Passing–Bablok Regression

The Bland–Altman plot is the preferred method for method comparison, directly quantifying whether two methods agree within clinically acceptable limits. Passing–Bablok regression is a useful supplement: it models the linear relationship between two methods and tests formally whether any systematic error is constant across the measurement range (constant bias, intercept $\neq 0$) or whether it grows with the magnitude being measured (proportional bias, slope $\neq 1$). This characterisation can help explain the pattern seen in the Bland–Altman plot and guide decisions about recalibration.

Why Ordinary Regression Fails

Standard least-squares regression minimises the sum of squared vertical distances from the data points to the fitted line. This is appropriate when the $x$-variable is measured without error and only the $y$-variable carries measurement uncertainty. In method comparison, $both$ variables are measurements subject to error. Applying ordinary regression assigns the role of "error-free" variable arbitrarily to one method, producing biased slope estimates — and the bias reverses direction depending on which method is placed on the $x$-axis. Two analysts using the same data can reach different conclusions simply by swapping the axes.

The Passing–Bablok Estimator

Passing–Bablok regression (Passing & Bablok, 1983) avoids all distributional assumptions. The slope $B$ is estimated as the median of all $\binom{n}{2}$ pairwise slopes:

$$ S_{ij} = \frac{y_j - y_i}{x_j - x_i} \quad \text{for all } i < j $$

with pairs where $S_{ij} = -1$ excluded (they correspond to observations aligned at 45° in a way that makes the slope undefined in this context). The intercept $A$ follows as:

$$ A = \text{median}(y_i - B \cdot x_i) $$

The result is the equation $y = A + Bx$, where $y$ is Method B and $x$ is Method A. An important property: the method is symmetric — swapping the axes yields a consistent result, which ordinary regression does not.

Confidence intervals for $A$ and $B$ are derived from the distribution of the ordered pairwise slopes, using a rank-based approach. No assumption of normality is required.

Interpreting the Slope and Intercept

Intercept $A = 0$, slope $B = 1$: No systematic difference between the methods — they are in perfect linear agreement.
Intercept $A \neq 0$, slope $B = 1$: Constant (systematic) bias — one method reads consistently higher or lower by a fixed amount regardless of the measured value.
Intercept $A = 0$, slope $B \neq 1$: Proportional bias — the disagreement grows with the magnitude; the two methods diverge as the measured quantity increases.
Both $A \neq 0$ and $B \neq 1$: A combination of constant and proportional bias.

If the 95% CI for $B$ includes 1 and the 95% CI for $A$ includes 0, no statistically significant systematic or proportional bias is detected. However, the absence of statistically significant bias does not by itself mean the methods can be used interchangeably — that judgement requires the Bland–Altman limits of agreement and a pre-specified clinical acceptance criterion $\Delta$.

The CUSUM Test for Linearity

Passing–Bablok regression assumes a linear relationship between the two methods. If the relationship is curved — for instance, because one method saturates at high values — the linear model is inappropriate. The CUSUM test (cumulative sum test) checks this assumption by examining whether the residuals from the fitted line show a systematic trend. If $p < 0.05$, linearity cannot be assumed and the regression should not be interpreted.

Sample Size Requirements

The Passing–Bablok estimator requires an adequate sample to produce reliable confidence intervals. A minimum of 30 to 50 paired observations is generally recommended; with fewer, the confidence intervals for slope and intercept become wide enough to be uninformative. For demanding applications — where narrow CIs are needed to rule out small biases — 90 or more observations may be necessary.

MedCalc's Passing–Bablok regression procedure reports the slope and intercept with 95% confidence intervals, the CUSUM test for linearity, and a scatter diagram with the regression line and identity line superimposed. It is best used alongside the Bland–Altman plot as a complementary analysis.