Passing–Bablok regression

{{Short description|Medical statistical method}}

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.{{cite journal |author=Passing H, Bablok W |title=A new biometrical procedure for testing the equality of measurements from two different analytical methods. Application of linear regression procedures for method comparison studies in Clinical Chemistry, Part I |journal= Journal of Clinical Chemistry and Clinical Biochemistry |volume=21 |issue=11 | year=1983 |pages=709–20 |pmid=6655447 |doi=10.1515/cclm.1983.21.11.709|s2cid=45557122 |url=http://edoc.hu-berlin.de/18452/11511 }}{{cite journal | author=Passing H, Bablok W |title= Comparison of several regression procedures for method comparison studies and determination of sample sizes. Application of linear regression procedures for method comparison studies in Clinical Chemistry, Part II | journal = Journal of Clinical Chemistry and Clinical Biochemistry | year= 1984 | volume = 22 | issue=6 | pages=431–45 |doi=10.1515/cclm.1984.22.6.431 | pmid =6481307 |s2cid= 26235878 |url=https://edoc.hu-berlin.de/bitstream/handle/18452/12448/schifferdecker.pdf?sequence=1}}{{cite journal |author=Bilić-Zulle L|title=Comparison of methods: Passing and Bablok regression |journal=Biochem Med |volume=21 |issue=1 | year=2011 |pages=49–52 |doi=10.11613/BM.2011.010 |pmid=22141206 |doi-access=free |url=http://hrcak.srce.hr/file/96718 }}{{cite journal | author=Dufey, F | title= Derivation of Passing–Bablok regression from Kendall's tau | journal = The International Journal of Biostatistics | volume= 16 | issue=2 | year =2020| doi =10.1515/ijb-2019-0157 |doi-access=free| pmid = 32780716 }}{{cite arXiv |eprint= 2202.08060|last1= Raymaekers|first1= Jakob|last2= Dufey|first2= Florian|title= Equivariant Passing-Bablok regression in quasilinear time|year= 2022|class= stat.ME}} 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 to create an approximately consistent estimator. The estimator is therefore close in spirit to the Theil-Sen estimator. The parameter $a$ is calculated by $a\; =\; \backslash operatorname\{median\}(\{y\_\{i\}-bx\_\{i\})\}$.

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.{{cite journal |author=Bablok W, Passing H, Bender R, Schneider B | title = A general regression procedure for method transformation. Application of linear regression procedures for method comparison studies in clinical chemistry, Part III | year=1988 | url = https://edoc.hu-berlin.de/bitstream/handle/18452/11842/cclm.1988.26.11.783.pdf?sequence=1 | pages=783–90 | journal = Journal of Clinical Chemistry and Clinical Biochemistry |volume=26| issue = 11 | doi=10.1515/cclm.1988.26.11.783 | pmid = 3235954 | s2cid = 4686716 }}

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.{{cite book|title=EP09-A3: Measurement Procedure Comparison and Bias Estimation Using Patient Samples; Approved Guideline|date=August 30, 2013|publisher=CLSI|isbn=978-1-56238-888-1|edition=Third}} 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.{{cite web |url=https://www.medcalc.org/manual/note-passingbablok.php |title=A note on Passing-Bablok regression |publisher=MedCalc Software bvba |access-date=19 October 2016 |quote=}}