---
title: "Heterosis, maternal effects and Gardner and Eberhart"
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{Heterosis, maternal effects and Gardner and Eberhart}
  %\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

When the parents are grown beside their crosses, the crosses can be compared with the parents: the heterosis of each
cross, and Gardner and Eberhart's split of the crosses into variety effects and heterosis. In a diallel with
reciprocals, the reciprocal effects split into maternal and non-maternal parts. This vignette takes these three in
turn.

# Heterosis

With the parents grown beside the crosses, `Heterosis()` gives for each cross the heterosis over the mid-parent
$MP = (P_i + P_j)/2$, the better parent and, if a check variety is named, the standard variety, in percent, each with a
$t$ test. With $B$ replicates the standard errors of the differences are $\sqrt{3MS_E/(2B)}$ for
$F_{ij} - MP$ and $\sqrt{2MS_E/B}$ for the other two.

```{r het}
het <-
  Heterosis(
      y      = Yield
    , Rep    = Rep
    , Cross1 = Cross1
    , Cross2 = Cross2
    , data   = GriffingData2
    , Check  = 3
    , Better = c("higher", "lower")[1]
    )
het$StdErr
head(x = het$Heterosis[order(-het$Heterosis$MP)], n = 6)
```

```{r het-plot, fig.height = 6}
plot(x = het, which = c("MP", "BP", "SH", "residuals", "qq")[1])
```

For traits where a lower value is better (days to flowering, disease score), `Better = "lower"` takes the lower
parent as the better one.

# Maternal and non-maternal reciprocal effects

Cockerham and Weir (1977) write the reciprocal effect of Methods 1 and 3 as $r_{ij} = m_i - m_j + n_{ij}$: a maternal
effect of each parent and a non-maternal remainder. `Griffing(..., Reciprocal = "split")` splits the reciprocal line
of the analysis of variance accordingly, on $P - 1$ and $(P - 1)(P - 2)/2$ degrees of freedom:

```{r recip}
rs <-
  Griffing(
      y          = Yield
    , Rep        = Rep
    , Cross1     = Cross1
    , Cross2     = Cross2
    , data       = GriffingData1
    , Method     = 1
    , Model      = 1
    , Reciprocal = c("combined", "split")[2]
    )
rs$ANOVA
round(x = rs$Maternal, digits = 3)
round(x = rs$StdErr[c("mi", "mi_mj", "nij")], digits = 3)
```

Here neither part is significant: the reciprocal differences of this diallel are small and show no maternal pattern.

# Gardner and Eberhart's Analyses II and III

For parents and one set of crosses, Gardner and Eberhart (1966) model the parents by variety effects $v_j$ and the
crosses by the mean of their parents plus heterosis. Analysis II splits the heterosis into average heterosis
$\bar{h}$, variety heterosis $h_j$ and specific heterosis $s_{jj'}$:

```{r ge2}
ge2 <-
  GardnerEberhart(
      y        = Yield
    , Rep      = Rep
    , Cross1   = Cross1
    , Cross2   = Cross2
    , data     = GriffingData2
    , Analysis = 2
    , Model    = 1
    )
ge2$ANOVA
ge2$Average
```

```{r ge2-plot}
plot(x = ge2, which = c("effects", "specific", "residuals", "qq")[1])
```

Analysis III keeps the variety effects for the parents and analyses the crosses by general and specific combining
ability, with average heterosis between the two; its gca and sca lines are Griffing's Method 4:

```{r ge3}
ge3 <-
  GardnerEberhart(
      y        = Yield
    , Rep      = Rep
    , Cross1   = Cross1
    , Cross2   = Cross2
    , data     = GriffingData2
    , Analysis = 3
    , Model    = 1
    )
ge3$ANOVA
```

# Summary

`Heterosis()` compares each cross with its parents and a check; `Griffing(..., Reciprocal = "split")` asks whether
the reciprocal differences come from the mother; `GardnerEberhart()` describes parents and crosses with one model of
variety effects and heterosis.

# References

Cockerham, C. C. and Weir, B. S. (1977). Quadratic analyses of reciprocal crosses. *Biometrics*, 33(1), 187--203.

Gardner, C. O. and Eberhart, S. A. (1966). Analysis and interpretation of the variety cross diallel and related
populations. *Biometrics*, 22(3), 439--452.

Murray, L. W., Ray, I. M., Dong, H. and Segovia-Lerma, A. (2003). Clarification and reevaluation of population-based
diallel analyses: Gardner and Eberhart Analyses II and III revisited. *Crop Science*, 43(6), 1930--1937.

Singh, R. K. and Chaudhary, B. D. (2004). *Biometrical Methods in Quantitative Genetic Analysis*. Kalyani Publishers,
New Delhi.