Why is a correlation coefficient the wrong way to judge agreement between two assay methods?

Why

Bland and Altman's 1986 Lancet paper argued that the correlation coefficient is no indicator of agreement. A new method that always reads exactly double the old one has a perfect correlation of 1 and terrible agreement. Correlation also rises simply because samples cover a wide range, which says nothing about how well the methods agree for any single sample.

The alternative is to plot the difference between methods against their mean for each sample. The mean difference estimates the bias. The limits of agreement, mean difference plus or minus 1.96 SD of the differences, show where about 95 percent of differences fall.

With 40 paired samples, a mean difference of 2.1 units and an SD of 3.0, the limits run from about -3.8 to 8.0 units. Whether that is acceptable is not a statistical question. Giavarina stresses that acceptable limits must be defined in advance from clinical or biological need.

The plot also reveals patterns a single number hides. If differences grow with concentration, the bias is proportional, and differences expressed as percentages or analyzed on a log scale are more informative. The limits are estimates too, so with small samples report their confidence intervals.