Table 1 Overview of sample size formulas

Lower boundary

\(N^{k}_{\text {lower}}\) = \(\left (\frac {q_{1-\alpha /2}+q_{1-\beta }}{\mu }\right)^{2}\left (\frac {\sigma ^{2}(k+1)^{2}}{k}\right)\)

Equal centres

\(N^{k}_{\mathrm {MC,E}}\) = \(\left (\frac {q_{1-\alpha /2}+q_{1-\beta }}{\mu }\right)^{2}\left (\frac {\sigma ^{2}(k+1)^{2}}{2k}+\sqrt {\frac {\sigma ^{4}(k+1)^{4}}{4 k^{2}} + \frac {\tau ^{2}(k+1)^{2}\mu ^{2} c\ \text {E}\left (\Delta _{1}^{2}|r_{1}\right)}{\left (q_{1-\alpha /2}+q_{1-\beta }\right)^{2}}}\,\right)\)

Unequal centres

\(N^{k}_{\mathrm {MC,U}}\)= \(\left (\frac {q_{1-\alpha /2}+q_{1-\beta }}{\mu }\right)^{2}\left (\frac {\sigma ^{2}(k+1)^{2}}{2k}+\sqrt {\frac {\sigma ^{4}(k+1)^{4}}{4 k^{2}} + \frac {\tau ^{2}(k+1)^{2}\mu ^{2} c\overline {\text {E}\left (\Delta _{1}^{2}|\cdot \right)}}{\left (q_{1-\alpha /2}+q_{1-\beta }\right)^{2}}}\,\right)\)

Upper boundary

\(N^{k}_{\text {upper}}\) = \(\left (\frac {q_{1-\alpha /2}+q_{1-\beta }}{\mu }\right)^{2}\left (\frac {\sigma ^{2}(k+1)^{2}}{2k}+\sqrt {\frac {\sigma ^{4}(k+1)^{4}}{4 k^{2}} + \frac {\tau ^{2}(k+1)^{2}\mu ^{2} c\ \text {E}\left (\Delta _{1}^{2}\middle |\frac {b}{k+1}\right)}{\left (q_{1-\alpha /2}+q_{1-\beta }\right)^{2}}}\,\right)\)