[1] "epoch" "mb" "bh" "bl" "nh" Applications in MANOVA
Multivariate Analysis of Variance
Getting a Visual on the Means?
We note that many of the means seem similar across epochs, but some measurements show more variation than others. This motivates a MANOVA to formally test for differences.
means_long <- means |>
pivot_longer(
cols = c(mb, bh, bl, nh),
names_to = "measurement",
values_to = "mean"
)
ggplot(means_long, aes(x = epoch, y = mean, group = 1, color = epoch)) +
geom_line() +
geom_point(size = 3) +
facet_wrap(~measurement) +
labs(
x = "Epoch",
y = "Mean measurement",
title = "Mean skull measurements across epochs" ) +
theme_minimal()
Getting a Visual on the Means Plot
Visualization
Mean Patterns Across Epochs
group_by(epoch)andsummarise()calculate each measurement’s mean within each epoch.pivot_longer()reshapes the summary forggplot(), which displays measurements side by side.- Compare patterns, but do not infer statistical significance from the plot alone.
# Calculate means by epoch
means_by_epoch <- skulls %>%
group_by(epoch) %>%
summarise(across(where(is.numeric), mean, na.rm = TRUE))
# Reshape for plotting
means_long <- means_by_epoch %>%
pivot_longer(cols = -epoch, names_to = "measurement", values_to = "mean_value")
# Create a vibrant, colorful plot
ggplot(means_long, aes(x = epoch, y = mean_value, fill = measurement)) +
geom_bar(stat = "identity", position = "dodge", alpha = 0.8) +
scale_fill_manual(values = c("mb" = "#FF6B6B", "bh" = "#4ECDC4",
"bl" = "#45B7D1", "nh" = "#96CEB4"),
labels = c("mb" = "Max Breadth ", "bh" = "Height ",
"bl" = "Length ", "nh" = "Nasal Height ")) +
labs(title = " Mean Skull Measurements by Epoch",
subtitle = "Colorful visualization of Egyptian skull evolution over time ",
x = " Historical Epoch",
y = " Mean Value (mm)",
fill = " Measurements") +
theme_minimal() +
theme(
axis.text.x = element_text(angle = 45, hjust = 1, size = 10),
plot.title = element_text(size = 16, hjust = 0.5, color = "#2C3E50"),
plot.subtitle = element_text(size = 12, hjust = 0.5, color = "#7F8C8D"),
legend.position = "bottom",
panel.grid.major = element_line(color = "#BDC3C7"),
panel.background = element_rect(fill = "#FFFFFF")
)
Visualization
Mean Patterns Across Epochs plot
Advanced Visualization
Correlation Heatmap, also known as a Correlation Matrix
# Create correlation matrix for skull measurements
skull_cor <- skulls %>%
select(mb, bh, bl, nh) %>%
cor()
# Convert to long format for ggplot
library(reshape2)
skull_cor_long <- melt(skull_cor)
# Create a stunning, colorful heatmap
ggplot(skull_cor_long, aes(Var1, Var2, fill = value)) +
geom_tile(color = "white", size = 1) +
geom_text(aes(label = paste0(round(value, 2),
ifelse(value > 0.6, " ",
ifelse(value > 0.4, " ", " ")))),
color = "white", size = 5, fontface = "bold") +
scale_fill_gradient2(low = "#3498DB", high = "#E74C3C", mid = "#9B59B6",
midpoint = 0.5, limit = c(0,1),
name = " Correlation\nStrength") +
labs(title = "Correlation Matrix of Skull Measurements ",
subtitle = "Exploring relationships between variables",
x = "Variables", y = "Variables") +
theme_minimal() +
theme(
plot.title = element_text(size = 16, hjust = 0.5, color = "#2C3E50"),
plot.subtitle = element_text(size = 12, hjust = 0.5, color = "#7F8C8D"),
axis.text = element_text(size = 12, color = "#2C3E50"),
legend.title = element_text(size = 10),
panel.grid = element_blank(),
axis.ticks = element_blank()
) +
coord_fixed()
Advanced Visualization
Correlation Heatmap, also known as a Correlation Matrix
Practical Exercise: Group Means Visualization
# Create vibrant boxplots for each measurement by epoch
skulls_long <- skulls %>%
pivot_longer(cols = c(mb, bh, bl, nh),
names_to = "measurement",
values_to = "value") %>%
mutate(measurement = factor(measurement,
labels = c("bh" = "Height ", "bl" = "Length ",
"mb" = "Max Breadth ", "nh" = "Nasal Height ")))
ggplot(skulls_long, aes(x = epoch, y = value, fill = epoch)) +
geom_boxplot(alpha = 0.8, outlier.color = "red", outlier.size = 2) +
geom_jitter(width = 0.3, alpha = 0.4, size = 0.8, color = "darkblue") +
facet_wrap(~ measurement, scales = "free_y", ncol = 2) +
scale_fill_manual(values = c("c4000BC" = "#FF6B6B", "c3300BC" = "#4ECDC4",
"c1850BC" = "#45B7D1", "c200BC" = "#96CEB4",
"cAD150" = "#FECA57")) +
labs(title = " Distribution of Skull Measurements by Epoch ",
subtitle = "Colorful exploration of measurement variations across time ",
x = " Historical Epoch",
y = " Measurement Value (mm)",
caption = " Boxplots show quartiles, dots show individual measurements") +
theme_minimal() +
theme(
axis.text.x = element_text(angle = 45, hjust = 1, size = 9),
plot.title = element_text(size = 16, hjust = 0.5, color = "#2C3E50"),
plot.subtitle = element_text(size = 12, hjust = 0.5, color = "#7F8C8D"),
plot.caption = element_text(size = 10, color = "#95A5A6"),
legend.position = "none",
strip.text = element_text(size = 12, face = "bold", color = "#34495E"),
panel.grid.minor = element_blank(),
panel.background = element_rect(fill = "#FAFAFA")
)
Practical Exercise: Group Means Visualization Plot
Quick Challenge 1
In
manova(cbind(mb, bh, bl, nh) ~ epoch, data = skulls),
what does cbind() specify?
Quick Challenge 2
A significant Pillai test for the skulls model supports which conclusion?
Quick Challenge 3
What is the purpose of
summary.aov(skull_fit) after a significant MANOVA?
Quick Challenge 4
What is the primary advantage of using MANOVA over multiple univariate ANOVAs?