described as follows: k = 2 factors (1 primary factor X1 and 1 blocking factor X2) L1 = 4 levels of factor X1 L2 = 3 levels of factor X2 n = 1 replication per cell N = L1 * L2 = 4 * 3 = 12 runs Before randomization, the design trials look like: Matrix representation An alternate way of summarizing the design trials would be to use a 4x3 matrix whose 4 rows are the levels of the treatment X1 and whose columns are the 3 levels of the blocking variable X2. The cells in the matrix have indices that match the X1, X2 combinations above. By extension, note that the trials for any K-factor randomized block design are simply the cell indices of a k dimensional matrix. Model The model for a randomized block design with one nuisance variable is where Yij is any observation for which X1 = i and X2 = j X1 is the primary factor X2 is the blocking factor μ is the general location parameter (i.e., the mean) Ti is the effect for being in treatment i (of factor X1) Bj is the effect for being in block j (of factor X2) Estimates Estimate for μ : = the average of all the data Estimate for Ti : with = average of all Y for which X1 = i. Estimate for Bj : with = average of all Y for which X2 = j. Generalizations Generalized randomized block designs (GRBD) allow tests of block-treatment interaction, and has exactly one blocking factor like the RCBD. Latin squares (and other row-column designs) have two blocking factors that are believed to have no interaction. Latin hypercube sampling Graeco-Latin squares Hyper-Graeco-Latin square designs Theoretical basis The theoretical basis of blocking is the following mathematical result. Given random variables, X and Y The difference between the treatment and the control can thus be given minimum variance (i.e. maximum precision) by maximising the covariance (or the correlation) between X and Y. See also Algebraic statistics Block design Combinatorial design Generalized randomized block design Glossary of experimental design Optimal design Paired difference test Dependent and independent variables Blockmodeling References Bibliography Pre-publication chapters are available on-line. Design of experiments