A nested genetic treatment structure was evaluated. The clones were obtained from different clonal families. Thus, it is possible to access family and clone within-family effects from the data.

For the analysis, in which c clones were sampled from s clonal families and evaluated together with p checks, the general form of the linear mixed model is presented in Equation (1). This model is suitable for both ST or MET data. For MET data analysis, appropriate (co)variance structures should be used to model the vectors of the family (u _s ) and clone within-family (u _c ) effects, aiming to account for the G×E:

where y ^{( N × 1)} is the vector of phenotypic observations, where N is the number of plots for ST data or plots by seasons for MET data; 1 ^{( N × 1)} is a vector in which all elements are unity; μ ^{(1 × 1)} is the intercept; τ _o ^{( o × 1)} is the vector of fixed effects, composed of check, environment, and check by environment interaction effects (the environment and check by environment interaction effects were only used in MET data analysis), associated with matrix the design X _o ^{( N × o )} (assuming full rank), where o is the number of fixed effects; u _s ^{( s × 1)} is the vector of random effects of family associated with the design matrix Z _s ^{( N × s )}, where s is the number of families for ST data or families by seasons for MET data; u _c ^{( cs × 1)} is the vector of random genotypic effects of clone within-family associated with the design matrix Z _c ^{( N × cs )}, where c is the number of clone within-family for ST data or clone within-family by seasons for MET data; u _b ^{( b × 1)} is the vector of random effects of block associated with the design matrix Z _b ^{( N × b )}, where b is the number of blocks for ST data or blocks by seasons for MET data; and e ^{( N × 1)} is the vector of random errors.

We assume that the u _c , u _s , u _b , and e vectors of random effects are mutually independent and distributed as multivariate Gaussian, with zero means and (co)variance matrices var(u _c ) = G _c , var(u _s ) = G _s , var(u _b ) = G _b , and var(e) = R. The structures of these (co)variance matrices are shown in Table 2 for ST (STMpF and STMwF) and MET (METMpF and METMwF) data analysis models, including or not the family effect, respectively. For STMwF and METMwF models, the vector of clone effects was called u _c′ . For the MET data analysis, the heterogeneity of variances of block and error effects was accommodated by the direct sum operation (⊕), whereas the heterogeneity of the variances and covariances of the G×E interaction for family and clone within-family effects was accommodated by the direct product operation (⊗). The (co)variance matrices of environments for family and clone within-family effects were modeled by an unstructured matrix with t(t + 1)/2 covariance parameters, where t is the number of trials.

The covariance parameters of models shown in Table 2 were estimated by the residual maximum likelihood (REML) method (Patterson and Thompson, 1971) and best linear unbiased predictions (BLUP) of the random effects by Henderson’s mixed model equations (Henderson et al., 1959) through the software Echidna Mixed Models (Gilmour, 2021) version 1.61. The graphic plots and other analyses discussed in the following sections were performed using the R software (R Core Team, 2021) and R packages base and ggplot2 (Wickham, 2016).

The inclusion of the family effect improved the goodness of fit of the models in all trials, on which the STMpF model showed lower AIC than the STMwF, and the only exception was for TTY from POP3(WHS) ( Table 3 ). In addition, the inclusion of family effect also increased the log-likelihood (ℓ) in all trials for both traits. Only for the TTY trait in the POP3(WHS) trial did this increment does not exceed 1.92 units (critical point for the detection of significant σ s 2 effect) ( Table 3 ). These results are reinforced by the results of the LRT test ( Table 4 ) and indicate the presence of genetic variability between families in most trials.

Clone ( σ c ' 2 , STMwF) and clone within-family ( σ c 2 , STMpF) variances were significant in all trials for both traits ( Table 4 ; Supplementary Material, Table S2 ), revealing the existence of genetic variability among clones. The contribution of clone within-family variance (C) to the total phenotypic variance was always lower than the clone effect (C′) ( Table 4 ). However, the magnitude of the difference between C′ and C was directly proportional to the contribution of family variance to the total genetic variation ( ρ S ). The amplitude of   ρ S for the SG trait (0.09 and 0.35) exceeded the amplitude for the TTY trait (0.03 and 0.26) ( Table 4 ).

