Two-by-two ANOVA: Power simulation of the interaction test
================
Last modified: 2026-01-09

## Application context

Effect of fertilizers

In an experiment, two fertilizers (A and B, each either low or high
dose) will be combined and the yield of peas (Y) in kg be observed. Goal
is to detect an increase of the Fertilizer-A effect by an additional 12
kg when combined with a high dose of Fertilizer B (interaction effect).

![](powersim-anova_files/figure-gfm/unnamed-chunk-1-1.png)<!-- -->

## Model

Assumptions

- $Y_{ijk} = \mu + \alpha_i + \beta_j + (\alpha\beta)_{ij} +  \varepsilon_{ijk}$
- $\varepsilon_{ijk} \sim N(0, \sigma^2) \text{ i.i.d.}$
- $i = 1, \dots, I$; $j = 1, \dots, J$; $k = 1, \dots, K$
- $\alpha_1 = \beta_1 := 0$

Hypothesis

- H$_0^{AB}\colon~ (\alpha\beta)_{ij} = 0 \text{ for all } i,j$

## Setup

``` r
set.seed(1704)
n <- 96
dat <- data.frame(
  A = factor(rep(1:2, each = n/2), labels = c("low", "high")),
  B = factor(rep(rep(1:2, each = n/4), 2), labels = c("low", "high"))
)
X <- model.matrix(~ A * B, dat)
unique(X)
```

    ##    (Intercept) Ahigh Bhigh Ahigh:Bhigh
    ## 1            1     0     0           0
    ## 25           1     0     1           0
    ## 49           1     1     0           0
    ## 73           1     1     1           1

``` r
beta <- c(mu = 30, a2 = 30, b2 = 5, ab22 = 12)
means <- X %*% beta

lattice::xyplot(I(means + rnorm(n, sd = 10)) ~ A, dat, groups = B,
                type = c("g", "p", "a"), auto.key = TRUE, ylab = "Yield (kg)")
```

![](powersim-anova_files/figure-gfm/unnamed-chunk-2-1.png)<!-- -->

## Parameter recovery

``` r
out <- replicate(2000, {
  y <- means + rnorm(n, sd = 10)   # y = mu + a + b + ab + e
  m <- aov(y ~ A * B, dat)
  c(coef(m), sigma = sigma(m))
})
boxplot(t(out))
```

![](powersim-anova_files/figure-gfm/unnamed-chunk-3-1.png)<!-- -->

## Power simulation

``` r
pval <- replicate(2000, {
  y <- means + rnorm(n, sd = 10)
  m <- aov(y ~ A * B, dat)
  summary(m)[[1]]$"Pr(>F)"[3]      # test of interaction
})
mean(pval < 0.05)
```

    ## [1] 0.817