Lead Concentration vs. Setback Distance Given Day-of-the-Week, Week, and Height
Below is a trellis display of lead concentration against setback distance given day-of-the-week (thu-wed), week (1-3), and height (3 values). There are 63 panels arranged into 31 columns and 3 rows. Each row conditions on a different value of height; as we go from bottom to top, the heights increase. The panels in each row are in time order because the panels first cycle through the days of the week and then through the weeks.
The display reveals much about the structure of the data. There is a strong interaction between height and setback distance. For the lowest height, lead decreases with setback. But for the middle value of height, lead typically first increases with setback and then decreases. For the highest height, lead occasionally has the increase-decrease pattern for about 1/3 of the days, most of them days with large concentrations, and is relatively stable for the remaining days. This behavior is consistent with air transport mechanisms. Lead is emitted at ground level from automobile tail pipes. The closest of the 9 monitors, the one with the lowest height and the closest setback, has the largest concentrations because it is close to the pollution source. From the source, the lead is carried laterally by the wind, spreading upward as it moves. This plume-like behavior can cause the concentrations to be relatively small at the higher monitors at the closest setback.
Barley Yield vs. Variety and Year Given Site
The following figure is a Trellis display of data from an agricultural field trial to study the crop barley. At six sites in Minnesota, ten varieties of barley were grown in each of two years. The data are the yields for all combinations of site, variety, and year, so there are 6 X 10 X 2 = 120 observations. Each panel in the figure displays the 20 yields at a single site.
The barley experiment was run in the 1930s. The data first appeared in a 1934 report published by the experimenters. Since then, the data have been analyzed and re-analyzed. R. A. Fisher presented the data for five of the sites in his classic book, The Design of Experiments. Publication in the book made the data famous, and many others subsequently analyzed the them, usually to illustrate a new statistical method.
Then in the early 1990s, the data were visualized by Trellis Graphics. The result was a big surprise. Through 60 years and many analyses, an important happening in the data had gone undetected. The above figure shows the happening, which occurs at Morris. For all other sites, 1931 produced a significantly higher overall yield than 1932. The reverse is true at Morris. But most importantly, the amount by which 1932 exceeds 1931 at Morris is similar to the amounts by which 1931 exceeds 1932 at the other sites. Either an extraordinary natural event, such as disease or a local weather anomaly, produced a strange coincidence, or the years for Morris were inadvertently reversed. More Trellis displays, a statistical modeling of the data, and some background checks on the experiment led to the conclusion that the data are in error. But it was Trellis displays such as the above figure that provided the ``Aha!'' which led to the conclusion.
The top panel graphs the yearly sunspot numbers from 1849 to 1924. The aspect ratio, the height of the data region of the graph divided by the width, is 1.0. An aspect ratio of 1.0 is what you might expect to see as a default in cases where aspect ratio has not been considered. But the graph fails to reveal an important property of the cycles. In the bottom panel, the data are graphed again, but this time the aspect ratio has been chosen by a trellis algorithm called banking to 45 degrees. Now the property is revealed. The sunspot cycles typically rise more rapidly than they fall; this behavior is pronounced for the cycles with high peaks, is less pronounced for those with medium peaks, and disappears for those cycles with the lowest peaks. In the top panel, the aspect ratio of 1.0 prevents an accurate visual decoding of the slopes of the line segments connecting successive observations. In the bottom panel, banking allows a more accurate visual decoding of the slopes.