Method agreement and limits of agreement

Let $(\mathrm{\Omega },\mathcal{F},\mathbb{P})$ be the declared probability model for a pre-specified target population of measurement conditions. Let $A$ and $B$ be real random variables giving quantitative measurements of the same target under the same stated conditions. The choice of target population and the scientific applicability of its probability law are model assumptions; this definition does not verify them.

Assume the paired difference $D:=A-B$ is square-integrable. It is measurable because measurable real functions are closed under finite sums and scalar multiples (Closure properties of measurable functions used by the integral). Since $0\le (\mathrm{\mid }D\mathrm{\mid }-1{)}^2$, we have $\mathrm{\mid }D\mathrm{\mid }\le (D^2+1)\mathrm{/}2$. Monotonicity, homogeneity, and additivity of the nonnegative integral, together with $\mathbb{P}(\mathrm{\Omega })=1$, give $\mathbb{E}[\mathrm{\mid }D\mathrm{\mid }]\le \frac{1}{2}\mathbb{E}[D^2]+\frac{1}{2}<\mathrm{\infty }$ (Monotonicity and nonnegative homogeneity of the nonnegative integral, Additivity of the nonnegative Lebesgue integral). Thus define the finite population bias ${\mu }_D:=\mathbb{E}[D]$. Put $H:=D-{\mu }_D$, which is measurable by the same closure result. Its square $H^2$ is measurable by Arithmetic and lattice operations preserve measurability whenever they are defined. Since $H^2\le 2D^2+2{\mu }_D^2,$ monotonicity, homogeneity, and additivity of the nonnegative integral give $\mathbb{E}[H^2]\le 2\mathbb{E}[D^2]+2{\mu }_D^2<\mathrm{\infty }$ (Monotonicity and nonnegative homogeneity of the nonnegative integral, Additivity of the nonnegative Lebesgue integral). Therefore ${\sigma }_D:=\sqrt{\mathrm{Var}(D)}=\sqrt{\mathbb{E}[H^2]}$ is a finite real standard deviation, since $\mathrm{Var}(D)=\mathbb{E}[H^2]$ by Moments, variance, and covariance on a probability space and every nonnegative real has a square root by Square roots exist: a unique $\sqrt{a}\ge 0$ with $(\sqrt{a}{)}^2=a$; the positives are $\{x^2:x\mathrm{\ne }0\}$. The sign convention is $D=A-B$, so positive bias means that $A$ reads larger than $B$ on average. The conventional rounded normal-reference limits of agreement are ${\mu }_D-1.96{\sigma }_D$ and ${\mu }_D+1.96{\sigma }_D$.

For an iid sample $D_1,\dots ,D_n$ of paired differences from the declared target population, with $n\ge 2$ and observed values $d_1,\dots ,d_n$, define $\bar{d}:=\frac{1}{n}{\sum }_{i=1}^n{d}_i,s_D:=\sqrt{\frac{1}{n-1}{\sum }_{i=1}^n(d_i-\bar{d}{)}^2}\mathrm{.}$ The sample plug-in limits are $\bar{d}\pm 1.96s_D$: they substitute the sample mean and sample standard deviation into the conventional population formula. They are point estimates; no unbiasedness, consistency, confidence coverage, or exact finite-sample coverage for either endpoint is asserted here.

Repeated measurements on one target are not iid merely because there are many recorded pairs. Their within-target dependence must be stated and modeled, for example with a covariance or random-effects model. When defined, correlation measures linear association and does not by itself establish agreement. Limits of agreement describe the distribution of differences; they do not decide whether those differences are acceptable for a particular use or whether either method is calibrated to a known truth.

The exact central normal quantile formula is established by a separate result on this page. The customary $1.96$ display is a rounded normal quantile convention.