A General Nested Split-Plot Analysis of Covariance

原文摘要

We present a general split-plot analysis of covariance (ANOCOVA) model for unbalanced data. The imbalance arises from assigning a different number of experimental units to the levels of the whole-plot factor A. The values of the covariable measurements on each experimental unit are allowed to vary across the levels of the split-plot factor B. Exact test statistics and confidence intervals are derived. Exact F test statistics for the A and B main effects and the A × B interaction each have one less numerator and denominator degree of freedom than for the corresponding analysis of variance (ANOVA). Reduced versions of the general model are given in which the lost numerator degrees of freedom are regained. Competing approximate tests are discussed, including generalized least squares. For the nested split-plot ANOVA model for balanced data, the numerator sums of squares for the A and B main effects and the A × B interaction, together with the A and B error sums of squares, add up to the total sum of squares. This is because the five subspaces from which these five sums of squares are generated, along with the vector of 1s, form an orthogonal decomposition of Euclidean space. Orthogonally projecting the covariate vector onto each of these subspaces creates the predictor vectors for our model. Our general split-plot ANOCOVA linear model is obtained by incorporating these five predictor vectors into the split-plot ANOVA linear model, each with its own coefficient. All ANOVA sums of squares for testing the A and B main effects and the A × B interaction under our general ANOCOVA model must be modified by removing that component of each sum of squares that is due to the predictor vector contained in the associated subspace, resulting in the loss of one degree of freedom for each sum of squares. For each test, the lost numerator degree of freedom can be regained by equating the coefficients of the two predictor vectors associated with the numerator and denominator subspaces for that test. With unbalanced data, the subspaces that generate the numerator sums of squares for the three tests are not mutually orthogonal. We give the necessary modification to our general model for balanced data and derive all analyses in this situation.

摘要来源：OpenAlex 索引摘要（稿件版本未注明）