Passing-Bablok Regression: Why OLS Is the Wrong Tool for Method Comparison

Comparing a new assay against a reference method is one of the most common statistical tasks in analytical chemistry and clinical laboratory science — and one of the most commonly done wrong. The default reach for most researchers is ordinary least squares (OLS) regression: plot the new method against the reference, fit a line, check the slope and intercept. OLS is the wrong tool for this job, for a specific, well-documented reason, and Passing-Bablok regression exists precisely to fix it.

Why Ordinary Least Squares Gets Method-Comparison Studies Wrong

Ordinary least squares regression rests on a core assumption: the independent variable (x, plotted on the horizontal axis) is measured without error, or with error small enough to ignore. OLS minimizes the vertical distances between each data point and the fitted line — the residuals in y — which only makes statistical sense if x itself is fixed or effectively error-free. That assumption holds in the textbook case OLS was built for: x is a controlled, precisely known input (a dose, a concentration prepared by dilution, a time point), and y is the noisy measurement.

A method-comparison study does not look like that. Both axes are measurements. A new assay and a reference method are both fallible instruments, each with its own imprecision, and there is no sense in which one of them is the “true,” error-free x-axis. Putting the reference method on the x-axis does not make its error disappear — it just gets ignored by the model.

The consequence is not merely cosmetic. When the predictor variable itself carries random measurement error, OLS produces a slope estimate that is biased toward zero — a well-established phenomenon in errors-in-variables regression sometimes called regression dilution or attenuation bias. In a method-comparison context this means an OLS-fitted slope will tend to underestimate any real proportional difference between the two methods, and the more error the reference method itself has, the worse the distortion. A method-comparison analysis that reports “no significant slope bias” purely because OLS attenuated a real slope toward 1 is not a clean bill of health; it is an artifact of using a model built for a different kind of data.

What Passing-Bablok Regression Does Instead

Passing-Bablok regression was introduced by Wolfgang Bablok and Heinrich Passing in 1983, specifically to address method-comparison data in clinical chemistry, where both measurement series are fallible. It is a non-parametric regression procedure that makes no assumption that either variable is measured without error, and no assumption about the distribution of the measurement errors on either axis.

Instead of minimizing residuals the way OLS does, Passing-Bablok computes the slope between every pair of data points in the sample, then takes the (shifted) median of that full set of pairwise slopes as the regression slope. The intercept follows from the median of the data points and the estimated slope. Because the method is built on medians of pairwise slopes rather than a sum-of-squares fit, it has two further practical advantages for method-comparison work:

• It is robust to outliers. A handful of unusual pairs shifts the estimate only modestly, because the median of many pairwise slopes is not pulled the way a least-squares fit is pulled by a single influential point.
• It does not require normally distributed differences. Because the estimator is rank-based rather than moment-based, skewed or non-normal error distributions — common in analytical chemistry, where measurement error often scales with concentration — do not violate a distributional assumption the way they would for OLS-based inference.

The trade-off is that Passing-Bablok assumes the underlying relationship between the two methods really is linear across the measured range; it estimates a straight line’s slope and intercept regardless, so a genuinely curved relationship needs to be checked separately (a CUSUM test for linearity is the standard companion check) before the slope/intercept output is treated as meaningful.

Interpreting the Slope and Intercept

A Passing-Bablok analysis reports a slope and an intercept, each with a 95% confidence interval, computed nonparametrically from the distribution of pairwise slopes rather than from a normal-theory formula. Reading the output means checking each interval against the value that would indicate perfect agreement:

• Slope. Perfect agreement implies a slope of 1. If the 95% confidence interval for the slope excludes 1, that is evidence of proportional bias — the two methods diverge more as the measured value increases (or decreases), for example if the new assay consistently reads 10% high relative to the reference across the range.
• Intercept. Perfect agreement implies an intercept of 0. If the 95% confidence interval for the intercept excludes 0, that is evidence of constant bias — a fixed offset that holds regardless of the magnitude being measured, for example if the new assay consistently reads 2 units high everywhere.

