# Clustering with Likert items and N/A option

I am currently evaluating the results of a 72 question survey with response levels from "strongly disagree" to "strongly agree". I would like to cluster the questions by response patterns using R (I'm using "clara").

Here's the rub: There is also a response option "N/A", because some questions are not relevant for some of the respondents, so they couldn't have an opinion of whether to agree or disagree with the premise of the question.

Currently, I have coded the agreement levels from -2 to 2, and "N/A" got a -3 just to see if everything works. It does, so this is not a coding question.

My question is: Do you know of a clever distance function that I could use in this situation to calculate more meaningful clusters? The goal would be to compare only the responses of those for whom the question is relevant. I don't think I can just drop the "N/A" because that would give me points of different dimensions, so both Euclidean and taxicab metric would not be happy.

EDIT 1: One possibility would be to apply a chi-squaresque metric that only compares the distributions of the desired responses, but that strikes me as too crude. EDIT 2: Another possibility would be to adapt the Euclidean or taxi-cab metric with appropriate weights so that "proper" responses would be given higher consideration.

PS: No, I cannot eliminate the N/As because I need to be able to calculate a "distance" between the response patterns of any two questions, so I can find out which groups of questions tended to be answered similarly throughout.

• You should only be applying your analysis to those respondents who were eligible to provide a meaningful response. In other words, your N/A responses should be removed from the analysis. After all if you had a survey skip pattern that asked the respondent's gender and then asked a follow-up pregnancy question (and marked N/A for me), it wouldn't make any sense to include the male N/A responses in your analysis. – StatsStudent Feb 21 '19 at 22:30
• This is an employee climate survey. Different questions on "work climate" have different levels of relevance to different constituents. Employees who mainly work in "A" will have no opinion on whether they think their input on matters concerning "B" is valued, hence "N/A". – Matthias Feb 22 '19 at 0:13
• Great, @Matthias. Then my comment applies directly to your situation. – StatsStudent Feb 22 '19 at 0:15
• I'm talking about finding clusters in the set of questions, where different questions are N/A to different constituents. How do you fathom I should eliminate the N/As globally? There is no guarantee that the left-over data vectors per question are of the same dimension. – Matthias Feb 22 '19 at 0:26

Let $$x_1, x_2$$ be two response vectors in $$\mathbb{R}^N$$, where $$N$$ is the number of responses. Let $$n$$ be the number of components where both $$x_1$$ and $$x_2$$ are not equal to "N/A". Let $$k$$ be the number of such components with absolute difference at most 1 (say).
Then define the measure of dissimilarity between $$x_1$$ and $$x_2$$ as $$\frac{k}{n}$$, so $$1-\frac{k}{n}$$ appears to give a reasonably good "metric." (As I said, I haven't checked in how far this violates the axioms for a metric.)