# Method comparison with repeated measurements -- Passing-Bablok regression for grouped data with errors in both variables

## TL;DR

This paper extends the Passing-Bablok regression to grouped data with errors in both variables, providing an unbiased estimator and a method comparison test with higher power for repeated measurements.

## Contribution

It introduces the Block-Passing-Bablok estimator for grouped data with errors in both variables and derives its asymptotic properties and variance.

## Key findings

- The estimator is asymptotically normally distributed.
- Variance depends on measurement overlap between groups.
- The method improves power in method comparison with repeated measurements.

## Abstract

The Passing-Bablok and Theil-Sen regression are closely related non-parametric methods to estimate the regression coefficients and build tests on the relationship between the dependent and independent variables. Both methods rely on the slopes of the connecting lines between pairwise measurements. While Theil and Sen assume no measurement errors in the independent variable, the method from Passing and Bablok accounts for errors in both variables. Here we consider the case where multiple, e.g. repeated, measurements with errors in both variables are available for m samples. We show that in this case the slopes between repeated measurements need to be excluded to obtain an unbiased estimate. We prove that the resulting Block-Passing-Bablok estimate for grouped data is asymptotically normally distributed. If measurements of the independent variable are without error the variance of the estimate equals the variance of the Theil-Sen method with tied ranks. If both variables are measured with imprecision the result depends on the fraction of measurements between groups that fall within the range of each other. Only if no overlap between measurements of different groups occurs the variance equals again the tied ranks version. Otherwise, the variance is smaller. We explicitly compute this variance and provide a method comparison test for data with repeated measurements based on the method from Passing and Bablok for independent measurements. If repeated measurements are considered this test has a higher power to detect the true relationship between two methods.