CC and Spearman’s correlation coefficient (r _S ) were used as comparison criterion for both selection strategies tested (u _c′ vs. u _gST and u _c vs. u _gST ). It was observed that, regardless of the selection strategy and trait, both coefficients showed an inverse relationship with ρ S , suggesting that an increase in genetic variability among families reduces the similarity of the u _c′ and u _c vectors with the u _gST vector in the ranking of clones. Furthermore, the magnitude of the correlations of the CC and r _S coefficients with ρ S was higher for the TTY (−0.82 and −0.78) compared with that for the SG (−0.72 and −0.71) ( Figure 2 ).

In general, both CC and r _S showed lower magnitude for the second selection strategy with family effect (u _c vs. u _gST ), which suggests a greater agreement of the u _c′ vector without family effect with u _gST in the clone ranking ( Figure 2 ; Supplementary Material , Tables S4, S5 ). Furthermore, independent of the selection strategy, an increment in CC and r _S can be observed with increasing heat stress for both traits in POP2 and POP3, except for the TTY in population POP3 ( Supplementary Material, Tables S4, S5 ).

Regardless of the trait, selection based on the vector of total genotypic effects of clone (u _gST ) was found to be greater than the selection based on the vector of clone effects within family (u _c ) ( Table 5 ). The efficiency was directly proportional to ρ S , showing that the increment in genetic variability among families increases the selective efficiency of clones. The correlation between efficiency and ρ S   was higher for SG (0.88) when compared with that for TTY (0.72) ( Figure 3 ).

On average, the accuracies r s ^ s , r c ^ c , and r g ^ g were higher for SG (0.80, 0.70, and 0.78, respectively) compared with that for TTY (0.71, 0.69 and 0.74), respectively. The difference between the accuracies r g ^ g and r c ^ c was higher for SG (0.78 and 0.70) than that for TTY (0.74 and 0.69), which resulted in higher average selective efficiency for the SG (11%) compared with that for TTY (7%) ( Table 5 ).

Variance component associated with family effect in the trial POP3(WHS) was not significant by the LRT test for TTY ( Table 4 ) and, thus, was not included in the MET analysis. The family effect (METMpF) did not improve the goodness of fit compared with the METMwF model for POP3 ( Table S5 ); thus, the MET analysis results were presented only for POP2 ( Table 6 ).

The variance estimates associated with the effects of clone ( σ c ' 2 , METMwF), family ( σ s 2 , METMpF), and clone within-family ( σ c 2 , METMpF), as well as their respective contributions to the phenotypic variance, were similar to those obtained in the ST analyses in all trials and traits. The variance components associated with clone, family, and clone within-family were higher than zero in all scenarios ( Table S6 ), confirming the results found in the ST analyses ( Tables 4 , 6 ; Supplementary Material , Tables S6, S7 ). It is worth noting that the values of ρ S   were also like those obtained in the ST analyses ( Tables 4 , 6 ).

Overall, variation was observed in the estimates of variance components σ c ' 2 , σ s 2 , and σ c 2 across the different seasons for both traits, indicating that populations that have G×E can be attributed to the interaction of a simple nature ( Supplementary Material, Table S8 ). Genetic correlation between seasons for the effects of clone, family, and clone within-family was positive in all scenarios for SG, with values higher than 0.50 in most cases. This suggests a low contribution of complex type G×E for this trait. In contrast, most genetic correlation estimates for TTY were lower than 0.50, suggesting a greater contribution of the complex G×E ( Supplementary Material, Tables S6, S7 ).

Regardless of the model and trait, the variance component estimates associated the effects of clone ( σ c ' 2 ) and clone within-family ( σ c 2 ) were always higher in the higher heat stress season (HHS) when compared with that in the no heat stress season (WHS), indicating an increase in genetic variability under extreme heat stress ( Supplementary Material, Table S1 ). The variance component associated with family effect ( σ s 2 ) showed a behavior inversely proportional to the increase of heat stress for the TTY trait, reducing about 50% with the increment of heat stress [11.32 (WHS), 7.32 (MHS), and 3.64 (HHS)] ( Supplementary Material, Table S3 ). Furthermore, regardless of the model adopted, we recorded reductions of 42% and 3% in the average of the clones in the POP2 population for the traits TTY and SG under HHS, respectively ( Supplementary Material, Table S3 ). It is worth noting that, considering the period from the beginning of tuberization (about 30 days after planting) until harvest, the average daily temperature exceeded 20°C on 64% of the days in the MHS season and 88% of the days in the HHS season (88%) ( Figure 1 ).

