--- title: "Regression and variance-based stability" author: "Muhammad Yaseen" date: last-modified format: html: self-contained: true number-sections: true toc: true toc-depth: 3 highlight-style: pygments code-tools: true html-math-method: mathjax vignette: > %\VignetteIndexEntry{Regression and variance-based stability} %\VignetteEngine{quarto::html} %\VignetteEncoding{UTF-8} execute: echo: true warning: false message: false --- {{< include _stability-macros.txt >}} ```{r setup, include = FALSE} knitr::opts_chunk$set(collapse = FALSE, comment = "", fig.width = 7, fig.height = 4.6, dev = if (knitr::is_latex_output()) "cairo_pdf" else if (requireNamespace("svglite", quietly = TRUE)) "svglite" else "png") if (requireNamespace("stability", quietly = TRUE)) library(stability) else pkgload::load_all(export_all = FALSE, helpers = FALSE, quiet = TRUE) ``` # Abstract Finlay and Wilkinson's regression, the variance measures, Shukla's tests with Kang's yield-stability statistic, Kang's rank-sum and Fox's top-third count. # Regression on the environment index ```{r reg} reg <- stab_reg( .data = ge_data , .y = Yield , .rep = Rep , .gen = Gen , .env = Env ) plot(x = reg, which = "lines") ``` The red line has slope 1. Lines steeper than it belong to genotypes that profit more than the average from good environments. # Variance measures ```{r measures} sm <- stab_measures( .data = ge_data , .y = Yield , .gen = Gen , .env = Env ) head(x = sm$StabMeasures[order(Ecov)], n = 6) plot(x = sm, which = "ecov") ``` ```{r measures-int} stab_interpret( x = sm , higher_better = TRUE , top = 5 ) ``` # Shukla's tests and Kang's yield-stability statistic ```{r par} sp <- stab_par( .data = ge_data , .y = Yield , .rep = Rep , .gen = Gen , .env = Env , alpha = 0.1 , .envCov = NULL ) sp$ANOVA head(x = sp$SimultSel, n = 10) plot(x = sp, which = "ys") ``` ```{r par-int} stab_interpret( x = sp , higher_better = TRUE , top = 5 ) ``` # Rank methods ```{r ranks} kg <- stab_kang( .data = ge_data , .y = Yield , .rep = Rep , .gen = Gen , .env = Env ) head(x = kg$Kang, n = 6) fx <- stab_fox( .data = ge_data , .y = Yield , .rep = Rep , .gen = Gen , .env = Env ) head(x = fx$Fox, n = 6) ``` # Safety-first and superiority Eskridge (1990) treats the choice of a cultivar as a decision under risk: the breeder accepts a small probability $\alpha$ of a disastrously low yield and prefers the genotype with the largest lower confidence limit, $$\ybar_{i\cdot} - Z_{1-\alpha}\,\sigma_i,$$ where $\sigma_i$ is the standard deviation of the genotype's yield under one of four definitions of stability. EV uses the variance over environments, FW the departure of Finlay and Wilkinson's slope from 1, SH Shukla's stability variance, and ER the slope with Eberhart and Russell's deviations. A small $\alpha$ weighs stability heavily, and $\alpha = 0.5$ gives the mean. ```{r eskridge} es <- stab_eskridge( .data = ge_data , .y = Yield , .rep = Rep , .gen = Gen , .env = Env , alpha = 0.05 ) head(x = es$Eskridge, n = 6) round(x = es$RankCor, digits = 2) plot(x = es, which = "ranks") stab_interpret( x = es , higher_better = TRUE , top = 5 ) ``` Annicchiarico's (1992) reliability index is the same idea on the relative scale (percent of the environment means), so that poor and rich environments weigh alike. Lin and Binns' (1988) superiority measure is the mean squared distance from the best genotype of every environment. ```{r reliability} rl <- stab_reliability( .data = ge_data , .y = Yield , .rep = Rep , .gen = Gen , .env = Env , alpha = 0.25 , higher_better = TRUE ) head(x = rl$Reliability, n = 6) su <- stab_superiority( .data = ge_data , .y = Yield , .rep = Rep , .gen = Gen , .env = Env , higher_better = TRUE ) head(x = su$Superiority, n = 6) plot(x = su) ``` # Nonparametric stability Huehn's (1979) $S^{(1)}$ and $S^{(2)}$ use only the ranks of the genotypes in each environment, after the genotype effect is removed. A stable genotype keeps its rank. Nassar and Huehn (1987) give tests for each genotype and for the whole set. ```{r huehn} hu <- stab_huehn( .data = ge_data , .y = Yield , .rep = Rep , .gen = Gen , .env = Env ) head(x = hu$Huehn, n = 6) hu$Tests stab_interpret( x = hu , higher_better = TRUE , top = 5 ) ``` # Summary The regression describes how genotypes respond, and the variance measures how much interaction each carries. Kang's statistics turn yield and stability into one selection rule. Eskridge's indices make the weight given to stability explicit through the risk $\alpha$ the grower accepts. # References Finlay, K. W. and Wilkinson, G. N. (1963). The analysis of adaptation in a plant-breeding programme. *Australian Journal of Agricultural Research*, 14(6), 742--754. Eberhart, S. A. and Russell, W. A. (1966). Stability parameters for comparing varieties. *Crop Science*, 6(1), 36--40. Wricke, G. (1962). Uber eine Methode zur Erfassung der okologischen Streubreite in Feldversuchen. *Zeitschrift fur Pflanzenzuchtung*, 47, 92--96. Shukla, G. K. (1972). Some statistical aspects of partitioning genotype-environmental components of variability. *Heredity*, 29(2), 237--245. Kang, M. S. and Pham, H. N. (1991). Simultaneous selection for high yielding and stable crop genotypes. *Agronomy Journal*, 83(1), 161--165. Kang, M. S. (1988). A rank-sum method for selecting high-yielding, stable corn genotypes. *Cereal Research Communications*, 16(1--2), 113--115. Eskridge, K. M. (1990). Selection of stable cultivars using a safety-first rule. *Crop Science*, 30(2), 369--374. Annicchiarico, P. (1992). Cultivar adaptation and recommendation from alfalfa trials in Northern Italy. *Journal of Genetics and Breeding*, 46, 269--278. Lin, C. S. and Binns, M. R. (1988). A superiority measure of cultivar performance for cultivar x location data. *Canadian Journal of Plant Science*, 68(1), 193--198. Huehn, M. (1979). Beitrage zur Erfassung der phanotypischen Stabilitat I. *EDV in Medizin und Biologie*, 10, 112--117. Nassar, R. and Huehn, M. (1987). Studies on estimation of phenotypic stability: tests of significance for nonparametric measures of phenotypic stability. *Biometrics*, 43(1), 45--53. Fox, P. N., Skovmand, B., Thompson, B. K., Braun, H. J. and Cormier, R. (1990). Yield and adaptation of hexaploid spring triticale. *Euphytica*, 47(1), 57--64.