Dynamical BLUP modeling with Robertson-Price updating of mean reaction
norm parameter values

A general reaction norm model of any order can be formulated as a linear
mixed model. From this follows that estimates of mean phenotypic traits
in a population (fixed effects), and predictions of individual additive
genetic deviations from mean reaction norm parameter values (random
effects), can be found from the best linear unbiased predictions (BLUP)
equation in matrix form. The resulting BLUP model is dynamical in the
sense that the incidence matrix varies with time. This leads to a
straightforward and multivariate alternative to infinite-dimensional and
random regression modeling of reaction norms. Based on such a BLUP
model, the within-generation changes in predicted mean reaction norm
parameter values can be found by use of the Robertson-Price identity,
applied on the predicted random effects. From this follows that the
between-generation change in the mean values are found from Robertson’s
secondary theorem of natural selection, applied on the predicted random
effects. This explains why and to which extent the variances of BLUP
random effects are underestimated, which is a well-known observation.
The dynamical BLUP model will thus produce the mean reaction norms over
time, which makes it possible to disentangle the microevolutionary and
plasticity components in for example climate change responses. The BLUP
responses will depend on the additive genetic relationship matrix A_t.
When A_t is an identity matrix, the results will be identical to the
results from a variant of the multivariate breeder’s equation, based on
the selection gradient with respect to the individual phenotypic trait
values. Parameters are assumed to be known and constant, but it is
discussed how they can be estimated by means of a prediction error
method. Generations are assumed to be non-overlapping, but adjustments
for overlapping generations can easily be done.

12 Apr 2023

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