Analysis of Unaided Vision Data using New Decomposition of Symmetry

Problem statement: First, this study considers how the structure of symmetry for probabilities is decomposed into two structures. Secondly, this study infers the structure of unknown probabilities which indicates how the right eye is better (or worse) than the left eye for three kinds of data on unaided distance vision of (1) women in Britain, (2) students in an university of Japan and (3) pupils in elementary schools in Tokyo, Japan. This study proposes a new decomposition of symmetry model for probabilities and analyzes these vision data using the decomposition. Approach: This study considers a new decomposition theorem that for the probabilities the symmetry model (indicates that the right eye vision is symmetric to the left eye vision) holds. Also this study analyzes the vision data using this decomposition. Results: From the statistical approach, we can see that (1) for the vision data of women, the right eye is better than the left eye and the mean of right eye is not equal to the mean of left eye, (2) for the vision data of students, the right eye is worse than the left eye and the mean of right eye is not equal to the mean of left eye and (3) for the vision data of pupils, the right eye is symmetric to the left eye and the mean of right eye is equal to the mean of left eye. Conclusion: When the symmetry model fits the data poorly, this new decomposition is useful for seeing which of decomposed two models influences stronger. We can see the structure of asymmetry for vision data in more details.