Clustering in 5-point Likert type dataset

Question

I want to cluster the data collected with a 5-point Likert scale. But I couldn't understand which method is more accurate to use. I searched the literature but couldn't find a clear answer. Can you help?

Answer 1

I'm assuming you have several variables and all are measured on a 5-point Likert scale.

There are no general results comparing clustering methods generally regarding "accuracy". Clustering always depends on the meaning of the data and the aim of clustering, also different structures in the data may require different methods, so we can't tell you what would be "best" for you to use. Note that there is an implicit assumption in much of statistics, which is that there is only one "true" best solution, and the task of statistical methodology is to find that, and the "best" methodology is the one that comes closest. In cluster analysis this cannot be assumed, as the same data can often legitimately and meaningfully be clustered in different ways.

A possibility is to define a distance measure between observations and then use distance-based clustering methods such as average or complete linkage or partitioning around medoids (PAM). These have different implicit concepts of what kinds of clusters can be found (see, e.g., my paper on clustering strategy and method choice, https://arxiv.org/abs/1503.02059), and in this way what you need depends on what the "cluster concept" is that you require.

You could score your observations as 1, 2, 3, 4, 5 and compute a Euclidean or Manhattan distance. Whether this is appropriate depends once more on the meaning of your data and your aims (ultimately your distance needs to capture what counts as "similar" or "distant" in your application). Some people will argue that Likert-scaled observations are on an ordinal and not on an interval scale, meaning that they are not quantitative, and that scoring your observations as above is inappropriate because this implicitly assumes that the distance between 1 and 2 is the same as the distance between 4 and 5, which cannot be taken for granted. Note however that any way of quantifying distances between observations (which in some way is required for any approach) implicitly imposes some kind of quantification, and existing methods specifically for for ordinal data are based on assumptions that are not always easier to justify than just treating the data as 1,2,3,4,5. One simple way of computing distances in which distances between categories are "estimated" from the data rather than imposed is to convert the data to (mid-)ranks, averaging ranks for observations in the same categories, and then use Euclidean or Manhattan distance. This will implicitly determine the distance between categories for each question based on the relative frequencies, imposing a larger distance between most frequent categories. Another possibility is assume a latent normally distributed random variable and the score the data by the resulting quantiles, which is somewhat similar but not identical to ranking.

More sophisticated approaches may model the data by lower dimensional underlying latent Gaussian variables, and then fit a mixture of such models. There are many alternatives, see for example https://www.tandfonline.com/doi/full/10.1080/10705511.2023.2250920, https://journal.r-project.org/archive/2021/RJ-2021-011/index.html, https://cran.r-project.org/web/packages/clustMD/index.html (this is what I'd probably try first, it is a mixture model advertised for mixed type data but should fit also purely ordinal data), https://cran.r-project.org/web/packages/CUB/index.html, answers here: Cluster analysis of ordinal variables (Likert scale).

I also recommend computing a distance, even if you eventually cluster your data in different ways, for running a multidimensional scaling for visualising your data. Sometimes (maybe rather often for Likert type data) you may find that the data look rather homogeneous, i.e., they don't clearly cluster, in which case you may decide to not run a clustering (or at least to not overinterpret any clustering you might compute). Alternatively (or additionally) this can be done using a correspondence analysis. Some people would cluster the outcome of a correspondence analysis or MDS using methods for clustering interval scaled data such as k-means or Gaussian mixtures, although this comes with a certain information loss.