Correspondence analysis provides a way to summarize categorical data in a reduced number of dimensions (Clausen, 1998; Greenacre, 2007). In that sense, it is very similar to principal components analysis. Principal components is an asymmetrical analysis. We use the correlations (or covariances) between the variables as a summary of the structure in the data. The principal components represent a way of describing the correlation matrix in fewer components than variables. The analysis is asymmetrical because we focus on the relationships between variables and use the principal components to compute scores for each of the observations in the new, reduced space.

In correspondence analysis, the data usually consist of counts of different kinds of things. They could be different artifact types from a variety of sites, strata, or features or they could be different elements in the composition of artifacts. Correspondence analysis is a symmetrical analysis because we adjust the data matrix by both the rows (observations) and the columns (variables) before conducting the analysis. As a result, we can project the observations into the space defined by the variables (as with principal components) or the variables into the space defined by the observations. We can also create biplots summarizing both views.

The adjustment of the data matrix is simply a modification of the Chi-square test that we covered in Chapter 9. In the Chi-square test we compute an expected value for a particular cell by multiplying the row sum by the column sum and dividing by the total sum. The difference between the observed and expected values is squared and divided by the expected value to get the Chi-square contribution for that cell. The sum of all the Chi-square contributions is the total Chi-square value that we use to see if the observed counts are significantly different from what we would expect by chance.

To perform a correspondence analysis, we modify that procedure slightly. First, we divide every value in the table by the sum of all the entries so that each cell represents the proportion of the total found in that cell. Then we compute the expected proportions using the row and column sums of the table of proportions.

Raw data comes in many sizes and shapes and occasionally they are the wrong sizes and shapes for what we want to do with them. In those situations, it can be useful to transform them before analysis. Transforming data is often useful to balance a non-symmetric distribution or to pull in outlying observations to reduce their influence in the analysis. Transformations can be applied down columns (e.g., standard scores to weight each variable equally) or across rows (e.g., percentages to weight each assemblage equally). In general, there are four data problems that can sometimes be resolved with transformations.

First, transformations can help to produce a distribution that is closer to a normal distribution, making it possible to use parametric statistical methods (such as t-tests). In this case, we are looking at the raw data distribution and using an order-preserving transformation that makes the data more symmetrical. The alternative to transforming the data is to use nonparametric tests that do not require a normal distribution or robust statistical methods that are not as influenced by extremely large or small values.

Second, transformations can make it possible to use simple linear regression to fit nonlinear relationships between two variables. Transforming one or both variables makes the relationship between them linear. The drawback with this approach is that the errors are transformed as well so that additive errors become multiplicative errors when using a log transform. The alternative to transformation is to use nonlinear regression.

Third, transformations can be used to weight variables equally so that differences in measurement scales or variance do not give some variables more influence than others in the analysis. This is particularly important when we are using the concept of “distance” between observations (Chapter 14).

Fourth, transformations can be used to control for size differences between assemblages or specimens that we want to exclude from the analysis in order to focus on shape or relationships between variables that are independent of differences in size. In this case the transformation is applied to the rows of the data. First, we will consider a collection of R functions that are useful for a number of purposes, including transformation.

An integral aspect of archaeological data is that they come from particular places. We often want to examine the distribution of artifacts, sites, or features over space and R provides a number of tools for this purpose. We may also be interested in the direction or orientation of the object, house, or feature. This chapter will cover some of the basics, but there are specialized R packages for mapping and for analyzing gridded and point data. If most of your analysis involves spatial data it may be easier to use a geographic information system (GIS) package, but R can handle shapefiles and other data structures that are produced by those packages and it provides extensive support for statistical analysis of spatial data. In this chapter we will cover directional statistics, creating simple distribution maps based on gridded or piece plotted data.

CIRCULAR OR DIRECTIONAL STATISTICS

Circular statistics include direction and orientation (Gaile and Burt, 1980; Jammalamadaka and Sengupta, 2001; Mardia and Jupp, 2000). If we are interested in the direction of something (for example burials or rock shelter openings), then we are using directional data. In general, this is recorded in degrees measured clockwise from north, but it can also include cyclical data where the cycle repeats daily, weekly, monthly, or yearly. In other cases, we are interested in the orientation of an elongated flake, blade, or bone fragment. Orientation can be defined as north/south or east/west so we are only using half of the circle since 0° and 180° or 90° and 270° are the same orientation. With bone fragments, for example, we usually cannot identify which end is the front and which is the back so we are working with orientation. With blades, we could define the platform end as the front in which case we could measure direction rather than orientation, but often only the orientation is recorded. The research question under consideration will help to make the decision between direction and orientation. Analytically, the first step with orientation data is to double each value and analyze it as directional data.

We have discussed the concepts of multivariate spaces and distances in earlier chapters. With discriminant analysis, we created a space that separated groups of observations. With principal components, we defined a multivariate space for the observations in fewer dimensions while losing the least amount of information. With correspondence analysis, we displayed observations and variables in terms of their Chi-square distances. This chapter is the first of two that focus on quantitative methods that start with a distance matrix and represent the distances between observations either in the form of a map (this chapter) or by grouping observations that are similar (Chapter 15). In both cases we generally start by computing a distance or similarity measure between pairs of observations.

The first part of the chapter describes different ways of defining distance and similarity. Different choices in the measurement of distance can have substantial influence on the results. The second part describes ways of analyzing distance matrices in order to represent them in the form of a map. If you have ever looked at a highway map, you may have noticed a triangular table of distances between major cities. What if you only had the table of distances? How would you go about reconstructing the map? Scaling methods were primarily developed in the field of psychology where data expressing perception or preferences are gathered directly in the form of a similarity matrix that reflects judgments regarding how similar pairs of stimuli are. In archaeology, the classic application is seriation, which attempts to represent variation in assemblages along a single dimension that may represent time (Chapter 17). The third part of the chapter illustrates how to compare two distance matrices. For example, if we have sites located in a region and collections of ceramics from those sites, how do we compare the geographic distances to ceramic assemblage distances?

DISTANCE, DISSIMILARITY, AND SIMILARITY

The term distance refers to a numeric score that indicates how close or far two observations are in terms of a set of variables. Larger distances mean the observations are less similar to one another. Dissimilarity measures are larger when two objects are more distant or different from one another. Similarity measures are larger when two objects are more like one another.