All pairwise differences between two vectors

Question

Suppose I have two vectors:

X1 <- c(44350, 38920, 37530, 42280, 37320, 36910, 35720, 31220, 33400, 40710, 43830, 37390, 32340, 30770, 35800, 40250, 31490, 40460, 33730, 35850, 35320, 37500, 35380, 40910, 29040, 33950)

X2 <- c(30390, 34170, 28910, 30660, 32510, 30540, 31990, 32380, 32110, 31260, 34670, 28240, 31840, 33350, 32150, 35640, 30730, 30280, 29420, 30990, 32880, 33280, 36960, 36990)

I am interested in counting all pairwise instances where a member of X1 < member of X2.

How would I do this in R for any two vectors?

Now, suppose I wish to combine both vectors, randomize, split into two vectors of length(X1) and length(X2) then count as above--creating a randomized distribution to compare against the initial count.

How would this be done?

Answer 1

Count of instances where a in X1 and b in X2 with a < b

(s <- sum(outer(X1,X2,`<`)))
# [1] 106

---

Combine the two

X <- c(X1,X2)

Shuffle, resplit, and retest many times

set.seed(1)

r <- replicate(1000000, {
  X <- sample(X)
  X1 <- head(X,length(X1)) 
  X2 <- tail(X,length(X2))
  sum(outer(X1,X2,`<`)) })

(p <- 1 - sum(s <= r)/length(r))
# [1] 1e-05

---

Note: It sounds like you might be looking for something like the Mann-Whitney test, which is under wilcox.test .

> wilcox.test(X2,X1)

        Wilcoxon rank sum test

data:  X2 and X1
W = 106, p-value = 2.858e-05
alternative hypothesis: true location shift is not equal to 0

Compare W = 106 to s = 106 above.