I am deriving covariance between haplodiploids to correct some misunderstandings in the literature (e.g. Liu & Smith, 2000) and to extend their work and work of others (eg. Grossman & Eisen, 1989; ...) for general estimation of genetic parameters and breeding values in haplodiploids.
Grossman & Eisen (1989, page 140) derive genetic covariance for sex-linked genes between two females i.e. between two diploids. This result should be the same as genetic covariance for autosomal genes between two diploid individuals, since females have two sex chromosomes i.e. the result should be $$ \inline Cov\left(g_{jk},g_{j'k'}\right)=a_{12}\sigma^2_a + d_{12}\sigma^2_d $$, where $$ \inline Cov\left(g_{jk},g_{j'k'}\right) $$ is genetic covariance between genetic values of individual 1 with genotype jk and individual 2 with genotype j'k', $$ \inline a_{12} $$ is the additive relationship coeffcient (see bellow), $$ \inline \sigma^2_a $$ is additive genetic variance, $$ \inline d_{12} $$ is the dominance relationship coeffcient (see bellow), and $$ \inline \sigma^2_d $$ is dominance variance. They show the following equation:
$$
Cov\left(g_{jk},g_{j'k'}\right)=\\
Cov(\alpha_j+\alpha_k+\delta_{jk},\\
\alpha_{j'}+\alpha_{k'}+\delta_{j'k'})\\
= Cov\left(\alpha_j, \alpha_{j'}\right) + \\
Cov\left(\alpha_j, \alpha_{k'}\right) + \\
Cov\left(\alpha_k, \alpha_{j'}\right) + \\
Cov\left(\alpha_k, \alpha_{k'}\right) + \\
Cov\left(\delta_{jk}, \delta_{j'k'}\right)\\
= \Pr\left(j \equiv j' \right)E\left(\alpha^2_j\right) + \\
\Pr\left(j \equiv k' \right)E\left(\alpha^2_j\right) + \\
\Pr\left(k \equiv j' \right)E\left(\alpha^2_j\right) + \\
\Pr\left(k \equiv k' \right)E\left(\alpha^2_j\right) + \\
\Pr\left(jk \equiv j'k'\right)E\left(\delta^2_{jk}\right)\\
= 4 r_{12}\left(\frac{1}{2}\sigma^2_a\right) + d_{12}\sigma^2_d\\
= 2 r_{12}\left(\sigma^2_a\right) + d_{12}\sigma^2_d\\
= a_{12}\left(\sigma^2_a\right) + d_{12}\sigma^2_d,
$$
where $$ \inline \equiv$$ means alleles identical by descent, $$ \inline \alpha_j $$ and $$ \inline \alpha_k $$ are additive genetic values of individual with genotype jk and $$ \inline \delta_{jk} $$ is dominance deviation.
This is not clear to me, since individual can have only one breeding (additive genetic) value, which is a sum of average effect of alleles. Are then $$ \inline \alpha_j $$ and $$ \inline \alpha_k $$ average effects of alleles? Note that they assumed that $$ \inline E\left(\alpha^2_j\right) = \frac{1}{2}\sigma^2_a $$. I think that this indicates that $$ \inline \alpha_j $$ and $$ \inline \alpha_k $$ are half the breeding values from parents of individual 1, which in turn is the average effect i.e. breeding value is a sum of average effects.
Followup: yes, the $$ \inline \alpha_j $$ and $$ \inline \alpha_k $$ are average effects of alleles! Lange (as well as others) have shown the derivation of genetic covariance in his book (Mathematical and statistical methods for genetic analysis, 2nd edition 2003, page 101).
additive relationship coeffcient = twice the coefficient of coancestry also called kinship or parentage (the probability that a random allele at a particular locus in individual 1 is identical by descent to a random allele at the same locus in individual 2)
dominance relationship coeffcient = the coefficient of fraternity (the probability that both alleles at a particular locus in individual 1 are identical by descent to alleles at the same locus in individual 2)
Liu, F.-H., Smith, S. M. (2000). Estimating quantitative genetic parameters in haplodiploid organisms. Heredity, 85(4):373-382.
http://dx.doi.org/10.1046/j.1365-2540.2000.00764.xGrossman, M., Eisen, E. J. (1989). Inbreeding, coancestry, and covariance between relatives for X-chromosomal loci. J. Hered., 89(2):137-142.
http://jhered.oxfordjournals.org/cgi/content/abstract/80/2/137
0.25 standard deviation units and the effects at each locus accounting for 1% to 4% of the phenotypic variance. We detected a variety of imprinting patterns, with paternal expression being the most common. These results indicate that genomic imprinting has small, but detectable, effects on the normal variation of complex traits in adults and is likely to be more common than usually thought."