Random regression of Hereford percentage intramuscular fat on geographical coordinates
Authors: Delgadillo Liberona Jose S , Langdon John M , Herring Andy D , Blackburn Harvey D , Speidel Scott E , Sanders Stacy , Riley David G 
Published: December 2019

Abstract
Accounting for genotype–environment interactions may improve genetic prediction and parameter estimation. The objective was to use random regression analyses to estimate variances and thereby heritability for intramuscular fat (IMF) across longitude and latitude coordinates within the continental United States. Records from the American Hereford Association (n = 169,440) were used. Analyses were first conducted using the continental United States in its entirety, and then as subdivided into two or four regions. Data were analyzed with an animal model, and linear and quadratic random regressions of additive genetic merit on longitude or latitude as covariate (separately). Subdivided data were analyzed with linear random regressions unique to regions. Regions were North and South separated at 40°N latitude, or West and East separated at 99°W longitude using longitude or latitude as covariate, respectively. Further subdivision to four regions included additional boundaries of 44.46° and 36.46°N latitude and 104.55° and 92.22°W longitude. The estimated heritability of IMF from the traditional model was 0.19 ± 0.004. Without regional subdivision of data, quadratic random regression had the best fit for the data based on likelihood ratio tests using longitude or latitude as covariate (P &lt; 0.01). Estimates of heritability from quadratic random regression on latitude ranged from 0.12 in the South to a high of 0.27 at the extreme Northern latitude. Estimates of heritability from quadratic random regression on longitude ranged from 0.17 in the middle of the parameter space (corresponding to the central United States) to 0.37; higher estimates were noted at the extremes, that is, the far West and East longitudes. Random regression analyses of data divided into regions were conducted with a linear coefficient, as increasing to a quadratic polynomial was never accomplished. Results from random regression on latitude in the East region were similar to results from analyses without regions (h2 ranged from 0.09 to 0.32); however, estimates of heritability in the West region had a lower range from South to North (0.14 to 0.27). Estimates of heritability from random regression on longitude with data divided into two regions were similar to those from analyses that did not include region. Estimates in the South region were somewhat lower and had a lower range (0.15 to 0.31) than those from the North region (0.19 to 0.47). When data were further subdivided, estimation of only a subset of covariances among random regression coefficients was possible, that is, within-region covariances of intercept and linear terms (latitude); those and covariances between all linear random regression coefficients were estimated when longitude was the covariate. Results from random regression analyses of data with four regions modeled produced very high estimates of heritability in low latitudes in the furthest West and high latitudes in the furthest East region, with approximate difference of 0.3 and 0.2 between estimates in the two West regions and the two East regions, respectively. Results from random regression on longitude indicated higher estimates of heritability in North region, especially at the furthest East longitudes of the most Northern region. There appeared to be substantial additive genetic variance differences, as well as estimates of heritability, that correspond to different geographical environments as modeled by random regressions on within-region latitude or longitude coordinates.