Bland-Altman Limits of Agreement Calculator
Bland-Altman Limits of Agreement Calculator
Turn a mean difference and the scatter around it into limits of agreement, with a confidence interval for the limits themselves — and see why a correlation coefficient of 0.99 tells you nothing about whether two methods agree.
Bland-Altman Limits of Agreement
Bias, SD, n → limitsA new assay against the current method on 50 patient samples: mean difference +2.0%, SD of the differences 4.0%, with an acceptable difference of 15% set before the study
Formula
limits of agreement = bias ± 1.96 × SDdifferences
SE of each limit ≈ √(3 SD² ⁄ n)
95% CI of each limit = limit ± 1.96 × SE
interchangeable when both limits, and their confidence interval, lie inside a difference decided in advance
- bias
- the mean of the paired differences — the systematic part of the disagreement. It is the part recalibration can remove, and the part a correlation coefficient cannot see at all
- SDdifferences
- the standard deviation of the paired differences. Not either method’s CV: it contains the imprecision of both, combined in quadrature, together with any genuine sample-specific interaction such as an interferent one method sees and the other does not
- 1.96
- the normal quantile for a 95% interval. Bland and Altman’s 1986 paper works throughout with 2 rather than 1.96, which widens the limits by about 2%; either is defensible provided you say which
- √(3 SD²⁄n)
- the standard error of a limit, from Bland and Altman. It has that form because a limit carries the uncertainty of the mean, SD²⁄n, plus 1.96² times the uncertainty of the SD itself, which is about SD²⁄2n: together 1 + 1.9208⁄2 ≈ 2.92, rounded to 3. Notice how little of the uncertainty comes from the mean
- the multiplier on the SE
- this page uses 1.96. Bland and Altman use the t distribution with n − 1 degrees of freedom, which is wider at small n: t is 2.093 at n = 20, 2.010 at n = 50 and 1.984 at n = 100. So the interval printed here is about 7% too narrow at n = 20 and about 1% too narrow at n = 100
- what the limits are not
- a pass mark. They describe how far apart the two methods are; they say nothing about whether that distance matters. The acceptable difference is a clinical judgement, and Bland and Altman are explicit that it should be written down before the comparison is run
- the assumptions
- the differences are approximately normally distributed, and their size and spread do not depend on the magnitude of the measurement. Both are checkable by eye on the difference plot, and both fail often enough that checking is not optional
Worked example
A new assay against the current method on 50 patient samples: mean difference +2.0%, SD of the differences 4.0%, with an acceptable difference of 15% set before the study
1.96 × 4.0 = 7.84%, so the limits of agreement are 2.0 ± 7.84
Limits = −5.84% to +9.84%. Ninety-five per cent of individual patient samples will differ between the two methods by somewhere in that range
The wider limit, taken as an absolute size, is 9.84%
Standard error of each limit = √(3 × 4.0² ⁄ 50) = √0.96 = 0.98%, so the 95% confidence interval of the upper limit is 9.84 ± 1.96 × 0.98 = 7.92% to 11.76%
Against the 15% acceptable difference: 9.84% passes, and so does 11.76%, the top of the limit's own confidence interval. The conclusion is not just that the methods agreed well enough but that 50 samples were enough to establish it
Compare what a correlation coefficient would have said. Bland's own demonstration: add 100 to every measurement by one method and r is unchanged at 0.94, while the bias becomes 100 and the methods agree on nothing. Correlation measures whether two variables move together, which two methods of measuring the same quantity always do
Change one thing — the acceptable difference to 10% — and the verdict inverts. 9.84% still looks like a pass, but the confidence interval reaches 11.76%, so the data do not support agreement at 10%. This is why the specification has to be fixed first
The same bias and SD, different sample sizes
| n | Limits of agreement | SE of a limit | 95% CI of the upper limit | t at 95% |
|---|---|---|---|---|
| 10 | −5.84% to 9.84% | 2.19% | 5.55% to 14.13% | 2.262 |
| 20 | −5.84% to 9.84% | 1.55% | 6.80% to 12.88% | 2.093 |
| 50 | −5.84% to 9.84% | 0.98% | 7.92% to 11.76% | 2.010 |
| 100 | −5.84% to 9.84% | 0.69% | 8.48% to 11.20% | 1.984 |
| 200 | −5.84% to 9.84% | 0.49% | 8.88% to 10.80% | 1.972 |
Where the disagreement comes from, and whether you can fix it
| Bias | SD of differences | Limits | Wider limit | What to do |
|---|---|---|---|---|
| 0% | 4.0% | −7.84% to 7.84% | 7.84% | Nothing is systematically wrong; the width is imprecision in the two methods |
| 2.0% | 4.0% | −5.84% to 9.84% | 9.84% | Recalibrating away the 2% bias would bring the wider limit to 7.84% |
| 5.0% | 4.0% | −2.84% to 12.84% | 12.84% | Bias now dominates. A traceability investigation is the first move, and it is the one move that can work |
| 2.0% | 2.0% | −1.92% to 5.92% | 5.92% | Halving the scatter halves the width. Usually much harder than removing a bias |
| 2.0% | 8.0% | −13.68% to 17.68% | 17.68% | Scatter dominates completely; removing the bias would change the wider limit by 2 points out of 17.7 |
| 9.0% | 1.0% | 7.04% to 10.96% | 10.96% | Both limits on the same side of zero — a pure, correctable offset that correlation would score as near-perfect agreement |
Why correlation cannot answer the question, and what the limits do and do not settle
Two methods are compared on fifty patient samples, the correlation coefficient comes back at 0.99, and the report concludes that the methods agree. It does not follow, and the reason is simple enough to be worth stating plainly: correlation measures whether two variables move together, and two methods of measuring the same quantity across a clinical range always move together. Martin Bland’s own demonstration is the cleanest available. Take a set of paired peak flow measurements with r = 0.94, add 100 to every reading by one of the methods, and the correlation is still 0.94 — while the two methods now disagree by 100 units on every single sample. Correlation is blind to bias by construction, it is inflated by a wide range of measurements, and it answers a question nobody asked.
The Bland-Altman analysis asks the question that was asked. Plot the difference between the two methods against their average, take the mean of the differences as the bias, and take the bias plus and minus 1.96 standard deviations of the differences as the limits of agreement. Those two numbers say that ninety-five per cent of individual samples will differ by somewhere between them. With a 2% bias and a 4% standard deviation of the differences, the limits are −5.8% to +9.8%: a single patient sample measured by both methods could plausibly come back nearly 10% higher on the new one. That is a statement about individual results, which is what a clinician has, rather than about the average behaviour of the two methods, which is what a regression line describes.
The limits themselves are an estimate, and the estimate of a limit is much less precise than people expect. Bland and Altman give the standard error of a limit as approximately the square root of three times the variance of the differences divided by n — most of that uncertainty coming from the standard deviation rather than from the mean. On fifty pairs that is 0.98%, so the upper limit of 9.84% carries a confidence interval of roughly 7.9% to 11.8%. On ten pairs the same limit ranges from 5.6% to 14.1%, which is the difference between a comfortable pass and a clear failure. Quote the interval. This page computes it with 1.96, whereas Bland and Altman use the t distribution with n − 1 degrees of freedom; at n = 20 that makes the honest interval about 7% wider than the one shown here, and by n = 100 the difference is under 1.5%.
What the limits cannot do is pass or fail anything. They describe the disagreement; whether that disagreement matters is a clinical judgement, and Bland and Altman say in the original paper that it “should be defined in advance to help in the interpretation of the method comparison”. A laboratory that computes the limits first and then decides what would have been acceptable has tested nothing. Write the acceptable difference into the protocol, derive it from biological variation or a regulatory limit or the decision the test supports, and then compare — and compare the confidence interval of the limit with it, not just the limit.
