Because of their high economic importance, growth traits in fish are under continuous improvement. For growth traits that are recorded at multiple time-points in life, the use of univariate and multivariate animal models is limited because of the variable and irregular timing of these measures. Thus, the univariate random regression model (RRM) was introduced for the genetic analysis of dynamic growth traits in fish breeding. We used a multivariate random regression model (MRRM) to analyze genetic changes in growth traits recorded at multiple time-point of genetically-improved farmed tilapia. Legendre polynomials of different orders were applied to characterize the influences of fixed and random effects on growth trajectories. The final MRRM was determined by optimizing the univariate RRM for the analyzed traits separately via penalizing adaptively the likelihood statistical criterion, which is superior to both the Akaike information criterion and the Bayesian information criterion. In the selected MRRM, the additive genetic effects were modeled by Legendre polynomials of three orders for body weight (BWE) and body length (BL) and of two orders for body depth (BD). By using the covariance functions of the MRRM, estimated heritabilities were between 0.086 and 0.628 for BWE, 0.155 and 0.556 for BL, and 0.056 and 0.607 for BD. Only heritabilities for BD measured from 60 to 140 days of age were consistently higher than those estimated by the univariate RRM. All genetic correlations between growth time-points exceeded 0.5 for either single or pairwise time-points. Moreover, correlations between early and late growth time-points were lower. Thus, for phenotypes that are measured repeatedly in aquaculture, an MRRM can enhance the efficiency of the comprehensive selection for BWE and the main morphological traits.

Background: Haplotypes combine the effects of several single nucleotide polymorphisms (SNPs) with high linkage disequilibrium, which benefit the genome-wide association analysis (GWAS). In the haplotype association analysis, both haplotype alleles and blocks are tested. Haplotype alleles can be inferred with the same statistics as SNPs in the linear mixed model, while blocks require the formulation of unified statistics to fit different genetic units, such as SNPs, haplotypes, and copy number variations. Results: Based on the FaST-LMM, the fastLmPure function in the R/RcppArmadillo package has been introduced to speed up genome-wide regression scans by a re-weighted least square estimation. When large or highly significant blocks are tested based on EMMAX, the genome-wide haplotype association analysis takes only one to two rounds of genome-wide regression scans. With a genomic dataset of 541,595 SNPs from 513 maize inbred lines, 90,770 haplotype blocks were constructed across the whole genome, and three types of markers (SNPs, haplotype alleles, and haplotype blocks) were genome-widely associated with 17 agronomic traits in maize using the software developed here. Conclusions: Two SNPs were identified for LNAE, four haplotype alleles for TMAL, LNAE, CD, and DTH, and only three blocks reached the significant level for TMAL, CD, and KNPR. Compared to the R/lm function, the computational time was reduced by ~ 10–15 times.