Bland-Altman plot: do two methods agree?

Paired readings from two methods, plotted as difference against average — with the bias, the limits of agreement, their confidence intervals, and a test for whether the disagreement depends on the magnitude.

Nothing you paste leaves this calculation. It is not stored, not logged and not sent anywhere else — the numbers are computed and returned, and that is all.

Why not just correlate the two methods?

Because correlation answers a question nobody asked.

Two methods measuring the same quantity across a range of samples are correlated by construction — big samples read big on both, small read small on both. That produces an r of 0.95 or better almost regardless of how badly the methods agree. A method reading consistently twenty per cent high correlates perfectly with the reference and should never be substituted for it.

Correlation measures whether two things move together. Agreement is whether they give the same answer. Bland and Altman's 1986 paper made the point that the medical literature was full of the first being reported as evidence for the second, and the plot below is what they proposed instead.

Reading the plot

Each point is one sample: its average across the two methods on the horizontal axis, and the difference between them on the vertical. That rotation is the whole trick — disagreement now has an axis of its own, and three things become visible at once.

Bias — the mean difference. A cloud sitting above zero means method A reads systematically higher. This part is correctable: if you know the offset you can subtract it.
Limits of agreement — bias ± 1.96 × the standard deviation of the differences. About 95% of differences between the two methods will fall inside them. This is the part you cannot correct away.
Proportional bias — a sloped cloud, meaning the methods disagree more at one end of the range than the other. When this is present a single pair of limits describes nothing, and the tool says so rather than quoting them anyway.

The question the plot cannot answer

Whether the limits of agreement are narrow enough is not a statistical question and no amount of analysis will settle it. It is a clinical or engineering judgement, and it has to be made before you look at the data.

Decide in advance what difference would change a decision — a dose, a disposition, a release. That figure is your acceptance criterion. Then compare the limits of agreement to it. Doing it the other way round, deciding after seeing the limits whether they seem acceptable, is how a method gets adopted on the strength of a plot that never tested anything.

Two honest caveats about the numbers

The limits use 1.96, the normal approximation, and so do the confidence intervals reported with them. For small samples this understates the interval — at seventeen pairs the correct t multiplier is about 2.12 rather than 1.96, so the true uncertainty around these limits is a little wider than shown.

The limits of agreement are themselves estimates from a sample, and their confidence intervals are wide. With under about fifty pairs the limits are known far less precisely than their two decimal places suggest, which is an argument for more samples rather than for quoting them more confidently.

Questions people ask about this

What is a Bland-Altman plot?

A plot of the difference between two measurement methods against their average, one point per sample. It shows the bias between the methods, how widely they disagree on individual samples, and whether that disagreement changes across the measuring range.

Why not use correlation to compare two methods?

Because two methods measuring the same quantity are correlated by construction, whatever their agreement. A method reading twenty per cent high correlates almost perfectly with the reference and must never be substituted for it. Correlation measures whether two things move together; agreement is whether they give the same answer.

How are the limits of agreement calculated?

The mean difference plus and minus 1.96 times the standard deviation of the differences. About 95% of differences between the two methods are expected to fall inside them. Note that 1.96 is the normal approximation — for small samples the correct t multiplier is larger, so the interval shown is slightly narrow.

What counts as acceptable agreement?

Not a statistical question. Decide in advance what size of difference would change a decision — a dose, a disposition, a release — and compare the limits of agreement to that. Deciding after seeing the limits whether they look acceptable is how a method gets adopted on a plot that tested nothing.

What is proportional bias?

When the difference between the methods depends on how large the measurement is — the cloud of points slopes rather than sitting level. A single pair of limits of agreement then describes nothing, because the methods disagree by different amounts at different ends of the range. This tool tests for it and says so when it is there.

How many samples do I need?

More than most comparisons use. The limits of agreement are themselves estimates and their confidence intervals are wide; below about fifty pairs they are known far less precisely than their decimal places suggest.

Want this to keep itself up to date?

A calculator answers for the data you pasted. A control chart answers for the data your line produced this morning — limits frozen at a baseline you locked, rules evaluated on every new reading, an alert when one trips.

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