The most widely used procedures to harness the power of a concomitant (nuisance) variable are block designs and analysis of covariance (ANCOVA). This study attempted to provide a scientific foundation on which to base decisions on whether to block or covary and how many blocks to be used if blocking is selected. Monte Carlo generated data were analyzed using one-way analysis of variance (ANOVA); two-block, four-block, and eight-block designs; and ANCOVA. Resulting empirical powers were entered into a repeated measures four-way factorial design with three factors representing different experimental conditions and one factor representing the five procedures being compared. The results indicated that the correlation coefficient between the concomitant and dependent variables was the critical factor to influence the choice. One-way ANOVA was the best choice when there was no relationship, blocking was preferred when the correlation was low, and ANCOVA achieved the highest power when the correlation was high. Block designs and ANCOVA became more powerful and the optimal number of blocks increased as the correlation coefficient, the number of treatments, and the number of subjects per treatment increased. Five appendixes provide four tables of supplemental information and computer codes used in the analysis. (Contains 5 tables and 44 references.) (Author/SLD)