Statistical shape models are powerful tools for describing anatomical structures and are increasingly being used in a wide variety of clinical and biological contexts. One of the promising applications of this technology is the testing of hypotheses that entail shape differences, and visualization of those differences between cohorts. Statistical testing of shapes, however, is difficult due the large numbers of degrees of freedom and challenge of obtaining sufficient numbers of subjects to ensure statistical power. To date, research in statistical shape modeling has focused mainly on the construction of representative models, and the field has not reached a consensus on the best approach to statistical hypothesis testing. This paper illustrates some problems inherent in the statistical analysis of high-dimensional shape models, and suggests a systematic approach to hypothesis testing that avoids those problems. The proposed framework is based on research in the factor analysis statistics literature, with permutation testing in the PCA space of the model, and dimensionality reduction via a a simulation-based analysis. We also describe two methods for visualizing group mean differences, first by direct visualization of the linear discriminant implicit in the hypothesis test metric, and second, by visualizing strain tensors from a deformation computed between the group means. We illustrate the proposed analysis and visualization framework on several clinical and biological datasets.