Linear models with a generalized AR(1) covariance structure for longitudinal and spatial data

Cross-sectional and longitudinal imaging studies are moving increasingly to the forefront of medical research due to their ability to characterize spatial and spatiotemporal features of biological structures across the lifespan. With Gaussian data, such designs require the general linear model for repeated measures data when standard multivariate techniques do not apply. A key advantage of this model lies in the flexibility of modeling the covariance of the outcome as well as the mean. Proper specification of the covariance model can be essential for the accurate estimation of and inference about the means and covariates of interest.;Many repeated measures settings have within-subject correlation decreasing exponentially in time or space. Even though observed correlations often decay at a much slower or much faster rate than the AR(1) structure dictates, it sees the most use among the variety of correlation patterns available. A three-parameter generalization of the continuous-time AR(1) structure, termed the generalized autoregressive (GAR) covariance structure, accommodates much slower and much faster correlation decay patterns. Special cases of the GAR model include the AR(1) and equal correlation (as in compound symmetry) models. The flexibility achieved with three parameters makes the GAR structure especially attractive for the High Dimension, Low Sample Size case so common in medical imaging and various kinds of "-omics" data. Excellent analytic and numerical properties help make the GAR model a valuable addition to the suite of parsimonious covariance structures for repeated measures data.;The accuracy of inference about the parameters of the GAR model in a moderately large sample context is assessed. The GAR covariance model is shown to be far more robust to misspecification in controlling fixed effect test size than the AR(1) model. It is as robust to misspecification as another comparable model, the damped exponential (DE), while possessing better statistical and convergence properties.;The GAR model is extended to the multivariate repeated measures context via the development of a Kronecker product GAR covariance structure. This structure allows modeling data in which the correlation between measurements for a given subject is induced by two factors (e.g., spatio-temporal data). A key advantage of the model lies in the ease of interpretation in terms of the independent contribution of every repeated factor to the overall within-subject covariance matrix. The proposed model allows for an imbalance in both dimensions across subjects.;Analyses of cross-sectional and longitudinal imaging data as well as strictly longitudinal data demonstrate the benefits of the proposed models. Simulation studies further illustrate the advantages of the methods. The demonstrated appeal of the models make it important to pursue a variety of unanswered questions, especially in the areas of small sample properties and covariance model robustness.