A hierarchical Bayesian model for a novel sparse partial diallel crossing design

Abstract

Partial diallel crossing designs are in common use among evolutionary geneticists, as well as among plant and animal breeders. When the goal is to make statements about populations represented by a given set of lines, it is desirable to maximize the number of lines sampled given a set number of crosses among them. We propose an augmented round-robin design that accomplishes this. We develop a hierarchical Bayesian model to estimate quantitative genetic parameters from our scheme. For example, we show how to partition genetic effects into specific and general combining abilities, and the method provides estimates of heritability, dominance, and genetic correlations in the face of complex and unbalanced designs. We test our approach with simulated and real data. We show that although the models slightly overestimate genetic variances, main effects are assessed accurately and precisely. We also illustrate how our approach allows the construction of posterior distributions of combinations of parameters by calculating narrow-sense heritability and a genetic correlation between activities of two enzymes.

Figure 1.—

Data structure. (A) The diallel table. Solid and shaded squares mark the crosses that we performed. The table is subdivided by population. The order of the lines is from left to right for the female axis and from top to bottom for the male axis; i.e., the top left corner of the table represents the cross of the A1 male to the A1 female. Within each population, the F₁ round-robin crosses between lines (off-diagonal) are depicted as solid squares, whereas the inbred lines themselves (diagonal) are shown as shaded squares. (B) Hierarchical levels of the data. For the “crosses” level, the solid arrow represents a “selfed” inbred line, dashed arrows represent within-population crosses, and dotted arrows show between-population crosses.

Figure 2.—

Accuracy and precision of standard deviation estimates. For each parameter, we show a pair of graphs. The top one plots the fractional difference between medians of estimated posterior distributions of a given parameter and true values (that is, the difference divided by the true value). The bottom plot shows coefficients of variation of the estimated posterior distributions (time-series SE divided by the true value; see methods). The box plots represent data across 500 simulated data sets with high heritability. Each plot represents results under six scenarios: 1 and 2 are for simulations without outliers and 3–6 are for those with outliers (see methods). 1 and 3 were analyzed using point estimates of slopes; 2 and 4 were analyzed modeling the uncertainty in slope values and assuming that replicates are normally distributed; and 5 and 6 were analyzed modeling the slopes, but with t₃ (5) and t₆ (6) distributions for replicates.

Figure 3.—

Accuracy and precision of narrow-sense heritability estimates. The plots are arranged and labeled as in Figure 2. Estimates are from high-heritability (A) and low-heritability (B) simulations.

Figure 4.—

Accuracy and precision of estimates of sample parameters. The box plots are arranged as in Figure 2. For line means (μ^line), we pooled information for all 60 lines across 500 simulated data sets. The scatterplot shows the relationship of fractional differences between true line means (μ^line) and the corresponding true population means (μ^pop) on the x-axis and the fractional difference between the posterior estimates of line means () and the corresponding true line means (μ^line) on the y-axis.