. 2008 Jul 9;66(4):210–222. doi: 10.1159/000143404

Table 2.

Components required for calculation of sample size for a tested marker which is (S1) a single disease susceptibility mutation (DSM); (S2) in LD with a single DSM; (S3) one of two DSMs; (S4) in LD with two DSMs

P(parent heterozygous | SAO)

P(parent transmits D | parent Dd, SAO)

Var(B_ij|H₁)

\begin{array}{l} h_{D} = \frac{p_{D} p_{d}}{K_{D}} {p_{D} f_{D D} + p_{d} f_{D} + p_{D} f_{D} + p_{d} f_{d}} \\ where \\ K_{D} = p_{D}^{2} f_{D D} + 2 p_{D} p_{d} f_{D} + p_{d}^{2} f_{d} \end{array}

τ_{D} = \frac{p_{D} f_{D D} + p_{d} f_{D}}{(p_{D} f_{D D} + p_{d} f_{D} + p_{D} f_{D} + p_{d} f_{d})}

σ_{D}^{2} = 1 - 2 h_{D} {(2 τ_{D} - 1)}^{2} + \frac{{(p_{D} p_{d})}^{2}}{h_{D} K_{D}} (f_{D D} - 2 f_{D} + f_{d})

\begin{array}{l} \begin{array}{l} h_{M} = \frac{1}{K_{D}} {(p_{D} f_{D D} + p_{d} f_{D}) 2 p_{D}^{2} p_{M \ D} p_{m \ d} \\ + (p_{D} f_{D D} + p_{d} f_{D} + p_{D} f_{D} + p_{d} f_{d}) p_{D} p_{d} (p_{M \ D} p_{m \ d} \\ + p_{M \ d} p_{m \ D}) + (p_{D} f_{D} + p_{d} f_{d}) 2 p_{d}^{2} p_{M \ d} p_{m \ D}} \end{array} \\ where \\ K_{D} = p_{D}^{2} f_{D D} + 2 p_{D} p_{d} f_{D} + p_{d}^{2} f_{d} \end{array}

\begin{array}{l} τ_{M} = \frac{p_{m}}{K_{D} h_{M}} {(p_{D} f_{D D} + p_{d} f_{D}) p_{M \ D} p_{D} \\ + (p_{D} f_{D} + p_{d} f_{d}) p_{M \ d} p_{d}} \end{array}

\begin{array}{l} σ_{M}^{2} = 1 - 2 h_{M} {(2 τ_{M} - 1)}^{2} + C [\frac{{(p_{M} P_{m})}^{2}}{h_{M} K_{D}}] \\ where \\ \begin{matrix} c = p_{D}^{2} f_{D D} {\frac{p_{M \ D}^{2}}{p_{M}^{2}} + \frac{p_{m \ D}^{2}}{p_{m}^{2}} - \frac{2 p_{M | D} p_{m \ D}}{p_{M} p_{m}}} \\ + 2 p_{D} p_{d} f_{D} {\frac{p_{M \ D} p_{M \ d}}{p_{M}^{2}} + \frac{p_{m \ D} p_{m \ d}}{p_{m}^{2}} - \frac{p_{M \ D} p_{m \ d}}{p_{M} p_{m}} - \frac{p_{M \ d} p_{m \ D}}{p_{M} p_{m}}} \\ + p_{d}^{2} f_{d} {\frac{p_{M \ d}^{2}}{p_{M}^{2}} + \frac{p_{m \ d}^{2}}{p_{m}^{2}} - \frac{2 p_{M \ d} p_{m \ d}}{p_{M} p_{m}}} \end{matrix} \end{array}

\begin{array}{l} \begin{matrix} h_{D E} = \frac{p_{D}}{K_{D E}} {p_{D} (1 - p_{D}) f_{D D} + p_{E} f_{D E} \\ + p_{E}^{2} f_{E E} + 2 p_{E} p_{d} f_{E} + p_{d} f_{D} + p_{d}^{2} f_{d}} \end{matrix} \\ where \\ \begin{array}{l} K_{D E} = p_{D}^{2} f_{D D} + 2 p_{D} p_{E} f_{D E} + p_{E}^{2} f_{E E} \\ + 2 p_{d} p_{D} f_{D} + 2 p_{d} p_{E} f_{E} + p_{d}^{2} f_{d} \end{array} \end{array}

