Proximities

Overview

A proximity is a measurement of the similarity or dissimilarity, broadly defined, of a pair of objects. If measured for all pairs of objects in a set (e.g. driving distances among a set of U.S. cities), the proximities are represented by an object-by-object proximity matrix, such as the following distance matrix:

A proximity is thought of as a similarity if the larger the value for a pair of objects, the closer or more alike we think they are. Examples of similarities are co-occurrences, interactions, statistical correlations and associations, social relations, and reciprocals of distances. A proximity is a dissimilarity if the smaller the value for a pair of objects, the closer or more alike we think of them. Examples are distances, differences, and reciprocals of similarities.

Proximities are normally symmetric, so that the proximity of object a to object b is the same as the proximity of object b to object a. For example, the distance from Boston to NY is 206 miles, and the distance from NY to Boston is also 206 miles. However, in the case of one-way streets, it is possible for distances to be non-symmetric.

There are two basic ways of obtaining proximity: directly (or dyadically) and indirectly (or monadically). Direct measures are obtained in the obvious way. For example, a direct measure of distance between cities is obtained by driving from one city to the other. A direct measure of interaction between two people is obtained by counting the number of times that they speak to each other over a given period.

Indirect measures are obtained by first measuring the objects on one or more attributes. This is recorded as a 2-way, 2-mode object-by-attribute matrix. The set of scores associated with an object or an attribute (that is, a row or a column of the data matrix) is called a profile. Then, a statistical measure of the similarity or dissimilarity of profiles is computed for each pair of objects or attributes (i.e., each pair of rows or columns of the data matrix).

In many situations, the objects are thought of as cases and the attributes are seen as variables.

Hundreds of measures are available. The choice of measure is determined in part by the type of data (see Figure 1). For categorical data, the typical measure is the match coefficient, which, for a given pair of objects, is simply the count of the number of times (attributes/columns) that one object has the exact same value as the other object. Typically, this count is then divided by the maximum possible, which is normally the total number of attributes/columns in the data (that both objects have non-missing values for). For example, if the people are objects

For quantitative data, two measures are commonly used, one a similarity measure (correlation) and the other a dissimilarity measure (euclidean distance). Typically, Euclidean distance is only used to measure proximities among cases (generally respondents), whereas correlation tends to be used to measure proximities among variables (generally attributes of the respondents).

The key issue in choosing a measure of proximity for quantitative data is what aspects of profiles we would like to the measure to attend to. Every profile can be said to possess three aspects: level, amplitude (or scatter), and pattern. Level refers to the general size of the numbers and is measured by the mean of all the values. Amplitude refers to the extremeness or variability of the numbers and is measured by the standard deviation. Pattern refers to the sequence of ups and downs in the values as we move from case to case. It is not measureable in isolation. We can ask whether two profiles have the same pattern, and even how different they are from each other, but there is no monadic measurement of pattern.

The Euclidean distance between two profiles is a function of differences in mean, differences in amplitude, and differences in pattern, all take together. Only if two profiles are the same across all three aspects will Euclidean distance say they are the same. In contrast, correlation ignores differences in level and amplitude, and pays attention only to differences in pattern. For example, if we were to measure the income in dollars of a sample of people, then change the units to thousands of dollars (so that $15,500 becomes $15.5), the amplitude of the variable would be reduced by a factor of 100, but the correlation between the two versions of income would be a perfect 1.0.

The reason why Euclidean distance is typically not used for comparing variables is that variables often have wildly different units of measurement. If we compare respondents' income (in dollars) with their education (in years), we will find a massive Euclidean distance between the variables, even if their patterns are identical (that is, when one variable is high relative to other cases, the other variable is high relative to other cases, and vice-versa).

So the only time we use Euclidean distances is when differences in scale (i.e., level and amplitude) are meaningful. For example, suppose our data consist of demographic information on a sample of individuals, arranged as a respondent-by-variable matrix. Each row of the matrix is a profile of m numbers, where m is the number of variables. We can evaluate the proximity (in this case, the distance) between any pair of rows. Now, consider what it means, for a moment, that the variables are the columns. A variable records the results of a measurement. For our purposes, in fact, it is useful to think of the variable as the measuring device itself. This means that it has its own scale, which determines the size and type of numbers it can have. For instance, the income measurer might yield numbers between 0 and 79 million, while another variable, the education measurer, might yield numbers from 0 to 30. The fact that income numbers are larger in general than the education numbers is not meaningful because the variables are measured on different scales. In order to compare columns we must adjust for or take account of differences in scale. But the row vectors are different. If one case has larger numbers in general then another case, this is because that case has more income, more education, etc. than the other case; it is not an artifact of differences in scale, because rows do not have scales: they are not even variables. In order to compute similarities or dissimilarities among rows, we do not need to (in fact, must not) try to adjust for differences in scale. Hence, euclidean distance is usually the right measure for comparing cases.

