Factor Analytic Research Articles

Linear and Nonlinear Indices of Score Accuracy and Item Effectiveness for Measures That Contain Locally Dependent Items.

The problem of local item dependencies (LIDs) is very common in personality and attitude measures, particularly in those that measure narrow-bandwidth dimensions. At the structural level, these dependencies can be modeled by using extended factor analytic (FA) solutions that include correlated residuals. However, the effects that LIDs have on the scores based on these extended solutions have received little attention so far. Here, we propose an approach to simple sum scores, designed to assess the impact of LIDs on the accuracy and effectiveness of the scores derived from extended FA solutions with correlated residuals. The proposal is structured at three levels-(a) total score, (b) bivariate-doublet, and (c) item-by-item deletion-and considers two types of FA models: the standard linear model and the nonlinear model for ordered-categorical item responses. The current proposal is implemented in SINRELEF.LD, an R package available through CRAN. The usefulness of the proposal for item analysis is illustrated with the data of 928 participants who completed the Family Involvement Questionnaire-High School Version (FIQ-HS). The results show not only the distortion that the doublets cause in the omega reliability estimate when local independency is assumed but also the loss of information/efficiency due to the local dependencies.

Comparison of additive main effect–multiplicative interaction model and factor analytic model for genotypes ordination from multi‐environment trials

AbstractAn additive main effects and multiplicative interaction (AMMI) model is used to explore the genotype × environment interaction (GEI) in complete multi‐environmental trials. This model orders genotypes (G) according to their performance across environments (E) on a vectorial plane generated by the first two axes of a principal component analysis (AMMI‐biplot). Alternatively, interaction terms can be regarded as random effects, which can be predicted from linear mixed models using a factor analytic (FA) covariance structure for the GEI terms. Here, an FA‐biplot was obtained by plotting the G and E scores derived from the FA mixed model with complete and incomplete data. The aim of this work was to compare AMMI‐biplot with FA‐biplot for balanced data and then show the impact of the imbalance on the FA‐biplot. The G ordinations were assessed in four scenarios generated using datasets of 3 consecutive years obtained from comparative wheat trials conducted under a complete random block design in different environments across the Argentine network of cultivar assessment. For each scenario, G with the lowest performance in the third year were deleted, one by one, from all sites to generate a scenario with missing G. Although we used different statistical procedures to obtain AMMI‐biplot and FA‐biplot, they showed the same interaction pattern in the case of up to 50% of G dropped from all E in the last year of the multiyear trials. We conclude that the FA‐biplot yields a robust G ordination even when with incomplete datasets.

Genotype by environment interaction and stability analysis of three agronomic traits in Kersting's groundnut (Macrotyloma geocarpum) using factor analytic modeling and environmental covariates

AbstractUnderstanding genotype by environment interaction (GEI) represents a challenge in Kersting's groundnut [Macrotyloma geocarpum (Harms) Maréchal and Baudet] breeding for selecting high‐performing and stable lines across environments. Here, we investigated GEI and stability in Kersting's groundnut using factor analytic (FA) based linear mixed models and environmental covariates. A total of 375 accessions were evaluated across 3 years (2017, 2018, and 2019) and two locations (Sékou and Savè) in Benin, generating five environments (E1, E2, E3, E4, and E5). The traits measured included days to 50% flowering (DFF), grain yield (YLD), and 100‐seed weight (HSW). The study generated multi‐environment values for grain yield and its components in Kersting's groundnut. The genetic correlations between pairs of environments ranged from −0.71 to 0.99. The genetic correlations between YLD and HSW indicated positive and moderate to high correlations in all environments. The FA analysis revealed that FA2 structure accounted for 93.9% of the genetic variability in DFF with factor 1 accounting for more than 90% of the environments variations. Two factors explained 87% of the genetic variance in grain yield, and 70% of the environments variability were clustered by factor 1. For HSW, two factors explained 85% of the genetic variance of the environments, and factor 1 accounted for 72.7%. Combining environmental covariates to FA models revealed that precipitation, temperature, and growth cycle duration were highly correlated to the environmental loadings of factor 1. Relative humidity and solar radiation showed moderate to high correlations with factor 2 loadings. Those covariates explained the high GEI among environments clustered by a given factor. Precipitations and temperatures affected the variations in grain yield. Finally, based on latent regression analysis, the accessions AF202, AF221, AF223, AF225, and AF256 were identified as accessions combining best performance for grain yield, early flowering, and 100‐seed weight, showing adaptability across environments and stability to some environments.

