We consider the problem of computing sums of squares in multifactor analysis of variance models with unequal but nonzero numbers per cell. Commonly used computing methods for obtaining main-effect and interaction hypothesis mean squares solve linear model equationsX′Xβ =X′Y. We present an algorithm based on Scheffé's method of contrasts that solves a smaller set of linear equations. It avoids subtraction of quadratic forms, which degrades floating-point accuracy and facilitates Scheffé's method of multiple comparisons. Selected efficiency calculations are presented.