no action need to be taken. 

Regarding the residuals, many an investigator would be tempted to cast out outliers the reader is advised to consult Section 1.5.5. If values are grouped (i.e. several values y, are measured at the same x), outlier tests can be applied to the individual group, however, blind reliance on a rule, such as y ,ean 2 5j, is Strongly discouraged. 

5 (A) resp. -0.75 1.3 (B)). If a horizontal line can be fitted between the confidence limits an interpretation X = f(y ) is impossible. It suffices in many cases to approximate the curves by straight lines (C). 

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