A while back my colleague Jim Frost wrote about applying statistics to decisions typically left to expert judgment; I was reminded of his post this week when I came across a new research study that takes a statistical technique commonly used in one discipline, and applies it in a new way.
The study, by paleontologist Zhijie Jack Tseng, looked at how the skulls of bone-cracking carnivores--modern-day hyenas--evolved. They may look like dogs, but hyenas in fact are more closely related to cats. However, some extinct dog species had skulls much like a hyena's.
Tseng analyzed data from 3D computer models of theoretical skulls, along with those of existing species, to test the hypotheses that specialized bone-cracking hyenas and dogs evolved similar skulls with similar biting capabilities, and that the adaptations are optimized from an engineering perspective.
This paper is well worth reading, and if you're into statistics and/or quality, you might notice how Tseng uses 3D surface plots and contour plots to explore his data and explain his findings. That struck me because I usually see these two types of graphs used in the analysis of Design of Experiments (DoE) data, when quality practitioners are trying to optimize a process or product.
Two other factors make this even more cool: Tseng used Minitab to create the surface plots (sweet!), and his paper and data are available to everyone who would like to work with them. When I contacted him to ask if he'd mind us using his data to demonstrate how to create a surface plot, he graciously assented and added, "In the spirit of open science and PLoS ONE's mission, the data are meant for uses exactly like the one you are planning for your blog."
So let's make (and manipulate) a surface plot in Minitab using the data from these theoretical bone-cracking skulls. If you don't already have it, download our 30-day trial of Minitab Statistical Software and follow along!
Creating a 3D Surface Plot
Three-dimensional surface plots help us see the potential relationship between three variables. Predictor variables are mapped on the x- and y-scales, and the response variable (z) is represented by a smooth surface (surface plot) or a grid (wireframe plot). Skull deepening and widening are major evolutionary patterns in convergent bone-cracking dogs and hyaenas, so Tseng used skull width-to-length and depth-to-length ratios as variables to examine optimized shapes for two functional properties: mechanical advantage (MA) and strain energy (SE).
So, here's the step-by-step breakdown of creating a 3D surface plot in Minitab. We're going to use it to look at the relationship between the ratio of skull depth to length (D:L), width to length (W:L), and skull-strain energy (SE), a measure of work efficiency.
- Download and open the worksheet containing the data.
- Choose Graph > 3D Surface Plot.
- Choose Surface, then click OK.
- In Z variable, enter SE (J). In Y variable, enter D:L. In X variable, enter W:L.
- Click Scale, then click the Gridlines tab.
- I'm going to leave them off, but if you like, you can use Show gridlines for, then check Z major ticks, Y major ticks, and X major ticks. Adding the gridlines helps you visualize the peaks and valleys of the surface and determine the corresponding x- and y-values.
- Click OK in each dialog box.
Minitab produces the following graph:
The "landscape" of the 3D surface plot is illuminated in places so that you can better see surface features, and you can change the position, color, and brightness of these lights to better display the data. You also can change the pattern and color of the surface. You can open the "Edit Surface" dialog box simply by double-clicking on the landscape. Here, I've tweaked the colors and lighting a bit to give more contrast:
Turn the Landscape Upside-Down
You may not want to go so far as to flip it, but rotating the graph to view the surface from different angles can help you visualize the peaks and valleys of the surface. You can rotate the graph around the X, Y, and Z axes, rotate the lights, and even zoom in with the 3D Graph Tools toolbar. (If you don't already see it, just choose Tools > Toolbars > 3D Graph Tools to make it appear.)
By rotating 3D surface and wireframe plots, you can view them from different angles, which often reveals interesting information. Changing these factors can help reveal different features of the data surface and dramatically impact what features are highlighted:
Off-Label Use of the Surface Plot?
Tseng notes that combining biomechanical analysis of the theoretical skulls and functional landscapes like the 3D surface plot is a novel approach to the study of convergent evolution, one that permits fossil species to be used in biomechanical simulations, and also provides comparative data about hypothesized form-function relationships. What did he find? He explained it this way in an interview:
What I found, using models of theoretical skulls and those from actual species, was that increasingly specialized dogs and hyenas did evolve stronger and more efficient skulls, but those skulls are only optimal in a rather limited range of possible variations in form. This indicates there are other factors restricting skull shape diversity, even in lineages with highly directional evolution towards biomechanically-demanding lifestyles...although the range of theoretical skull shapes I generated included forms that resemble real carnivore skulls, the actual distribution of carnivoran species in this theoretical space is quite restricted. It shows how seemingly plausible skull shapes nevertheless do not exist in nature (at least among the carnivores that I studied).
In addition to 3D surface plots, Tseng used contour plots to help visualize his theoretical landscapes. In my next post, I'll show how to create and manipulate those types of graphs in Minitab. Meanwhile, please be sure to check out his paper for the full details on Tseng's research: