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

  • Alison B. SmithEmail author
  • Lauren M. Borg
  • Beverley J. Gogel
  • Brian R. Cullis


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.


Average information algorithm Factor analytic models Factorial treatment structure Linear mixed model 


Supplementary material

13253_2019_362_MOESM1_ESM.r (6 kb)
Supplementary material 1 (R 6 KB)


  1. Bailey, R. A. (2008), “Design of Comparative Experiments, Cambridge Series in Statistical and Probabilistic Mathematics,” Cambridge University Press, Cambridge.Google Scholar
  2. Butler, D., Cullis, B., Gilmour, A., and Gogel, B. J. (2009), “ASReml-R Reference Manual, Version 3. Training and Development Series, No. QE02001,” Queensland Department of Primary Industries.Google Scholar
  3. Butler, D. G., Cullis, B. R., Gilmour, A. R., and Thompson, R. (2018), “ASReml-R Reference Manual, Version 4,” University of Wollongong. Available at
  4. Gelfand, A. E., Diggle, P. J., Fuentes, M., and Guttorp, P. (2010), “Handbook of Spatial Statistics. Handbooks of Modern Statistical Methods”, Chapman and Hall/CRC, Boca Raton.Google Scholar
  5. Gilmour, A. R., Thompson, R., and Cullis, B. R. (1995), “Average Information REML: An Efficient Algorithm for Variance Parameter Estimation in Linear Mixed Models,” Biometrics, 51, 1440–1450.CrossRefzbMATHGoogle Scholar
  6. Gogel, B., Smith, A., and Cullis, B. (2018), “Comparison of a One- and Two-Stage Mixed Model Analysis of Australia’s National Variety Trial Southern Region wheat data”’ Euphytica, 214, 1–21.Google Scholar
  7. GRDC. (2017),“ Blackleg Management Guide: 2017 Spring Variety Ratings”, Fact sheet, Grains Research & Development Corporation.Google Scholar
  8. Kelly, A. M., Cullis, B. R., Gilmour, A. R., Eccleston, J. A., and Thompson, R. (2009), “Estimation in a Multiplicative Mixed Model Involving a Genetic Relationship Matrix,” Genetics Selection Evolution, 41, 1186–1297.CrossRefGoogle Scholar
  9. Oakey, H., Verbyla, A. P., Cullis, B. R., Pitchford, W. S., and Kuchel, H. (2006), Joint Modeling of Additive and Non-additive Genetic Line Effects in Single Field Trials,” Theoretical and Applied Genetics, 113, 809–819.CrossRefGoogle Scholar
  10. Sivasithamparam, K., Barbetti, M., and Li, H. (2005), “Recurring Challenges from Necrotrophic Fungal Plant Pathogen: A Case Study with Leptosphaeria maculans (Causal Agent of Blackleg Disease in Brassicas) in Western Australia,” Annals of Botany, 96, 363–377.Google Scholar
  11. Smith, A., Cullis, B. R., and Thompson, R. (2001), “Analysing Variety by Environment Data Using Multiplicative Mixed Models and Adjustments for Spatial Field Trend,” Biometrics, 57, 1138–1147.MathSciNetCrossRefzbMATHGoogle Scholar
  12. ——— (2005), “The Analysis of Crop Cultivar Breeding and Evaluation Trials: An Overview of Current Mixed Model Approaches,” Journal of Agricultural Science, 143, 449–462.CrossRefGoogle Scholar
  13. Smith, A. B., and Cullis, B. R. (2018), “Plant Breeding Selection Tools Built on Factor Analytic Mixed Models for Multi-environment Trial Data,” Euphytica, 214, 1–19.CrossRefGoogle Scholar
  14. Thompson, R., Cullis, B., Smith, A., and Gilmour, A. (2003), “A Sparse Implementation of the Average Information Algorithm for Factor Analytic and Reduced Rank Variance Models,” Australian and New Zealand Journal of Statistics, 45, 445–459.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© International Biometric Society 2019

Authors and Affiliations

  • Alison B. Smith
    • 1
    Email author
  • Lauren M. Borg
    • 1
  • Beverley J. Gogel
    • 1
  • Brian R. Cullis
    • 1
  1. 1.Centre for Bioinformatics and Biometrics, National Institute for Applied Statistics Research AustraliaUniversity of WollongongWollongongAustralia

Personalised recommendations