Marginal Projected Multivariate Linear Models for Clustered Angular Data

Abstract

Among the diverse frameworks that have been proposed for regression analysis of angular data, the projected multivariate linear model provides a particularly appealing and tractable methodology. In this model, the observed directional responses are assumed to correspond to the angles formed by latent bivariate normal random vectors that are assumed to depend upon covariates through a linear model. This implies an angular normal distribution for the observed angles, and incorporates a regression structure through a familiar and convenient relationship. In this paper we extend this methodology to accommodate clustered data (e.g., longitudinal or repeated measures data) by formulating a marginal version of the model and basing estimation on an EM-like algorithm in which correlation among withincluster responses is taken into account by incorporating a working correlation matrix into the M step. A sandwich estimator is used for the parameter estimates' covariance matrix. The methodology is motivated and illustrated using an example involving clustered measurements of microbril angle on loblolly pine (Pinus taeda L.) Simulation studies are presented that evaluate the finite sample properties of the proposed fitting method. In addition, the relationship between within-cluster correlation on the latent Euclidean vectors and the corresponding correlation structure for the observed angles is explored.

A random‐effects model for clustered circular data

“…A cluster effect can be introduced in a projected normal model by expressing u j = b j + e j , j = 1,2 where ( b 1 , b 2 ) are independent N (0, ψ ) variables representing a cluster effect while the unit errors ( e 1 , e 2 ) have independent N (0,1 − ψ ) distributions (Nuñez‐Antonio & Gutiérrez‐Peña, ; Hall & Shen, ). Considering Equation , y is approximately equal to a sum featuring a fixed effect, a cluster effect, and an experimental error, with the β ‐parameters appearing in all three terms.…”