Measure of distance between two survey responses

Question

I've found some survey data where respondents answer 63 question by giving a response for each question between 0-10 (0 for strong disagreement, 10 for strong agreement). So I can view every respondent as a integer point in $[0,10]^{63}$.

Question: is there a "standard" way to measure the distance between two respondents?

Something like the taxicab distance between them seems somewhat reasonable, but likewise so does any $L^p$-distance. Though these wouldn't account for questions that contributed the most of least variation in responses, for example.

My intended application is using this to measure distances between points to experiment with persistent homology.

Answer 1

If you have Likert-10 data for two groups, you could compare the two groups using a two-sample Wilcoxon rank sum test. Hypothetical data and test below with 200 subjects in each group.

Data and summary: The p vectors show relative 'popularities' of the score from 0 to 10 in each group.

set.seed(2020)
x1 = sample(1:10, 200, repl=T, p=c(1,2,3,4,5,6,4,3,2,1))
x2 = sample(1:10, 200, repl=T, p=c(1,1,2,3,4,5,6,7,6,5))

table(x1)
x1
 1  2  3  4  5  6  7  8  9 10 
 6 13 21 32 26 37 29 15 12  9 
table(x2)
x2
 1  2  3  4  5  6  7  8  9 10 
 7  5  5 19 23 23 29 42 25 22 

summary(x1)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
   1.00    4.00    6.00    5.48    7.00   10.00 
summary(x2)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  1.000   5.000   7.000   6.725   8.000  10.000 

Wilcoxon test shows a highly significant difference between the two groups with a P-value near $0.$

wilcox.test(x1,x2)

         Wilcoxon rank sum test 
         with continuity correction

data:  x1 and x2
W = 13606, p-value = 2.524e-08
alternative hypothesis: 
  true location shift is not equal to 0

'Notched' boxplots. Here are boxplots of the two samples of Likert-10 scores. That the 'notches' in the sides of the boxes do not overlap suggests that the two groups differ.

boxplot(x1,x2, notch=T, col="skyblue2")