Towards a Sweetpotato Genomic-Enabled Breeding:

Optimizing Two-Stage Analysis of Multi-Environment Augmented Trials

Saulo Chaves

San Diego, USA
January 2026

Objectives

To provide a brief overview about the importance of sweetpotato breeding in developing countries

To demonstrate the importance of genomic selection as a feasible strategy in sweetpotato breeding

To show the utility of a proper two-stage analysis when facing complex situations

Content

Background
- Sweetpotato breeding
- Genomic selection
Population
- Material and experimental design
Models and outputs
- Two-stage models
- Single-stage models
Wrap-up

Background

Sweetpotato

Crop Trust

High production and consumption in Sub-Saharan Africa
- 3rd most important food crop in 7 Eastern and Central African countries
- 4th in 6 Southern African countries
- 8th in 4 of Western African countries

Sweetpotato

Davies and Starr

Tolerates marginal growing conditions
Provides more edible energy per hectare per day than wheat, rice and cassava
A valuable source of vitamins A, B, C, and E

Sweetpotato

Orange-fleshed sweetpotato

“Just 125 g of fresh roots from most orange-fleshed varieties contain enough beta-carotene to provide the daily pro-vitamin A needs of a preschooler”

CIP

Sweetpotato breeding

Low et al. (2020)

In addition to increasing the nutrient rate of newly developed varieties, we still have to worry about how this variety will perform, and if it will be resistant to the main pests and diseases. This is why performing a recurrent procedure such as conducting a breeding program is important. CIP’s standard procedure is illustrated in this figure. In the initial stages, there are the observational trials (OTs), which are composed of several (thousands) of clones in unreplicated trials. As the breeding program proceeds, the selected clones are tested in trials with more replications, in more environments. At the end of this process, clones are selected for both variety development and population improvement.

This other figure illustrates the so-called “accelerated breeding scheme”, which prioritizes multi-location trials rather than multi-year trials, as a measure to decrease generation interval.

Genomic selection

Wallace et al. (2018)

Modelling strategies

Single-stage model

\[ \boldsymbol{y} = \boldsymbol{X}\boldsymbol{b} + \boldsymbol{Z}\boldsymbol{g} + \boldsymbol{\varepsilon} \]

Plot-level multi-environment model
- Gold-standard practice
- No loss of information

Two-stage model

Single-environment model
- Obtain adjusted means: \(\boldsymbol{y} = \boldsymbol{X}_1\boldsymbol{b} + \boldsymbol{X}_2\boldsymbol{g} + \boldsymbol{\varepsilon}\)
Weighted multi-environment model
- Obtain GEBVs: \(\boldsymbol{\bar{y}} = \boldsymbol{X}\boldsymbol{e} + \boldsymbol{Z}\boldsymbol{g} + \boldsymbol{\varepsilon}\)

Modelling strategies

If single-stage models are gold standard, why using two-stage models?
- Lack of computational resource
- Data availability (e.g.: historical data)
- Complexity of single-stage models

Hypothesis and objective

Considering that the efficiency of the two-stage approach relies heavily on the quality of the adjusted means obtained in the first stage, our goal was to test the hypothesis that dBLUPs or pedigree-based dBLUPs (dABLUPs) would be more appropriate as inputs for second-stage models than BLUEs

Population

1,138 test clones
- Two pools: partial diallel
- ~20 parents each
39 parents
8 checks
Trait: storage root yield

Genotyping

3,120 (2,367 after filtering) DArTag panel (Zhao et al., 2024)

Models and outputs

Single-environment analyses

Here is the linear mixed model we used in the first stage. Since we were dealing with plot-level data, we considered the design-related effects and performed spatial adjustment in the residual.