Genomic Prediction from Multi-Environment Trials of Wheat Breeding.

Genomic prediction relates a set of markers to variability in observed phenotypes of cultivars and allows for the prediction of phenotypes or breeding values of genotypes on unobserved individuals. Most genomic prediction approaches predict breeding values based solely on additive effects. However, the economic value of wheat lines is not only influenced by their additive component but also encompasses a non-additive part (e.g., additive × additive epistasis interaction). In this study, genomic prediction models were implemented in three target populations of environments (TPE) in South Asia. Four models that incorporate genotype × environment interaction (G × E) and genotype × genotype (GG) were tested: Factor Analytic (FA), FA with genomic relationship matrix (FA + G), FA with epistatic relationship matrix (FA + GG), and FA with both genomic and epistatic relationship matrices (FA + G + GG). Results show that the FA + G and FA + G + GG models displayed the best and a similar performance across all tests, leading us to infer that the FA + G model effectively captures certain epistatic effects. The wheat lines tested in sites in different TPE were predicted with different precisions depending on the cross-validation employed. In general, the best prediction accuracy was obtained when some lines were observed in some sites of particular TPEs and the worse genomic prediction was observed when wheat lines were never observed in any site of one TPE.

GIS-FA: an approach to integrating thematic maps, factor-analytic, and envirotyping for cultivar targeting.

We propose an "enviromics" prediction model for recommending cultivars based on thematic maps aimed at decision-makers. Parsimonious methods that capture genotype-by-environment interaction (GEI) in multi-environment trials (MET) are important in breeding programs. Understanding the causes and factors of GEI allows the utilization of genotype adaptations in the target population of environments through environmental features and factor-analytic (FA) models. Here, we present a novel predictive breeding approach called GIS-FA, which integrates geographic information systems (GIS) techniques, FA models, partial least squares (PLS) regression, and enviromics to predict phenotypic performance in untested environments. The GIS-FA approach enables: (i) the prediction of the phenotypic performance of tested genotypes in untested environments, (ii) the selection of the best-ranking genotypes based on their overall performance and stability using the FA selection tools, and (iii) the creation of thematic maps showing overall or pairwise performance and stability for decision-making. We exemplify the usage of the GIS-FA approach using two datasets of rice [Oryza sativa (L.)] and soybean [Glycine max (L.) Merr.] in MET spread over tropical areas. In summary, our novel predictive method allows the identification of new breeding scenarios by pinpointing groups of environments where genotypes demonstrate superior predicted performance. It also facilitates and optimizes cultivar recommendations by utilizing thematic maps.

Application of factor analytic and spatial mixed models for the analysis of multi-environment trials in common bean (Phaseolus vulgaris L.) in Ethiopia.

