Matrix manipulation and interpretation of data related to AIDS transmission

Abstract

This research describes how a currently available spreadsheet program can be set up to analyze sociographic data. An algorithm developed by Jamrich (1960) is presented, along with both the mathematical and the spreadsheet equations. An example of the application of the algorithm and equations to AIDS transmission data is presented.Ever since Moreno (1934) introduced sociometry to the behavioral sciences, it has been utilized in many types of investigations. In Moreno's original conception, sociometric choices were represented by a sociograph. In this representation, a circle usually indicated an individual and arrows connecting individuals indicated the choices that were made. Arrows pointing away from a subject, toward another individual, usually meant that this individual was chosen by the subject. By using appropriate symbols, we may indicate that A chooses B, rejects B, or ignores B. The problem with sociographs was that they were not unique representations of the data. A sociograph may be drawn in many ways, and, when a number of relationships are considered, we may obtain a number of crossing lines that confound interpretation.One of the areas of refinement in sociometry has been the development of techniques for the matrix representation of sociometric data. A matrix is well suited for this purpose. The matrix can be arranged with the subjects listed in the same order, both down the side and across the top. If we designate the rows as the choices made by the subjects, we may represent each sociometric choice made by an individual by putting a "1" in the column associated with the person he/she chooses; otherwise, we leave the cell blank. If we sum down the columns, we have the number of sociometric choices received by the individual. Unlike the sociograph, this representation of the data is unique. Many elaborate schemes for the utilization of sociometric data have been derived (Beum & Brundage, 1950;Glanzer & Glaser, 1959). Some of them, like Beum and Brundage's procedure, depend on the hierarchical rearrangement of a matrix. Another approach has involved raising matrices to some power. This technique was introduced by Festinger (1949) and elaborated on by Luce and Perry (1949) and by Luce (1950). A modification of the Festinger procedure has been suggested by Jamrich (1960).Correspondence should be addressed to Michael Yost, Office of the President, Trinity University, 715 Stadium Drive, San Antonio, TX 78212.Jamrich reasoned that in a squared matrix we have second-order relationships, which can be interpreted as second-order communications. In other words, Tom contacts Bob through an intermediary whom we might call Sam. Jamrich argued that in sociometric situations, if we utilize the squared matrix and then add back the original matrix to it, we will have a matrix that weights the choices by the status of the chooser. If the subject receives a nomination from someone who is highly chosen, the sociological implications of this choice are probably different from the implications associated with a nomina...