Theoretical and Applied Genetics

, Volume 131, Issue 4, pp 851–860 | Cite as

Computation of the inverse additive relationship matrix for autopolyploid and multiple-ploidy populations

  • Matthew G. Hamilton
  • Richard J. Kerr
Original Article


Key message

Rules to generate the inverse additive relationship matrix (A −1 ) are defined to enable the adoption restricted maximum likelihood (REML) and best linear unbiased prediction (BLUP) in autopolyploid populations with multiple ploidy levels.


Many important agronomic, horticultural, ornamental, forestry, and aquaculture species are autopolyploids. However, the adoption of restricted maximum likelihood (REML), for estimating co/variance components, and best linear unbiased prediction (BLUP), for predicting breeding values, has been hampered in autopolyploid breeding by the absence of an appropriate means of generating the inverse additive relationship matrix (A −1 ). This paper defines rules to generate the A −1 of autopolyploid populations comprised of individuals of the same or different ploidy-levels, including populations exhibiting (1) odd-numbered ploidy levels (e.g. triploids), (2) sex-based differences in the probability that gametic genes are identical by descent and (3) somatic chromosome doubling. Inbreeding, due to double reduction, in autopolyploid founders in the absence of mating among relatives is also accounted for. A previously defined approach is modified, whereby rules are initially defined to build an inverse matrix of kinship coefficients (K −1 ), which is then used to generate A −1 . An R package (polyAinv; to implement these rules has been developed and examples of analyses provided. The adoption of REML and BLUP methods made possible by these new rules has the potential to provide further insights into the quantitative genetic architecture of autopolyploid and multiple-ploidy populations, improve estimates of breeding values, and increase genetic gains made through recurrent selection.



Funding for this work was provided by the CSIRO Agriculture and Food project ‘Genomics platforms to assist applied aquaculture breeding’ (AgSIP53). We thank reviewers for their helpful comments.

Compliance with ethical standards

Conflict of interest

On behalf of both authors, the corresponding author states that there is no conflict of interest.

Supplementary material

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.CSIRO AquacultureCSIRO Agriculture and FoodHobartAustralia
  2. 2.PlantPlan Genetics Pty LtdUniversity of TasmaniaHobartAustralia

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