Common bean (Phaseolus vulgaris L.) is one of the most important grain legumes consumed globally, especially in Ethiopia, for its edible seeds, cash crops, and supply of protein for farmers. Efficient statistical methods must be employed for the evaluation of common bean varieties to accurately select superior varieties that contribute to agricultural productivity. The objective of this study was to identify promising large mottled bean varieties through analysis of multi-environment trials (MET) data using multiplicative spatial mixed models. In this study, 16-18 large mottled common bean varieties, including one check, were sown across nine growing environments in Ethiopia using lattice and alpha lattice designs, with three replications laid out in a square or rectangular (row by column) array of plots, respectively during the main cropping season from 2015 to 2018. We present a linear mixed model analysis that integrates spatial and factor analytic (FA) models, and the heritability measure was used to evaluate the efficiency of these models with the conventional analysis. The analysis of the spatial model, and more significantly, the spatial+FA model, revealed a notable enhancement in heritability. With the exception of a trial conducted at Kobo, a genotype DAP 292, found to be good performing for days to flowering and maturity, but for yield only across four clusters of trials, C2, C3, C5 and C7, formed with trials of relatively high genetic variance. Across these clusters, the yield advantage of this variety over the check ranged from 10-32%. This genotype also has a yield that is somewhat comparable to the check across the remaining clusters. Overall, both the spatial and factor analytic models proved to be effective approaches for analyzing the data in this study. The analysis of multi-environment trial data through the use of more efficient statistical models can provide a more robust platform for evaluating common bean varieties with greater confidence in selecting superior varieties across a range of environments. Hence, scaling up the use of this efficient analysis method is indispensable for enhancing the selection of superior varieties.

Genotype x environment interaction and climate sensitivity in growth and wood density of European larch

European larch is an ecologically and economically important tree species planted across a wide range of environments. Understanding its response to the rapidly changing climate is essential to developing efficient gene resource management strategies in line with the assisted migration methodology. In this study, we quantified the genotype x environment interaction (GxE) of a larch provenance in a multi-environmental trial analysis using factor analytic (FA) modeling. We analyzed the mean annual height increment (MAI-H), the mean annual diameter increment (MAI-DBH), and wood density (PP). Additionally, we evaluated the effect of several environmental variables on these traits. The genetic correlations between sites varied from −0.53 to 0.99, −0.87 to 0.99, and −0.17 to 0.99 for MAI-H, MAI-DBH, and PP, respectively. This indicates the presence of low to high GxE. The growth traits showed more GxE signal than wood density. Narrow-sense heritability estimates were 0.41, 0.16, and 0.35 for MAI-H, MAI-DBH, and PP, respectively. The FA modeling of order 2 explained 77.4%, 67.9%, and 77.3% of the genetic variance for MAI-H, MAI-DBH, and PP, respectively. For MAI-H, the latent regression plots for the predicted site means indicate a significant positive correlation with the first factor loading. At the individual genotype level, the latent regression plots indicated high levels of GxE, especially for growth traits. Significant and relatively high correlations were found between several EVs and the factor loadings of the FA models, suggesting their possible influence on the genotypic performance across environments. Regarding MAI-H, elevation, averaged monthly mean temperature, precipitation seasonality, and stand age were correlated with the first factor loading. Annual temperature range with both the first and second factor loadings. For MAI-DBH, the mean diurnal range was correlated with the first factor loading. In the case of PP, we found that elevation was correlated with the second factor loading. Overall, this approach successfully modeled the GxE and identified environmental variables likely affecting it. Therefore, we conclude that this methodology is a valuable tool for selecting suitable genotypes adapted to specific environmental conditions.

Image-based phenomic prediction can provide valuable decision support in wheat breeding

Key messageGenotype-by-environment interactions of secondary traits based on high-throughput field phenotyping are less complex than those of target traits, allowing for a phenomic selection in unreplicated early generation trials.Traditionally, breeders’ selection decisions in early generations are largely based on visual observations in the field. With the advent of affordable genome sequencing and high-throughput phenotyping technologies, enhancing breeders’ ratings with such information became attractive. In this research, it is hypothesized that GtimesE interactions of secondary traits (i.e., growth dynamics’ traits) are less complex than those of related target traits (e.g., yield). Thus, phenomic selection (PS) may allow selecting for genotypes with beneficial response-pattern in a defined population of environments. A set of 45 winter wheat varieties was grown at 5 year-sites and analyzed with linear and factor-analytic (FA) mixed models to estimate GtimesE interactions of secondary and target traits. The dynamic development of drone-derived plant height, leaf area and tiller density estimations was used to estimate the timing of key stages, quantities at defined time points and temperature dose–response curve parameters. Most of these secondary traits and grain protein content showed little GtimesE interactions. In contrast, the modeling of GtimesE for yield required a FA model with two factors. A trained PS model predicted overall yield performance, yield stability and grain protein content with correlations of 0.43, 0.30 and 0.34. While these accuracies are modest and do not outperform well-trained GS models, PS additionally provided insights into the physiological basis of target traits. An ideotype was identified that potentially avoids the negative pleiotropic effects between yield and protein content.

