Analysis of three-dimensional grids: cubes and cubic coefficients

It is widely believed that third-order coefficients cannot be estimated from eight or nine data in cubical array. Equations that are exact on first, second, and third powers of trilinear data can be used for the purpose. The estimates are compared to the values obtained by Taylor expansions of typical generating functions. First-order coefficients are obtained from higher-order terms by the method of least squares. On monotonic data, they are often closer to the true values than main-effect approximations. The accuracy of the coefficients may be sufficient to interest experimentalists.