Motivating Scenario:
You are digging deeper into your summaries of associations between two categorical variables.
Learning Goals: By the end of this subchapter, you should be able to:
Explain the intuition of the multiplication rule for independent events
Calculate and explain the covariance as a deviation from expectations under independence. Again you should be able to do this with basic math and with R code.
library(dplyr)
library(readr)
library(ggplot2)
ril_link <- "https://raw.githubusercontent.com/ybrandvain/datasets/refs/heads/master/clarkia_rils.csv"
gc_rils <- readr::read_csv(ril_link) |>
rename(visits = mean_visits)|>
mutate(visited = ifelse(visits > 0,"some_visits","no_visits")) |>
dplyr::filter(location == "GC", !is.na(prop_hybrid), ! is.na(visits),!is.na(petal_color))|>
select(petal_color, visits, visited, prop_hybrid)In the previous section we calculated UNCONDITIONAL (also known as marginal) proportions (n_case / n_total) of two binary categorical variables. Let;s call these proportions:
\(P_\text{pink}= n_\text{pink} / n_\text{plants}\)
\(P_\text{some visits}= n_\text{some visits} / n_\text{plants}\)
gc_rils |>
summarise(prop_pink = mean(petal_color == "pink"),
prop_visited = mean(visited == "some_visits")) | prop_pink | prop_visited |
|---|---|
| 0.538 | 0.264 |
We also saw that proportions of one outcome can depend on the value of another variable. We quantified this as a CONDITIONAL PROPORTION:
\(P_\text{some visits | pink}= n_\text{some visits | pink} / n_\text{pink plants}\)
\(P_\text{some visits | white}= n_\text{some visits | white} / n_\text{white plants}\)
These differed substantially. Pink plants were way more likely to be visited than white plants.
gc_rils |>
summarise(n_pink = sum(petal_color == "pink"),
n_white = sum(petal_color == "white"),
p_visit_pink = sum(petal_color == "pink" & visited == "some_visits") / n_pink ,
p_visit_white = sum(petal_color == "white" & visited == "some_visits") / n_white) | n_pink | n_white | p_visit_pink | p_visit_white |
|---|---|---|---|
| 49 | 42 | 0.449 | 0.048 |
A common quantification of the association between two binary variables is the covariance. This quantification follows some simple mathematical logic:
If two categorical variables, A & B are independent, the probability of observing both A and B (aka, \(P_{AB}\)) should equal the product of their unconditional probabilities: \[P_{AB \mid \text{independence}} = P_A \times P_B\]
The covariance is the deviation from this expectation: \[COV_{AB} = P_{AB} - P_A \times P_B\]
So for the association between petal color and being visited, the covariance is:
\[COV_{\text{visited, pink}} =P_{\text{visited and pink}} - P_{\text{visited}} \times P_{\text{pink}}\]
We can find this in R:
gc_rils |>
summarise(p_pink_and_visited = mean(petal_color == "pink" & visited == "some_visits"),
p_pink = mean(petal_color == "pink"),
p_visited = mean(visited == "some_visits"),
cov_pink_and_visited = p_pink_and_visited - p_pink * p_visited)| p_pink_and_visited | p_pink | p_visited | cov_pink_and_visited |
|---|---|---|---|
| 0.2418 | 0.5385 | 0.2637 | 0.0997 |
Alternatively, R can calculate the covariance with the cov() function. This only works if the values are numbers or logical (TRUE [1] / FALSE [0]). Note the slight mismatch between our calculation due to “Bessel’s correction” (see fyi box, below).
gc_rils |>
summarise(cov_pink_visited = cov(petal_color == "pink", visited == "some_visits"))| cov_pink_visited |
|---|
| 0.1009 |
Our answer differ slightly from one another. We used a denominator of \(n\) (which calcualtes the population covariance), while the sample covariance has a denominator of \(n-1\) (this is known as Bessel’s correction. We will soon see that although the later is more appropriate – their difference is negligible when n is large. We can convert the populaiton covariance to the sample covaraiance by multiplying by \(\frac{n}{n-1}\). See the next section for a discussion of the difference between populaiton parameters and sample estimates.
Covariance gives us a numerical measure of how far our data deviate from what we’d expect under independence. In this case, the value is 0.10 — but is that meaningful? It turns out a covariance is difficult to interpret as strong or weak. The correlation, r, describes both the sign and strength of the association between variables. An absolute value of r close to 1 means that x almost perfectly predicts y, while a value close to zero means x has no information about y. The sign of r describes if x and y increase (\(r > 0\)), or decrease (\(r <0\)) with one another.
We can convert a covariance to an effect size, known as a correlation (or r), by dividing the covariance by the product of standard deviations.
\[r_{x,y}= \frac{cov_{x,y}}{s_x s_y}\]
We have previously seen the equation for a standard deviation: \[s_x = \sqrt{\frac{\sum{(x_i-\bar{x})^2 }}{n-1}}\].
Again this simplification arises when we ignore Bessel’s correction and divide by n instead of n − 1.
For a binary variable, this equation equals \(p_x (1-p_x)\). So in R:
gc_rils |>
summarise(var_pink = mean(petal_color == "pink") * (1 - mean(petal_color == "pink")),
var_visited = mean(visited == "some_visits") * (1 - mean(visited == "some_visits")),
cov_pink_visited = cov(petal_color == "pink", visited == "some_visits"),
cor_pink_visited = cov_pink_visited / (sqrt(var_pink) * sqrt(var_visited ))) | var_pink | var_visited | cov_pink_visited | cor_pink_visited |
|---|---|---|---|
| 0.2485 | 0.1942 | 0.1009 | 0.4591 |
This is a strong correlation!
gc_rils |>
summarise(cor_pink_visited = cor(petal_color == "pink", visited == "some_visits"))| cor_pink_visited |
|---|
| 0.4541 |
Two population genetic summaries tend to confuse genetics students – deviations from Hardwy-Weinberg Equilibrium, and Linkage disequilibrium. Both can be naturally considered as covarances between two binary variables.
Hardy-Weinberg Equilbrium (HWE)
According to the Hardy-Weinberg Equilbrium principle, states that under anumerous assumptions – which essentially ensure that an individual’s two alleles at a locus are chosen indepednently – geneotypes at a locus are simply two random samples of alleles from a population,
We can think of this as a covariance between alleles inherited from mom and dad at the same locus.
Linkage Disequilibrium (LD)
Linkage disquilibrium quantifies the statistical asscoiation between alleles at different loci. The most standard quantification of linkage disequilbrium is \(D = Cov(A,B) = p_{AB} - p_A p_B\). Where A and B asre alleles at two different loci. Although usually quantified as a covariance, other measures of LD include a correlation in allele frequencies between loci.