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How MDS Works

You’ll sometimes see multidimensional scaling called smallest-space analysis. That’s because MDS programs work out the best spatial representation of a set of objects that are represented by a set of similarities. Suppose, for example, that you measure the distance, in miles, among three cities, A, B, and C. The matrix for these cities is in the inside box of table 16.3.

Clearly, cities A and C are closer to one another than are A and B, or B and C. You can represent this with a triangle, as in figure 16.4a.

In other words, we can place points A, B, and C on a plane in some position relative to each other. The distance between A and B is longer than that between A and C (reflecting the difference between 40 and 50 miles); and the distance between B and C is longer than that between A and C (reflecting the difference between 40 and 80 miles).

A rule in graph theory says that you can plot the relations among any set of relations

FIGURE 16.3.

Two-dimensional MDS for 18 fruits sorted by one informant.

in n-1 dimensions, where n is the number relations. With three cities, there are three relations: AB, AC, and BC, so it’s easy to plot all the distances in proper proportion to one another in a two-dimensional graph. In fact, figure 16.4a contains precisely the same information as the inside box of table 16.3, but in graphic form. You can see in figure 16.4a that the physical distance (in inches) between B and C is twice that of A and C.

If we add a fourth city, things get considerably more complicated. With four items,

Table 16.3 Matrix of Distances among Four Cities

A

B

C

D

A

X

50

40

110

B

X

80

45

C

X

115

D

X

FIGURE 16.4.

Two-dimensional plot of the distance among three cities (a) and among four cities (b).

there are six relations to cope with: AB, AC, AD, BC, BD, and CD. These relations are shown in the large box of table 16.3. We can plot the relations among these six pairs perfectly in a graph of n — 1 = 5 dimensions, but what would we do with a 5-dimensional graph? Instead, we try 2 dimensions and see if the distortion is acceptable.

MDS programs produce a statistic that measures this distortion, or ‘‘stress,’’ as it’s called, which tells us how far off the graph is from one that would be perfectly proportional. The lower the stress, the better the solution. Table 16.4 shows the road distance, in miles, between all pairs of nine cities in the United States. Most researchers will accept a stress of <0.15 in an MDS graph. The MDS graph produced from the set of relations in table 16.4, and shown in figure 16.5, has a stress of about zero because table 16.4 contains metric data—reasonably accurate measures of a physical reality. In this case, it’s distance between points on a map (box 16.3).

Table 16.4 Distances between Nine U.S. Cities (in Miles)

BOS

NY

DC

MIA

CHI

SEA

SF

LA DEN

Boston

0

NY

206

0

DC

429

233

0

Miami

1504

1308

1075

0

Chicago

963

802

671

1329

0

Seattle

2976

2815

2684

3273

2013

0

SF

3095

2934

2799

3053

2142

808

0

LA

2979

2786

2631

2687

2054

1131

379

0

Denver

1949

1771

1616

2037

996

1037

1235

1059 0

SOURCE:

Anthropac 4.0 and Anthropac 4.0 Methods Guide, by S.

P Borgatti, 1992a, 1992b. Reprinted with

permission of the author.

Figure 16.5 looks suspiciously like a map of the United States. All nine cities are placed in proper juxtaposition to one another, but the map looks sort of upside-down and backward. If we could only flip the map over from left to right and from bottom to top. . . . Multidimensional scaling programs are notoriously unconcerned with details like this. So

BOX 16.3

TWO KINDS OF PROXIMITIES

The numbers in table 16.4 are dissimilarities, not similarities. Recall from chapter 15 that in a similarity matrix, bigger numbers mean that pairs of things are closer to each other—more like each other—and smaller numbers mean that things are farther apart—less like each other. In a dissimilarity matrix, bigger numbers means that pairs of things are farther apart—less like each other—and smaller numbers mean that things are closer to each other—more like each other. And recall, too, that similarity and dissimilarity matrices are known collectively as proximity matrices because they tell you how close or far apart things are.

FIGURE 16.5.

Two-dimensional MDS solution for the numbers in table 16.4.

long as they get the juxtaposition right, they’re finished. Figure 16.5 shows that the program got it right. (Obviously, you can rotate any MDS graph through some angle about any axis and it will still have the same meaning. Think about a map of the surface of the Earth from the perspective of someone who is looking out from 100 miles inside the Earth. It would be the same map that you’re accustomed to, but flipped.)

This means that you can use MDS to create outline maps of your own. To map a village, you would measure the distance between all pairs of some set of points, like a church, a well, a school, and so on. A set of 10 points means 45 pairs and 45 measurements. The more pairs of points you measure the more accurate the map will be, but, as you can see from figure 16.5, even a set of nine points (and 36 measurements) gets you a basic outline map of the United States (Further Reading: multidimensional scaling).

 
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