Considering the confidence interval estimation of LOAs has symmetry of *μ* and –*μ* (*μ *≥ 0) and the sample size estimations of these two situations should be the same, we just discuss the situation when *μ *≥ 0. According to the statistical inference principle of Bland–Altman limits of agreement, we can separate total type I error $(\mathrm{\alpha})$ into two parts which are both $\mathrm{\alpha}/2$. Similarly, we can separate total type II error $(\mathrm{\beta})$ into two parts. One is the first type II error $({\mathrm{\beta}}_{1})$ of the upper limit value of LOAs and the other is the second type II error $({\mathrm{\beta}}_{2})$ of the lower limit value of LOAs (Figure 1.) [13].

Figure 1: Decomposition diagram of type I error (*α*) and type II error (*β*) for Bland and Altman method. Total type I error (*α*) consisted of two parts which are both *α*/2. Similarly, total type II error (*β*) consisted of two parts. One is the first type II error (*β*_{1}) of the upper limit value of LOAs and the other is the second type II error (*β*_{2}) of the lower limit value of LOAs.

We can get a direct estimate of ${\mathrm{\beta}}_{1}$ and ${\mathrm{\beta}}_{2}$:

$\begin{array}{rl}& \begin{array}{ccc}\stackrel{\u203e}{D}& +& {z}_{{}_{1-\mathrm{\gamma}/2}}{S}_{D}-{z}_{{\mathrm{\beta}}_{1}}s{e}_{LOAs}=\mathrm{\delta}+{z}_{\mathrm{\alpha}/2}s{e}_{LOAs}\\ \stackrel{\u203e}{D}& -& {z}_{{}_{1-\mathrm{\gamma}/2}}{S}_{D}+{z}_{{\mathrm{\beta}}_{2}}s{e}_{LOAs}=-\mathrm{\delta}-{z}_{\mathrm{\alpha}/2}s{e}_{LOAs}\\ {\mathrm{\beta}}_{1}& =& \mathrm{\Phi}\left(\frac{\stackrel{\u203e}{D}+{z}_{1-\mathrm{\gamma}/2}{S}_{D}-\mathrm{\delta}}{s{e}_{LOAs}}-{z}_{\mathrm{\alpha}/2}\right)\\ {\mathrm{\beta}}_{2}& =& \mathrm{\Phi}\left(\frac{-\stackrel{\u203e}{D}+{z}_{1-\mathrm{\gamma}/2}{S}_{D}-\mathrm{\delta}}{s{e}_{LOAs}}-{z}_{\mathrm{\alpha}/2}\right)\end{array}\\ & \end{array}$

where $\mathrm{\Phi}(\bullet )$ is defined as the cumulative density function of standard normal distribution $N(0,1)$, $s{e}_{LOA\mathrm{s}}$ is the standard error of the lower limit or upper limit of LOAs, $s{e}_{LOAs}={S}_{D}\sqrt{\frac{1}{n}+\frac{{z}_{1-\mathrm{\gamma}/2}^{2}}{2(n-1)}}$, $\mathrm{\delta}$ is the maximum allowable difference that can be accepted clinically, it needs to be defined in advance.

According to the statistical distribution theory, it is best to calculate the type II error $(\mathrm{\beta})$ under the assumption of a non-central *t*-distribution [14], that is:

${\mathrm{\beta}}_{1}=1-probt\left[{t}_{1-\mathrm{\alpha}/2,n-1},n-1,{\mathrm{\tau}}_{1}\right]$(3)

${\mathrm{\beta}}_{2}=1-probt\left[{t}_{1-\mathrm{\alpha}/2,n-1},n-1,{\mathrm{\tau}}_{2}\right]$(4)

where $probt\left[\bullet ,n-1,{\mathrm{\tau}}_{1}\right]$ denotes the cumulative distribution function of a Student’s non-central *t*-distribution with $n-1$ degrees of freedom and non-centrality parameter ${\mathrm{\tau}}_{1}$.

The non-centrality parameters ${\mathrm{\tau}}_{1}$ and ${\mathrm{\tau}}_{2}$ are non-central parameters defined as

${\mathrm{\tau}}_{1}=\frac{\mathrm{\delta}-\stackrel{\u02c9}{D}-{z}_{1-\mathrm{\gamma}/2}{S}_{D}}{{S}_{D}\sqrt{\frac{1}{n}+\frac{{z}_{1-\mathrm{\gamma}/2}^{2}}{2(n-1)}}}.$

${\mathrm{\tau}}_{2}=\frac{\mathrm{\delta}+\stackrel{\u02c9}{D}-{z}_{1-\mathrm{\gamma}/2}{S}_{D}}{{S}_{D}\sqrt{\frac{1}{n}+\frac{{z}_{1-\mathrm{\gamma}/2}^{2}}{2(n-1)}}}.$

We can get an estimate of the power:

$\begin{array}{ccc}power& =& 1-\mathrm{\beta}=1-\left({\mathrm{\beta}}_{1}+{\mathrm{\beta}}_{2}\right)\\ & =& probt\left({t}_{1-\mathrm{\alpha}/2,n-1},n-1,{\mathrm{\tau}}_{1}\right)+probt\left({t}_{1-\mathrm{\alpha}/2,n-1},n-1,{\mathrm{\tau}}_{2}\right)\end{array}$(5)

When $\mathrm{\mu}=0$, the sample size calculation can be written as follows:

$n=\frac{(2+{z}_{1-\mathrm{\gamma}/2}^{2}){[tinv(1-\mathrm{\beta}/2,n-1,{t}_{1-\mathrm{\alpha}/2,n-1})]}^{2}{S}_{D}^{2}}{2{({z}_{1-\mathrm{\gamma}/2}{S}_{D}-\mathrm{\delta})}^{2}}$(6)

where $tinv(1-\mathrm{\beta}/2,n-1,{t}_{1-\mathrm{\alpha}/2,n-1})$ is defined as the inverse of a Student’s non-central *t*-distribution.

In eq. (6), $tinv(1-\mathrm{\beta}/2,n-1,{t}_{1-\mathrm{\alpha}/2,n-1})$ is related to sample size $(n)$, we need to use iterative method to calculate sample size. Firstly we replace non-central *t*-distribution quantile with standard normal distribution quantile to obtain the initial value $({n}_{0})$, and then iterate step-by-step until $n$ reaches a stable value.

When $\mathrm{\mu}>0$, we firstly calculate by eq. (6) to obtain an initial value $({n}_{1})$, then calculate by eq. (5) to achieve the power. If the estimated power is close enough to the pre-specified power then $n$ is the sample size that we want to estimate. Otherwise we make ${n}_{1}$ equal to ${n}_{1}+1$ and judge whether the estimated power is close enough the pre-specified power. Repeat the procedure above until be closest to but greater than the pre-specified power. summaries reasonable estimates of the sample size using eqs (5) and (6) for various standardized difference limits $(\mathrm{\mu}/\mathrm{\sigma})$, different standardized agreement limits $(\mathrm{\delta}/\mathrm{\sigma})$, and different type II error $(\mathrm{\beta})$ assuming that the data are well-behaved.

can be a reference for clinical researchers to estimate the sample size in the agreement assessment trial between two methods of measurement.

Table 1: Sample size $(n)$ for Bland–Altman method with non-central t-distribution for different standardized difference limits $(\mathrm{\mu}/\mathrm{\sigma})$, different standardized agreement limits $(\mathrm{\delta}/\mathrm{\sigma})$, and different type II error $(\mathrm{\beta})$. $(\mathrm{\alpha}=0.05)$.

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