# Comparing Distributions

**R on kieranhealy.org**, and kindly contributed to R-bloggers]. (You can report issue about the content on this page here)

Want to share your content on R-bloggers? click here if you have a blog, or here if you don't.

When we want to see how something varies across categories, the *trellis* or *small multiple* plot is a good friend. We repeatedly draw the same graph once for each category, lining them up in a way that makes them comparable. Here’s an example from my book, using the `gapminder`

data, which provides a cross-national time series of GDP per capita for many countries.

1 2 3 4 5 6 7 8 9 10 11 |
library(tidyverse) library(gapminder) p <- ggplot(data = gapminder, mapping = aes(x = year, y = gdpPercap)) p + geom_line(color="gray70", aes(group = country)) + geom_smooth(size = 1.1, method = "loess", se = FALSE) + scale_y_log10(labels=scales::dollar) + facet_wrap(~ continent, ncol = 5) + labs(x = "Year", y = "GDP per capita", title = "GDP per capita on Five Continents", subtitle = "Individual countries shown in gray, trend in blue.") |

Sometimes, we’re interested in comparing distributions across categories in something like this way. In particular, I’m interested in cases where we want to compare a distribution to some reference category, as when we look at subpopulations in comparison to an overall distribution.

## Generate some population and subgroup data

Say we have some number of observed units, e.g., three thousand “counties” or whatever. Each county has some population. Across all counties, the population is distributed log-normally. Within counties, the populations are divided into three groups. The particular proportions of groups A, B, and C will vary across counties but always sum to one within each county.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 |
## Keep track of labels for as_labeller() functions in plots later. grp_names <- c(`a` = "Group A", `b` = "Group B", `c` = "Group C", `pop_a` = "Group A", `pop_b` = "Group B", `pop_c` = "Group C", `pop_total` = "Total", `A` = "Group A", `B` = "Group B", `C` = "Group C") set.seed(1243098) N <- 3e3 alphas <- c(1.5, 0.9, 2) pop <- round(rlnorm(N, meanlog = 10.3, sdlog = 1.49), 0) df <- as_tibble(gtools::rdirichlet(N, alphas), .name_repair = "unique") %>% rename_with(~ c("a", "b", "c")) %>% rowid_to_column("unit") %>% add_column(pop_total = pop) %>% mutate(across(a:c, .fns = list(pop = ~ round(.x * pop_total, 0) + 1), .names = "{fn}_{col}")) df ## # A tibble: 3,000 × 8 ## unit a b c pop_total pop_a pop_b pop_c ## <int> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> ## 1 1 0.156 0.196 0.648 6467 1008 1269 4194 ## 2 2 0.288 0.154 0.558 211075 60729 32579 117771 ## 3 3 0.165 0.391 0.443 128243 21186 50184 56876 ## 4 4 0.294 0.124 0.582 76843 22561 9555 44730 ## 5 5 0.301 0.146 0.553 12178 3671 1780 6730 ## 6 6 0.397 0.148 0.455 2707 1076 401 1232 ## 7 7 0.364 0.258 0.378 143261 52112 36987 54166 ## 8 8 0.859 0.0375 0.103 61109 52517 2290 6305 ## 9 9 0.129 0.477 0.394 61718 7968 29433 24320 ## 10 10 0.185 0.182 0.632 3217 597 588 2035 ## # … with 2,990 more rows |

In the tibble we’ve just made up, `unit`

is our county, `a`

, `b`

, and `c`

are the proportions of the groups within each county, and the `pop_`

columns are the total populations and the subgroup populations. We make a vector of 3,000 populations using `rlnorm`

and plausible values based on the mean and standard deviations of the logged population of actual US counties. A call to `rdirichlet`

produces the matrix of subgroup proportions where each row sums to one. Then we multiply the populations by their respective proportions, and now we have a world of three thousand counties, each with some population that we’ve also broken out by group.

We can look at the distribution of group populations across counties:

1 2 3 4 5 6 7 8 9 10 11 12 13 |
df %>% pivot_longer(a:c) %>% ggplot() + geom_area(mapping = aes(x = value, y = ..count.., color = name, fill = name), stat = "bin", bins = 20, size = 0.5) + scale_fill_manual(values = alpha(my_oka, 0.7)) + scale_color_manual(values = alpha(my_oka, 1)) + guides(color = "none", fill = "none") + labs(x = "Logged Population", y = "Count", title = "Subgroup distribution across units") + facet_wrap(~ name, nrow = 1, labeller = as_labeller(grp_names)) |

From now on let’s just work with the population counts.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 |
df <- df %>% select(unit, pop_a:pop_c, pop_total) df ## # A tibble: 3,000 × 5 ## unit pop_a pop_b pop_c pop_total ## <int> <dbl> <dbl> <dbl> <dbl> ## 1 1 1008 1269 4194 6467 ## 2 2 60729 32579 117771 211075 ## 3 3 21186 50184 56876 128243 ## 4 4 22561 9555 44730 76843 ## 5 5 3671 1780 6730 12178 ## 6 6 1076 401 1232 2707 ## 7 7 52112 36987 54166 143261 ## 8 8 52517 2290 6305 61109 ## 9 9 7968 29433 24320 61718 ## 10 10 597 588 2035 3217 ## # … with 2,990 more rows |