Euclidean Distance

Euclidean distance is defined as the square root of the sum of squared differences between two profiles. For example, the Euclidean distance between profiles A and B below is 30 (1+1+1+1+0+4+16+1+1+4).

Note that Euclidean distance is not clearly bounded -- it runs from zero (when the profiles are identical) to an unknown maximum. Furthermore, it is sensitive to the scale of numbers (the level and amplitude). If we were to add 10 to every value in profile A, or multiply every value by 10, the Euclidean distance between the profiles would increase.

Pearson Correlation

The correlation between profiles X and Y is defined as follows:

where µ_X and µ_Y are the means of X and Y respectively, and _X and _Y are the standard deviations of X and Y. The numerator of the equation is called the covariance of X and Y, and is the difference between the mean of the product of X and Y subtracted from the product of the means. Note that if X and Y are standardized, they will each have a mean of 0 and a standard deviation of 1, so the formula reduces to:

Whereas euclidean distance was the sum of squared differences, correlation is basically the average product. There is a further relationship between the two. If we expand the formula for euclidean distance, we get this:

But if X and Y are standardized, the sums x² and y² are both equal to n. That leaves xy as the only non-constant term, just as it was in the reduced formula for the correlation coefficient. Thus, for standardized data, we can write the correlation between X^* and Y^* in terms of the squared distance between them:

Hence, for standardized data (where level and amplitude differences are removed), correlation is a simple linear transformation of Euclidean distance squared.

Step-by-Step

1. Collect data for a person-by-method matrix which contains a 1 if a given person has used a given statistical method, and 0 otherwise. Here is a hypothetical example of such a matrix:

Note that these data could be treated as either categorical or quantitative. Furthermore, although the rows appear to be cases and the columns variables, the units of measurement are the same across the board. Consequently, we can use all three measures discussed above.

2. Enter the data into an ascii file called STATMETH.DAT using the following format:

3. Import the data as a dataset called STATMETH.

Proximities Among Persons

1. Choose TOOLS>SIMILARITIES from the menu. Fill in the input form as shown below:

To run the program, press F10. The result should be the following matrix:

As you can see, the similarity between Bill and John is given as 0.57. This is because Bill and John give exactly the same answer (whether "1" or "0" makes no difference) on 4 out of 7 questions, which is 57% to two decimal places.

2. Choose TOOLS>SIMILARITIES from the menu. Fill in the input form as shown below (note change of measure to CORRELATION):

To run the program, press F10. The result should be the following matrix:

Note that the numbers have changed, but the pattern is fairly similar. Looking at Bill's correlations with others we see they are highest with John and Jan, lowest (in fact, negative) with Mary and Don, and in between with Sally. The same is true in the similarity matrix obtained using the match coefficient.

3. Choose TOOLS>DISSIMILARITIES from the menu (note change to dissimilarities). Fill in the input form as shown below.

To run the program, press F10. The result should be the following matrix:

Note that the numbers have not only changed but reversed. Looking at Bill's proximities with others we see they are lowest with John and Jan, highest with Mary and Don, and in between with Sally. This is exactly the opposite of our previous two results.

Proximities Among Methods

To compute proximities among each pair of methods, just repeat the process above, but change "ROWS" to "COLUMNS" in every case. The result in each case will be a method-by-method proximity matrix. For example, in the case of the matches coefficient, the matrix will give the extent to which each pair of methods was used by exactly the same individuals.

References

Methodological

Liebetrau. Measures of Association. Sage.

SPSS, Inc. "Proximities." SPSS Reference Guide. Pp. 550-562.

Applications

Boster & Johnson 1989. "Form or function: A comparison of expert and novice judgements of similarity among fish." American Anthropologist 91:866-889.

Byrne & Forline 1993. "Brazilian emic use of physical cues to ascribe social race identity" Unpublished manuscript, University of Florida.