Evaluation of finger millet (Eleusine coracana (L.) Gaertn.) in multi-environment trials using enhanced statistical models.

Spatial variation and genotype by environment (GxE) interaction are common in varietal selection field trials and pose a significant challenge for plant breeders when comparing the genetic potential of different varieties. Efficient statistical methods must be employed for the evaluation of finger millet breeding trials to accurately select superior varieties that contribute to agricultural productivity. The objective of this study was to improve selection strategies in finger millet breeding in Ethiopia through modeling of spatial field trends and the GxE interaction. A dataset of seven multi-environment trials (MET) conducted in randomized complete block design (RCBD) with two replications laid out in rectangle (row x column) arrays of plots was used in this study. The results revealed that, under the linear mixed model, the spatial and factor analytic (FA) models were efficient methods of data analysis for this study, and this was demonstrated with evidence of heritability measure. We found two clusters of correlated environments that helped to select superior and stable varieties through ranking average Best Linear Unbiased Predictors (BLUPs) within clusters. The first cluster was chosen because it contained a greater number of environments with high heritability. Based on this cluster, Bako-09, 203439, 203325, and 203347 were the top four varieties with relatively high yield performance and stability across correlated environments. Hence, scaling up the use of this efficient analysis method will improve the selection of superior finger millet varieties.

Psychometrics vs neurochemistry: A controversy around mobility-like scales of temperament

There is a disagreement between the factor-analytic (FA) approach in differential psychology and findings in neuroscience in identifying mobility-like temperament traits (i.e. speed of integration of actions that include plasticity, tempo, impulsivity). Due to their entanglement with “energetic” traits, mobility-like traits never emerge as independent dimensions in FA models of temperament and personality. This paper points out that there are, however, well-documented neurochemical biomarkers of these traits, which are distinct from biomarkers of “energetic” traits. To highlight this controversy, this paper reports the results from three studies conducted on English-Canadian, Russian, and Portuguese-Brazilian samples. The studies confirmed the correspondence between similar scales of two models developed independently in two distinct branches of the Pavlovian tradition. In all three samples and two inventories, there were strong positive correlations between mobility-like and endurance-like scales. By psychometric standards, these scales should be viewed as parts of one dimension, but this would be contrary to the evidence from neuroscience pointing to their different biomarkers. Moreover, our examination of PTS and STQ-77 temperament profiles associated with polymathy demonstrated the benefits of mobility-like traits. The disagreement between psychometric and neurochemical perspectives shows the limitations of relying on FA in deriving models of differential psychology.

Genomic prediction for grain yield in a barley breeding program using genotype × environment interaction clusters

AbstractGenotype × environment interaction (GEI) is one of the key factors affecting breeding value estimation accuracy for agronomic traits in plant breeding. Measures of GEI include fitting prediction models with various kernels to capture the variance resulting from GEI, and characterizing trials into megaenvironment (ME) clusters within which breeding values can be estimated to remove the main GEI effects. However, many of the current approaches require observations of common genotypes across all trials, which is unavailable in most breeding programs. Our study introduces two methods that can be implemented on unbalanced data to categorize trials into clusters, where both need a correlation matrix between trials: one estimated via a factor analytic (FA) model and another estimated via weather variables. The methods were tested using empirical barley (Hordeum vulgare L.) yield data in a commercial breeding program from 102 trials over 5 yr spread across multiple locations in Australia. Leave‐one‐year‐out cross‐validation achieved comparable predictive accuracies using either trials or clusters as the observed variable in GEI FA models (max. 0.45), which was higher than the accuracy achieved using the non‐GEI model (0.37). In the random cross‐validations, accuracies achieved within clusters (0.42–0.64) were mostly comparable with those achieved in the full population (0.62). In the within‐cluster validations, higher predictive accuracies were achieved when the training population was from the same cluster (mean 0.22) than outside of the cluster (mean 0.16). Our proposed methods of characterizing multienvironment trials into clusters provides a novel way to define training populations by reducing the variance resulting from GEI and could be implemented in any plant breeding program.