Here’s what our population totals look like across groups, including the total:

1 2 3 4 5 6 7 8 9 10 11 |
df %>% pivot_longer(pop_a:pop_total) %>% group_by(name) %>% summarize(total = sum(value)) %>% ggplot(mapping = aes(x = total, y = name, fill = name)) + geom_col() + guides(fill = "none") + scale_fill_manual(values = alpha(c( my_oka[1:3], "gray40"), 0.9)) + scale_x_continuous(labels = scales::label_number_si()) + scale_y_discrete(labels = as_labeller(grp_names)) + labs(y = NULL, x = "Population") |

## Single panels

Now we can plot the group-level population distributions across counties. Again, we
want to compare group distributions to one another and to the overall population
distribution by county. A single-panel histogram showing all four distributions isn’t very satisfactory. Even though we’re using `alpha`

to make the columns semi-transparent, it’s still very muddy.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 |
df %>% pivot_longer(cols = pop_a:pop_total) %>% ggplot() + geom_histogram(mapping = aes(x = log(value), y = ..count.., color = name, fill = name), stat = "bin", bins = 20, size = 0.5, alpha = 0.7, position = "identity") + scale_color_manual(values = alpha(c( my_oka[1:3], "gray40"), 1), labels = as_labeller(grp_names)) + scale_fill_manual(values = alpha(c( my_oka[1:3], "gray40"), 0.6), labels = as_labeller(grp_names)) + labs(x = "Logged Population", y = "Count", color = "Group", fill = "Group", title = "Comparing Subgroups: Histograms", subtitle = "Overall distribution shown in gray") |

If we use a `geom_density()`

rather than `geom_histogram()`

we’ll generate kernel density estimates for each group. These look a little better, but not great.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 |
df %>% pivot_longer(cols = pop_a:pop_total) %>% ggplot() + geom_density(mapping = aes(x = log(value), color = name, fill = name), alpha = 0.5) + scale_color_manual(values = alpha(c( my_oka[1:3], "gray40"), 1), labels = as_labeller(grp_names)) + scale_fill_manual(values = alpha(c( my_oka[1:3], "gray40"), 0.6), labels = as_labeller(grp_names)) + labs(x = "Logged Population", y = "Density", title = "Comparing Subgroups: Density", color = "Group", fill = "Group") |

Better, but still not great. A very serviceable compromise that has many of the virtues of a small multiple but has the advantage of keeping things in one panel is a ridgeline plot, courtesy of `geom_ridgeline()`

from Claus Wilke’s `ggridges`

package:

1 2 3 4 5 6 7 8 9 10 11 12 |
df %>% pivot_longer(cols = pop_a:pop_total) %>% ggplot() + geom_density_ridges(mapping = aes(x = log(value + 1), y = name, fill = name), color = "white") + scale_fill_manual(values = alpha(c( my_oka[1:3], "gray40"), 0.7)) + scale_y_discrete(labels = as_labeller(grp_names)) + guides(color = "none", fill = "none") + labs(x = "Logged Population", y = NULL, title = "Comparing Total and Subgroups: Ridgelines") + theme_ridges(font_family = "Myriad Pro SemiCondensed") |

Ridgeline plots look good and scale pretty well when there are larger numbers of categories to put on the vertical axis, especially if there’s a reasonable amount of structure in the data, such as a trend or sequence in the distributions. They can be slightly inefficient in terms of space with smaller numbers of categories. When the number of groups gets large they work best in a very tall and narrow aspect ratio that can be hard to integrate into a page.

## Histograms with a reference distribution

Like with the Gapminder plot, we can facet our plot so that every subgroup gets its own
panel. But instead of having the Total population be its own panel, we will put it inside each group’s panel as a reference point. This allows us to compare the group to the overall population, and also makes eyeballing differences between the group distributions a little easier. To do this, we’re going to need to have some way to put the total population distribution into every panel. The trick is to hold on to the total population by only pivoting the subgroups to long format. That leaves us with repeated
values for the total population, `pop_total`

, like this:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 |
df %>% pivot_longer(cols = pop_a:pop_c) ## # A tibble: 9,000 × 4 ## unit pop_total name value ## <int> <dbl> <chr> <dbl> ## 1 1 6467 pop_a 1008 ## 2 1 6467 pop_b 1269 ## 3 1 6467 pop_c 4194 ## 4 2 211075 pop_a 60729 ## 5 2 211075 pop_b 32579 ## 6 2 211075 pop_c 117771 ## 7 3 128243 pop_a 21186 ## 8 3 128243 pop_b 50184 ## 9 3 128243 pop_c 56876 ## 10 4 76843 pop_a 22561 ## # … with 8,990 more rows |

