Using `lme()` to fit the Environmental Variance mixed model to genotype experiments

Published at December 4, 2025 ·  11 min read

Yield stability is a key aspect in the selection of crop genotypes. Its definition is not entirely straightforward (see, for example, Annichiarico, 2002), but, in simple terms, it refers to the ability of a crop to maintain its yield potential across different environments, helping farmers safeguard their income. Several statistical indicators of stability have been proposed (see, e.g., Mohammadi, 2008). In this post, however, I will focus on the so-called environmental variance, which represents the portion of phenotypic variance attributable to environmental (non-genetic) factors and is measured as the overall variance across environments for each genotype.

Using `lme()` to fit the Stability Variance mixed model to genotype experiments

Published at December 2, 2025 ·  11 min read

Yield stability is a fundamental aspect of the selection of crop genotypes. Its definition is rather complex (see, for example, Annichiarico, 2002), but, in simple terms, it represents the ability of a crop to maintain its potential yield level across environments, which helps farmers preserve their income. Several statistical indicators of stability exist (see, e.g., Mohammadi, 2008) and, in this post, I would like to concentrate on the so-called stability variance, that is, for a specific genotype, the amount of yield variability across different environments, after correcting for the additive effects of each environment, which are common to all genotypes under investigation (Shukla, 1972).

Meta-analysis for a single study. Is it possible?

Published at July 21, 2022 ·  12 min read

We all know that the word meta-analysis encompasses a body of statistical techniques to combine quantitative evidence from several independent studies. However, I have recently discovered that meta-analytic methods can also be used to analyse the results of a single research project. That happened a few months ago, when I was reading a paper from Damesa et al. (2017), where the authors describe some interesting methods of data analyses for multi-environment genotype experiments. These authors gave a few nice examples with related SAS code, that is rooted in mixed models. As an R enthusiast, I was willing to reproduce their analyses with R, but I could not succeed, until I realised that I could make use of the package ‘metafor’ and its bunch of meta-analityc methods.

Building ANOVA-models for long-term experiments in agriculture

Published at August 20, 2020 ·  29 min read

This is the follow-up of a manuscript that we (some colleagues and I) have published in 2016 in the European Journal of Agronomy (Onofri et al., 2016). I thought that it might be a good idea to rework some concepts to make them less formal, simpler to follow and more closely related to the implementation with R. Please, be patient: this lesson may be longer than usual.

First question: what are long-term experiments? Agricultural experiments have to deal with long-term effects of cropping practices. Think about fertilisation: certain types of organic fertilisers may give effects on soil fertility, which are only observed after a relatively high number of years (say: 10-15). In order to observe those long-term effects, we need to plan Long Term Experiments (LTEs), wherein each plot is regarded as a small cropping system, with the selected combination of rotation, fertilisation, weed control and other cropping practices. Due to the fact that yield and other relevant variables are repeatedly recorded over time, LTEs represent a particular class of multi-environment experiments with repeated measures.

Fitting complex mixed models with nlme. Example #5

Published at June 5, 2020 ·  14 min read

Joint Regression is a very old, but, nonetheless, useful technique. It is widely known that the yield of a genotype in different environments depends on environmental covariates, such as the amount of rainfall in some critical periods of time. Apart from rain, also temperature, wind, solar radiation, air humidity and soil characteristics may concur to characterise a certain environment as good or bad and, ultimately, to determine yield potential.

Early in the 60s, several authors proposed that the yield of genotypes is expressed as a function of an environmental index \(e_j\), measuring the yield potential of each environment \(j\) (Finlay and Wilkinson, 1963; Eberhart and Russel, 1966; Perkins and Jinks, 1968). For example, for a genotype \(i\), we could write: