Introduction to Multidimensional Scaling
The goal of multidimensional scaling (MDS) is to elucidate the psychological relationship in a multidimensional space of stimuli in which the underlying dimensions and the number of dimensions are unknown. What is a multidimensional psychological space? Let's look at an example to see if the concept can become clearer:
Havana is like Beijing (both are capitals of communist countries) and
Havana is like Miami (both are warm, sunny, tropical cities) however
Havana is not like Beijing.
We judge the similarities between (and among) objects based on many different characteristics that vary along different dimensions (both continuously and categorically). Along a psychological dimension that is based on the perception of politics, Havana and Beijing may be similar. Along a different dimension corresponding to impressions of climate, Havana and Miami are similar. Cities can be judged on many dimensions. The goal of MDS is to discover these dimensions and the relationships of stimuli along these dimensions. But it must be remembered that the psychological dimensions may not be the same as the measurable or observable dimensions in the physical world.
Proximity refers to a number which indicates how similar or different two objects are, or are perceived to be, or any measure of this kind. Judgments of similarity/dissimilarity are most frequently used.
Here is an obvious example to show what the MDS algorithm does:
Here are the distances between ten cities in the U.S. placed in a matrix. Notice that the matrix is really a triangle matrix that is repeated symmetrically across the diagonal. The distances are the proximities can be thought of as creating a dissimilarity matrix.
When the data is run through an MDS algorithm using 2 dimensions it constructs a map based on the proximities. Since there is no error in the data, it reconstructs a map that shows the relative locations of the cities. Move the mouse over the figure and you will see how we can interpret the map.
Notice that the dimensions, east/west and north/south, do not align with the dimensions from the configuration. The algorithm uses its own rules for constructing the coordinates.However, because the data has no error, the interpreted dimensions are orthoginal (at right angles). This may not be the case for other data.
Here is another example:
In this experiment, 150 subjects were presented with pairs of letters in Morse code. The task was to say whether the two signals were the same or different. Because the subjects were unfamiliar with Morse code, they confused signals that they found to be similar. The data formed a confusion matrix. The more times signals were confused, the greater the number inthe cell that represented that pair of signals. So you can think of the confusion matrix as a similarity matrix (larger values indicate greater similarity).
Presented in the figure is the configuration based on a two dimensional MDS analysis. See if you can interpret the configuration. Then click on the figure to see if your interpretation corresponds to the experimenter's.
Notice how this the interpretation of this configuration is more qualitative and does not have specific numerical axes.
OK, here is one more example. These are the mean raings from 18 subjects who judges the simialrity of all 66 (12*11/2) pairs of nations on a 9 point scale ranging from 1-"very different" to 9-"very similar".
Roll the mouse over the configuration to see the interpretation of the experimenters. (This was done a number of years ago so the results may reflect a previous time in history.) Again, the interpreted dimensions are not numerical but more qualitative and the dimensions do not correspond to the cartesian dimensions of the configuration. The experimenters made the interpreted dimensions orthoginal, but that may not necessarily be the truth, especially if these dimensions are not truly independent (or even correctly inferred).
Now let's move on to theory.