We present intrinsic and extrinsic methods for studying anatomical coordinates in order to perform statistical inference on random physiological signals F across clinical populations. In both intrinsic and extrinsic methods, we introduce generalized partition functions of the coordinates, psi(x), x epsilon M, which are used to construct a random field of F on M as statistical model. In the intrinsic analysis, such partition functions are built intrinsically for individual anatomical coordinate based on Courant's theorem on nodal analysis via self-adjoint linear elliptic differential operators. In contrast, the extrinsic method needs only one set of partition functions for a template coordinate system, and then applies to each anatomical coordinate system via diffeomorphic transformation. For illustration, we apply both intrinsic and extrinsic methods to a clinical study: cortical thickness variation of the left cingulate gyrus in schizophrenia. Both methods show that the left cingulate gyrus tends to become thinner in schizophrenia relative to the healthy control population. However, the intrinsic method increases the statistical power.