Algebraic solutions to the connectivity problem for mm-way layouts: interaction-contrast aliasing

An m-way layout, or design, is called complete if there is at least one observation per cell. Without any side conditions the parameters are not estimable but contrasts may be estimable and they are all estimable for the complete layout. For m=2m=2 the condition for estimability is the well-known connectivity condition: one can “walk” from any row to any other row, stepping on columns. The case m>2m>2 remains unsolved in some sense. The principal method used here is the Gröbner basis (G-basis) method introduced by Pistone and Wynn [1996. Generalised confounding with gröber bases. Biometrica 83, 653–666]. The problem is set up, using indicator functions, and necessary and sufficient conditions given for full estimability and various constructions using the G-basis method.