Data Analysis for Graduate Research – Describing Relationships between variables

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

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?

Watch your margins

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

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

Show the code

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

[1] 0.8712018

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

Strong relationship
Moderate relationship
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 , 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.

Describing Relationships between variables

Bivariate Association between 2 variables

Naming conventions

Types of combinations

C ~ C

Watch your margins

Watch your margins

Q ~ C

Q ~ Q