# A note on Passing-Bablok regression

Passing & Bablok (1983) have presented a statistical method for non-parametric regression analysis suitable for method comparison studies. The procedure is symmetrical and is robust in the presence of one or few outliers. But probably its main advantage is that is (or rather - seems to be) easy to interpret, which might explain its widespread use in method comparison studies.

In short, 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
median of all slopes of the straight lines between any two points, excluding
lines for which *b* = 0 or *b* = ∞. The parameter
*a* is calculated by *a* = median { y_{i} -* b* x_{i}
}. For both *a* and *b* the 95% confidence intervals (CI) are
calculated. The Passing-Bablok procedure is valid only when a linear
relationship exist between x and y, which can be assessed by a cusum test (Passing & Bablok ,1983).

The results then 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. Passing & Bablok (1983) make a
stronger statement "If we accept both *b*=1 and *a*=0 we can infer
y*=x* or, in other words, both methods are identical". 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.

The equation y = *a* + *b* x defines the regression
line, but not all observations lie on that line. In fact, observations are not
defined by the equation y = *a* + *b* x but rather by y = *a* + *b*
x + *e*, with *e* (the residual) being the difference of the observed
y value with the value predicted by the regression equation for the
corresponding x value. Passing & Bablok (1983) suggest that residuals should
be calculated as the orthogonal distance to the fitted line; this is merely a linear transformation of the residuals as defined above.

The residuals represent the remaining variation after correcting for systematic and proportional differences.

Since the procedure supposes a linear relationship the residuals should show a random pattern and should be close to a normal distribution. Therefore 95% of the residuals should lie in the interval ± 1.96 times the residual standard deviation. This interval then defines the random differences between the two laboratory methods.

With intercept close to 0 and slope close to 1, the residuals plot shows a remarkable correspondence with the bias plot (Bland & Altman 1986; Krouwer 2008) and its interpretation can be similar.

Passing-Bablok regression residuals plot. Data from Bland & Altman (1986). Residual standard deviation = 22.46.

Bias plot with limits of agreement lines. Differences are plotted against method 1 for sake of comparison with the residuals plot above.

The bias plot shows the differences that can be expected between two methods, but what differences are acceptable?

Jensen & Kjelgaard-Hansen (2010) give two approaches to
define acceptable differences between two methods. In the first approach the
combined inherent imprecision of both methods is calculated (CV^{2}_{method1}
+ CV^{2}_{method2})^{1/2}, or in case of duplicate
measurements [(CV^{2}_{method1} /2)+ (CV^{2}_{method2})/2)]^{1/2}.
In the second approach acceptance limits are based on analytical quality
specifications such as for example reported by the Clinical Laboratory
Improvement Amendments (CLIA). A third approach might be to base acceptance
limits on clinical requirements. If the observed random differences are too
small to influence diagnosis and treatment, these differences may be acceptable
and the two laboratory methods can be considered comparable.

With regards to sample size, it is obvious that when sample size becomes smaller, the CIs become larger (wider) and the probability that 0 is in the CI of the intercept and 1 is in the CI of the slope increases. The result is that method comparison studies based on small sample sizes are biased to the conclusion that the laboratory methods are comparable. Sample sizes for Passing-Bablok regression should therefore not be small. Ludbrook (2010) cites a sample size of at least 50. More detailed estimates of suitable sample sizes are given by Passing & Bablok (1984).

We conclude that the statement of Passing & Bablok (1983) "If we accept both *b*=1 and *a*=0 we can infer
y*=x* or, in other words, both methods are identical", is wrong and that the two laboratory methods are not comparable when the condifence interval of the residuals is large.

We recommend that Passing-Bablok regression for method comparison studies should not be limited to estimation of intercept and slope with their 95% CI. With the intercept being a measure of systematic difference, the slope being a measure of proportional difference, the residuals represent random differences and should be included in the statistical analysis and in the method comparison report.

## References

- Bland JM, Altman DG (1986) Statistical method for assessing agreement between two methods of clinical measurement. The Lancet i:307-310. [Abstract]
- Jensen AL, Kjelgaard-Hansen M (2010) Diagnostic test valiation. In: Weiss D, Wardrop KJ, editors. Schalm's Veterinay Hematology, 6
^{th}ed. Ames: Wiley-Blackwell; p. 1027-1033. - Krouwer JS (2008) Why Bland-Altman plots should use
*X*, not (*Y*+*X*)/2 when*X*is a reference method. Statistics in Medicine 27:778-780. [Abstract] - Ludbrook J (2010) Linear regression analysis for comparing two measures or methods of measurement: but which regression? Clinical and Experimental Pharmacology & Physiology 37:692-699. [Abstract]
- Passing H, Bablok W (1983) 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 of Clinical Chemistry & Clinical Biochemistry 21:709-720. [Abstract]
- Passing H, Bablok W (1984) 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 of Clinical Chemistry & Clinical Biochemistry 22:431-445. [Abstract]