When we draw the plot, we first call on `geom_histogram()`

to draw the
distribution of the total population, setting the color to gray. Then we
call it again, separately, to draw the subgroups. Finally we facet on
the subgroup names. This leaves us with a faceted plot where each panel
shows one subpopulation’s distribution and, for reference behind it, the
overall population distribution.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 |
df %>% pivot_longer(cols = pop_a:pop_c) %>% ggplot() + geom_histogram(mapping = aes(x = log(pop_total), y = ..count..), bins = 20, alpha = 0.7, fill = "gray40", size = 0.5) + geom_histogram(mapping = aes(x = log(value), y = ..count.., color = name, fill = name), stat = "bin", bins = 20, size = 0.5, alpha = 0.7) + scale_fill_okabe_ito() + scale_color_okabe_ito() + guides(color = "none", fill = "none") + labs(x = "Logged Population", y = "Count", title = "Comparing Subgroups: Histograms", subtitle = "Overall distribution shown in gray") + facet_wrap(~ name, nrow = 1, labeller = as_labeller(grp_names)) |

This is a handy trick. We’ll use it repeatedly in the remaining figures, as we look at different ways of drawing the same comparison.

While putting the reference distribution behind the subgroup distribution is nice, the way the layering works produces an overlap that some viewers find difficult to read. It seems like a third distribution (the darker color created by the overlapping area) has appeared along with the two we’re interested in. We can avoid this by taking advantage of the underused `geom_step()`

and its `direction`

argument. We can tell `geom_step()`

to use a binning method (`stat = "bin"`

) that’s the same as `geom_histogram()`

. Here we’re also using the computed `..density..`

value rather than `..count..`

, but we could use `..count..`

just fine as well.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 |
df %>% pivot_longer(cols = pop_a:pop_c) %>% ggplot() + geom_histogram(mapping = aes(x = log(value), y = ..density.., color = name, fill = name), stat = "bin", bins = 20, size = 0.5, alpha = 0.7) + geom_step(mapping = aes(x = log(pop_total), y = ..density..), bins = 20, alpha = 0.9, color = "gray30", size = 0.6, stat = "bin", direction = "mid") + scale_fill_manual(values = alpha(my_oka, 0.8)) + scale_color_manual(values = alpha(my_oka, 1)) + guides(color = "none", fill = "none") + labs(x = "Logged Population", y = "Density", title = "Comparing Subgroups: Histograms", subtitle = "Overall distribution shown in outline") + facet_wrap(~ name, nrow = 1, labeller = as_labeller(grp_names)) |

With `geom_step()`

, we get a histogram with just its outline drawn. This works quite well, I think. Because we’re just drawing the outline, we call it *after* we’ve drawn our histograms, so that it sits in a layer on top of them.

## Frequency polygons

A final option, half way between histograms and smoothed kernel density estimates, is to use filled and open *frequency polygons*. Like `geom_histogram()`

, these use `stat_bin()`

behind the scenes but rather than columns they draw filled areas (`geom_area`

) or lines (`geom_freqpoly`

). The code is essentially the same as `geom_histogram`

otherwise. We switch back to counts on the y-axis here.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 |
df %>% pivot_longer(cols = pop_a:pop_c) %>% ggplot() + geom_area(mapping = aes(x = log(value), y = ..count.., color = name, fill = name), stat = "bin", bins = 20, size = 0.5) + geom_freqpoly(mapping = aes(x = log(pop_total), y = ..count..), bins = 20, color = "gray20", size = 0.5) + scale_fill_manual(values = alpha(my_oka, 0.7)) + scale_color_manual(values = alpha(my_oka, 1)) + guides(color = "none", fill = "none") + labs(x = "Logged Population", y = "Count", title = "Comparing Subgroups: Frequency Polygons", subtitle = "Overall distribution shown in outline") + facet_wrap(~ name, nrow = 1, labeller = as_labeller(grp_names)) |

We can do the same thing with kernel densities, of course:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 |
df %>% pivot_longer(cols = pop_a:pop_c) %>% ggplot() + geom_density(mapping = aes(x = log(value), color = name, fill = name), size = 0.5) + geom_density(mapping = aes(x = log(pop_total)), color = "gray20", size = 0.5) + scale_fill_manual(values = alpha(my_oka, 0.7)) + scale_color_manual(values = alpha(my_oka, 1)) + guides(color = "none", fill = "none") + labs(x = "Logged Population", y = "Density", title = "Comparing Subgroups: Kernel Densities", subtitle = "Overall distribution shown in outline") + facet_wrap(~ name, nrow = 1, labeller = as_labeller(grp_names)) |

While these look good, kernel densities can be a little tricker for people to interpret than straightforward bin-and-count histograms. So it’s nice to have the frequency polygon as an option to use when you just want to show counts on the y-axis.

The full code for this post is available on GitHub.

**leave a comment**for the author, please follow the link and comment on their blog:

**R on kieranhealy.org**.

R-bloggers.com offers

**daily e-mail updates**about R news and tutorials about learning R and many other topics. Click here if you're looking to post or find an R/data-science job.

Want to share your content on R-bloggers? click here if you have a blog, or here if you don't.