by Ryan Parker, Research Statistician Developer, SAS
Note: To fit models using the mixed model personality, select Analyze > Fit Model and then select Mixed Model from the Personality list.
With JMP Pro 12, we have improved fitting mixed models, with noticeable changes coming to models with spatial covariance structures. These changes are detailed below.
Fitting models with a spatial covariance requires a high computing cost, particularly as the number of observations increases. For JMP Pro 12, the algorithm we use to fit models in the Fit Mixed platform has been improved by Chris Gotwalt, the Director of Statistical Research and Development at JMP. His work further reduced the number of computations required to fit all models in Fit Mixed. This has greatly reduced the time required to fit spatial models in particular.
In our testing, we have experienced anywhere from a 5 to 15 times increased speed when fitting these spatial models. In practical terms, this means we have reduced fits that took more than an hour to complete before to now being completed in just over 10 minutes.
JMP Pro 12 provides the ability to create variograms, visual tools to examine and diagnose the existence of spatial correlation. The variogram is available when you fit an isotropic spatial model -- an AR(1) or one of Power, Exponential, Gaussian, or Spherical with or without a nugget). The variogram in Figure 1 visualizes the change in covariance as observation locations move apart in time or space.
Variogram for examining the existence of spatial correlation
The variogram plots the semivariance (half the variance) of the difference between observations at two locations versus the distance between the locations. As depicted in Figure 1, we can see that observations close together have strong correlation, hence a small semivariance.
This correlation decreases as distance increases, leading the semivariance to increase. The semivariance no longer increases at what is known as the range, the distance at which observations have little to no correlation. In the variogram above, the distance is approximately 0.25.
The maximum value of the semivariance is known as the sill, and in most cases with the models fit in JMP Pro, this is simply the variance of the observations. In the presence of non-spatial error, a nugget can be included to estimate this variability. The nugget effect is depicted in the variogram shown in Figure 2.
Figure 2 Nugget effect shown in a variogram
The semivariance is zero at distance zero, but the semivariance then jumps up to the nugget as soon as the distance between locations is positive. In the variogram above, the nugget is approximately 0.5.
The variogram is also available when you fit a residual-only model. You can access the variogram option under Marginal Model Inference (Figure 3).
Variogram option for residual-only models
After selecting this option and specifying the column(s) to use for computing distance in time or space, the empirical variogram is shown. The variogram red triangle menu allows you to fit one or more spatial models to this empirical variogram (Figure 4).
Selecting all spatial models
Figure 5 shows the spatial models.
All spatial models
You can use the variogram shown in Figure 5 to determine if a spatial covariance should be added to the residual only model. Under the absence of spatial correlation, the empirical variogram would likely appear flat, suggesting no change in correlation over distance.
In Figure 5, we see that there appears to be an increase in the empirical variogram as distance increases, and all of the models may be suitable. After using the Fit Mixed platform to formally fit a model to each one of these correlation structures, we find that the Exponential model is preferred because it has the smallest corrected Akaike Information Criterion (AICc) and Bayesian Information Criterion(BIC). This means that the variogram should be used to identify possible models that can then be formally fit with Fit Mixed. With these fits, you should use an AICc/BIC comparison to choose the best model.