Describing Relationships between variables

Robin Donatello

2024-09-23

Bivariate Association between 2 variables

Naming conventions

Response	Explanatory
y	x
outcome	predictor
dependent variable	independent variable
	covariate
	feature

Model notation: \(y \sim x\)

Types of combinations

Categorical response and categorical explanatory variable. (C ~ C)
Quantitative response and categorical explanatory variable. (Q ~ C)
Quantitative response and quantitative explanatory variable. (Q ~ Q)

The \(Q\) and \(C\) notation is internal to this class. Not universal.

library(tidyverse); library(ggpubr); library(gtsummary)
library(palmerpenguins);  library(sjPlot)
pen <- penguins

C ~ C

Categorical Response vs Categorical Explanatory

Two way tables (cross-tabs)
Plots

Show the code

table(pen$species, pen$island)

           
            Biscoe Dream Torgersen
  Adelie        44    56        52
  Chinstrap      0    68         0
  Gentoo       124     0         0

Show the code

table(pen$species, pen$island) |> 
  prop.table(margin=2) |> round(2)

           
            Biscoe Dream Torgersen
  Adelie      0.26  0.45      1.00
  Chinstrap   0.00  0.55      0.00
  Gentoo      0.74  0.00      0.00

All of the 52 penguins on Torgersen island are the Adelie species.
74% of penguins on Biscoe island are Gentoo.

Show the code

plot_xtab(pen$island, grp=pen$species, show.total = FALSE)

Do these percents match the table?

Watch your margins

Always double check your work

One of the most common places for a mistake when creating a plot or a table between two categorical variables is by not paying close attention to the choice of denominator. And then also confirming the interpretation matches the table, which matches the plot.

What % of penguins on each island are Gentoo?
What % of Gentoo penguins are on each Island?

Row Percents

Show the code

table(pen$species, pen$island) |> 
  prop.table(margin=1) |> round(2)

           
            Biscoe Dream Torgersen
  Adelie      0.29  0.37      0.34
  Chinstrap   0.00  1.00      0.00
  Gentoo      1.00  0.00      0.00

29% of Adelie penguins are on Biscoe Island.

Column Percents

Show the code

table(pen$species, pen$island) |> 
  prop.table(margin=2) |> round(2)

           
            Biscoe Dream Torgersen
  Adelie      0.26  0.45      1.00
  Chinstrap   0.00  0.55      0.00
  Gentoo      0.74  0.00      0.00

74% of penguins on Biscoe island are Gentoo.

Distribution of Islands within each species

Show the code

pen %>% select(island, species) %>%
  tbl_summary(by = "species")

Characteristic	Adelie, N = 152¹	Chinstrap, N = 68¹	Gentoo, N = 124¹
island
Biscoe	44 (29%)	0 (0%)	124 (100%)
Dream	56 (37%)	68 (100%)	0 (0%)
Torgersen	52 (34%)	0 (0%)	0 (0%)
¹ n (%)

Distribution of Species on each Island

Show the code

pen %>% select(island, species) %>%
  tbl_summary(by = "island")

Characteristic	Biscoe, N = 168¹	Dream, N = 124¹	Torgersen, N = 52¹
species
Adelie	44 (26%)	56 (45%)	52 (100%)
Chinstrap	0 (0%)	68 (55%)	0 (0%)
Gentoo	124 (74%)	0 (0%)	0 (0%)
¹ n (%)

Distribution of islands for each species

Show the code

plot_xtab(pen$species, grp=pen$island, 
          margin = "row", show.total = FALSE)

29% of Adelie penguins are on Biscoe Island.

Distribution of species on each island.

Show the code

plot_xtab(pen$island, grp=pen$species, 
          margin = "row", show.total = FALSE)

74% of penguins on Biscoe island are Gentoo.

Q ~ C

Quantitative Response vs Categorical Explanatory

Grouped summary statistics
gtsummary
Plots
Plot2
Description

n, nean, median, sd, IQR of the quantitative variable for each level of the categorical level.

