A Diamond-Shaped Equiponderant Graphical Display of the Effects of Two Categorical Predictors on Continuous Outcomes
Journal article by Xiuhong Li, Jennifer M. Buechner, Patrick M. Tarwater, Alvaro Munoz; The American Statistician, Vol. 57, 2003
Journal Article Excerpt
A diamond-shaped equiponderant graphical display of the effects of two categorical predictors on continuous outcomes. by Xiuhong Li , Jennifer M. Buechner , Patrick M. Tarwater , Alvaro Munoz 1. INTRODUCTION Three-dimensional (3-D) bar graphs are commonly used in biomedical studies to portray how two categorical variables (predictors, risk factors) jointly contribute to an outcome (Klag et al. 1996; Huang et al. 1997; Farahmand et al. 2000). However, most 3-D bar graphs fail to achieve the desired feature of equally representing the relationships between the outcome variable and each of two predictors, including fixing (conditioning on) one predictor to examine the relationship of the other predictor and the outcome. In addition, 3-D bar graphs are prone to misinterpretation and misperception, and are limited to data that exhibit nonoverlapping trends (i.e., none of the outcomes in joint categories is concealed by others), as addressed by Cleveland and McGill (1984), Wilkinson (1999), and Harris (1999). To overcome these shortcomings, two-dimensional (2-D) alternatives have been used, such as mosaic (Hartigan and Kleiner 1981, 1984; Friendly 1994; Wilkinson 1999), grouped bar graphs (Tufte 1983), grouped dot plots and framed rectangle charts (Cleveland and McGill 1984), and Trellis display (Becker, Cleveland, and Shyu 1996). However, none of these 2-D displays can equally present the relationships between a continuous outcome and each of two categorical predictors in a single plot. This article proposes a graphing methodology that projects 3-D bar graphs into 2-D whereby the third dimension is replaced with a polygon whose area and middle vertical and horizontal lengths represent the outcome. The proposed graphical representation is invariant to rotations and avoids outcomes in categories being concealed by others. Therefore, our method circumvents limitations of both 3-D bar graphs and current 2-D alternatives, while preserving a desired feature of 3-D bar graphs. 2. METHODS The key idea in our proposal is to replace the parallelepiped volume in 3-D bar graphs with a polygon in each cell of the 2-D grid defined by categories of the two predictors. To achieve equal representation for two predictors, we choose to use square cells rotated 45[degrees] clockwise to construct the grid resembling a diamond (Figure 1). The result is a diamond square cell that not only produces a more aesthetically pleasing graph, but also facilitates the correct display of data because it gives both predictors equal importance. Once the square cells are established, the volumes of the parallelepipeds are compressed into the areas of a polygon within each cell (i.e., three dimensions are projected down to two dimensions). In addition, given that the perception of areas is more difficult than the perception of lengths (Cleveland and McGill 1984), we aim to select a polygon whose middle vertical and horizontal lengths also represent the outcome. [FIGURE 1 OMITTED] Our proposal is suitable for a variety of data types (e.g., proportions, incidence rates, relative risks). To facilitate the description of our proposal we use the case where the outcome is proportional data. First of all, to achieve equal representation of the two predictors, we restrict attention to polygons centered around the middle of the squares in the grid defined by the categories of two predictors. Figure 1 shows 2-D representations of proportional data (p = 0.10, 0.25, 0.50, and 0.75) using five methods (A, B, C, D, and E). The upper left-hand panel corresponds to simply providing the values of the percentages of the areas of the four square cells in each panel occupied by the shaded polygons. Each of the five methods provides a graphical representation that is invariant to rotation and avoids outcomes being concealed by others. We used S-Plus to develop a function for drawing the square cells of the grid and the shaded areas defining the polygons. In our function, p represents the proportion describing the outcome in a given cell and l is the length of the diagonal for each square cell. The shaded polygons in Figure 1 are constructed based on three attributes: (1) the middle vertical length of the polygons (V (p)); (2) the middle horizontal length of the polygons (H(p)); and (3) the horizontal length of the top and bottom of the polygons (h(p)). The areas of the shaded polygons relative to the area of the square cell are given by V(p)[H(p) + h(p)].Table 1 illustrates the characteristics of the shaded polygons for all five methods in Figure 1 when l is assumed to be one. Although it is apparent from Table 1 that all five methods correctly display the shaded polygons' ... |
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