---
title: "Line by tester and North Carolina designs"
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

Not every crossing scheme is a diallel. In a line $\times$ tester experiment each of $L$ lines is crossed with each of
$T$ testers; the North Carolina designs of Comstock and Robinson estimate additive and dominance variance from
half-sib and full-sib families. Both are analysed from the plot data of a randomized complete block experiment, with
the same result objects, plots and tables as the diallel analyses.

# Line $\times$ tester analysis

`LineTester()` gives Kempthorne's analysis of $L$ lines crossed with $T$ testers: the split of the crosses into lines,
testers and their interaction with the proportional contribution of each, the gca and sca effects with standard
errors, and the genetic variances; `Inbreeding` is the inbreeding coefficient $F$ of the parents.

```{r lt}
lt <-
  LineTester(
      y          = Yield
    , Rep        = Rep
    , Line       = Line
    , Tester     = Tester
    , data       = LineTesterData
    , Inbreeding = 1
    )
lt
round(x = lt$GCA.Lines, digits = 2)
round(x = lt$StdErr, digits = 3)
```

```{r lt-plot}
plot(x = lt, which = c("gca", "sca", "contribution", "biplot", "residuals", "qq")[1])
```

# North Carolina designs

`NCDesign()` analyses the three designs of Comstock and Robinson: females nested in males (Design I), males crossed
with females (Design II), and $F_2$ males backcrossed to both inbred parents (Design III). Each gives the additive and
dominance variances and the average degree of dominance $\sqrt{2\sigma^2_D/\sigma^2_A}$.

```{r nc}
nc <- lapply(X = 1:3, FUN = function(k) {
  NCDesign(
      y          = Yield
    , Rep        = Rep
    , Male       = Male
    , Female     = Female
    , data       = NCData[NCData$Design == k, ]
    , Design     = k
    , Inbreeding = 0
    )
})
nc[[2]]$ANOVA
round(
    x      = sapply(
        X   = nc
      , FUN = function(z) z$Genetic.Components[c("Additive", "Dominance", "Degree of dominance"),
                                                                       "Components"]
      )
  , digits = 3
  )
```

# Summary

`LineTester()` gives combining ability and the contribution of lines, testers and their interaction; `NCDesign()`
gives the additive and dominance variances of Designs I, II and III.

# References

Comstock, R. E. and Robinson, H. F. (1948). The components of genetic variance in populations of biparental
progenies and their use in estimating the average degree of dominance. *Biometrics*, 4(4), 254--266.

Kempthorne, O. (1957). *An Introduction to Genetic Statistics*. John Wiley and Sons, New York.

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