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I have following questions that were answered by respondents of a survey:

Question 1: Should we save more water? (1 - 5 Likert Scale, Strongly Disagree / Agree)

Question 2: I do my best to avoid wasting water (1 - 5 Likert Scale, Strongly Disagree / Agree)

How would you statistically measure this relationship?

Looking at the data, I see a difference between desired global state and personal contribution. I thought about a correlation test.

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This is a case where you have to think carefully about what you actually want to find out. Your question seems to hint at two different things: 1) "delta" and "difference", and 2) "relationship".

You can use correlation to see if there is a correlation, or "relationship" between the answers to one question and the other. Kendall tau-b might be the most desireable for Likert-type item data because of the high rate of ties [citation needed].

In the following example, the Questions are perfectly correlated even though there is a large difference in the absolute values.

Note that the values here have to be paired for each respondent.

Another possibility is the linear-by-linear test described by Agresti. This test assumes the response categories are equally spaced, or that the spacing is known. (Not shown).

Question1 = c(1,1,1,1,2,2,2,2)
Question2 = c(4,4,4,4,5,5,5,5)

cor.test( ~ Question1 + Question2, method="kendall")

   ### Kendall's rank correlation tau
   ###
   ### z = 2.6458, p-value = 0.008151
   ###
   ### tau 
   ### 1

To assess the idea of "delta" or "difference", you might look at the sign test or the Wilcoxon signed-rank test. The latter test assumes that the categories are equally spaced.

Each of these tests looks at if the observations in one sample are larger than observations in the other, but I recommend understanding the hypotheses tested and the assumptions of each before proceeding.

Again, the observations have to be paired by respondent.

if(!require(BSDA)){install.packages("BSDA")}

Question1 = c(1,1,1,1,2,2,2,2)
Question2 = c(4,4,4,4,5,5,5,5)

library(BSDA)

SIGN.test(Question1, Question2)

   ### Dependent-samples Sign-Test
   ###
   ### S = 0, p-value = 0.007813
   ### 
   ### sample estimates:
   ### median of x-y 

wilcox.test(Question1, Question2, paired=TRUE)

   ### Wilcoxon signed rank test with continuity correction
   ###
   ### V = 0, p-value = 0.005962
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