| # Senate Voting: Scoring Senators We consider the data matrix containing the votes of the Senators in the 2004-2006 US Senate (2004-2006) introduced here. ## Scoring function We are interested in assigning a ‘‘score’’ to each Senator, and thus represent all the Senators as a single value on a line, using the method described here. We will project the data along a vector in the ‘‘bill’’ space, which is . That is, we are going to form linear combinations of the bills, so that the votes for each Senator are reduced to a single number, or ‘‘score’’. Our scoring function takes the form where, is the -vector of average votes (across Senators). Here, wihtout loss of generality, the direction is normalized (). ## Computing the scores To compute the scores, we first center the data: where is the vector of ones in . Then, we compute the (row) vector of scores: ## Example: behavior with respect to average bill Choosing to be a (normalized) vector of ones: (Our normalization ensures that the Euclidean norm of is , and allows to compare several directions. The scaling factor is actually irrelevant here.) The direction corresponds to trying to understand the voting behaviors of the Senators in terms a single, synthetic, ‘‘average’’ bill. The scores we get this way correspond to an ‘‘extremism/conformism index’’, since Senators with a low score tend to vote along the average (on the ‘‘average’’ bill), while those with a high score would tend to vote opposite to it. With that choice, here is what we get:
| Average vs. extreme scores on average bill: This image shows the values of the projections of the Senator’s votes (that is, with average across Senators removed) on the (normalized) ‘‘average bill’’ direction defined above. The party affiliation of each Senator is shown, with names of Democrats in blue and those of Republicans in red. Based on this crude analysis, we could be led to conclude that the former tend to vote more according to the average, while those of the former tend to vote less according to the average.