Modelling temporal genetic and spatio‐temporal residual effects for high‐throughput phenotyping data*

SummaryHigh‐throughput phenomics data are being collected in both the laboratory and the field. The data are often collected at many time points and there may be spatial variation in the laboratory or field that impacts on the growth of the plants, and that may influence the traits of interest. Modelling the genetic effects is of primary interest in such studies, but these effects might be biased if non‐genetic effects present in the experiment are ignored. With data that are collected both in time and space, there may be a need to jointly model these multi‐dimensional non‐genetic effects. Thus both modelling of genetic effects over time and non‐genetic effects over time and space in a one‐stage analysis is considered. An experiment that involves field phenomics data with four dimensions, two in space and two in time, provides the vehicle to examine the models. Factor analytic (FA) models are often used for genetic effects for different environments to provide reliable estimates of genetic variances and correlations. As the time dimension defines the environments, FA models are examined for the phenomics data. Reduced rank tensor smoothing splines are presented as a possible approach for modelling the spatio‐temporal effects, although an additional term is included for heterogeneity over the two time dimensions. This approach is feasible, although very time‐consuming. The process of model selection for the genetic effects is presented including tests, information criteria and diagnostics. Comparisons of more simplistic models are made with the reduced rank tensor spline. This also shows the interplay between the genetic and residual models in model selection.

Genotype-by-environment interaction and yield stability analysis of biomass sorghum hybrids using factor analytic models and environmental covariates

Abstract Biomass sorghum has emerged as an alternative crop for biofuel and bioelectricity production. Fresh biomass yield (FBY) is a quantitative trait highly correlated with the calorific power of energy sorghum cultivars, but also highly affected by the environment. The main goal of this study was to investigate the genotype-by-environment interaction (G × E) and the stability of sorghum hybrids evaluated for FBY across different locations and years, using factor analytic (FA) mixed models and environmental covariates. Pairwise genetic correlations between environments ranged from -0.21 to 0.99, indicating the existence of null to high G × E. The FA analysis unveiled that solely three factors explained more than 79% of the genetic variance, and that more than 60% of the environments were clustered in the first factor. Moderate correlations were found between some environmental covariates and the loadings of FA models for environments, suggesting the possible factors to explain the high G × E between environments clustered in a given factor. For example: precipitation, minimum temperature and speed wind were correlated to the environmental loadings of factor 1; minimum temperature, solar radiation and altitude to factor 2; and crop growth cycle to factor 3. The latent regression analysis was used to identify hybrids more responsive to a set of environments, as well as hybrids specifically adapted to a given environment. Finally, FA models can be successfully used to identify the main environmental factors affecting G × E, such as minimum temperature, precipitation, solar radiation, crop growth cycle and altitude.

A comparison of factor‐analytic and equal diagonal factor‐analytic models in multi‐location trials analyses

AbstractThe two most promising models for analyzing multi‐location trials (MLT) include the factor‐analytic (FA) and equal diagonal factor‐analytic (FA1) models. Here, we compare the two models by analyzing four datasets performed for the corn (Zea mays L.) performance trails in Northeastern and Northern China. The FA model contained more parameters for the variance–covariance of variety effects than the FA1 model. In data fitting, the effective (parameter value estimated >0) parameter number difference between the two models was not as large as in theory and was reduced with an increasing number of factors. In goodness‐of‐fit for trial data, the two models were competitive based on both the Akaike Information Criterion (AIC) and the likelihood‐ratio test (LRT). There were no notable differences between the two models in the estimates, rankings, standard errors of estimates, test efficiencies of contrasts for variety main (mean over all locations) effects, and discrimination power for varieties. The FA models were superior to FA1 models in the standard error of estimates and test efficiency of contrasts for variety simple effects. Therefore, the FA model was suggested for the analysis of MLTs. A general recommendation for the selection of either the FA or FA1 models using AIC or LRT cannot be made from this analysis.