Show the code

pen %>% group_by(species) %>% 
  summarize(n=n(), 
            mean   = mean(bill_depth_mm, na.rm = TRUE), 
            median = median(bill_depth_mm, na.rm = TRUE), 
            sd     = sd(bill_depth_mm, na.rm = TRUE), 
            IQR    = IQR(bill_depth_mm, na.rm = TRUE))

# A tibble: 3 × 6
  species       n  mean median    sd   IQR
  <fct>     <int> <dbl>  <dbl> <dbl> <dbl>
1 Adelie      152  18.3   18.4 1.22   1.5 
2 Chinstrap    68  18.4   18.4 1.14   1.90
3 Gentoo      124  15.0   15   0.981  1.5

Gentoo penguins have lower average bill depth compared to Adelie or Chinstrap (15.0mm vs 18.3 and 18.4mm respectively). Chinstrap however have a larger IQR at 1.9 compared to 1.5 for the others.

Similarly nice tables, but defaults to median and IQR.

Show the code

pen %>% select(bill_depth_mm, species) %>%
  tbl_summary(by = "species")

Characteristic	Adelie, N = 152¹	Chinstrap, N = 68¹	Gentoo, N = 124¹
bill_depth_mm	18.40 (17.50, 19.00)	18.45 (17.50, 19.40)	15.00 (14.20, 15.70)
Unknown	1	0	1
¹ Median (IQR)

Have to manually specify you want mean & sd.

Show the code

pen %>% select(bill_depth_mm, species) %>%
  tbl_summary(by = "species", 
              statistic = list(all_continuous() ~ "{mean} ({sd})"))

Characteristic	Adelie, N = 152¹	Chinstrap, N = 68¹	Gentoo, N = 124¹
bill_depth_mm	18.35 (1.22)	18.42 (1.14)	14.98 (0.98)
Unknown	1	0	1
¹ Mean (SD)

Overlaid density plots

Show the code

gghistogram(pen, 
    x = "bill_depth_mm", fill = "species", 
    add_density = TRUE, add="mean")

Side by side boxplots

Show the code

ggviolin(pen, 
  x="species", y = "bill_depth_mm", 
  color = "species", add = c("mean", "boxplot"))

The distribution of bill depth are fairly normal for each species, with some higher end values causing a slight right skew for Adelie and Gentoo.

More code but nice alternative to violins.

Show the code

library(ggdist) # for the "half-violin" plot (stat_slab)
ggplot(pen, aes(x=bill_depth_mm, y=species, fill=species)) + 
      stat_slab(alpha=.5, justification = 0) + 
      geom_boxplot(width = .2,  outlier.shape = NA) + 
      geom_jitter(alpha = 0.5, height = 0.05) +
      stat_summary(fun="mean", geom="point", col="red", size=4, pch=17) + 
      theme_bw() + 
      labs(x="Bill depth (mm)", y = "Species") + 
      theme(legend.position = "none")

Discuss shape, spread and center using comparative language.

The distribution of bill depth are fairly normal for each species, with some higher end values causing a slight right skew for Adelie and Gentoo. Gentoo penguins have lower average bill depth compared to Adelie or Chinstrap (15.0mm vs 18.3 and 18.4mm respectively). Chinstrap however have a larger IQR at 1.9 compared to 1.5 for the others.

Q ~ Q

Quantitative Response vs Quantitative Explanatory

Summary statistic
Plots
Description

Show the code

cor(pen$flipper_length_mm, pen$body_mass_g, 
    use="pairwise.complete.obs")

[1] 0.8712018

The correlation coefficient \(r\) is a measure of the strength and direction of a linear relationship between two variables.

\(|r| > 0.6\) Strong relationship
\(0.4 \leq |r| < 0.6\) Moderate relationship
\(|r| < 0.4\) Weak to no relationship

Note: These values are guidelines and subject to variability within different disciplines ref

Show the code

ggscatter(pen, 
  x="flipper_length_mm", y = "body_mass_g")

Show the code

ggscatter(pen, 
  x="flipper_length_mm", y = "body_mass_g", 
  add = "loess", conf.int = TRUE)

The relationship between flipper length and body mass in penguins is relatively linear, but there may be possible clustering on a third variable. There appears to be two groups below and above a flipper length of about 205mm.

Must include the direction (positive or negative), the strength quantified using \(r\), and the form (linear vs non-linear)

The penguin flipper length (mm) has a strong, positive, and reasonably linear correlation with body mass (g), r=0.87.