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Linear Algebra and Legislative Voting Behavior: Rice's Indices

L legislative voting behavior has been extensively researched by behavioral scientists using a variety of quantitative indices and mathematical techniques. Some of these indices and techniques have been used for decades, but their essentially mathematical properties have been subjects of debate or have not been thoroughly explored. Stuart Rice's index of cohesion and index of likeness, for example, have been used continuously since the 1920s but have rarely been investigated from a mathematical point of view.' Despite continuous use since the 1920s, Rice's indices suffer from a well-known flaw, viz., legislators who neither vote aye nor vote nay are omitted. This shortcoming hardly needs elaboration, but two aspects deserve notice. (i) Empirically, the number of omitted legislators may be rather large in relation to the total size of the party or other subset of legislators being studied. An extreme example is furnished by the 1910 British House of Commons. During the session of July 18, on division number 106, the vote was 49 ayes, 1 nay, and approximately 620 MPs did not vote. (There were 670 seats in the House of Commons, but a few were vacant as a result of deaths, resignations, etc.) The low turnout may be amusingly explained by the fact that the division occurred at 3 A.M. in the morning of July 19. During the 1910 Parliament, however,