---
title: "Griffing's methods with fixed effects"
author: "Muhammad Yaseen"
date: last-modified
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---

{{< 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

Griffing's four methods with the fixed-effects model (Model 1): the analysis of variance, the gca, sca and
reciprocal effects, their standard errors and how to compare parents and crosses.

# The four methods on one data set

The data sets for Methods 2 to 4 are subsets of the complete diallel, so the methods can be compared directly.

```{r methods}
fits <-
  lapply(
      X   = 1:4
    , FUN = function(m) Griffing(
          y          = Yield
        , Rep        = Rep
        , Cross1     = Cross1
        , Cross2     = Cross2
        , data       = get(paste0("GriffingData", m))
        , Method     = m
        , Model      = 1
        , Reciprocal = c("combined", "split")[1]
        )
    )
names(fits) <- paste("Method", 1:4)
do.call(
    what = rbind
  , args = lapply(X = fits, FUN = function(f) f$ANOVA[c("gca", "sca"), "Mean Sq"])
  )
```

# Method 1: parents, $F_1$ and reciprocals

With $a_i = Y_{i\cdot} + Y_{\cdot i}$ and $Y_{\cdot\cdot}$ the grand total,
$$
SS_g = \frac{1}{2P}\sum_i a_i^2 - \frac{2}{P^2}Y_{\cdot\cdot}^2, \qquad
\hat{g}_i = \frac{a_i}{2P} - \frac{Y_{\cdot\cdot}}{P^2}, \qquad
\hat{R}_{ij} = \frac{Y_{ij} - Y_{ji}}{2} .
$$

```{r m1}
f1 <- fits[["Method 1"]]
knitr::kable(x = f1$ANOVA, digits = 3)
knitr::kable(x = f1$Genetic.Components, digits = 4)
```

The table `Effects` keeps the layout of earlier versions: gca effects on the diagonal, sca effects above it and
reciprocal effects below it. The full matrices are in `SCA` and `Reciprocal`:

```{r m1-effects}
round(x = f1$Effects, digits = 2)
round(x = f1$SCA[1:4, 1:4], digits = 2)
```

# Standard errors and comparisons

`StdErr` gives the standard errors of single effects and of differences between effects; for Method 1
$\Var(\hat{g}_i) = (P-1)\sigma^2_m/(2P^2)$ and $\Var(\hat{g}_i - \hat{g}_j) = \sigma^2_m/P$.

```{r se}
round(x = f1$StdErr, digits = 3)
t_gca <- f1$GCA / f1$StdErr[["gi"]]
round(x = t_gca, digits = 2)
```

A parent's gca effect is significant at the 5% level when $|t|$ exceeds the $t$ quantile on the error degrees of
freedom:

```{r tcrit}
qt(p = 0.975, df = f1$ANOVA["Error", "Df"])
```

# Predicting crosses from general combining ability

Under the fixed model a cross is predicted by $\hat{\mu} + \hat{g}_i + \hat{g}_j$ from the gca effects alone (the
prediction for a cross not yet made) or by adding $\hat{S}_{ij} + \hat{R}_{ij}$ (which reproduces the observed
mean). The gap between the two is the specific combining ability of the cross:

```{r predict}
pr_g <- predict(object = f1, newdata = NULL, type = c("gca", "full")[1])
pr_f <- predict(object = f1, newdata = NULL, type = c("gca", "full")[2])
pr <- merge(x = pr_g, y = pr_f, by = c("Cross1", "Cross2"), suffixes = c("_gca", "_full"))
head(pr[order(-predicted_full)], 8)
```

# Methods 2, 3 and 4

```{r m234}
knitr::kable(x = fits[["Method 2"]]$ANOVA, digits = 3)
knitr::kable(x = fits[["Method 3"]]$ANOVA, digits = 3)
knitr::kable(x = fits[["Method 4"]]$ANOVA, digits = 3)
round(
    x      = rbind(
        Method1 = fits[[1]]$GCA
      , Method2 = fits[[2]]$GCA
      , Method3 = fits[[3]]$GCA
      , Method4 = fits[[4]]$GCA
      )
  , digits = 2
  )
```

The best general combiners (parents 3 and 7) and the poorest (parent 8) are the same in all four methods; including
the parents themselves (Methods 1, 2) or the reciprocals (Methods 1, 3) changes the size of the effects and the order
of the parents in between.

# Plots

```{r plots}
plot(fits[["Method 2"]])
```

# Summary

Model 1 gives the effects of each parent and cross with their standard errors; the `Effects` table keeps the layout
of earlier versions and `GCA`, `SCA`, `Reciprocal` give the effects in full.

# 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.

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