Correspondence Analysis and West Mexico Archaeology: Ceramics from the Long-Glassow Collection

2: Correspondence Analysis of Archaeological Abundance Matrices

67 Chapter Two Correspondence Analysis of Archaeological Abundance Matrices By Jan de Leeuw Introduction vvvCorrespondence analysis (CA) is a technique used to analyze data matrices of nonnegative numbers. CA is related to principal component analysis (PCA) and multidimensional scaling (MDS), that is, it is a form of proximity analysis. CA is most frequently applied to rectangular tables of frequencies, also known as cross tables or contingency tables, although applications to binary incidence or presence-absence matrices are also quite common. The most often used statistical technique for analyzing cross tables computes and tests some measure of independence or homogeneity, such as chi-square. In the analysis of independence we investigate whether the body of the table is the product of the marginals. Or, if one prefers an asymmetric formulation, if the rows of the table differ only because they have different row totals (and the columns only differ because they have different column totals). Pearson’s chi-square and related measures quantify how different an observed table is from an expected table, based on the row and column totals. Pearson residuals are used to investigate deviations from independence. CA supplements this classical chi-square analysis because it makes both a decomposition and a graphical representation of the deviations from independence. Chapter two / 68 History CA has a complicated history, both in statistics and in archaeology. The prehistory of CA, starting with work by Pearson around 1900 and ending with the reinvention of the technique by Fisher and Guttman around 1940, is discussed in De Leeuw 1983. The technique has been re-reinvented under many different names, in many different countries, and in many scientific disciplines. New reincarnations still continue to appear, although at a slower pace than before, in the data mining and data analysis literature. Beh 2004 is a recent comprehensive bibliographic review. The history of CA in archaeology is discussed by Baxter (1994:133– 39). Although the literature contains some earlier applications of CA to archaeological examples, the credit for the introduction of the technique to archaeologists usually goes to Bølviken and others (1982). Early applications almost without exception came from archaeologists in Continental Europe, under the influence, no doubt, of the French analyse des données school and the leadership of Benzécri (1973a, 1973b). A good overview of these various Continental archaeological applications of CA is found in, for example, Müller and Zimmerman 1997. It is clear from Baxter’s discussion that archaeologists in Continental Europe were ahead of archaeologists in Great Britain, who came on board around 1990. Orton (1999:32), one of the deans of quantitative archaeology in Britain, argues that CA was the most important technique introduced into archaeology in the 1980s. From Britain archaeological CA migrated to the United States, where it arrived shortly before 2000. Duff (1996:90) indicates in an influential article from the mid1990s that CA was “not well established in Americanist literature.” And very recently, Smith and Neiman (2007:55) have concurred: “CA has a long history of use by archaeologists in continental Europe but its use by Americanist archaeologists is both more recent and rare.” There are several possible reasons why CA did not rapidly become popular in archaeology in Britain and the United States. Most importantly , perhaps, archaeological methodologists tend to look to statisticians for guidance, and in statistics CA was not really known until about 1980, despite the work of Hill (1974). Except in France, of course, but French statistics was relatively isolated from that of the mainstream . The dominant multivariate techniques applied in archaeology were MDS and PCA (sometimes in the disguise of factor analysis). The Jan de Leeuw / 69 most influential work in the area in the seventies was Hodson et al. 1971, which concentrated on the MDS techniques of Boneva, Kendall, and Kruskal. These are all forms of proximity analysis, but they differ from CA in various ways. In a pioneering article, LeBlanc (1975:22) predicted, “Proximity analysis seems to hold a great deal of promise and will in all probability supplant all other seriation methods.” If we interpret this prediction narrowly, in terms of the methods that were available in 1975, it turned out to be incorrect, for reasons that are quite obvious in hindsight. Data, in archaeology and elsewhere, come in many different forms. Sometimes we deal with cross tables, sometimes with incidence matrices , and sometimes with multivariate data that describe archaeological objects in terms of...