Spatial analyses of nerve patterns and global testing approaches

Doctoral thesis, 2024

This thesis consists of two parts: statistical analyses of and models for epidermal nerve fibers (ENFs), and extensions of global envelope tests. In the first part, the main goal was to improve our understanding of the ENF structure changes that occur as a result of nerve damage due to neurological disorders, such as diabetic neuropathy. For this purpose, different stochastic models were proposed. The ENF data were treated as point patterns in three-dimensional boxes, and samples from subjects suffering from diabetic neuropathy and healthy volunteers were considered. In Paper I, we introduced a new summary that measured the volume of the epidermis covered by the nerves and examined second-order properties of the underlying processes. Further, a three-dimensional point process model for the nerve structure was developed. The two-dimensional version of the model captured the planar spatial structure. However, the complete model could not capture the attraction between the nerve fiber endings in the data. Therefore, in Paper II a pairwise interaction Markov model allowing neighboring nerve endings to interact was proposed. In Paper III, we considered the two-dimensional projections of the ENF patterns and developed spatial thinning models to study the nerve death process. Insights from our analyses indicated that nerve mortality is guided by a biological process that favors the removal of isolated nerve trees. The goodness-of-fit of the models in Papers I-III was evaluated using global envelope tests. In the second part, we extended the global envelope tests for quantile regression and for comparison of distributions of $n$ samples. In Paper IV, we proposed non-parametric, permutation based global tests, that allowed for simultaneous inference of the quantile regression process. In Paper V, we proposed graphical $n$-sample tests for correspondence of distributions based on the global envelope testing framework. Further, we presented a detailed discussion regarding the graphical interpretation of the test results for each suggested test statistic.