The exploratory factor analysis showed mean communality of 0.91, 0.91, and 0.87 for the respective vectors u _c′ , u _c , and u _gMET , respectively, indicating that the three factors were sufficient to explain more than 87% of the relationship between seasons. Regardless of the effect, Factor 1 represented the three seasons for SG, Factor 2 represented the WHS and MHS seasons for TTY, and Factor 3 represented only the HHS season ( Supplementary Material, Table S8 ).

After obtaining the FAI-BLUP index scores, which included both traits and all seasons, the Czarnowski’s coefficients (CC) and Spearman’s correlation (r _S ) were utilized to compare the two selection strategies tested (u _c′ vs. u _gMET and u _c vs. u _gMET ). Similarly, to what was observed for the ST analysis, both CC and r _S showed lower magnitude for the second selection strategy (u _c vs. u _gMET ), which suggests closer concordance of the vector u _c′ with u _gMET in the ranking of the clones ( Table 7 ).

Similar to the ST results, independent of trait or season, the selection based on the vector of genotypic effect of clones in the MET (u _gMET ) was higher than the selection based on the vector of clone within-family in the MET (u _c ) ( Table 8 ). It is worth noting that, because of the relationship between the season’s pairs, the accuracy associated with the within-family clone effect was increased in all seasons for both traits, although the accuracy associated the family effect was maintained or was increased ( Table 8 ).

For the TTY trait, efficiency estimates were progressively reduced with increasing heat stress levels [1.09 (WHS), 1.06 (MHS), and 1.02 (HHS)], although, for the SG trait, efficiencies were similar in all seasons. On average, the accuracies r s ^ s , r c ^ c , and r g ^ g were higher for the SG trait (0.82, 0.70, and 0.78) in comparison with the TTY trait (0.70, 0.71, and 0.75), respectively. Furthermore, the magnitude of the difference between the accuracies r g ^ g and r c ^ c was higher for the SG trait (0.78 and 0.70) detriment of TTY (0.75 and 0.71), which resulted in higher average selective efficiency for the SG trait (12%) when compared to the TTY trait (6%) ( Table 8 ).

The results presented here indicate that adjusting for the structure created by family effects can improve the estimation of genetic values. However, it is important to highlight that phenotyping individual data in the early stages of the process may not be compatible with certain breeding pipelines. For example, in certain programs, the first-year evaluation takes place in the field, with mass selection, and using single-hill trials (one seed potato). In this case, the extra phenotyping and modeling of the data would incur additional resources, and the feasibility of such change needs to be evaluated on a case-by-case basis. One technology that potentially can be used to facilitate data collection in the early stages of the program would be the use of aerial images for phenotyping tubers that are dug out of the ground but left in the field (Matias et al., 2020). In contrast, in cases where the early stages are already established with larger plots and randomization of clones, the inclusion of the family effect in the quantitative genetics model would come at no additional effort in data collection.

The genetic progress obtained through conventional potato breeding is slow due to the complexity of the tetrassomic inheritance, the elevated level of heterozygosity of the genitors, and the large number of traits to be assessed (~40) (Haynes et al., 2012; Slater et al., 2014b; Gopal, 2015; Bradshaw, 2017). Potato-breeding programs generate thousands of seedlings annually to overcome these challenges (Stich and Van Inghelandt, 2018). Furthermore, initial stage of potato breeding has low availability of seeds because of the slow tuber multiplication rates, which restricts the use of repetitions and plot size (Paget et al., 2017). These factors contribute to lower experimental precision and selection accuracy.

Typically, the clones being evaluated come from various families, which are often created through biparental crossings. Therefore, the selection process involves making comparisons not only between clones from the same family but also between those from different families. Thus, the effect of clones, and its variance component, are confounded with the family effect. To address this problem and achieve higher selection accuracy, is the use of a nested model, on which the effect of clones is nested within the family effect (Family/Clone = Family + Family + Clone). According to Piepho et al. (2008), models with this structure can be advantageous as they implicitly account for the genetic relatedness of clones. It is important to highlight that including the family effect enhances the model performance due to a better representation of the families, once the clones within a family work as repetitions.

Resende et al. (2016) conducted a study on the effectiveness of the nested structure in a dry bean breeding program using progenies from various populations. They emphasized the significance of the population effect, as the use of the nested model resulted in 20 times more repetitions for each population compared to the progenies. Resulted from an increase in the selective efficiency of progenies. Some papers that considered the effect of populations are reported on bean (Resende et al., 2016; Paula et al., 2020) and soybean (Pereira et al., 2017; Volpato et al., 2018) crops. Thus, the nested structure allows obtaining a more accurate BLUP of the family, which enables a more reliable selection of the best clones. In addition, it allows for the selection of the best families, which enables the evaluation of a greater number of clones per family and increased plot size, thereby increasing the probability of obtaining superior clones.

