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I have data with 4 observations and 24 variables. To understand the underlying relationship I performed Multi-Dimensional Scaling (MDS), and got a plot like this:

enter image description here

Now the issue is with the correct interpretation of the plot. I understand the two axes (i.e., the x-axis and y-axis) imply the variation in data along the two principal components. But, my specific doubts are:

  1. Should I infer that points 1 and 3 vary along dimension-1 only and there is no relationship between them in dimension-2?
  2. Similarly, should I infer points 1 and 2 along dimension-2 only?
  3. How should I explain the relationship of point 4 with the rest of the points?
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  • $\begingroup$ Unclear what you're asking. What are your specific concerns? What makes you fear that you cannot interpret an MDS plot like a usual scatterplot? $\endgroup$ – ttnphns Nov 13 '15 at 8:38
  • $\begingroup$ I admit that I am not interpreting this as a usual scatter plot. I thought that plotting data from two principal axis might need some different interpretation. So, should I take it exactly as a scatter plot while interpreting ? $\endgroup$ – Haroon Rashid Nov 13 '15 at 11:33
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Despite having 24 original variables, you can perfectly fit the distances amongst your data with 3 dimensions because you have only 4 points. It is possible that your points lie exactly on a 2D plane through the original 24D space, but that is incredibly unlikely, in my opinion. It is reasonable to imagine that the variation on the third dimension is inconsequential and/or unreliable, but I don't have any information about that. I am assuming that there is a third dimension that isn't represented in your plot.

  1. You can infer that 1 and 3 do not vary on dimension 2, but you have no information here about whether they vary on dimension 3. Thus, you cannot necessarily assume that they vary on dimension 1 only.
  2. Likewise, you can infer that 1 and 2 do not vary on dimension 1, but again you have no information about whether they vary on dimension 3. So, you cannot necessarily assume that they vary on dimension 2 only.
  3. Point 4 differs from 1, 2, and 3 on both dimensions 1 and 2. Its relationship to them on dimension 3 is unknown. Ignoring dimension 3 for a moment, you could think of point 4 as the medoid of your dataset.
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Sorry to necro, but found this through a search and thought I could help others.

The correct answer is that there is no interpretability to the MDS1 and MDS2 dimensions with respect to your original 24-space points.

This is because MDS performs a nonparametric transformations from the original 24-space into 2-space.

The only interpretation that you can take from the resulting plot is from the distances between points. The further away two points are the more dissimilar they are in 24-space, and conversely the closer two points are the more similar they are in 24-space.

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