Detection of marker–QTL associations by studying change in marker frequencies with selection

Abstract

The value of selective genotyping for the detection of QTL has already been studied from a theoretical point of view but with the assumption of a negligible contribution $(r^{2}_{P})$ of the QTL to the phenotypic variance. For predicting change in gene frequency, we show that this assumption is only valid for $r^{2}_{P}$ less than 0.05 and for a proportion selected higher than 1%. Therefore, we develop a study of the optimization of selective genotyping without assumption on QTL effect, with selection either of both tails (bidirectional genotyping or BSG) or only one tail (unidirectional genotyping or USG). For a given population size of phenotyped plants the optimal proportion selected for selective genotyping is around 30% for each tail. For the same investment as in ANOVA, by investing more in phenotyping than in genotyping when the cost ratio of genotyping to phenotyping is higher than 1, the optimal proportion selected appears to be between 10 and 20% for each tail. It is mainly affected by the cost ratio and decreases when the cost ratio increases. At this optimum, BSG is competitive with ANOVA, or even more powerful, when the cost ratio is higher than 1. USG can also be competitive when the cost ratio is higher than 2. Using experimental data from two populations of about 300 F4 inbred families of maize, it was verified that BSG at the optimum gives the same results as ANOVA or is better whereas USG is less powerful or equivalent.

Appendix. Derivation of the power

Power of t-test for the comparison of means

The t-test for comparison of the two marker genotype means ${\overline{\text{MM}}}$ and ${\overline{\text{mm}},}$ can be written

$$ t = \frac{{\overline{{\rm MM}} - \overline{{\rm mm}}}}{{2{\sqrt {\sigma _{\rm W}^{2} /N}}}} = \frac{{(\alpha _{U} - \alpha _{L})}}{{2{\sqrt {\sigma _{\rm W}^{2} /N}}}} + \frac{{a^{*}}}{{{\sqrt {\sigma _{W}^{2} /N}}}} $$

where ɛU (ɛL) are random variation due to sampling and ${a^{*} /{\sqrt {\sigma _{\rm W}^{2} /N}} = r_{\rm P} {\sqrt N}/{\sqrt {(1 - r_{\rm P}^{2})}}= {E}(t) = c}$ is the noncentrality parameter, i.e. the expected value of test knowing that the H ₀ assumption of absence of difference between the two means is wrong. Then, the probability β associated with risk II, i.e. to conclude on the absence of difference knowing that they is a difference, is (Dagnélie 1975)

$$ \beta = \Phi (u') - \Phi (u) \, \hbox{and the power is}\, P = 1-\beta = 1-[\Phi (u') - \Phi (u)] $$

where Φ is the normal distribution with variance 1, and u’ and u′′ define an interval centered on −c: u′ = u − c, u′′ = −u − c, u being defined for threshold 1−α/2 (Φ (u) = 1−α/2). This approach gives the same results as Charcosset and Gallais (1996).

Power u-test for the comparison of marker frequencies

The u-test for the difference in marker frequency in the situation of BSG, can be written $u = \frac{{P_{U} - P_{L} }}{{{\sqrt {0.5/N_{{\text{S}}} } }}} = \frac{{{\left( {\varepsilon _{U} - \varepsilon _{L} } \right)}}}{{{\sqrt {0.5/N_{{\text{S}}} } }}} + \frac{{2\Delta p}}{{{\sqrt {0.5/N_{{\text{S}}} } }}}$ where ɛ _U (ɛ _L ) are random variation in gene frequency due to sampling and ${2\Delta p/{\sqrt {0.5/N_{\rm S}}}= {E}(u) = c}$ is the noncentrality parameter, i.e. the expected value of test knowing that the H ₀ assumption of absence of difference between p _U and p _L is wrong. Then, from this noncentrality parameter, the probability β associated with risk II, is derived as above. For USG, the noncentrality parameter is ${\Delta p/{\sqrt {0.25/N_{\rm S}}}.}$