# A Complete Guide: The 2×4 Factorial Design

A 2×4 factorial design is a type of experimental design that allows researchers to understand the effects of two independent variables on a single dependent variable.

In this type of design, one independent variable has two levels and the other independent variable has four levels. For example, suppose a botanist wants to understand the effects of sunlight (none vs. low vs. medium vs. high) and watering frequency (daily vs. weekly) on the growth of a certain species of plant. This is an example of a 2×4 factorial design because there are two independent variables, one having two levels and the other having four levels:

• Independent variable #1: Sunlight
• Levels: None, Low, Medium, High
• Independent variable #2: Watering Frequency
• Levels: Daily, Weekly

And there is one dependent variable: Plant growth.

## The Purpose of a 2×4 Factorial Design

A 2×4 factorial design allows you to analyze the following effects:

Main Effects: These are the effects that just one independent variable has on the dependent variable.

For example, in our previous scenario we could analyze the following main effects:

• Main effect of sunlight on plant growth.
• Mean growth of all plants that received no sunlight.
• Mean growth of all plants that received low sunlight.
• Mean growth of all plants that received medium sunlight.
• Mean growth of all plants that received high sunlight.
• Main effect of watering frequency on plant growth.
• Mean growth of all plants that were watered daily.
• Mean growth of all plants that were watered weekly.

Interaction Effects: These occur when the effect that one independent variable has on the dependent variable depends on the level of the other independent variable.

For example, in our previous scenario we could analyze the following interaction effects:

• Does the effect of sunlight on plant growth depend on watering frequency?
• Does the effect of watering frequency on plant growth depend on the amount of sunlight?

## How to Analyze a 2×4 Factorial Design

We can perform a two-way ANOVA to formally test whether or not the independent variables have a statistically significant relationship with the dependent variable.

For example, the following code shows how to perform a two-way ANOVA for our hypothetical plant scenario in R:

```#make this example reproducible
set.seed(0)

#create data
df <- data.frame(sunlight = rep(c('None', 'Low', 'Medium', 'High'), each=10, times=2),
water = rep(c('Daily', 'Weekly'), each=40, times=2),
growth = c(rnorm(10, 8, 2), rnorm(10, 8, 3), rnorm(10, 13, 2),
rnorm(10, 14, 3), rnorm(10, 10, 4), rnorm(10, 12, 3),
rnorm(10, 13, 2), rnorm(10, 14, 4)))

#fit the two-way ANOVA model
model <- aov(growth ~ sunlight * water, data = df)

#view the model output
summary(model)

Df Sum Sq Mean Sq F value   Pr(>F)
sunlight         3  744.1  248.04   34.16  < 2e-16 ***
water            1   43.1   43.05    5.93    0.016 *
sunlight:water   3  195.8   65.27    8.99 1.61e-05 ***
Residuals      152 1103.5    7.26
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1```

Here’s how to interpret the output of the ANOVA:

Main Effect #1 (Sunlight): The p-value associated with sunlight is <2e-16. Since this is less than .05, this means sunlight exposure has a statistically significant effect on plant growth.

Main Effect #2 (Water): The p-value associated with water is .016. Since this is less than .05, this means watering frequency also has a statistically significant effect on plant growth.

Interaction Effect: The p-value for the interaction between sunlight and water is .000061. Since this is less than .05, this means there is an interaction effect between sunlight and water.