forced_oneway_anova.qmd

---
title: "Forced one-way ANOVA"
author: "Clay Ford"
format:
  html:
    embed-resources: true

---


## Generate data with an interaction

Notice there are two factor variables: `x1` and `x2`. But also notice I use the `interaction()` function to create a single variable called `x12`.

```{r}
n <- 200
x1 <- sample(x = letters[1:2], size = n, replace = TRUE)
x2 <- sample(x = LETTERS[4:5], size = n, replace = TRUE)
y <- 10 + (x1=="b")*3 + (x2=="E")*4 + (x1=="b")*(x2=="E")*-4 + 
  rnorm(n = n, mean = 0, sd = 3)
d <- data.frame(y, x1, x2, x12 = interaction(x1, x2)) 
# show a few rows
d[sample(200, size = 10),]
```

## Fit model using main effects and interaction

Notice the last line of output, the omnibus F test. Null is all coefficients (except intercept) are 0.

```{r}
m <- lm(y ~ x1*x2, data = d)
summary(m)
```

## Fit model using only the x12 variable

This is basically a one-way ANOVA using a variable that is an interaction of two factor variables. Notice the last line of output, the omnibus F test, is equivalent to the previous model.

```{r}
m2 <- lm(y ~ x12, data = d)
summary(m2)
```

## pairwise comparisons

Posthoc pairwise comparisons are identical for both models.

```{r}
library(emmeans)
emmeans(m, specs = c("x1", "x2")) |>
  contrast("pairwise")
emmeans(m2, specs = c("x12")) |>
  contrast("pairwise")
```

## Non-parametric Krusak-Wallis

We can use this approach with Kruskal-Wallis:

```{r}
kruskal.test(y ~ x12, data = d)
```