The utilization of a nested structure presents a significant advantage in providing precise estimations of genetic parameters. This is because the variance components within and between families are not confounded, which can happen in models that do not take family effects into account ( Supplementary Material, Note S1 ). In the former, the heritability of clones is overestimated due to the variance component of family, whereas, in the nested models, the family structure gives a better estimate of the clone effect ( Supplementary Material, Note S1 ). The effects of clone and clone within-family are only equivalent when the variance component of families is zero or close to zero. Pereira et al. (2017) indicates that incorporating the population effect into the estimation of genetic and non-genetic components in soybean breeding data offers a more accurate and realistic methodology.

Higher accuracy was observed using the vector of total genotypic effects (u _gST ), obtained by combining the vectors of family effects (u _s ) and clone within-family (u _c ), and it was directly proportional to the contribution of the variance component associated with family in the total genetic variation ( ρ S ). Resende et al. (2016) in a simulation study showed that the selection accuracy increases in function of the increased contribution of the between-population variance component to the total genetic variation. Therefore, although, in potato breeding, the genetic variance between families is lower than the genetic variance within families, the inclusion of the effect of family can increase the selection accuracy because the heritability of families has been, in general, higher than the heritability of clone within-family (Diniz et al., 2006; Melo et al., 2011). Greater average RE recorded for the SG trait is associated with the higher mean accuracy for the effect of families (0.80) and the higher ρ S mean (0.23).

The utilization of the family effect in potato breeding, which models the nested effect of clones within the family, has been shown to be advantageous. This methodology has led to an improvement in the accuracy of predicting the genotypic values of clones and has achieved a greater degree of RE for the two traits that were studied. The increase in accuracy achieved by the nested model does not reflect increased costs in cases where the data are already recorded. We expect that animal modeling properly accounting for additive and dominance effects would also result in more efficient results. However, the results showed that family effect inclusion is a simpler approach, mainly in cases where the mating design is not complete.

Understanding and effectively managing the G×E interaction is essential for achieving long-term improvements in plant breeding programs because the success of a new cultivar depends on its improved performance on different traits (e.g., TTY and SG) while also presenting good adaptability and stability. The G×E assumes a particular importance for potato breeding, mainly under tropical conditions, because heat stress limits yield and quality in the hottest periods of the year (Fleisher et al., 2017; Fernandes Filho et al., 2021; Patiño-Torres et al., 2021).

Potato crop generally presents better performance in regions with temperate climate, with average temperatures between 5°C and 21°C (Haverkort and Verhagen, 2008). With temperature exceeding 21°C, heat stress significantly reduces tuber yield and quality, due to a series of physiological changes, including an increase in tuber disorders (e.g., internal heat necrosis, knobs, and tuber chain), reduced plant growth, increased respiration rate, greater allocation of dry biomass to leaves at the expense of the tubers, and the reduction in photosynthetic pigments (Lambert et al., 2006; Haverkort and Verhagen, 2008; Hancock et al., 2014; Rykaczewska, 2015; Patiño-Torres et al., 2021). Thus, the selection of heat-tolerant clones is vital to increase potato tuber yield and quality.

Brazil potato season is carried out in three distinct seasons: dry (January to March), winter (April to July), and water (August to December) seasons. Temperatures above the critical threshold, 21°C, are commonly recorded in the dry and water seasons (Andrade et al., 2021). Fernandes Filho et al. (2021) classified these seasons according to their level of heat stress: MHS, WHS, and HHS. One of the major limiting factors for Brazilian potato yield is the use of cultivars that are poorly adapted for tropical conditions and hot temperatures.

The Universidade Federal de Lavras’s potato-breeding program has worked intensively developing heat-tolerant clones (Benites and Pinto, 2011; Figueiredo et al., 2015; Fernandes Filho et al., 2021; Patiño-Torres et al., 2021). To determine which clones, have high heat tolerance, they are tested under varying levels of heat stress. Only clones that perform well under mild and high temperatures are selected (Benites and Pinto, 2011; Rykaczewska, 2015; Fernandes Filho et al., 2021; Patiño-Torres et al., 2021). Furthermore, Andrade et al. (2021) reported that environments with a higher level of heat stress showed a greater capacity to differentiate clones according to their performance.