Analyzing the performance of corn in China using a factor‐analytic variance‐covariance structure with multiple factors

AbstractFor effective use of the factor‐analytic (FA) model in analyzing data from multi‐environmental trials (METs), we undertook an empirical study of data sets from the corn (Zea mays L.) performance trails in China to compare all FA models with possible number of factors. We found that for the variety main effects, the estimate, ranking, and test efficiency of contrast were compatible among the FA models with various numbers of factors. For the variety simple effects, the effect estimate, and hence the ranking, changed; the test efficiency of contrast and discrimination power increased with increasing number of factors as the number of factors was smaller than a specific number. With consideration of both test efficiency of contrast and discrimination power of varieties for the trials analyzed, about 10 could be chosen as the optimal number of factors of the FA model approach.

Modeling Genotype × Environment Interaction Using a Factor Analytic Model of On‐Farm Wheat Trials in the Yaqui Valley of Mexico

On‐farm trials of bread and durum wheat in the Yaqui Valley region of southern Sonora, Mexico, were established for three cropping seasons (2012, 2013, and 2015) using the management practices implemented by farmers. The trials comprised bread and durum wheats that were sown together under two regimes: full irrigation and reduced irrigation. The experiments were replicated and unbalanced, as several bread wheat and durum wheat lines were not repeated during the 3 yr. We studied the interaction between bread and durum wheats and environments (farmer‐irrigation‐year combinations). To model the crossover interaction (COI) and the non‐COI components of the genotype × environment interaction (G×E) between the wheat lines and the environments, we used a linear mixed model with the Factor Analytic (FA) model, a parsimonious model that is similar to the multiple regression of lines on environments based on latent variables. In this case, we modeled the combined effects of the wheat lines and their interactions with the farmer‐irrigation‐year combinations. Results show the separation of the dynamic (unpredictable) component of the interaction (year) from the more static component of the interaction due to farmer and irrigation level. The FA model offers a useful alternative for modeling interactions in agronomy‐breeding experiments to dissect and account for complex interactions that are common in agriculture experiments. Furthermore, stable wheat lines across all environments were also detected, as well the environments that caused most of the interaction.Core Ideas In agriculture is important to study genotype × environment interaction (G×E) interaction and especially the interaction between bread and durum wheats and farmer–irrigation–year environmental combinations. The linear mixed model with the Factor Analytic (FA) is efficient for dissenting the main factors affecting G×E.

Spatial analysis in multi-environment trials of malt barley in Ethiopia

Selection of superior genotypes and measuring heritability are some of the basic objectives of plant breeding. For this purpose, plant breeders grow crops across environments. Understanding the pattern of response across environments is an integral component of selection of superior and stable genotypes. The objective of this study was to improve selection strategies in barley breeding of Ethiopia through modeling spatial field trend. A set of multi-environment trials (MET) data from the national variety trial series conducted over four years, was taken from the Ethiopian Barley Breeding Programme, spanning stages from early generation to national variety trial testing for yield, was used in this study. The trials were analysed in a linear mixed model framework. Then, fitting a one-stage model for MET data, including a correlated spatial process for field trend within each trial, and combining a factor analytic (FA) model for genotype by environment interaction was conducted. The genetic correlations from this MET analysis were then used to cluster the environments based on their similarity. Performance of genotypes across these environmental clusters indicate broad (Bekoji-2005 and Bekoji-2004) and specific adaptation (Sgonder-2007 and Sgonder-2006) of genotype to certain types of environments. In addition, analysis of this historical MET data shed light on how breeding programme design can be improved to capture responses across the target population of environments, as it can inform the adequacy of the current number of barley grown areas in Ethiopia and the improvement in measuring heritability. Key words: Barley MET, heritability, linear mixed model

Bayesian factor analytic model: An approach in multiple environment trials.

