Passing–Bablok regression

Passing–Bablok regression is a method from robust statistics for nonparametric regression analysis suitable for method comparison studies introduced by Wolfgang Bablok and Heinrich Passing in 1983. The procedure is adapted to fit linear errors-in-variables models. It is symmetrical and is robust in the presence of one or few outliers.

The Passing-Bablok procedure fits the parameters $$a$$ and $$b$$ of the linear equation $$y = a + b * x$$ using non-parametric methods. The coefficient $$b$$ is calculated by taking the shifted median of all slopes of the straight lines between any two points, disregarding lines for which the points are identical or $$b = -1$$. The median is shifted based on the number of slopes where $$b < -1$$ to create an approximately consistent estimator. The estimator is therefore close in spirit to the Theil-Sen estimator. The parameter $$a$$ is calculated by $$a = \operatorname{median}({y_{i}-bx_{i})}$$.

In 1986, Passing and Bablok extended their method introducing an equivariant extension for method transformation which also works when the slope $$ b $$ is far from 1. It may be considered a robust version of reduced major axis regression. The slope estimator $$b$$ is the median of the absolute values of all pairwise slopes.

The original algorithm is rather slow for larger data sets as its computational complexity is $$O(n^2)$$. However, fast quasilinear algorithms of complexity $$O(n$$ ln $$n)$$ have been devised.

Passing and Bablok define a method for calculating a 95% confidence interval (CI) for both $$a$$ and $$b$$ in their original paper, which was later refined, though bootstrapping the parameters is the preferred method for in vitro diagnostics (IVD) when using patient samples. The Passing-Bablok procedure is valid only when a linear relationship exists between $$x$$ and $$y$$, which can be assessed by a CUSUM test. Further assumptions include the error ratio to be proportional to the slope $$b$$ and the similarity of the error distributions of the $$ x $$ and $$ y $$ distributions. The results are interpreted as follows. If 0 is in the CI of $$a$$, and 1 is in the CI of $$b$$, the two methods are comparable within the investigated concentration range. If 0 is not in the CI of $$a$$ there is a systematic difference and if 1 is not in the CI of $$b$$ then there is a proportional difference between the two methods.

However, the use of Passing–Bablok regression in method comparison studies has been criticized because it ignores random differences between methods.