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eduKateSG · Why Mathematics? · Passing–Bablok Regression, Pairwise Slopes and Robust Method Comparison
Start with the question, audit the worked example, or jump to the assumptions and practice. Every chapter stays open for uninterrupted reading.
Understand the question
Define the quantity and follow the mechanism.Route 2 · Chapters 4–7
Reproduce the example
Audit the numbers one step at a time.Route 3 · Chapters 8–13
Check assumptions
See where design and calibration matter.Route 4 · Chapters 14–18
Explore applications
Connect the method to a concrete investigation.Route 5 · Chapters 19–25
Study and verify
Practise, inspect failure cases and read further.
Full chapter index · Practice and checking · Mathematics Learning Hub
Why is mathematics important when two measurement methods should follow a linear relationship but ordinary regression assumptions are doubtful? Passing–Bablok regression estimates a method-comparison line from the ordered set of pairwise slopes. It is nonparametric in a specific sense, resistant to some outliers and designed to examine constant and proportional differences without pretending that one axis is error-free.
This chapter uses six original pairs: X = 1, 2, 3, 4, 5, 6 and Y = 3, 5, 7, 9, 11, 14. Five points lie on Y = 2X + 1; the final point is one unit higher. Ten of the fifteen pairwise slopes equal two, so the worked median slope is two and the median intercept is one. The clean result lets us inspect the algorithm, its assumptions and its limits.
Full chapter index
Chapters 1–3 · Understand the question
Chapters 4–7 · Reproduce the example
Chapters 8–13 · Check assumptions
Chapters 14–18 · Explore applications
Chapters 19–25 · Study and verify
If two methods were identical on the same units, the target line would be Y = X. Method comparison often asks whether an estimated intercept is compatible with zero and slope with one, within practical tolerances.
Our methods instead have a conversion relationship near Y = 2X + 1. The purpose is to recover that relationship despite one modest departure.
Passing–Bablok uses ranks and medians of pairwise slopes rather than minimising squared vertical residuals.
The pairs are (1,3), (2,5), (3,7), (4,9), (5,11) and (6,14). The first five fit exactly on slope two, intercept one.
The sixth point would be 13 on that line but is observed as 14. It is a synthetic illustration, not a real instrument claim.
Plot the pairs and the identity line before computing anything. The graph immediately shows strong linear increase and non-identity scaling.
| Item | X | Y | Y − (2X + 1) |
|---|---|---|---|
| 1 | 1 | 3 | 0 |
| 2 | 2 | 5 | 0 |
| 3 | 3 | 7 | 0 |
| 4 | 4 | 9 | 0 |
| 5 | 5 | 11 | 0 |
| 6 | 6 | 14 | 1 |
Six points generate 6 choose 2 = 15 unordered pairs. For i < j with different X values, calculate (Yj − Yi)/(Xj − Xi).
Among the first five points, there are 5 choose 2 = 10 slopes, all exactly two. The five slopes connecting the final point to earlier points are 11/5, 9/4, 7/3, 5/2 and 3.
The slope list therefore contains ten twos and five larger values.
Sort the fifteen slopes. The first ten positions are all two. The remaining values are 2.2, 2.25, about 2.333, 2.5 and 3.
With no special negative-slope adjustment needed in this positive example, the middle eighth value is two. The estimated slope is therefore two.
Official implementations include rules for slopes of −1, infinite slopes and the shifted median used in method comparison. Hand calculations should follow the exact documented algorithm rather than assume every median-of-slopes method is identical.
| Source pairs | Number | Slopes |
|---|---|---|
| Among first five | 10 | 2 each |
| Point 6 with earlier points | 5 | 11/5, 9/4, 7/3, 5/2, 3 |
| Median position | 8th | 2 |
For slope two, compute Yi − 2Xi for each point. The six values are 1, 1, 1, 1, 1 and 2.
Their median is one. The fitted line is Y = 1 + 2X.
This line captures the dominant relationship while the final point retains a residual of one.
Identity requires intercept zero and slope one. Our fitted line has intercept one and slope two, so the methods are not interchangeable without conversion.
The intercept describes a constant component and the slope a proportional component within the linear model. Confidence intervals are needed for population claims.
Even if intervals included zero and one, practical equivalence would require predefined tolerances and adequate precision.
Did you know that adding one observation creates n new pairwise comparisons when a dataset grows from n to n + 1? Pairwise-slope methods expand quickly.
The slopes are not independent observations because many share points. The algorithm uses their ordering, but inference must respect the underlying six pairs.
Do not calculate a confidence interval as if fifteen slopes came from fifteen independent experiments.
The five-point line dominates the median despite the final departure. That demonstrates resistance to one moderate outlier.
Robust does not mean immune. Several influential points, clustered contamination or leverage at the range extremes can change the result.