The strategy mentioned above for selecting heat-tolerant clones requires clones to be evaluated in two or more contrasting environments regarding heat stress. According to Smith et al. (2001), to better evaluate G×E, it is more realistic to use models that consider variance heterogeneity and genetic covariances when analyzing MET data. Cullis et al. (1998) and Smith et al. (2001) reported a significant reduction in the log-likelihood REML when homogenous variance was assumed for the G×E. In multiplicative models for MET data, the interaction of simple nature is accounted by the heterogeneity of genetic variances along the environments, whereas the complex interaction is accounted by the heterogeneity of genetic correlations between pairs of environments (Crossa et al., 2004; Eeuwijk et al., 2016). According to Eeuwijk et al. (2016), when the genetic correlations between two environments are high, it means that there is a smaller percentage of complex interaction involved.

MET data analysis can be carry out in one or two stages (Cullis et al., 1998; Smith et al., 2001; Smith et al., 2015; Gogel et al., 2018). Although the two-stage analysis is often employed, under unbalance (varying number of genotypes between environments or varying number of repetitions between and within environments), a common condition in the early stage of potato-breeding programs, the MET data analysis in one stage is more efficient (Cullis et al., 1998; Smith et al., 2001; Paget et al., 2017; Gogel et al., 2018).

Although the analysis of MET data is advantageous because of the advantage of the interrelationship between environments, it is increasing selective accuracy and allows a better interpretation of the G×E interaction (Smith et al., 2001; Kelly et al., 2007). The family effect, naturally present in the initial stage of potato-breeding programs, has also been neglected in this type of analysis (Fernandes Filho et al., 2021). However, one can combine the advantages cited above with those described in Section 4.1. Making MET analysis a powerful tool for clone selection in the early stages of potato-breeding programs, especially under tropical conditions, it is opportune to highlight that the recovery of inter-environmental information was more pronounced for the effect of clones within-family because of the higher magnitude of genetic correlations between seasons. This corroborates with the lower magnitude of the selective efficiency in MET analysis comparison to ST analysis. Selective efficiency for the TTY trait progressively reduced with increasing heat stress levels, whereas the selective efficiency for the SG trait was maintained practically constant. These results ratify the importance of environments with higher heat stress levels to discriminate potato clones for the TTY trait, as well as the greater contribution of the family effect to the selection of superior clones for the SG trait.

The unstructured model can be used in cases when only a few trials are included, due to the smaller number of parameters that need to be estimated, avoiding the use of factor analysis (Smith et al., 2001; Melo et al., 2020; Fernandes Filho et al., 2021). However, the use of the unstructured model implies the prediction of genotypic effects for each trial (Kelly et al., 2007; Eeuwijk et al., 2016; Melo et al., 2020; Fernandes Filho et al., 2021). Hence, it is desirable to use a selection index to capture the G×E interaction and to combine MET analyses realized for multiple traits (Kelly et al., 2007; Melo et al., 2020; Fernandes Filho et al., 2021).

There are many options for selection indexes (Mendes et al., 2009; Rocha et al., 2018; Yan and Frégeau-Reid, 2018; Melo et al., 2020). Among the different indexes, the FAI-BLUP index stands out, because it can incorporate data from various environments and traits without the need for weights. It also avoids issues with multicollinearity and makes it easier to understand the G×E interaction through exploratory factor analysis (Rocha et al., 2018; Oliveira et al., 2019). In the present study, factor analysis showed a superior performance in summarizing the relationships among environments (WHS, MHS, and HHS) and traits (TTY and SG) for each effect (u _c′ , u _c , and u _gMET ). In addition, the factor analysis showed that the SG trait had a lower level of interaction because all seasons were grouped under Factor 1. On the other hand, the TTY trait had a higher level of interaction as the WHS and HHS seasons were grouped under distinct factors. Thus, including the effect of families in the MET analysis is a helpful strategy to increase the accuracy of superior clone selection.

Finally, the inclusion of the family effect increased the selective efficiency of clones in ST and MET selection on schemes through an increment in the accuracy of the total genotypic value. On average, the selective efficiency of clones was 11% and 7% for ST and 12% and 6% in MET for the traits SG and TTY, respectively. An expressive reduction of the family effect under heat stress for TTY and of lower magnitude for SG was observed.

Thus, the results of the present work suggest that the inclusion of the family effect in clone selection models, in the initial stage of potato-breeding programs, is desirable because it contributes to increasing the selective efficiency of clones without generating additional costs, especially for the SG trait.