Inequalities for the precision of the path of steepest ascent in response surface methodology

The method of steepest ascent plays an important role in Response Surface Methodology giving a direction of further experimentation after an initial factorial design. To determine whether the path of steepest ascent has been determined precisely enough the fraction of all possible directions that are included in the confidence 'cone' around the fitted path of steepest ascent can be calculated. In this paper we give some background and provide an expression for the fraction in terms of the radius of a projected cap of the unit hypersphere onto the plane through the origin and normal to the path of steepest ascent and extend the problem to cover heterogeneous regression estimates. In this case the projection of the cap is no longer spherical but is a hyperellipsoid and we give some inequalities that provide bounds for the fraction that should be practically useful.