Two assumptions are doing quiet work throughout. The differences are taken to be approximately normally distributed, and their size and spread are taken not to depend on the magnitude of the measurement. Both are visible on the difference plot and both fail often. If the plot fans out — differences proportional to concentration, which is the usual pattern for an immunoassay across three decades — a single pair of limits averages tight agreement at one end with none at the other, and the remedy is to analyse the logarithms of the measurements, which Bland and Altman recommend as the only transformation whose back-transformed differences read naturally, as ratios. If instead the differences drift with the mean without fanning, regress the difference on the average and quote limits that vary with concentration. If a handful of points sit far from the rest, find out what those samples were before letting them widen the limits for everyone else.
Frequently asked questions
How do you calculate Bland-Altman limits of agreement?
Take the difference between the two methods for every paired sample. The mean of those differences is the bias, and the limits of agreement are the bias plus and minus 1.96 times the standard deviation of the differences. With a bias of 2.0% and an SD of differences of 4.0%, the limits are −5.84% to +9.84%.
What is the confidence interval for the limits of agreement?
Bland and Altman give the standard error of each limit as approximately the square root of 3 SD²/n, where SD is the standard deviation of the differences. The interval is the limit plus and minus that standard error multiplied by t with n − 1 degrees of freedom; this page uses 1.96 instead, which is about 7% too narrow at n = 20 and around 1% too narrow at n = 100. On 50 pairs with an SD of 4%, the standard error is 0.98% and an upper limit of 9.84% has a confidence interval of 7.9% to 11.8%.
Why is a correlation coefficient not a measure of agreement?
Because it measures whether two variables move together, not whether they give the same answer. Adding a constant to every measurement by one method leaves the correlation completely unchanged while destroying the agreement, and r also rises simply because the samples span a wide range. A correlation of 0.99 is compatible with a clinically unacceptable bias, which is why Bland and Altman wrote that r “measures the strength of a relation between two variables, not the agreement between them”.
Do the limits of agreement tell me whether two methods are interchangeable?
Only once you have said what interchangeable means. The limits describe how far apart the methods are; the acceptable difference is a clinical judgement that has to be set in advance, from biological variation, a regulatory specification, or the decision the test supports. Then the test is whether both limits — and preferably the confidence interval of the wider one — lie inside that difference.
What if the differences get bigger as the measurements get bigger?
Then a single pair of limits describes neither end of the range and should not be quoted. Look at the difference plot. If the spread is proportional to the mean, analyse the logarithms of the measurements and back-transform, which turns the limits into ratios rather than absolute differences. If the differences merely drift with the mean, regress the difference on the average and report limits that change with concentration.
How many paired samples do I need?
Enough that the confidence interval of the limits is narrower than the decision you are making with them. Because the standard error of a limit falls as the square root of n, going from 10 pairs to 50 halves the interval and going from 50 to 200 halves it again. CLSI EP09c method comparison studies commonly use 40 or more paired patient samples, and 40 is a reasonable floor where the samples span the measuring interval.
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References
- Bland JM, Altman DG. Statistical methods for assessing agreement between two methods of clinical measurement. Lancet. 1986;1(8476):307–310.
- Bland JM, Altman DG. Measuring agreement in method comparison studies. Stat Methods Med Res. 1999;8(2):135–160.
- Bland JM, Altman DG. Comparing methods of measurement: why plotting difference against standard method is misleading. Lancet. 1995;346(8982):1085–1087.
- CLSI EP09c. Measurement Procedure Comparison and Bias Estimation Using Patient Samples. 3rd ed. Clinical and Laboratory Standards Institute; 2018.
- Bilić-Zulle L. Comparison of methods: Passing and Bablok regression. Biochem Med (Zagreb). 2011;21(1):49–52.
Medical Disclaimer: The tools and content provided here are for educational and reference purposes only. They are not intended to substitute for professional medical advice, diagnosis, or treatment. Clinical decisions should always be based on the comprehensive assessment of a qualified healthcare professional.