We fitted three types of model. The first one had the genotypic effects as fixed, generating BLUEs. This is the standard procedure in terms of two-stage analysis. In the second one, we considered the genotypic effect as random, but did not use any relationship information. This model produces BLUPs. Note that we cannot use BLUPs directly in the second stage due to double shrinkage. Thus, we had to perform a deregression strategy. The last model also considered the genotypic effect as random, but in this case, we partitioned it into additive and non-additive effects, with the additive being modelled using the numerator relationship matrix, built from the pedigree. From these effects, we obtained a deregressed ABLUP.

\[ \boldsymbol y = \boldsymbol 1 \mu + \boldsymbol{X}_1 \boldsymbol b + \boldsymbol{X}_2 \boldsymbol t + \boldsymbol{Z}_1 \boldsymbol g + \boldsymbol {Z}_2 \boldsymbol r + \boldsymbol Z_3 \boldsymbol c + \boldsymbol Z_4 \boldsymbol{rc} + \boldsymbol \varepsilon \]

\(\boldsymbol g\) as fixed \(\to\) BLUEs
\(\boldsymbol g\) as random with \(\boldsymbol g \sim \mathcal N \left(\boldsymbol 0, \sigma^2_g \boldsymbol I_V \right) \to\) dBLUPs
\(\boldsymbol g\) as random with \(\boldsymbol g \sim \mathcal N \left(\boldsymbol 0, \begin{bmatrix} \sigma^2_a \color{red}{\boldsymbol A} & \boldsymbol 0 \\ \boldsymbol 0 & \sigma^2_{na} \color{red}{\boldsymbol I_V} \end{bmatrix} \right) \to\) dABLUPs
- \(\boldsymbol g^\prime = \left[ \boldsymbol a \; \boldsymbol {na} \right]\) and \(\boldsymbol Z_1 = \left[\boldsymbol Z_{1_a} \; \boldsymbol Z_{1_{na}} \right]\)

Second-stage model

\[ \boldsymbol y^\star = \boldsymbol 1 \mu + \boldsymbol X \boldsymbol e + \boldsymbol Z \boldsymbol g + \boldsymbol \varepsilon \]

Full weight (FW): \(\boldsymbol{ \varepsilon} \sim \mathcal N \left(\boldsymbol 0, \boldsymbol \Omega^{-1} \right)\)
Diagonal weight (DW): \(\boldsymbol{ \varepsilon} \sim \mathcal N \left[\boldsymbol 0, D\left(\boldsymbol \Omega^{-1}\right) \right]\)
\(\boldsymbol g\) partitioned into additive and non-additive: \(\boldsymbol g^\prime = \left[ \boldsymbol a \; \boldsymbol {na} \right]\)
- Additive: Factor analytic covariance structure: \(\boldsymbol a \sim \mathcal{N}[ \boldsymbol 0, \underbrace{\left(\boldsymbol \Lambda \boldsymbol \Lambda^\prime + \boldsymbol \Psi\right)}_{\boldsymbol \Sigma_g} \otimes \boldsymbol G]\)

Single-stage model

\[ \boldsymbol y = \boldsymbol X \boldsymbol f + \boldsymbol{Z}_1 \boldsymbol g + \boldsymbol {Z}_2 \boldsymbol r + \boldsymbol Z_3 \boldsymbol c + \boldsymbol Z_4 \boldsymbol{rc} + \boldsymbol \varepsilon \]

\(\boldsymbol f \to\) fixed effects
\(\boldsymbol g^\prime = \left[ \boldsymbol a \; \boldsymbol {na} \right]\)
- Additive: factor analytic covariance structure
Block-diagonal covariance structure for design-related effects, spatial adjustments and residuals

Factor Analytic model selection

Genotype-by-environment interaction

Partitioning the genotype-by-environment interaction (Bančič et al., 2024)

Selection

Cross-validations

More important than changing the entry is providing proper weighting!

Cross-validations

Particularizing the predictions by pool did not harm the model’s performance and had less computational cost!

Wrap-up

Key messages

Despite the possible loss of information, two-stage models are still useful
- Early-stage multi-environment unreplicated trials
A proper weighting is crucial for obtaining correct results from two-stage models
- Models weighted using the full matrix performed better than those weighted using diagonal weights
Is possible to simplify models focusing or per pool predictions

Acknowledgments

Funding and management:

Bill & Melinda Gates Foundation
CGIAR
CIP
UFV
ESALQ-USP

People:

UFV: Guilherme da Silva Pereira, Kaio Olimpio G. Dias
CIP: Reuben Ssali, Bert de Boeck, Thiago Mendes, Hannele Lindqvist-Kreuze, Hugo Campos
ESALQ-USP: José Tiago Barroso Chagas
NCSU: Craig Yencho

Thank you!

📧 saulochaves@usp.br