Bland–Altman Plot

When a new measurement method — a point-of-care analyser, a novel immunoassay, a wearable sensor — is evaluated against an established reference, the natural but incorrect instinct is to compute a correlation coefficient. A high Pearson $r$ is taken as evidence of agreement. This logic is wrong in a fundamental way: correlation measures the strength of a linear relationship, not the closeness of agreement. Two methods that are perfectly correlated but where one consistently reads twice as high as the other will produce $r = 1.0$ while disagreeing at every single observation. Correlation answers the wrong question.

The correct question is: $how large are the differences between the two methods, and are those differences small enough to be clinically acceptable?$ The Bland–Altman plot (Bland & Altman, 1986) answers this directly and is the standard graphical method for assessing agreement between two methods of clinical measurement.

Construction of the Plot

For each subject, two quantities are computed:

The difference $d_i = A_i - B_i$ between the two methods.
The mean $m_i = (A_i + B_i) / 2$ of the two methods.

The differences are plotted on the vertical axis against the means on the horizontal axis. The mean is used as the horizontal axis rather than one method alone because it is the best available estimate of the true (unknown) value, and using either method's value alone would create an artificial correlation between the axis variable and the plotted differences.

Bland–Altman plot showing differences (New Method minus Reference) on the y-axis against the mean of the two methods on the x-axis. A solid blue line marks the bias of +1.5. Red dashed lines mark the upper limit of agreement (+9.7) and lower limit of agreement (−6.7). Grey bands show the 95% confidence intervals for each line. One open red circle marks an outlier above the upper limit of agreement.

Three horizontal reference lines are drawn:

The mean difference (bias) — the central line.
The upper limit of agreement — the mean difference plus 1.96 times the standard deviation of the differences.
The lower limit of agreement — the mean difference minus 1.96 times the standard deviation of the differences.

Bias and Limits of Agreement

Let $\bar{d}$ be the mean of the $n$ individual differences and $s_d$ be their standard deviation. The bias is simply $\bar{d}$ — the average systematic error of the new method relative to the reference. The limits of agreement (LoA) are:

$$ \text{LoA} = \bar{d} \pm 1.96 \, s_d $$

Under the assumption that the differences are approximately normally distributed, about 95% of individual differences between the two methods will fall within the limits of agreement. In the example above, with $\bar{d} = +1.5$ and $s_d = 4.2$, the limits span from $-6.7$ to $+9.7$: for a typical subject, the new method will read somewhere between 6.7 units lower and 9.7 units higher than the reference.

If the differences are not normally distributed, a non-parametric approach is used instead: the limits of agreement are defined as the 2.5th and 97.5th percentiles of the observed differences.

Confidence Intervals and the Acceptable Difference Δ

The bias and limits of agreement are estimated from a sample and are therefore uncertain. MedCalc reports 95% confidence intervals for all three lines — shown as grey bands in the figure. These CIs communicate how precisely the study has pinned down the true bias and the true limits.

Agreement can only be declared adequate if the limits of agreement are clinically acceptable. This requires pre-specifying a maximum allowable difference $\Delta$ — the largest disagreement between the two methods that would still be clinically harmless. The two methods can be considered interchangeable when both limits of agreement, including their confidence intervals, lie within $[-\Delta, +\Delta]$: that is, when the upper 95% CI limit of the upper LoA does not exceed $+\Delta$, and the lower 95% CI limit of the lower LoA is not below $-\Delta$.

Example: For serum potassium, a clinically acceptable difference might be $\Delta = 0.5$ mmol/L. If the upper limit of agreement is $+0.45$ mmol/L with a 95% CI extending to $+0.52$ mmol/L, the CI exceeds $\Delta$ and agreement cannot be declared adequate — despite the point estimate being within bounds. The confidence interval, not the point estimate alone, determines the verdict.

What to Look for in the Plot

Beyond the summary statistics, the plot itself reveals patterns that numbers alone cannot:

Systematic bias: If the mean difference is far from zero and its 95% CI excludes zero, one method reads consistently higher than the other across the entire measurement range.
Proportional bias: If the differences increase or decrease systematically as the mean increases — showing a trend in the scatter — one method performs relatively better at low values and relatively worse at high values, or vice versa. A regression line through the differences detects this formally.
Heteroscedasticity: If the spread of the differences fans out as the mean increases, the constant-SD assumption behind the LoA formula is violated. In this case MedCalc can fit regression-based limits of agreement that widen or narrow with the magnitude of the measurement.
Outliers: Individual points far outside the limits of agreement warrant investigation — they may represent transcription errors, sample mix-ups, or genuine biological extremes.

The Regression Approach for Proportional Bias

When both proportional bias and heteroscedasticity are present, the fixed limits of agreement are replaced by curves. MedCalc fits two regression models: the differences $D$ on the means $A$, and the absolute residuals $R$ on the means. The resulting limits of agreement are:

$$ \text{LoA}(A) = (\hat{b}_0 \pm 2.46\,\hat{c}_0) + (\hat{b}_1 \pm 2.46\,\hat{c}_1) \cdot A $$

where $\hat{b}_0, \hat{b}_1$ are coefficients from the difference–mean regression and $\hat{c}_0, \hat{c}_1$ are coefficients from the residual–mean regression. The limits now vary with the magnitude $A$, correctly reflecting that measurement uncertainty is not the same across the range.

Relationship to Regression-Based Method Comparison

The Bland–Altman plot is the primary and preferred tool for method comparison. It is graphically transparent, clinically intuitive, and directly addresses whether the methods can be used interchangeably.

Regression-based methods — Passing–Bablok and Deming regression — model the relationship between the two methods as a straight line and test whether the slope equals 1 and the intercept equals 0. They are useful as a supplement when the nature of the systematic error needs to be characterised more precisely: a slope different from 1 signals proportional bias; an intercept different from 0 signals constant bias. They do not, however, directly answer whether the methods agree within clinically acceptable limits — that remains the domain of the Bland–Altman plot.

MedCalc's Bland–Altman plot procedure produces the difference plot with bias, limits of agreement, and their 95% confidence intervals. Regression-based limits of agreement for heteroscedastic data are also available.