Dimensional Graphics
The cornerstone of our dimensional thinking is the work on Trellis graphics by William Cleveland, a statistician with Bell Labs in the 80s and 90s (see http://stat.bell-labs.com/wsc/). Subsequent interest has spawned other names for the Trellis approach, including panel, lattice and small multiples. Trellis graph patterns, reminiscent of a garden trellis, depict relationships between two variables (x and y) conditioned on one or more dimensional attributes, and derived from the need to visually investigate complex multivariate relationships with a response variable. Each panel of the trellis represents the graph of a relationship between x and y (the response variable) for a single combination of the conditioning variables. An example of reasonable trellis candidate is the relationship between cholesterol level (x) and the odds of cardiovascular disease (y), conditioned on age category, sex and race. The trellis in this case would depict a separate panel graph for each combination of age category, sex and race that occurred in the data. The individual plots can be many types, including scatter, xy plot, curve plot, quantile, strip plot, and dot plot. A defining characteristic of the Trellis approach is that of common scales for each panel in the overall graph, thus allowing consistent visual comparison between panels. Along with the commonality in panel scales is the flexibility to control the layout of panels to rows, columns and pages, promoting clear graphical comparisons.
In addition to Trellis graphs, we deploy other techniques to help achieve a dimensional effect. Ordering attributes by value levels can promote dimensionality. Superpositioning or overlay, where multiple graphs are combined in a single panel, is a useful dimensional approach, as is a strategic use of colors and shadings to denote different attribute levels. 3-D graphs, often abused in practice, can be invaluable in establishing multivariate patterns. And finally, sometimes opposing views of the same graph, perhaps differing only in plot shading, can provide dimensional insight.
There are several commercial graphics-only packages that support many of the concepts of dimensional graphics detailed here. Tableau Software Version 2.0 supports dimensionality, as does ADVIZOR Solutions Visual Discovery and Spotfire DXP. The commercial statistical package S-Plus from Insightful, and its open source half-brother, R, provide support for all concepts presented in this column and offer extensible programming and data management capabilities as well as comprehensive sets of statistical graphics procedures. The programs and graphs produced for this column were developed in freely available R using the lattice and scatterplot3d libraries. In R as in S-Plus, graphics programming consists of scripts housing straightforward calls to functions with extensive parameter options. The supporting data structures are readily built with R functions and language.
The data used to demonstrate the dimensional concepts consists of stock portfolio returns derived from work at the Center for Research on Security Prices (CRSP) at the University of Chicago and readily accessible from the Web site of Ken French, professor of finance at the Tuck School of Business, Dartmouth College http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/index.html. Professor French, often in tandem with Eugene Fama, professor of finance at the University of Chicago Graduate School of Business, has made seminal contributions to the field of financial economics over the last 30 years. We use data from two files downloaded from French's site for this column: "6 Portfolios Formed on Size and Book-to-Market (2x3)," and "25 Portfolios Formed on Size and Book-to-Market (5x5)." Each file now contains 80 years of portfolio return data. For each file, the x and y variables are date and monthly percentage portfolio return respectively; the dimensions are size, book-to-market value, and weighting. Our charter is to use dimensional graphing techniques to shed light on the performance of these portfolios measured as wealth accumulation or growth of an initial $1 investment over time. We settle on examining the latest 10-year returns for each portfolio reviewed, contrasting performance over time by dimensions weighting, size and book-to-market-value, searching for snippets of intelligence.
A basic lattice xy plot of the 2x3 data is given in Figure 1a. For this illustration, we limit attention to cap-weighted returns. There are six panels in the graph, one for each size-value combination. As is hopefully apparent from the panel strips, the first row represents the latest 10-year returns of large portfolios; the second the returns of small. Similarly, the first column represents growth portfolios, the second neutral and the third value. Note the common scales for each panel, promoting easy comparison across portfolios. Note also the bubble for each growth portfolio reflecting, not surprisingly, the whims of the Internet frenzy time period. These 10-year returns suggest that value trumps growth and small trumps large - findings perhaps coincidental here but consistent with established wisdom. Figure 1b modifies the layout to one row by six columns, offering a second viewpoint for comparison. It is a straightforward parameter change to modify panel layouts across rows, columns and pages, allowing different juxtapositions of the data. Though we show only the current 10-year returns, the functions developed can produce return graphs for any time frames requested.
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