Is the relation between keratometric astigmatism and refractive astigmatism linear?

Javal's rule, which states that the total astigmatism of the eye--or refractive astigmatism--is equal to 1.25 (keratometric astigmatism) - 0.50 x 90, implies a linear relation between the two entities. Data published by Grosvenor, Quintero, and Perrigin (Am J Optom Physiol Opt 1988;65:292-7) show that for eyes having no more than 2.50 D of keratometric astigmatism, the slope of the regression line for refractive astigmatism plotted against keratometric astigmatism is less than 1.00, rather than the 1.25 required by Javal's rule. When data compiled for eyes having greater amounts of keratometric astigmatism are plotted in the same manner, the slope of the regression line is greater than 1.00, suggesting that the relation between keratometric and refractive astigmatism may be non-linear. However when data for these highly astigmatic eyes are plotted in combination with data published by Grosvenor, Quintero, and Perrigin, they are adequately characterized by a straight-line relation.