Inspect the plot, residuals and data provenance. Never use “robust” as permission to skip quality checks.
Passing–Bablok assumes a linear relationship for method comparison. A curved calibration cannot be summarised honestly by one slope and intercept.
The NIST implementation describes a modified cusum check for departure from linearity. A non-rejection is not proof of perfect linearity, especially in a small sample.
Collect enough points across the operating range and examine residual structure.
The method-comparison procedure is intended for strongly positively related methods. If measurements are weakly related or decreasing, the usual interpretation may not apply.
Confirm that both variables measure the same quantity in compatible direction. Reverse-coded scales should be harmonised only with a documented meaning.
The algorithm’s special treatment of negative slopes is another reason to use a verified implementation.
If two points share the same X value, their pairwise slope can be undefined or infinite depending on Y. If both X and Y tie, the slope is 0/0 and is omitted.
Official software defines practical handling for these cases. Large numbers of ties can affect the estimate and interval.
Preserve raw precision rather than adding arbitrary jitter solely to make denominators nonzero.
Like Deming regression, Passing–Bablok is used when both methods contain measurement error. It does not require a known variance ratio in the same way.
That advantage comes with its own assumptions and efficiency tradeoffs. The choice should follow the measurement process, error structure and study aim.
Compare methods through design and sensitivity, not a universal ranking.
Point estimates alone are insufficient. Passing–Bablok procedures use the ordering of slopes to form a slope interval, then derive intercept uncertainty.
With six pairs, intervals are likely wide. The clean slope two should not be mistaken for high precision.
Software version, confidence level and handling of ties should be reported.
Slope and intercept can be translated into predicted difference at a meaningful X value. At X = 5, the fitted Y is 11, compared with identity value 5.
For methods on different known scales, conversion may be expected. If they should share units, such a difference is unacceptable.
State the target relationship before interpreting coefficients.
The final pair (6,14) is not automatically an error. It might represent random variation, a transcription problem or a genuine range-specific issue.
Check source records and conditions. If corrected, retain an audit trail. If genuine, show sensitivity with and without it.
Deleting a point because it changes the fit undermines the robustness the method is meant to provide.
Pairwise slopes depend on where points lie. A narrow range produces unstable slopes when X differences are small.
Include low, middle and high values relevant to use, with attention to decision thresholds. Avoid extrapolation beyond observed values.
A well-spread design often contributes more than a complicated formula applied to convenience data.
Deming uses a variance-ratio model and squared scaled distances. Passing–Bablok uses an ordered pairwise-slope construction and no declared lambda.
Deming can be efficient under its assumptions; Passing–Bablok can resist some outliers and distributional departures. Neither proves interchangeability alone.
Use difference plots and practical bias limits alongside either regression.
Define methods, units, identity or conversion target, range and tolerance. Collect paired independent items under comparable conditions.
Plot data, compute the verified Passing–Bablok fit and its intervals, check linearity and inspect residuals and outliers. Evaluate bias at important concentrations.
Validate any conversion on new pairs. Preserve software settings and raw data.
Write all fifteen slopes in a triangular list. Verify ten slopes among the first five points equal two and calculate the five slopes involving point six.
Sort them and select the eighth slope. Then calculate six intercept candidates and their median.
This exercise joins combinations, fractions, medians, coordinate geometry and measurement reasoning without relying on a black box.
- Passing–Bablok is not ordinary least squares with a different name.
- Robustness does not remove outliers or assumptions.
- A fitted line does not prove methods agree.
- Pairwise slopes are not independent replicates.
- A confidence interval containing the identity values does not prove equivalence.
- Strong correlation does not establish interchangeability.
The model must be interpreted against units, tolerance and the intended relationship.
Using the six pairs:
- confirm there are fifteen slopes;
- list the ten slopes equal to two;
- calculate 11/5, 9/4, 7/3, 5/2 and 3;
- verify the median slope is two;
- calculate intercept candidates 1, 1, 1, 1, 1 and 2; and
- verify median intercept one.
Then change the final Y from 14 to 20. Determine whether one extreme point changes the median slope and explain why.
Students can compare two measuring methods and calculate pairwise slopes by hand for a small dataset. Drawing the lines between pairs makes the median idea concrete.
Parents can ask why the middle slope may resist one unusual reading and why that does not make the unusual reading irrelevant.
The activity develops problem-solving skills, numeracy and the habit of matching a mathematical method to how data were produced.
Is Passing–Bablok the same as Theil–Sen regression?
They are related through pairwise slopes, but the method-comparison algorithm includes specific ordering and adjustment rules. Use the intended implementation.
Does it need a variance ratio?
It does not use a Deming-style known lambda, but it has other assumptions including linear, strongly positive relation.
Can it handle ties?
