Saturday, November 13, 2004

Information-Theoretic Dependency Analysis

If there were only one criterion in one dimension with which to differentiate the conservatives and liberals out of the four Myers-Briggs factors, then the summary results would be sufficient to answer the question. However, people are not so single dimensional; we expect that the political affiliation decision to be a more complex function of personality.

Let’s first loosen the condition that the differentiation must be a single boundary in one dimension. Let’s allow that along this dimension there could be clusters of liberal and clusters of conservative respondents separated by multiple boundaries. So instead of a measure of central tendency such as the mean or correlation, we need a more general metric of dependency to determine which dimensions of personality tend to separate respondents according to their political affiliation. The measure of choice is the mutual information (MI) between the distributions of the respondents' political affiliation and the respondents' Myers-Briggs scores in each dimension. These totals are shown below where the number of bins is chosen to equally resolve all Myers-Briggs totals for our sample down to the individual question level:

DimensionBinsMI (nats)
Focus 18 0.096
Processing 16 0.142
Decision Making 18 0.176
Organizing 19 0.143
Age 19 0.106
Gender 2 0.074

The objective is to be able to assign each of the uniformly distributed bins to a certain political affiliation of the four different possibilities (weak/strong and conservative/liberal). To do this we want to use that information that has the closest one-to-one relationship with the distribution of the respondent’s political affiliation. This is the dimension with the highest MI value. The total information to be covered in the output dimension of political affiliation is 1.14 nats (natural units). So, while the decision making dimension is still the most potentialy descriptive of political affiliation, it is by no means comprehensive. It is possible that there is complementary information in another dimension that will yield a better discriminant.

Let's now expand the investigation into two dimensions while reducing the number of bins in each dimension to nine (gender is included with two bins) in order that the proportion of total bins to data records remain above three. The first 6 of the 15 combinations in rank order are:

DimensionsMI (nats)
(Processing; Decision Making) 0.391
(Decision Making; Age) 0.379
(Focus; Decision Making) 0.376
(Decision Making; Organizing) 0.363
(Focus; Processing)0.363
(Focus; Organizing) 0.357

For a more direct comparison with the single dimension results let's limit the grid to 4x4 for a total of 16 bins:

DimensionsMI (nats)
(Processing; Decision Making) 0.196
(Processing; Gender) 0.180
(Decision Making; Organizing) 0.173
(Decision Making; Age) 0.157
(Decision Making; Gender) 0.150
(Focus; Decision Making) 0.144
(Organizing; Gender) 0.136
(Focus; Organizing) 0.120
(Focus; Processing)0.117

It is clear that a better differentiation in general can be had with certain combinations of two dimensions rather than just one. In this case the dimensions of processing and decision making should provide us with significant information by which to distinguish many conservatives from liberals. The plot below demonstrates this fact (red diamonds = conservatives; blue squares = liberals):

Compare the simplicity and efficiency of that discriminant to the dual case: