Given is an investigation of experimental plans that require a minimum number of assemblies, minimal multidimensional designs (MMD's) and minimal augmented multidimensional designs (MAMD's). Such designs are incomplete factorials which allow for only some two-factor interactions. An analysis of incomplete factorials with and without interactions is developed from a reduced form of the normal equations.

C_ip_i = Q_i

where p_i is a vector of estimates of factor i effects and C_i is the coefficient matrix for pi and Qi is a vector of transformed observations. The general forms for C_i and Q_i are presented.

The construction of MMD's and MAMD's is made possible from results obtained on connected designs. A definition of a connected design where two-factor interactions are assumed leads to a procedure for"connecting" experimental plans. This procedure provides a way of adding assemblies to a design in order to estimate contrasts not originally estimable in the design. Using this augmentation procedure and the minimum number of assemblies to be added, MAMD's may be constructed. MMD's follow by sequentially augmenting with the minimum number of assemblies, m-factor designs, m = 1,2,...,m*, whose total number of factor level combinations are a minimum. A method for finding MMD's and MAMD's which are optimal for one factor or for a set of two-factors is then presented as well as some examples of MMD's and MAMD's with and without two-factor interactions. Data is generated and analyzed for a particular design which is both a MAMD and a MMD, and a discussion of this design's optimality is also given.