A simple non-iterative procedure is described for obtaining missing value estimates by solving a set of simultaneous linear equations that can be written directly. The method is derived for the two-way crossed classification and results for theP-way crossed classification are also given. Following the procedure of Yates (1933), estimates of the missing values are obtained such that their residuals are zero, thereby minimizing the error mean square. For theP-way crossed classifications the error mean square may be a pooled mean square of higher ordered interactions. The procedure is applicable to other designs as well.