 Rotatable design

A rotatable design for a specified regression model has variance of a fitted regression mean depending on $x$ only through its Euclidean distance from the design center. It concerns prediction precision, not rotational symmetry of the mean response itself.

For a full quadratic regression model in $d\ge 2$ variables, consider all $2^d$ factorial corners, the $2d$ axial points $\pm \alpha e_i$, and center runs giving a nonsingular information matrix of an experimental design. Sign symmetry cancels odd moments and permutation symmetry makes the second coordinate moments equal. Fourth-order rotational balance requires, for distinct $i,j$,The factorial corners contribute $2^d$ to both sums, axial runs contribute $2{\alpha }^4$ only to the first, and center runs contribute zero. Thus $2^d+2{\alpha }^4=3\cdot 2^d$, givingFor $d=2$, $\alpha =\sqrt{2}$; for $d=3$, $\alpha =8^{1/4}$. Center replication can estimate pure error but does not change this fourth-moment equation. If a fractional factorial portion is used, its own moments and estimability must be checked; simply substituting the original factor count into the full-factorial formula is not justified. In one dimension rotational invariance only means reflection symmetry, so the distinct-coordinate fourth-moment equation does not impose the displayed choice.