Significance and impotence: towards a balanced view of the null and the alternative hypotheses in marker selection for plant breeding

The isolation of DNA markers that are linked to interesting genes helps plant breeders to select parent plants that transmit useful traits to future generations. Such `marker-assisted breeding and selection' heavily leans on statistical testing of associations between markers and a well-chosen trait. Statistical association analysis is guided by classical p-values or the false discovery rate and thus relies predominantly on the null hypothesis. The main concern of plant breeders, however, is to avoid missing an important alternative. To judge evidence from this perspective, we complement the traditional p-value with a one-sided `alternative p-value' which summarizes evidence against a target alternative in the direction of the null hypothesis. This p-value measures `impotence' as opposed to significance: how likely is it to observe an outcome as extreme as or more extreme than the one that was observed when data stem from the alternative? We show how a graphical inspection of both p-values can guide marker selection when the null and the alternative hypotheses have a comparable importance. We derive formal decision tools with balanced properties yielding different rejection regions for different markers. We apply our approach to study rye-grass plants.