Statistical inference on mean dioptric power: hypothesis testing and confidence regions

Abstract

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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