  In this paper, we review widely used statistical analysis frameworks for data
defined along cortical and subcortical surfaces that have been developed in
last two decades. The cerebral cortex has the topology of a 2D highly
convoluted sheet. For data obtained along curved non-Euclidean surfaces,
traditional statistical analysis and smoothing techniques based on the
Euclidean metric structure are inefficient. To increase the signal-to-noise
ratio (SNR) and to boost the sensitivity of the analysis, it is necessary to
smooth out noisy surface data. However, this requires smoothing data on curved
cortical manifolds and assigning smoothing weights based on the geodesic
distance along the surface. Thus, many cortical surface data analysis
frameworks are differential geometric in nature. The smoothed surface data is
then treated as smooth random fields and statistical inferences can be
performed within Keith Worsley's random field theory. The methods described in
this paper are illustrated with the hippocampus surface data set. Using this
case study, we will determine if there is an effect of family income on the
growth of hippocampus in children in detail. There are a total of 124 children
and 82 of them have repeat magnetic resonance images (MRI) two years later.
