# Passing and Bablok regression

Passing and Bablok regression is used to compare two measurement methods. It is available in Excel using the XLSTAT add-on statistical software.

## What is Passing and Bablok regression?

**Passing and Bablok** (1983) developed a **regression** method that allows **comparing two measurement methods** (for example, two techniques for measuring concentration of an analyte), which overcomes the assumptions of the classical single linear regression that are inappropriate for this application. As a reminder the assumptions of the OLS regression are

- The explanatory variable, X in the model y(i)=a+b.x(i)+ε(i), is deterministic (no measurement error),
- The dependent variable Y follows a normal distribution with expectation aX
- The variance of the measurement error is constant.

Furthermore, extreme values can highly influence the model.

Passing and Bablok proposed a method which overcomes these assumptions: the two variables are assumed to have a random part (representing the measurement error and the distribution of the element being measured in medium) without needing to make assumption about their distribution, except that they both have the same distribution. We then define:

- y(i) = a+b.x(i)+ ξ(i)
- x(i) = A+B.y(i)+ η(i)

Where ξ and η follow the same distribution. The Passing and Bablok method allows calculating the a and b coefficients (from which we deduce A and B using B=1/b and A=-a/b) as well as a confidence interval around these values. The study of these values helps comparing the methods. If they are very close, b is close to 1 and a is close to 0.

Passing and Bablok also suggested a linearity test to verify that the relationship between the two measurement methods is stable over the interval of interest. This test is based on a CUSUM statistic that follows a Kolmogorov distribution. XLSTAT provides the statistic, the critical value for the significance level chosen by the user, and the p-value associated with the statistic.

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