Fourfold Displays for ‘ggplot2’

Fourfold displays

The fourfold display is a special case of a radial diagram, or “polar area chart”, designed for \(2 \times 2\) (or \(2 \times 2 \times k\)) tables (Fienberg, 1975; Friendly, 1994a, 1994b). The frequency \(n_{ij}\) in each cell is shown by a quarter circle whose radius is proportional to \(\sqrt{n_{ij}}\), so that its area is proportional to the cell count.

Like a pie chart, it uses segments of a circle to show frequencies; unlike a pie chart, it keeps the angles of the segments constant and varies the radius. This is the \(2 \times 2\) cousin of the graphic form used by Florence Nightingale (Nightingale, 1858). Friendly & Andrews (2021) describe the rich history of radial diagrams.

The main purpose of the display is to show the sample odds ratio (\(\hat{\theta}\)). This gives a number that compares the odds of an event (accepted!) happening in one group (men) to the odds of it happening in another group (women).

\[\hat{\theta} = \frac{n_{11} / n_{12}}{n_{21} / n_{22}} = \frac{n_{11} \, n_{22}}{n_{12} \, n_{21}} .\]

An association between the two variables (\(\theta \neq 1\)) appears as a tendency for one pair of diagonally opposite cells to be larger than the other pair. The fourfold plot shows the association and the odds ratio visually as follows:

• Color and a diagonal line show the direction of association: the relatively larger diagonal pair is drawn in blue (more than expected under independence) with a line through the center along that diagonal, and the other pair in red.

• Confidence rings around each quadrant give a visual test of independence, \(H_0: \theta = 1\): in a standardized display, the rings for adjacent quadrants overlap if and only if the observed counts are consistent with the null hypothesis.

Example: Berkeley admissions

The classic example, UCBAdmissions gives applicants to the six largest graduate departments at Berkeley in 1973, classified by admission and gender (Bickel et al., 1975). geom_fourfold() currently works on data in long form: map the two-level variables to x (drawn left to right) and y (drawn top to bottom), and the cell frequencies to weight. Without faceting, rows for the same cell are summed, so this pools over departments:

ucb <- as.data.frame(UCBAdmissions)
berkeley <- ggplot(ucb, aes(x = Admit, y = Gender, weight = Freq)) +
  theme_fourfold()

The unstandardized display (std = "ind.max") shows the raw frequencies, with the largest cell drawn at full size. The sample odds ratio, Odds(Admit | Male) / Odds(Admit | Female), is \((1198 / 1493) / (557 / 1278) = 1.84\): male applicants were nearly twice as likely to be admitted.

berkeley + geom_fourfold(std = "ind.max")

These comparisons are difficult because there are more male applicants than female, and because more applicants were rejected than admitted. In the unstandardized display the confidence rings have no interpretation as a test of \(H_0: \theta = 1\).

The data can be standardized to make the comparison easier while preserving the odds ratio. With margin = 1 the rows (here, Gender) are equated, so each quadrant shows the proportion of men or women who were admitted or rejected: 44.5% of male applicants were admitted, compared with 30.4% of female applicants. Now the rings for each upper quadrant overlap those of the quadrant below it if the odds ratio does not differ from 1.

berkeley + geom_fourfold(margin = 1)

The default, margin = c(1, 2), standardizes both margins. The four quadrants then align horizontally and vertically when the odds ratio is 1, regardless of the marginal frequencies, so the association between admission and gender can be seen without the influence of the overall admission rate or of the numbers of men and women who applied. This fully standardized display is usually the most useful form.

berkeley + geom_fourfold()

The quadrants do not align, and the 95% confidence rings, computed from the interval \(1.62 \le \theta \le 2.09\), do not overlap: the odds ratio differs significantly from 1, apparent evidence of gender bias in favor of men.

Stratified \(2 \times 2 \times k\) tables

In a \(2 \times 2 \times k\) table the third variable often defines strata, and the question is whether the association between the first two is the same in each stratum. Make one fourfold display per stratum with facet_wrap() or facet_grid(); standardizing the margins within each panel allows the pattern of association to be compared across strata even when their marginal frequencies differ.

berkeley +
  geom_fourfold() +
  facet_wrap(vars(Dept), ncol = 3, labeller = label_both)

Surprisingly, for five of the six departments the odds of admission are about the same for men and women. Only Department A differs, and in the opposite direction: there women were about 2.86 times as likely as men to be admitted (\(1 / 0.349\)). The more intense colors mark the one panel whose odds ratio differs significantly from 1. With several strata, geom_fourfold() adjusts the p-values that control this emphasis for multiple testing, by default using Holm’s method via stats::p.adjust(); the confidence rings themselves are not adjusted.

This reversal of the aggregate result is an example of Simpson’s paradox, and it shows why an analysis of a table collapsed over a third variable can mislead. Men and women applied to departments in different proportions, and women applied in greater numbers to departments with low admission rates.