A method can show either kind of bias alone, both together, or neither. A slope confidence interval that comfortably contains 1 and an intercept confidence interval that comfortably contains 0 is the result that supports treating the two methods as interchangeable across the studied range — but that conclusion is about the linear relationship specifically, not about the full picture of agreement (see the next section). Software implementations differ slightly in exactly how the confidence intervals are computed, but the interpretation rule — check whether 1 and 0 fall inside the respective intervals — is the same regardless of the specific package used.

Passing-Bablok and Bland-Altman: Complementary, Not Competing

Passing-Bablok regression and CASRAI’s guide to Bland-Altman plots are frequently confused for alternatives to each other. They are not: they answer different questions about the same pair of methods, and a thorough method-comparison analysis typically reports both.

• Passing-Bablok asks whether the relationship between the two methods is linear with slope 1 and intercept 0 — it characterizes the structure of any bias (proportional vs. constant) across the measurement range.
• Bland-Altman asks how large the actual disagreement is, in the original measurement units, and visualizes it directly as a difference plotted against the mean of the two methods, with limits of agreement showing where about 95% of individual disagreements fall.

A method pair can show a Passing-Bablok slope indistinguishable from 1 and an intercept indistinguishable from 0, and still have limits of agreement too wide to be clinically or analytically acceptable — the linear relationship is unbiased, but the scatter around it is large. Conversely, a small but statistically significant proportional bias detected by Passing-Bablok may still fall within limits of agreement that are acceptable for the intended use. Neither method substitutes for the other: Passing-Bablok characterizes the bias structure, and Bland-Altman quantifies and visualizes the resulting disagreement.

When OLS or Deming Regression Is Still the Right Tool

Not every two-variable regression in a laboratory context is a method-comparison problem. OLS remains the correct tool when the x-variable genuinely is fixed or effectively error-free — a calibration curve built from precisely prepared standard concentrations, a dose-response relationship, or any design where one axis is a controlled input rather than a second fallible measurement. Deming regression is a related errors-in-variables alternative to Passing-Bablok worth knowing about: like Passing-Bablok, it accounts for error in both variables, but it does so parametrically, requiring an estimate of the ratio of the two methods’ error variances and an assumption that errors are normally distributed — assumptions Passing-Bablok avoids. In practice, Passing-Bablok’s non-parametric, outlier-robust design is why it is the more commonly recommended default for method-comparison studies in clinical chemistry and laboratory medicine specifically, with Deming regression as the parametric alternative when the error-variance ratio is well characterized and normality is a reasonable assumption.

Frequently Asked Questions

Does Passing-Bablok regression require a large sample size?

It works with smaller samples than many parametric alternatives because it does not depend on distributional assumptions, but because the confidence intervals are built from the distribution of pairwise slopes, precision improves with more data points, the same way any regression’s confidence interval narrows with sample size. Method-comparison guidelines in clinical chemistry commonly recommend at least 40 paired samples spanning the full measurement range of clinical interest.

Can Passing-Bablok regression be computed by hand?

In principle yes for a small dataset — it only requires computing every pairwise slope and taking a shifted median — but in practice it is done with statistical software (R’s mcr package, MedCalc, Analyse-it, and similar clinical-chemistry statistics packages all implement it directly), since the number of pairwise slopes grows quickly with sample size and the confidence-interval calculation involves a jackknife-based procedure not practical to do manually.

What does it mean if the Passing-Bablok slope and intercept both indicate bias?

It means the disagreement between the two methods has both a proportional component (bias that scales with the measured value) and a constant component (a fixed offset) across the range studied. Both should be reported and considered together when deciding whether the new method can substitute for the reference, since correcting for one alone would not make the two methods interchangeable.

References

• Passing H, Bablok W. 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. J Clin Chem Clin Biochem. 1983;21(11):709–720. PMID 6655447.
• Bland JM, Altman DG. Statistical methods for assessing agreement between two methods of clinical measurement. The Lancet. 1986;1(8476):307–310. DOI 10.1016/S0140-6736(86)90837-8.