Abstract
Applied statistics as an independent field of study can be said to have been recognized as such roughly a century ago. A large part of this recognition can be attributed to analysis of variance, a method which has been of paramount importance for much of the scientific and industrial progress throughout the 20th century. However, as the complexity of experiments and data has increased during this time, so has the difficulty in carrying out an analysis of variance. Often it is difficult to understand how statistical parameters correspond to real-world quantities; to relate statistical methods to one another; to distinguish between the practical relevance and statistical significance of an analysis; and to recognize the computational demands of a chosen method. The work herein contributes to the field of applied statistics by identifying a methodology that addresses these challenges. This method is then applied to large-scale simulated climate projections in a way that is computationally efficient and that provides clear and intuitive conclusions on what explanatory factors contribute to the overall variability of the data. The work begins with viewing analysis of variance as a more general procedure than it is traditionally considered. By placing it into a more broadly defined hierarchical Bayesian context, the ability to draw connections between fixed and random effects within a single procedure is facilitated. To present results, novel numerical and graphical summaries that communicate practical relevance, as well as the statistical significance, of variance parameters are then developed. Novel summaries for general multivariate cases are also proposed. Existing methods that achieve these goals typically rely on Markov chain Monte Carlo methods. Whereas here a class of conjugate distributions for variance parameters is derived, which allow for a factorization of the posterior distribution and which obviate the need for MCMC methods, thus reducing computational demands. For cases involving spatial geostatistical models, approximation methods that further alleviate computational demands are also developed. These approximations are closely examined, and shown to produce accurate representations of the original quantities. The culmination of these efforts is in the spatial statistical analysis of a set of simulated climate projections. A single simulated climate projection is carried out using a distinct mathematical climate model, and an assumed rate of future greenhouse gas emissions. Thus, the sources of variability examined in the data are the climate model and the emissions scenario, their interaction, both as a constant as well as over decadal time increments. The magnitude of the variability for each source is then estimated, together with the uncertainty of these estimates. While the method is original, the results agree with those found in previous peer-reviewed works.
Geinitz, Steven Richard