Similarity measures for more than 2 variables

Question

If I have two binary variables, I can determine the similarity of these variables quite easily with different similarity measures, e.g. with the Jaccard similarity measure:

$J = \frac{M_{11}}{M_{01} + M_{10} + M_{11}}$

Example in R:

# Example data
N <- 1000
x1 <- rbinom(N, 1, 0.5)
x2 <- rbinom(N, 1, 0.5)

# Jaccard similarity measure
a <- sum(x1 == 1 & x2 == 1)
b <- sum(x1 == 1 & x2 == 0)
c <- sum(x1 == 0 & x2 == 1)

jacc <- a / (a + b + c)
jacc

However, I have a group of 1.000 binary variables and want to determine the similarity of the whole group.

Question: What is the best way to determine the similarity of more than 2 binary variables?

One idea is to measure the similarity for each pairwise combination and then take the average. You can find an example of this procedure below:

# Example data
N <- 1000 # Observations
N_vec <- 1000 # Amount of vectors
x <- rbinom(N * N_vec, 1, 0.5)
mat_x <- matrix(x, ncol = N_vec)
list_x <- split(mat_x, rep(1:ncol(mat_x), each = nrow(mat_x)))

# Function for calculation of Jaccard similarity
fun_jacc <- function(v1, v2) {

  a <- sum(v1 == 1 & v2 == 1)
  b <- sum(v1 == 1 & v2 == 0)
  c <- sum(v1 == 0 & v2 == 1)

  jacc <- a / (a + b + c)
  return(jacc)
}

# Apply function to all combinations (takes approx. 1 min to calculate)
mat_jacc <- sapply(list_x, function(x) sapply(list_x, function(y) fun_jacc(x,y)))
mat_jacc[upper.tri(mat_jacc)] <- NA
diag(mat_jacc) <- NA
vec_jacc <- as.vector(mat_jacc)
vec_jacc <- vec_jacc[!is.na(vec_jacc)]
median(vec_jacc)

This is very inefficient though and I am also not sure if this is theoretically the best way to measure the similarity of such a group of variables.

UPDATE: According to user43849's suggestion the dissimilarity could be calculated with Sorensen's multiple-site dissimilarity:

# Example data
N <- 1000 # Observations
N_vec <- 1000 # Amount of vectors
x <- rbinom(N * N_vec, 1, 0.5)
mat_x <- matrix(x, ncol = N_vec)

# Multiple-site dissimilarity according to Sorensen
library("betapart")
beta.multi(t(mat_x), index.family = "sor")$beta.SOR # Vectors are not similar --> almost 1

Answer 1

This answer will draw heavily on the ecological literature, where Jaccard and other (dis)similarity measures are commonly used to quantify the compositional (dis)similarity between species assemblages at different sites. The single best reference is Baselga (2013) Multiple site dissimilarity quantifies compositional heterogeneity among several sites, while average pairwise dissimilarity may be misleading, which is freely available here.

Basically, there are several approaches to quantifying higher-order dissimilarities (higher-order than pairwise). One is to average the pairwise dissimilarities for all pairs in the sample. This metric generally performs poorly for a variety of reasons, detailed in Baselga (2013). Another possibility is to find the average distance from an observation to the multivariate centroid.

There is an explicit generalization of the Sorensen index to more than two observations. Recall that the Sorensen index is $\frac{2ab}{a+b}$ where a is the number of species (ones in your case) in sample A, b is the number of species in sample B, and ab is the number of species shared by samples A and B (i.e. the dot product). The three-site generalization, formulated by Diserud and Odegaard (2007) and discussed by Chao et al (2012) is $\frac{3}{2}\frac{ab+ac+bc-abc}{a+b+c}$. Consult Diserud and Odegaard (2007) for the motivation behind this metric as well as extensions to $N>3$. The references in Baselga (2013) will also point you to a multi-site generalization of the Simpson index, as well as R packages to compute the multi-site Sorensen and Simpson metrics.

Some researchers have also found it useful to examine the average number of species shared by $i$ sites, where $i$ ranges from $2$ to $N$. This reveals some interesting scaling properties and unites a variety of concepts for different values of $i$. The key reference here is Hui and McGeoch (2014) available for free here. This paper also has an associated R package called 'zetadiv'.

Answer 2

A possible direction is to represent the problem as a graph. The variables will be the nodes and the edges will be strong enough pair wise correlation.

You can define many correlation measures based on the graph, and you should find the one that suites you most. In most common graph correlations, the denser the graph, the more correlated it is.

Possible measures might be:

• The number (or ratio) of nodes (variables) with and edges (a strong correlation with some variable). It is easy to compute and gives you the number of independent variables. A variation on this metric is choosing only nodes with at least few edges.
• The number of edges. This metric give higher weight to the number of correlations but might give an high score to a graph with many not connected variables.
• The number (or ratio) of connected components. This measure is a bit more complex but it captures better our common definition os similarity. If a variable is connected to a group of variables, they tend to behave as a single unit, regardless of the number of variables. You can use dbscan in order to get the graph structure but other graph algorithms might fit your needs also.

Answer 3

Similarity is always between two items. It can then be extended to more (say three) items by first creating single representation for two items and then finding its similarity with the third item. So, when you have 1000 items you can ask the question how similar or far is any given vector from the representation of these 1000 vectors. This representation can be the mean of these 1000 vectors or it could be anything based on how you define it.