Implementations specify how undefined, infinite and special slopes are handled. Many ties require careful interpretation.
Does slope one and intercept zero prove agreement?
No. Uncertainty, difference spread and practical tolerances still matter.
The primary paper is Passing and Bablok, A New Biometrical Procedure for Testing the Equality of Measurements from Two Different Analytical Methods, 1983, DOI 10.1515/cclm.1983.21.11.709. The NIST Passing–Bablok documentation specifies pairwise slopes, special cases, shifted median and linearity guidance.
Continue with the Mathematics Learning Hub and Deming regression documentation to compare modelling assumptions.
Verify pairing, units, operating range, positive linear relation, complete slope rules, ties, intervals, residuals, outliers and practical tolerance. Keep method conversion separate from method agreement.
The importance of mathematics is that a complicated calibration question can be made auditable through combinations and medians while its assumptions remain visible.
Extended practice laboratory
Practice case 1: Exact line
Setup: All points satisfy Y equals 2X plus 1. Task: Calculate slopes.
Check: Every finite pairwise slope is two and every intercept candidate is one. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer.
Practice case 2: One high point
Setup: Raise the final Y by one. Task: Test resistance.
Check: The median stays two because ten of fifteen slopes remain two. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer.
Practice case 3: Many outliers
Setup: Alter three points. Task: Question robustness.
Check: Contamination can dominate pairwise slopes; robust never means invulnerable. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer.
Practice case 4: Six points
Setup: Count pairwise slopes. Task: Use combinations.
Check: Six choose two gives fifteen. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer.
Practice case 5: Ten points
Setup: Count again. Task: Assess computation.
Check: Ten choose two gives forty-five dependent slopes. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer.
Practice case 6: Tied X
Setup: Two X values match but Y differs. Task: Handle the slope.
Check: The pair is infinite under documented implementation rules. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer.
Practice case 7: Tied pair
Setup: Both X and Y match. Task: Handle 0 over 0.
Check: The undefined slope is omitted according to the algorithm. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer.
Practice case 8: Slope minus one
Setup: A pair produces exactly minus one. Task: Check method rules.
Check: Passing–Bablok method-comparison implementations apply a special omission rule. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer.
Practice case 9: Negative slopes
Setup: Some slopes fall below minus one. Task: Use verified software.
Check: The shifted-median index depends on their count. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer.
Practice case 10: Weak association
Setup: Points show little positive relation. Task: Assess suitability.
Check: The method-comparison assumptions are doubtful. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer.
Practice case 11: Curved relation
Setup: Residuals form an arc. Task: Change strategy.
Check: A linear line cannot represent the calibration across range. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer.
Practice case 12: Narrow range
Setup: X differences are tiny. Task: Predict slope noise.
Check: Small denominators amplify measurement variation. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer.
Practice case 13: Wide range
Setup: Add well-spaced valid pairs. Task: Explain benefit.
Check: Range supports slope estimation when it reflects intended use. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer.
Practice case 14: Identity target
Setup: The methods should match. Task: Interpret parameters.
Check: Compare slope with one and intercept with zero using intervals and tolerance. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer.
Practice case 15: Known conversion
Setup: Y should equal two X plus one. Task: Set the target.
Check: Agreement is assessed against that documented line rather than identity. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer.
Practice case 16: Outlier removal
Setup: Delete the final point after seeing it. Task: Assess bias.
Check: Use an independent error rule and report sensitivity. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer.
Practice case 17: Deming comparison
Setup: Both methods are noisy. Task: Choose assumptions.
Check: Deming needs a variance ratio; Passing–Bablok uses slope ordering and other assumptions. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer.
Practice case 18: Theil–Sen confusion
Setup: Software offers a median-slope fit. Task: Verify definition.
Check: Related estimators may handle negative slopes and intercepts differently. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer.
Practice case 19: Intervals
Setup: Point slope equals two. Task: Discuss precision.
Check: A small sample can give wide intervals despite a tidy estimate. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer.
Practice case 20: Threshold bias
Setup: A decision occurs at X equals five. Task: Evaluate the fitted value.
Check: Compare predicted Y and its uncertainty with the operational target. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer.
Practice case 21: Repeated items
Setup: Several pairs come from one specimen. Task: Handle dependence.
Check: Inference must respect clusters instead of counting every row as independent. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer.
Practice case 22: Order drift
Setup: Method Y is always later. Task: Improve design.
Check: Balance timing so drift is not attributed to calibration. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer.
Practice case 23: Validation
Setup: Apply the conversion to new pairs. Task: Prevent optimism.
Check: Independent validation tests transfer beyond the fitted sample. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer.
Practice case 24: Audit file
Setup: Another analyst repeats the procedure. Task: Save essentials.
Check: Provide pairs, units, implementation rules, confidence level, software version and residual checks. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer.