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eduKateSG · Why Mathematics? · Lin’s Concordance Correlation Coefficient, Precision and Identity-Line Accuracy
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 19–23
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 instruments, reviewers or systems produce numerical readings that are strongly correlated but do not actually agree? Lin’s concordance correlation coefficient, usually called the CCC, combines precision with accuracy relative to the 45-degree identity line. It can reveal a constant shift or scale mismatch that Pearson correlation alone may overlook.
This chapter builds the coefficient from six original paired readings. Pearson correlation is exactly one because every second reading is the first plus one. Lin’s CCC is only 70/73 because the pairs lie on a parallel line above equality. The difference teaches a central lesson for mathematics in science, engineering, AI evaluation and everyday measurement: association is not interchangeability.
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 19–23 · Study and verify
If two methods agree perfectly, every paired point lies on y = x. A reading of 10 from method X is accompanied by 10 from method Y, and the same is true across the entire range.
Pearson correlation asks whether points follow a straight increasing or decreasing pattern. It does not require that pattern to be the identity line. The points can lie exactly on y = x + 1 and still have correlation one.
Lin’s CCC brings location and scale into the comparison. It rewards points that are both tightly aligned and close to equality. This makes the geometry of agreement visible.
The synthetic X readings are 10, 12, 14, 16, 18 and 20. The Y readings are 11, 13, 15, 17, 19 and 21. Every Y value exceeds its paired X by one unit.
The X mean is 15 and the Y mean is 16. Both series have the same spread. A scatterplot is a perfect straight line, but it is displaced upward.
These numbers are a teaching construction, not observations from a real school, device or learner. In a real method-comparison study, units, pairing, range and data-collection conditions must be documented before any coefficient is interpreted.
| Item | X | Y | Y − X |
|---|---|---|---|
| 1 | 10 | 11 | 1 |
| 2 | 12 | 13 | 1 |
| 3 | 14 | 15 | 1 |
| 4 | 16 | 17 | 1 |
| 5 | 18 | 19 | 1 |
| 6 | 20 | 21 | 1 |
The X deviations from 15 are −5, −3, −1, 1, 3 and 5. Their squared sum is 70. Using population-style moments with denominator six gives variance 70/6 = 35/3.
The Y deviations from 16 are identical, so Y has the same variance 35/3. The paired cross-products also sum to 70, giving covariance 35/3.
Using sample denominator five would multiply covariance and both variances by the same factor, but the mean-difference term requires a consistent CCC definition. State the software convention and reproduce its result.
Pearson r is covariance divided by the product of the two standard deviations. Here covariance and both variances are 35/3, so r = 1.
That is correct. The two series have perfect linear association. Knowing X determines Y exactly through Y = X + 1.
Yet method Y is consistently one unit higher. If the methods are meant to be interchangeable, that shift matters. Correlation answers a relationship question; agreement adds an equality question.
One common CCC formula is 2sxy divided by sx² + sy² + (mean X − mean Y)². Substitute sxy = 35/3, both variances 35/3 and the squared mean difference 1.
The numerator is 70/3. The denominator is 70/3 + 1 = 73/3. The CCC is therefore 70/73, approximately 0.9589.
The value remains high because the one-unit bias is small relative to the data spread. It is below one because the identity line is missed systematically.
| Quantity | Value |
|---|---|
| Mean X | 15 |
| Mean Y | 16 |
| Variance X | 35/3 |
| Variance Y | 35/3 |
| Covariance | 35/3 |
| Pearson r | 1 |
| Lin CCC | 70/73 |
Lin’s CCC can be written as Pearson r multiplied by a bias-correction factor Cb. In this example r = 1 and Cb = 70/73, so their product is 70/73.
The precision component asks how closely points follow a line. The accuracy component asks how far that line’s location and scale depart from equality.
This decomposition is valuable in calibration. A low r suggests scatter; a lower Cb with high r suggests a systematic location or scale mismatch. Different fixes may be needed.
Did you know that the same one-unit shift can look less important to CCC when the observed range becomes much wider? The mean difference stays one, while variance grows.
That is not a mathematical error. It means CCC judges bias relative to spread. A one-degree difference across temperatures from 0 to 100 is different from the same difference across 20 to 22.
Always report units and practical tolerance. A high coefficient does not prove the absolute difference is acceptable for a decision.
Constant bias shifts the line through the intercept, as in Y = X + 1. Proportional bias changes slope, as in Y = 1.1X. Both can reduce concordance.
The CCC combines these effects into one summary. To diagnose them, also inspect a scatterplot, an equality line and a difference plot or regression model.
A single coefficient is useful for comparison and monitoring, but the graph tells the reader where disagreement occurs.
The identity line assumes X and Y use the same units and meaning. Comparing metres with centimetres without conversion creates an artificial scale mismatch.
Convert units first using a documented rule. Do not transform one method merely to maximise CCC unless recalibration is the stated purpose and validated on new data.
In education, numerical scores may also have different meanings despite similar ranges. A raw score and a scaled score are not interchangeable just because both run from 0 to 100.
Each X reading must correspond to the same item, person or specimen as its Y reading. Sorting both columns separately can create a high correlation while destroying the actual pairs.
Keep a stable identifier and measurement time. If repeated readings exist, decide whether to average, model repeated measures or analyse within-item replicates.
Mathematics depends on data structure. CCC cannot recover pairing that was lost during preparation.
One reading per method mixes repeatability and between-item variation. Replicates can estimate within-method noise and help distinguish random scatter from systematic bias.
Do not treat repeated readings from one item as independent new items. A model for repeated measures or a cluster-aware interval may be required.
The intended generalisation matters: new items, new operators, new instruments or new days can introduce different sources of variation.
One extreme pair can inflate the range, change means and variances, and alter CCC substantially. Inspect the raw plot before deleting anything.
An outlier may be a recording error, a device failure or a genuine difficult case. Removal needs an independent rule and a sensitivity analysis.
Report results with and without a justified correction when the conclusion depends on one observation. Quiet deletion turns uncertainty into hidden selection.
The coefficient describes agreement; it does not by itself supply the best transformation from one method to another. A calibration problem may need regression with error in both variables.
Deming or Passing–Bablok regression can examine intercept and slope. Bland–Altman analysis can display differences and limits. Each answers a related but distinct question.
Use a toolkit rather than asking one statistic to prove every aspect of comparability.
A narrow sample range can lower correlation and CCC even when absolute differences are small. A very wide range can raise them while important local errors remain.
Select items that represent the intended operating range, especially around decision thresholds. State any deliberate oversampling.
Comparing CCC values across studies with different ranges is therefore risky. Show the range, distribution and units alongside the coefficient.
CHAPTER 15 OF 23 · Explore applications
15. Uncertainty belongs beside the estimate
The exact 70/73 is a property of our six synthetic pairs. A real CCC estimate varies with the sample. Confidence intervals can use methods appropriate to the coefficient and design.
Six pairs would usually provide limited precision. Clustered or repeated data require intervals that respect dependence.
Avoid rigid adjective labels without uncertainty and consequences. A coefficient near 0.96 can still be inadequate if allowed error is tiny.
Define interchangeability and tolerance before collecting data. Measure the same items under comparable conditions and across the relevant range.
Plot Y against X with the identity line. Calculate means, spreads, Pearson r and CCC. Add a difference plot and method-comparison regression when needed.
Investigate bias, repeatability, outliers and threshold behaviour. Validate any recalibration on new paired observations.
List the six deviations and verify their squared sum is 70. Show the covariance also equals 35/3 and Pearson r equals one.
Insert the one-unit mean difference into the CCC formula to obtain 70/73. Then add one to every Y value again and observe how the squared mean difference becomes four.
The exercise joins algebra, statistics, geometry and measurement reasoning. It shows why “moves together” and “gives the same answer” are different claims.
- Perfect correlation does not guarantee agreement.
- A high CCC does not prove differences are practically acceptable.
- CCC does not identify the cause of disagreement.
- Different units must be reconciled before using an identity-line measure.
- A wide measurement range can influence the coefficient.
- Agreement does not establish validity against truth.
The most useful correction is simple: always view the paired plot and the units before reading the decimal.
Using X = 10, 12, 14, 16, 18, 20 and Y = X + 1:
- verify means 15 and 16;
- verify both variances and covariance are 35/3;
- confirm r = 1;
- calculate CCC = 70/73; and
- identify the constant bias.
Next replace Y by 2X. Pearson correlation remains one. Calculate the new means and variances, then explain why concordance with y = x becomes much weaker.
Parents can connect this idea to two clocks that gain time together. Their readings may rise in perfect correlation while one remains five minutes fast.
Students can compare two rulers, sensors or scoring methods. Record pairs, use consistent units and decide an acceptable difference before computing a statistic.
This habit supports numeracy, science experiments, engineering, computing and mathematics in careers because it links calculations to an explicit decision.
Is CCC the same as Pearson correlation?
No. CCC incorporates departure from the equality line as well as linear precision.
Can CCC be negative?
Yes. Negative association or severe discordance can produce negative values.