\begin{array}{l} τ_{D E} = \\ \begin{matrix} \frac{(p_{D} f_{D D} + p_{E} f_{D E} + p_{d} f_{D}) (1 - p_{D})}{{p_{D} (1 - p_{D}) f_{D D} + p_{E} f_{D E} + p_{E}^{2} f_{E E} +} \\ 2 p_{E} p_{d} f_{E} + p_{d} f_{D} + p_{d}^{2} f_{d}} \end{matrix} \end{array}

\begin{array}{l} σ_{D E}^{2} = 1 - 2 h_{D E} {(2 τ_{D E} - 1)}^{2} + C {\frac{p_{D}^{2} {(1 - p_{D})}^{2}}{h_{D E} K_{D E}}} \\ where \\ \begin{array}{l} C = f_{D D} + \frac{p_{E}^{2} f_{E E}}{{(1 - p_{D})}^{2}} + \frac{2 p_{d} p_{E} f_{E}}{{(1 - p_{D})}^{2}} \\ + \frac{p_{d}^{2} f_{d}}{{(1 - p_{D})}^{2}} - \frac{2 p_{E} f_{D E}}{(1 - p_{D})} - \frac{2 p_{d} f_{D}}{(1 - p_{D})} \end{array} \end{array}

\begin{matrix} \begin{matrix} h_{M, D E} = \frac{1}{K_{D E}} { \\ F_{1} (D D, D E, D d) [F_{2} (D, D) + F_{2} (D, E) + F_{2} (D, d)] \\ \begin{array}{l} + F_{1} (D E, E E, d E) [F_{2} (D, E) + F_{2} (E, E) + F_{2} (d, E)] \\ + F_{1} (D d, d E, d d) [F_{2} (D, d) + F_{2} (d, E) + F_{2} (d, d)]} \end{array} \end{matrix} \\ where, for genotypes W, Y, Z, and alleles A, B, \\ \begin{array}{l} F_{1} (W, Y, Z) = p_{D} f_{W} + p_{E} f_{Y} + p_{d} f_{z} \\ F_{2} (A, B) = p_{A} p_{B} (p_{M | A} p_{m | B} + p_{M | B} p_{m | A}) \end{array} \end{matrix}

\begin{array}{l} \begin{matrix} τ_{M, D E} = \frac{p_{m}}{K_{D E} h_{M, D E}} {F_{1} (D D, D E, D d) p_{D} p_{M / D} \\ + F_{1} (D E, E E, d E) p_{E} p_{M / E} \\ F_{1} (D d, d E, d d) p_{d} p_{M / d}}, \end{matrix} \\ and \\ F_{1} (W, Y, Z) as defined for h_{M, D E} \end{array}

\begin{array}{l} σ_{M, D E}^{2} = 1 - 2 h_{M, D E} {(2 τ_{M, D E} - 1)}^{2} + C [\frac{{(p_{M} p_{m})}^{2}}{h_{M, D E} K_{D}}] \\ where \\ \begin{matrix} C = F_{1} (D D, D E, D d) [F_{3} (D, D) + F_{3} (D, E) - F_{3} (D, d)] \\ + F_{1} (D E, E E, d E) [F_{3} (D, E) + F_{3} (E, E) + F_{3} (d, E)] \\ + F 1 (D d, d E, d d) [F_{3} (D, d) + F_{3} (d, E) + F_{3} (d, d)] \end{matrix} \\ and \\ F_{1} (W, Y, Z), F_{2} (A, B) as defined for h_{M, D E}, \\ F_{3} (A, B) = \frac{p_{A} p_{B}}{p_{M}^{2} p_{m}^{2}} {(p_{M \ A} p_{m} - p_{m | A} p_{M}) \times (p_{M \ B} p_{m} - p_{m \ B} p_{M})} \end{array}