An Alternate Approach to Missing Value Estimation

Abstract

Occasionally, with a designed experiment, certain observations may be missing for some reason. An accepted procedure in such a situation is to estimate these missing values from the remaining data, making slight changes in the analysis to account for this action. A good description of this technique is given in Reference 1. As proposed by Yates, Reference 2, missing values are inserted by minimizing the residual sum of squares. That is, unknown quantities are inserted when values are missing, and the analysis is carried out as usual to yield a residual sum of squares. This residual will be a function of the unknown quantities. It is then differentiated with respect to each of the unknown quantities, and the resulting system of equations is solved for these unknowns. These solutions are inserted in place of the missing observations. For many standard experimental plans, missing value formulas for a single missing observation have been derived. A good listing of such formulas is given in Reference 1. In application, when more than one observation is missing, it is often simpler to use such standard formulas together with an iterative estimation procedure, rather than to proceed as in the preceding paragraph. With this iterative procedure, initial estimates are assigned all the missing observations but one, which is then estimated by the applicable formula. Using this value, and the initial estimates of all but one of the remaining quantities, the second missing value is found by the formula. This procedure is followed until all estimates show no significant change from one cycle to the next. The number of cycles necessary before convergence is attained depends heavily upon the choice of the initial estimates and, of course, on the user's definition of convergence in a given application. With this iteration technique, the problem of finding several missing observations reduces to the problem of finding one such value. Whether by use of formula, or by the direct approach, it is generally accepted that the criterion for choosing a missing value is the minimization of the residual sum of squares as proposed by Yates, Reference 2. It is not generally recognized that this is equivalent to choosing the missing value to make the model fit perfectly at this point, that is, to make the particular residual equal to zero. A proof of this equivalence is contained in the appendix. It has been the author's experience that recognition of this fact leads to two advantages: 1. When teaching missing value estimation, the derivation of missing value formulas is simpler for certain patterned experiments using this approach. 2. In some applications, this method of estimating missing values leads to a very simple solution. This is especially valuable when standard missing value formulas are not readily available.