Suppose that a polynomial of order d is to be fitted to data from a designed experiment and that \(V(\cdot)\) is any selected variance function. For example, if \(\hat y(\)x) is the fitted value at a location \(x=(x_ 1,x_ 2,...,x_ k)'\) in the k-dimensional predictor space, \(V\{\hat y(x)\}\), which we shall also write \(V(\hat y)\), denotes the usual variance function. Design rotatability of order d is the property that \(V\{\hat y(x)\}\) has spherical contours in the x space.
Here, a more general definition, Schläflian rotatability, is made with respect to any \(V(\cdot)\), by requiring that the contours of \(V(\cdot)\) are spherical in the space of the model predictors, specified in the usual Schläflian form.
It is shown that Schläflian rotatability for \(V(\hat y)\) implies ordinary rotatability, and vice versa. The selection of a particular Schläflian variance profile for \(V(\cdot)\) for a second order rotatable design implies a choice of the number of center points and the consequences of this are explored.