Projected multivariate linear models for directional data

We consider the spherically projected multivariate linear (SPML) model for directional data in the general d-dimensional case. This model treats directional observations as projections onto the unit sphere of unobserved responses from a multivariate linear model. We show that maximum likelihood estimates for the model are readily computed using iterative methods such as the Newton-Raphson and EM algorithms. Under the model we develop three tests of hypotheses, namely, a test for equal mean directions of several populations assuming the same unknown concentration of the populations, a test for the assumption of a common concentration, and a test for equal mean vectors. The performance of the tests in terms of both size and power is investigated through simulations. An example with three-dimensional anthropological data is presented to demonstrate the application of the tests. We also introduce the one-way random effects model for directional data. We consider four different methods for computing the MLEs of the parameters, including Markov chain Monte Carlo EM (MCMCEM) and Gauss-Hermite approximation methods. We also define a general mixed effects model for directional data, including the special case of a random intercept model, and show how estimation can be done using an MCMCEM algorithm. A formula for the mean resultant length of the projected normal distribution, which is the underlying distribution for the SPML model, for the general d-dimensional case is presented.