Bottom Line Up Front
It’s called a Two-way ANOVA, because we are exploring two independent factor variables difference in means (variance) against a single response variable.
If an interaction.plot shows crossing lines for a
two-way analysis, it is likely to have an effect on the
response in concert.
Overview
In the original
post on this the author walks through how to create an interaction
plot using ANOVA in R. I wanted to re-post this as it falls under
statistical and optimization concerns in the case of
n variables and to curate this information accordingly.
“To find out if the means of three or more independent groups that have been divided based on two factors differ, a two-way ANOVA is performed.
When we want to determine whether two distinct factors have an impact on a certain response variable, we employ a two-way ANOVA.”
Data
set.seed(123)
# Two independent variables: Factors
# One response variable: Numeric
#
# 60 observations, 30 for gender duplication, 20 for exercise duplication.
#
data <- data.frame(gender = rep(c("Male", "Female"),
each = 30, times = 1),
exercise = rep(c("None", "Light", "Intense"),
each = 10, times = 2),
weight_loss = c(runif(10, -3, 3),
runif(10, 0, 5),
runif(10, 5, 9),
runif(10, -4, 2),
runif(10, 0, 3),
runif(10, 3, 8)))
# Sample from our data
head(data) gender exercise weight_loss
1 Male None -1.2745349
2 Male None 1.7298308
3 Male None -0.5461385
4 Male None 2.2981044
5 Male None 2.6428037
6 Male None -2.7266610e.grid = expand.grid(gender = c("Male", "Female"),
exercise = c("None", "Light", "Intense"))
# All independent factor combinations
e.grid gender exercise
1 Male None
2 Female None
3 Male Light
4 Female Light
5 Male Intense
6 Female Intense# Number of groups
nrow(e.grid)[1] 6Fit ANOVA
For our experiment, we are interested in how gender and exercise impact weight loss.
“Let’s say researchers want to know if gender and activity volume affect weight loss.
They enlist 30 men and 30 women to take part in an experiment where 10 of each gender are randomly assigned to follow a program of either no activity, light exercise, or severe exercise for one month in order to test this.
To see the interaction impact between exercise and gender, use the following procedures to generate a data frame in R, run a two-way ANOVA, and create an interactive graphic.”
Df Sum Sq Mean Sq F value Pr(>F)
gender 1 43.7 43.72 19.032 5.83e-05 ***
exercise 2 438.9 219.43 95.515 < 2e-16 ***
gender:exercise 2 2.9 1.46 0.634 0.535
Residuals 54 124.1 2.30
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1At a glance, we can see that independently, gender or
exercise are significant, that is one or the other.
However, together there is no significant variance in weight loss. So we
can see that gender as an individual factor is significant toward weight
loss (up or down, will require regression or interaction plots). Same
goes for exericse. However together, the influence is not strong, that
is they don’t together influence weight loss
significantly. Another way to think about it is that their product
doesn’t offer any additional information. Let’s take a look at the
interaction plot to see if this can validate the things the model
discovered.
ANOVA Assumptions
This is just a little gotcha when using ANOVA.
- *Independence of variables**: The two variables for testing should be independent of each other. One should not affect the other, or else it could result in skewness . This means that one cannot use the two-way ANOVA test in settings with categorical variables.
- Homoscedasticity: In a two-way ANOVA test, the variance should be homogeneous. The variation around the mean for each set of data should not vary significantly for all the groups.
- Normal distribution of variables: The two variables in a two-way ANOVA test should have a normal distribution . When plotted individually, each should have a bell curve . If the data does not meet this criterion, one could attempt statistical data transformation to achieve the desired result.
library(ggplot2)
ggplot(data = data) +
geom_histogram(aes(x = weight_loss, fill=gender)) +
facet_grid(gender ~ exercise, scales="free") +
xlab("Weight Loss (Centered and Scaled)") +
ylab("Count") + ggtitle("Glance for Weight Loss Normality") +
theme_bw()
Interaction Plot
Now let’s take a look and see
“An interaction plot is the most effective tool for spotting and comprehending the effects of interactions between two variables.
This particular plot style shows the values of the first factor on the x-axis and the fitted values of the response variable on the y-axis.
The values of the second component of interest are depicted by the lines in the plot.”
interaction.plot(x.factor = data$exercise, #x-axis variable
trace.factor = data$gender, #variable for lines
response = data$weight_loss, #y-axis variable
fun = median, #metric to plot
main = "Weight Loss Two-way ANOVA Interaction",
ylab = "Weight Loss",
xlab = "Exercise Intensity",
col = c("pink", "blue"),
lty = 1, #line type
lwd = 2, #line width
trace.label = "Gender"
)
“In general, there is no interaction effect if the two lines on the interaction plot are parallel. However, there will probably be an interaction effect if the lines cross.
As we can see in this plot, there is no intersection between the lines for men and women, suggesting that there is no interaction between the variables of exercise intensity and gender.
This is consistent with the p-value in the ANOVA table’s output, which indicated that the interaction term in the ANOVA model was not statistically significant.”
Referneces
https://www.r-bloggers.com/2022/07/how-to-create-an-interaction-plot-in-r/