One of the main challenges in plant breeding programs is the efficient quantification of the genotype-by-environment interaction (GEI). The presence of significant GEI may create difficulties for breeders in the selection and recommendation of superior genotypes for a wide environmental network. Among the diverse statistical procedures developed for this purpose, we highlight those based on mixed models and factor analysis that are called factor analytic (FA) models. However, some inferential issues are related to the factor analytic model, such as Heywood cases that make the model non-identifiable. Moreover, the representation of the loads and factors in the conventional biplot does not involve any measurement of uncertainty. In this work, we propose dealing with the FA model using the Bayesian framework with direct sampling of factor loadings via spectral decomposition; this guarantees identifiability in the estimation process and eliminates the need for the rotationality of factor loadings or imposition of any ad hoc constraints. We used simulated and real data to illustrate the method’s application in multi-environment trials (MET) and to compare it with traditional FA mixed models on controlled unbalancing. In general, the Bayesian FA model was robust under different simulated unbalanced levels, presenting the superior predictive ability of missing data when compared to competing models, such as those based on FA mixed models. In addition, for some scenarios, the classical FA mixed model failed in estimating the full FA model, illustrating the parametric problems of convergence in these models. Our results suggest that Bayesian factorial models might be successfully used in plant breeding for MET analysis.

Estimation of Factor Analytic Mixed Models for the Analysis of Multi-treatment Multi-environment Trial Data

An efficient and widely used method of analysis for multi-environment trial (MET) data in plant improvement programs involves a linear mixed model with a factor analytic (FA) model for the variety by environment effects. The variance structure is generally constructed as the kronecker product of two matrices that relate to the variety and environment dimensions, respectively. In many applications, the FA variance structure is assumed for the environment dimension and either an identity matrix or a known matrix, such as the numerator relationship matrix, is used for the variety dimension. The factor analytic linear mixed model can be fitted to large and complex MET datasets using the sparse formulation of the average information algorithm of Thompson et al. (Aust N Z J Stat 45:445–459, 2003) or the extension provided by Kelly et al. (Genet Sel Evo, 41:1186–1297, 2009) for the case of a known non-identity matrix for the variety dimension. In this paper, we present a sparse formulation of the average information algorithm for a more general separable variance structure where all components are parametric and one component has an FA structure. The approach is illustrated using a large and highly unbalanced MET dataset where there is a factorial treatment structure. Supplementary materials accompanying this paper appear online.

Genetic Covariance of Environments in the Potato National Chip Processing Trial

The National Chip Processing Trial is a collaborative effort between public breeding programs and the potato (Solanum tuberosum L.) industry to identify new clones with broad adaptation. The objective of this study was to investigate the genetic covariance of trial locations, based on 337 clones evaluated in 10 states from 2011 to 2016. Three models were considered: (I) assuming a uniform genetic correlation between locations within a year, (II) using a factor‐analytic (FA) model of the total genetic covariance of environments (location–year combinations), and (III) using a FA model of the additive genetic covariance based on 5278 single‐nucleotide polymorphism (SNP) markers. With Model I, the genetic correlation between locations was 0.50 for vine maturity, 0.54 for tuber yield, and 0.72 for specific gravity. The Akaike information criterion decreased as model complexity increased, from Models I to II to III, for maturity and yield but not specific gravity. Of the 10 states in the dataset, Florida stood out for having environments with substantial (up to 88%) genetic variance unexplained by the latent factors. Linear discriminants (LD) of the factor loadings were used to visualize the genetic correlation between locations. For vine maturity, LD1 separated Florida from the other locations, and LD2 separated the remaining southern locations from the northern ones. For yield, LD1 separated Texas from the other locations. This study has created a foundation for the design of more efficient trialing and selection programs for the US potato community.