Residual covariance structure of the associative model

Builds the known matrix D such that the residual covariance of the associative model is exactly s2_ED * I + s2_ES * D, with one row per RECORD. Pass it to kernel() as the covariance of a record-level term and s2_ES becomes an ordinary variance component, estimated jointly and reported with a standard error.

Usage

associative_matrix(pen, id, labels = NULL, dilution = 0)

Arguments

pen

pen (group) label of each record. A missing pen, NA, NaN, Inf, -Inf, blank text or the text "NaN", is refused with its row, as in every fitter that takes indirect()

id

animal of each record, used to count DISTINCT animals per pen, which is the n of the model and the same count the indirect() incidence uses; a missing id is refused the same way

labels

row and column names for the returned matrix, one per record; defaults to the record position. These are the levels the kernel() term will match on, so they must be the same values as the column named in the formula; a missing label (NA, NaN, Inf, -Inf, blank text or the text "NaN") is refused with its row, as in a level column of model()

dilution

the same d as in indirect(dilution = d): each mate's social environmental deviation enters with weight (n - 1)^(-d), so the block of a pen becomes (n - 1)^(-2d) [I + (n - 2) J]. Keep it equal to the formula's value; with a mismatch the residual describes a different model from the genetic side.

Value

a square matrix, records by records, positive semi-definite, with the attribute br_associative holding dilution: h2() and t2() read a kernel() that carries it as this residual structure and not as a relationship matrix. Indexing the matrix (D[o, o]) or inverting it drops the attribute; with every pen of 2 animals, or dilution = 0.5, the diagonal alone does not tell it from a relationship matrix, so pass it as returned, built on the rows of the fit, or restore the attribute with attr(D, "br_associative") <- dilution

Details

The block of a pen of n animals is I + (n - 2) J, from var(e_i) = s2_ED + (n - 1) s2_ES and cov(e_i, e_j) = (n - 2) s2_ES for pen mates. Its eigenvalues are (n - 1)^2 once and 1 with multiplicity n - 1, so it is positive semi-definite for every n, and a pen of one animal gives a block of exactly zero. The covariance GROWS with pen size, which is why a single random(pen) variance cannot stand in for it unless every pen has the same size.

Unlike the weights route of indirect_residual(), this one does NOT need pens of different sizes: from n >= 3 the off-diagonal (n - 2) s2_ES is a signature the identity does not have, and the two components separate on a balanced design. At n = 2 exactly the block is the identity and nothing separates, whatever the number of pens.

THE BINDING CONSTRAINT IS RELATEDNESS, NOT PEN SIZE. The social GENETIC effect contributes Z_S A Z_S' * s2_AS, and when pen mates are mutually unrelated and non-inbred the A restricted to a pen is the identity, so Z_S A Z_S' is exactly this same D: the genetic and the environmental social components are then perfectly ALIASED and only their sum is estimable. Measured, at 192 animals in pens of 8 with a planted s2_AS = 0.12 and s2_ES = 0.25: with families kept apart, so that no pen held two half sibs, the fit returned s2_AS = -0.02 and s2_ES = 0.39, whose sum 0.369 reproduces the planted 0.37 while the split is meaningless; with the same animals rearranged so that half sibs share pens, s2_ES came back at 0.21. Separation needs relatives sharing pens and families split across pens. Pen-size variation does not buy it; family structure does.

Two more things the design has to avoid. A free random(pen) alongside this term spans the same pair of matrices on a balanced design and is a rank deficiency, not a refinement. And a pen that is fixed, or nested inside the contemporary group, absorbs the within-pen contrasts that carry the covariance, leaving only the diagonal slope in n_i, which is what indirect_residual() already fits.

The derivation assumes one record per animal within a pen: with two records of the same animal in one pen the direct deviation eps_D is shared between them and the block is no longer I + (n - 2) J. That case is refused rather than approximated.

References

Bijma, P., Muir, W.M. & Van Arendonk, J.A.M. (2007). Multilevel selection 1: quantitative genetics of inheritance and response to selection. Genetics 175:277-288.

Bijma, P. (2010). Multilevel selection 4: modeling the relationship of indirect genetic effects and group size. Genetics 186:1029-1031.