Back to posting after a long weekend and more than enough rugby coverage to last a few years. Anyway, back to linear models, where we usually assume normality, independence and homogeneous variances. In most statistics courses we live in a fantasy world where we meet all of the assumptions, but in real life—and trees and forests are no exceptions—there are plenty of occasions when we can badly deviate from one or more assumptions. In this post I present a simple example, where we have a number of clones (genetically identical copies of a tree), which had between 2 and 4 cores extracted, and each core was assessed for acoustic velocity (we care about it because it is inversely related to longitudinal shrinkage and its square is proportional to wood stiffness) every two millimeters. This small dataset is only a pilot for a much larger study currently underway.

At this stage I will ignore any relationship between the clones and focus on the core assessements. Let’s think for a moment; we have five replicates (which restrict the randomization) and four clones (A, B, C and D). We have (mostly) 2 to 4 cores (cylindrical pieces of wood covering from tree pith to cambium) within each tree, and we have longitudinal assessments for each core. I would have the expectation that, at least, successive assessments for each core are not independent; that is, assessments that are closer together are more similar than those that are farther apart. How does the data look like? The trellis plot shows trees using a Clone:Rep notation:

Incidentally, cores from Clone C in replicate four were damaged, so I dropped them from this example (real life is unbalanced as well!). Just in case, distance is in mm from the tree pith and velocity in m/s. Now we will fit an analysis that totally ignores any relationship between the successive assessments:

Incidentally, our assessment setup looks like this. The nice thing of having good technicians (Nigel made the tool frame), collaborating with other departments (Electrical Engineering, Michael and students designed the electronics and software for signal processing) and other universities (Uni of Auckland, where Paul—who cored the trees and ran the machine—works) is that one gets involved in really cool projects.

What happens if we actually allow for an autoregressive process?

In ASReml-R this would look like (for the same results, but many times faster):

Oops! What happened to the significance of Clone and its interaction with distance? The perils of ignoring the independence assumption. But, wait, isn’t an AR(1) process too simplistic to model the autocorrelation (as pointed out by D.J. Keenan when criticizing IPCC’s models and discussing Richard Mueller’s new BEST project models)? In this case, probably not, as we have a mostly increasing response, where we have a clue of the processes driving the change and with far less noise than climate data.

Could we improve upon this model? Sure! We could add heterogeneous variances, explore non-linearities, take into account the genetic relationship between the trees, run the whole thing in asreml (so it is faster), etc. Nevertheless, at this point you can get an idea of some of the issues (or should I call them niceties?) involved in the analysis of experiments.