Linearity is the assumption that the relationship between the methods is linear.
The regression procedures used in method comparison studies assume the relationship between the methods is linear. A CUSUM is a measure of the linearity, defined as a running sum of the number of observations above and below the fitted regression line. When the relationship is linear it is expected the points above and below the line are randomly scattered, and the CUSUM statistic is small. Clusters of points on one side of the regression line produce a large CUSUM statistic.
A formal hypothesis test for linearity is based on the largest CUSUM statistic and the Kolmogorov-Smirnov test. The null hypothesis states that the relationship is linear, against the alternative hypothesis that it is not linear. When the test p-value is small, you can reject the null hypothesis and conclude that the relationship is nonlinear.
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