Because a fourfold display keeps the angle of each cell fixed, corresponding cells are always in the same position in every panel. An array of fourfold displays therefore supports controlled comparison, against a standard with other things held constant, much better than an array of pie charts. Standardizing the margins adds a further standard for comparison while preserving the odds ratio, the quantity the display is designed to show.

Confidence rings

Each quadrant of a fourfold display is drawn with a pair of thin rings, one just inside and one just outside its edge. Together they show the range of tables that are consistent with the data at the confidence level given by conf_level (0.95 by default). They are constructed in three steps.

1. An interval for the odds ratio. Inference is simplest on the log scale. The log odds ratio, \(\hat{\psi} = \log \hat{\theta}\), has an approximate standard error

\[\hat{s}(\hat{\psi}) = \sqrt{\frac{1}{n_{11}} + \frac{1}{n_{12}} + \frac{1}{n_{21}} + \frac{1}{n_{22}}} ,\]

so an approximate \(1 - \alpha\) confidence interval for \(\psi\) is \(\hat{\psi} \pm z_{1 - \alpha/2} \, \hat{s}(\hat{\psi})\). Exponentiating its end points, \(\hat{\psi}_l\) and \(\hat{\psi}_u\), gives the interval \(\{\exp(\hat{\psi}_l), \exp(\hat{\psi}_u)\}\) for \(\theta\). (If any cell is zero, 0.5 is first added to every cell, as in vcd::fourfold().) For the pooled Berkeley data, \(\hat{\psi} = 0.610\) and \(\hat{s} = 0.0639\), so the 95% interval for the odds ratio runs from 1.624 to 2.087.

2. A table for each end of the interval. Each limit is turned into the \(2 \times 2\) table that has that odds ratio. When both margins are equated, such a table has the symmetric form

\[\begin{bmatrix} p & 1 - p \\ 1 - p & p \end{bmatrix} ,\]

whose odds ratio is \(\theta = p^2 / (1 - p)^2\). Solving for \(p\) gives \(p = \sqrt{\theta} / (1 + \sqrt{\theta})\). More generally, the table can be found that keeps the observed row and column totals and has the required odds ratio; with the margins fixed, the whole table follows from a single cell, which is the root of a quadratic equation.

3. Draw the tables like the data. Each limiting table is standardized in the same way as the observed table (std, margin) and drawn as outlines over the quadrants. For the Berkeley data this gives:

Every row has the observed margins, 2691 male and 1835 female applicants, of whom 1755 were admitted. Only the balance among the cells changes.

Customizing the display

berkeley +
  geom_fourfold(shape = "square") +
  facet_wrap(vars(Dept), ncol = 3, labeller = label_both)

berkeley +
  geom_fourfold(shape = "square", counts = "outside") +
  facet_wrap(vars(Dept), ncol = 3, labeller = label_both)

berkeley +
  geom_fourfold(extended = FALSE) +
  facet_wrap(vars(Dept), ncol = 3, labeller = label_both)

fourfold_palette()
#> [1] "#99CCFF" "#6699CC" "#FFA0A0" "#A0A0FF" "#FF0000" "#000080"

okabe_ito <- c(
  "#56B4E9", "#0072B2", # extended = FALSE: sky blue, blue
  "#F2CF7F", "#AAD9F3", # not significant: orange, sky blue (lightened)
  "#D55E00", "#0072B2"  # significant: vermilion, blue
)

berkeley +
  geom_fourfold(palette = okabe_ito) +
  facet_wrap(vars(Dept), ncol = 3, labeller = label_both)

berkeley +
  geom_fourfold(palette = fourfold_palette("okabe-ito")) +
  facet_wrap(vars(Dept), ncol = 3, labeller = label_both)

berkeley +
  geom_fourfold(
    diagonal.length = 0.05,
    diagonal.fill = "limegreen",
    diagonal.width = 1.5,
    color = "grey30"
  )

applicants <- margin.table(UCBAdmissions, 3)
dept_labels <- sprintf("%s (n = %d)", names(applicants), applicants)
names(dept_labels) <- names(applicants)

ggplot(ucb, aes(x = Admit, y = Gender, weight = Freq)) +
  geom_fourfold() +
  facet_wrap(vars(Dept), ncol = 3, labeller = labeller(Dept = dept_labels)) +
  labs(
    title = "UC Berkeley Admissions, 1973",
    subtitle = "Only department A shows a significant association",
    caption = "Data from Bickel et al. (1975)"
  ) +
  theme_fourfold(base_family = "serif") +
  theme(
    plot.title.position = "plot",
    plot.title = element_text(hjust = 0),
    plot.subtitle = element_text(
      hjust = 0,
      color = "grey30",
      margin = margin(b = 10)
    ),
    plot.caption = element_text(
      hjust = 1,
      color = "grey30",
      size = rel(0.8),
      margin = margin(t = 10)
    )
  )