Does a high CCC mean methods are interchangeable?
Not automatically. Practical tolerances, difference patterns, uncertainty and operating range also matter.
Should the data be paired?
Yes. The coefficient concerns corresponding readings on the same units of analysis.
Lin’s primary paper is A Concordance Correlation Coefficient to Evaluate Reproducibility, Biometrics 45(1), 1989, DOI 10.2307/2532051. The NIST correlation documentation helps separate linear correlation from agreement.
Use the full chapter index above for the measurement-agreement sequence, then continue to Kendall’s W for rank concordance, and the Mathematics Learning Hub.
Verify pairing, units, identity-line meaning, sample range, means, variances, covariance, interval method and practical tolerance. Report the scatterplot and useful component summaries. Do not describe association as interchangeability or a coefficient as proof of validity.
The importance of mathematics is that it separates two questions that everyday language often blends: whether measurements move together and whether they give the same answer.
Extended practice laboratory
Practice case 1: Perfect equality
Setup: Set Y equal to X for every pair. Task: Calculate r and CCC.
Check: Both equal one when spread is positive. 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: Constant shift
Setup: Add two units to every Y value. Task: Track r and CCC.
Check: Correlation stays one while the squared mean difference reduces concordance. 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: Scale shift
Setup: Set Y equal to twice X. Task: Compare association and identity agreement.
Check: Pearson remains one but CCC falls because means and variances miss equality. 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: Unit conversion
Setup: Record X in metres and Y in centimetres. Task: Prepare the data.
Check: Convert units before treating y equals x as the agreement 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 5: Zero spread
Setup: Give every X the same value. Task: Check definability.
Check: Correlation and CCC can become undefined because variance vanishes. 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: Wide range
Setup: Add distant valid pairs on the same shifted line. Task: Predict CCC.
Check: Bias becomes smaller relative to spread, so the coefficient may rise. 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: Narrow range
Setup: Keep only middle values. Task: Predict stability.
Check: Reduced variance can make the same absolute bias more prominent. 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: Outlier
Setup: Move one Y value far away. Task: Audit influence.
Check: Means, variance, covariance and CCC all change; investigate before exclusion. 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: Reordered Y
Setup: Sort Y independently of X. Task: Identify the error.
Check: Pairing is destroyed even though marginal values stay identical. 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: Negative relation
Setup: Let Y decrease as X rises. Task: Interpret sign.
Check: Negative covariance can produce negative concordance and rejects interchangeability. 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: Repeated items
Setup: Measure each item three times. Task: Choose an analysis unit.
Check: Use a repeated-measures approach rather than treating replicates as independent items. 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: Threshold sample
Setup: Add many pairs near a decision cut-off. Task: State the benefit.
Check: Local agreement becomes visible where consequences are greatest. 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: Different constructs
Setup: Compare raw marks with scaled grades. Task: Assess identity meaning.
Check: Similar numbers do not create a common measurement scale. 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: Calibration
Setup: Subtract the estimated constant bias from Y. Task: Validate properly.
Check: Recalculate on new pairs to avoid claiming success on the calibration 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 15: Pearson only
Setup: A report gives r equals 0.99. Task: Request missing evidence.
Check: Ask for identity-line plot, differences, units and practical 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 16: High CCC
Setup: CCC is 0.98. Task: Avoid overclaiming.
Check: Absolute errors may still exceed the allowed decision threshold. 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: Low CCC
Setup: CCC is modest in a narrow range. Task: Inspect context.
Check: Small absolute differences can coexist with limited between-item variance. 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: Clustered students
Setup: Several scores come from each learner. Task: Plan uncertainty.
Check: Resample or model at learner level when dependence matters. 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: Time drift
Setup: Method Y is always measured later. Task: Protect design.
Check: Balance order or control time so drift is not mistaken for method bias. 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: Missing pair
Setup: Y is absent for difficult items. Task: Define the subset.
Check: Complete cases may be selective and need explicit reporting. 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: Bias factor
Setup: Write CCC as r times Cb. Task: Diagnose the example.
Check: Here r is one and all loss comes from the bias-correction factor. 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: Tolerance
Setup: Allow differences of at most 0.2 units. Task: Interpret 70/73.
Check: A high coefficient cannot override an absolute tolerance that the one-unit shift violates. 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: New operator
Setup: A different person repeats Y. Task: Name the change.
Check: The study now includes operator variation and may not match the original estimand. 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: Reproducibility
Setup: Another analyst checks the result. Task: List inputs.
Check: Provide paired values, units, formula convention, software version and interval method. 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.