Missing Values in Factorial Experiments

Abstract

METHODS of estimating missing values have been discussed by Goulden¹. Some applications to factorial experiments were presented by Cochran and Cox², but they do not give a general formula. If there is one replication of an a × b × c × … f factorial experiment, with n factors A, B, C, … F at a, b, c, … f different levels respectively, then it may be shown by minimizing the highest-order interaction sum of squares that a suitable estimate, y, of a missing value is given by: where (G) is the sum of the available observations, that is, the incomplete grand total, (A) is the sum of the bc … f − 1 observations for the level of A which contains the missing value, or the incomplete A total, (AB) is the incomplete A × B total, … and (BC … F) is the sum of a − 1 observations. There will be 2ⁿ − 1 terms on the right-hand side. For two factors this reduces to the well-known randomized-block formula: