---
title: "Theory and notation"
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{Theory and notation}
  %\VignetteEngine{quarto::html}
  %\VignetteEncoding{UTF-8}
execute:
  echo: true
  warning: false
  message: false
---

{{< include _DiallelAnalysisR-macros.txt >}}

```{r setup, include = FALSE}
knitr::opts_chunk$set(collapse = FALSE, comment = "", fig.width = 7, fig.height = 4.2,
                      dev = if (knitr::is_latex_output()) "cairo_pdf" else
                        if (requireNamespace("svglite", quietly = TRUE)) "svglite" else "png")
if (requireNamespace("DiallelAnalysisR", quietly = TRUE)) library(DiallelAnalysisR) else
  pkgload::load_all(export_all = FALSE, helpers = FALSE, quiet = TRUE)
```

# Abstract

The models behind the analyses, the notation used in the help pages and vignettes, and the expected mean squares
that turn mean squares into variance components.

# Notation

Scalars are upper-case letters in normal face, indices lower-case italic, vectors lower-case bold and matrices
upper-case bold; elements of vectors and matrices keep their letter in normal face.

```{r notation}
nt <-
  da_notation(
      format = c("table", "path", "tex")[1]
    )
nt[, symbol := paste0("$", symbol, "$")]
nt[, argument := ifelse(test = nzchar(argument), yes = paste0("`", argument, "`"), no = "")]
knitr::kable(x = nt[, .(symbol, kind, meaning, argument)])
```

# The diallel model

With $P$ parents in $B$ replicates, the mean of cross $i \times j$ (female $i$, male $j$) is
$$
Y_{ij} = \mu + g_i + g_j + S_{ij} + R_{ij} + \bar{e}_{ij}, \qquad S_{ij} = S_{ji}, \quad R_{ij} = -R_{ji},
$$
where $\vg$ holds the general combining ability (gca) effects of the parents, $\mS$ the specific combining ability
(sca) effects of the crosses and $\mR$ the reciprocal effects (Griffing 1956). The error of a cell mean has variance
$\sigma^2_m = \sigma^2/B$, estimated by $MS_E/B$ from the block analysis of the plot data.

# Least squares

The vector of cell means splits into orthogonal parts: the mean, the gca part, the sca part and the reciprocal part.
The sums of squares of the analysis of variance are the squared lengths of these parts, and Griffing's formulas are
their closed forms. The package computes the closed forms and its tests check them against the explicit projection
for every method and several numbers of parents (vignette 13).

# Fixed and random models

In the fixed model the effects are constants and are estimated with their standard errors. In the random model
$g_i$, $S_{ij}$ and $R_{ij}$ are independent with variances $\sigma^2_g$, $\sigma^2_s$, $\sigma^2_r$ and the mean
squares have expectations (Method 1)
$$
\E(MS_g) = \sigma^2_m + \frac{2(P-1)}{P}\sigma^2_s + 2P\sigma^2_g, \quad
\E(MS_s) = \sigma^2_m + \frac{2(P^2-P+1)}{P^2}\sigma^2_s, \quad
\E(MS_r) = \sigma^2_m + 2\sigma^2_r .
$$
Solving these equations with the observed mean squares gives the variance components; for non-inbred parents without
epistasis $\sigma^2_A = 2\sigma^2_g$ and $\sigma^2_D = \sigma^2_s$.

```{r ems}
P <- 8
rbind(
    Method1 = c(gca = 2 * P, sca_in_gca = 2 * (P - 1) / P, sca = 2 * (P^2 - P + 1) / P^2)
  , Method2 = c(P + 2, 1, 1)
  , Method3 = c(2 * (P - 2), 2, 2)
  , Method4 = c(P - 2, 1, 1)
  )
```

# Hayman's model

Hayman (1954) works with the arrays of the diallel table (the progeny of one parent) after averaging the reciprocal
crosses. The variance $V_r$ of array $r$ and its covariance $W_r$ with the parents lie, under the additive-dominance
model, on a line of unit slope; the components $D$ (additive), $H_1$ and $H_2$ (dominance), $F$ and $h^2$ follow from
the variances and covariances of the arrays (vignette 06).

# Summary

All analyses rest on the same table of cell means and the same error mean square; the fixed model estimates effects,
the random model variance components, and Hayman's approach the components of gene action.

# References

Griffing, B. (1956). Concept of general and specific combining ability in relation to diallel crossing systems.
*Australian Journal of Biological Sciences*, 9(4), 463--493.

Hayman, B. I. (1954). The theory and analysis of diallel crosses. *Genetics*, 39(6), 789--809.

Hayman, B. I. (1954). The analysis of variance of diallel tables. *Biometrics*, 10(2), 235--244.

Mather, K. and Jinks, J. L. (1982). *Biometrical Genetics*, 3rd edition. Chapman and Hall, London.