Problem: Health-risk behaviors contribute to the leading causes of morbidity and mortality among youth and adults in the United States. In addition, significant health disparities exist among demographic subgroups of youth defined by sex, race/ethnicity, and grade in school and between sexual minority and nonsexual minority youth. Population-based data on the most important healthrelated behaviors at the national, state, and local levels can be used to help monitor the effectiveness of public health interventions designed to protect and promote the health of youth at the national, state, and local levels.

Every dioptric power of the usual form sphere/cylinder x axis may be represented by means of a point in three-dimensional space. Graphical representation of data in this manner is important for statistical analysis. In particular, graphical representation may be used to display confidence regions about a mean power, for example. A disadvantage of these representations, however, is that simply changing the reference meridian for cylinder axes changes distances in the space and, therefore, changes the shapes of the confidence regions. Because the shapes define the nature of the variation and give, in particular, the principal components of variation, a researcher who happens to measure the orientation of cylinder axes from the 20 degree meridian, for example, instead of the conventional horizontal meridian, could be led to different statistical conclusions. The implication is that such conclusions are unlikely to have much physical meaning. A representation is described here in which distances and shapes do not depend on the meridian that happens to be chosen as reference. Each dioptric power is represented by a point in Euclidean 3-space. Several examples of graphical representation are given. The spherical powers occupy a particular line in the space, the Jackson crossed cylinders occupy a plane, the cylindrical powers occupy a cone, and so on, for all types of conventional dioptric power. These lines and surfaces are illustrated. The statistical implications are discussed briefly. The representation satisfies the requirements of the statistics and is proposed as the standard one for future use.

We have cloned a homolog of Pax-6 in Xenopus laevis. Its deduced amino acid sequence has a 95% overall identity with Pax-6 homologs in other vertebrates. It is expressed early in development in cells fated to form the eye and parts of the forebrain, hindbrain, and spinal cord. It has two phases of expression in the eye. In the early phase, from stage 12.5 to stage 33/34, Xenopus Pax-6 is expressed throughout the developing retina. In the late phase, after stage 33/34, it is excluded from mature cells in the outer half of the retina and from cells in the ciliary marginal zone, remaining only in amacrine and ganglion cells. Misexpression of Pax-6 early in development results in axial defects, but no specific eye phenotype is observed. Targeted misexpression in the retina at later stages does not result in any significant bias toward formation of amacrine or ganglion cells or away from photoreceptors. Ectopic expression of the proneural gene NeuroD alters the pattern of Pax-6, substantially reducing its expression in the eye field and later reducing or eliminating the eye itself. Our results show that Pax-6 expression appears to be necessary, but not sufficient, for eye formation in Xenopus.

It has not hitherto been possible to apply formal methods of statistical analysis to data on dioptric powers. The solution to the basic statistical problem is now provided in this paper. Recognition of the matric-variate nature of dioptric power allows calculation of sample means and variance-covariances. These in turn can be used to calculate a statistic for testing hypotheses on population means and for obtaining confidence regions for those means. In a graphical representation of dioptric power the confidence region turns out to be an ellipsoid centred on the mean of the sample of dioptric powers. The theory is illustrated by means of numerical examples. Singularity of the variance-covariance matrix may occur especially when the sample is small. When it does occur it is the cause of some difficulty in applying the statistics. Nevertheless singularity is rare in practical situations and can usually be avoided simply by increasing the size of the sample. Singularity, therefore, is not treated fully in this paper. Dioptric power is essentially four-dimensional in character but in practice a three-dimensional subspace is almost always sufficient. To avoid the difficulty of having to represent four-dimensional shapes and to avoid the complication of singularity (which is the rule rather than the exception in practice in four-space) only the common three-dimensional problem is considered in detail.

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