Linear models for diallel crosses: a review with R functions

Abstract

Key message A new R-software procedure for fixed/random Diallel models was developed. We eased the diallel schemes approach by considering them as specific cases with different parameterisations of a general linear model. Abstract Diallel experiments are based on a set of possible crosses between some homozygous (inbred) lines. For these experiments, six main diallel models are available in literature, to quantify genetic effects, such as general combining ability (GCA), specific combining ability (SCA), reciprocal (maternal) effects and heterosis. Those models tend to be presented as separate entities, to be fitted by using specialised software. In this manuscript, we reinforce the idea that diallel models should be better regarded as specific cases (different parameterisations) of a general linear model and might be fitted with general purpose software facilities, as used for all other types of linear models. We start from the estimation of fixed genetical effects within the R environment and try to bridge the gap between diallel models, linear models and ordinary least squares estimation (OLS). First, we review the main diallel models in literature. Second, we build a set of tools to enable geneticists, plant/animal breeders and students to fit diallel models by using the most widely known R functions for OLS fitting, i.e. the ‘lm()’ function and related methods. Here, we give three examples to show how diallel models can be built by using the typical process of GLMs and fitted, inspected and processed as all other types of linear models in R. Finally, we give a fourth example to show how our tools can be also used to fit random/mixed effect diallel models in the Bayesian framework.

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Genotype × <math xmlns="http://www.w3.org/1998/Math/MathML" altimg="urn:x-wiley:00029122:media:ajb216091:ajb216091-math-0001" wiley:location="equation/ajb216091-math-0001.png"><mrow><mrow><mo>\unicode{x000D7}</mo></mrow></mrow></math> environment interaction obscures genetic sources of variation in seed size in Dithyrea californica but provides the opportunity for selection on phenotypic plasticity

“…where Y ijk is the individual seed size produced by the combination of dam j and sire k, μ is the grand mean of seed size within a diallel, G j and G k are the GCAs of dam j and sire k, S jk is the SCA of the jth and kth parent combination, RG j and RG k are the RGCAs of the jth and the kth parents, RS jk is the RSCA of the jth and kth parent combination, and E ijk is the residual variance. We analyzed the data using the R packages lmDiallel (Onofri et al, 2021) and sommer package to estimate random effects of variance components (Covarrubias-Pazaran, 2016). We first constructed design matrices that indicated the identity of dams and sires for each pedigreed seed with the lmDiallel package functions GCA, SCA, RGCA, and RSCA.…”