```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE)
```

### Applied design and analysis of an RCBD   

An experiment was run in South Australia to compare for 107 different wheat varieties, in a randomized complete block design with 3 blocks.

```{r eval=TRUE, echo=FALSE}
t_rcbd <- data.frame(Source = c("Blocks", "Genotype", "Error", "Total"),
                          df = c("b-1 = 2", "g-1 = 106", "N-b-g+1 = 221", "N-1 = 329"))

knitr::kable(t_rcbd, caption = "ANOVA table for the RCBD")
```


```{r message=FALSE, warning=FALSE}
library(tidyverse)
library(agridat)
library(lme4)
library(emmeans)

dat_wheat_rcbd <- agridat::gilmour.serpentine

dat_wheat_rcbd %>% 
  ggplot(aes(col, row))+
  geom_tile(aes(fill = rep))
```

```{r}
# block as fixed
m_block.as.fixed <- lm(yield ~ gen + rep, data = dat_wheat_rcbd)

# block as random
m_block.as.random <- lme4::lmer(yield ~ gen + (1|rep), data = dat_wheat_rcbd)
```

```{r}
summary(m_block.as.fixed)
```

```{r}
sigma(m_block.as.fixed)^2
```

```{r}
summary(m_block.as.random)
```


```{r}
sigma2.fixed <- sigma(m_block.as.fixed)^2
sigma2.random <- sigma(m_block.as.random)^2
sigma2_b.random <- as.data.frame(VarCorr(m_block.as.random))[1, 'vcov']
b <- 3
```

```{r}
# se mean for the fixed model
sqrt(sigma2.fixed/b)
```

```{r}
# se mean for the random model
sqrt((sigma2.random+sigma2_b.random)/b)
```


```{r}
emmeans(m_block.as.fixed, ~gen)[1:10,]
emmeans(m_block.as.random, ~gen)[1:10,]

emmeans(m_block.as.fixed, ~gen, contr = list(c(1, -1, rep(0, n_distinct(dat_wheat_rcbd$gen)-2))))$contr
emmeans(m_block.as.random, ~gen, contr = list(c(1, -1, rep(0, n_distinct(dat_wheat_rcbd$gen)-2))))$contr
```

```{r}
sqrt(2*sigma2.fixed/b)
```
```{r}
sqrt(2*sigma2.random/b)

```