Matrix Preliminaries and General Linear Model

Abstract

Volume I of the text was devoted to a study of various models with the common feature that the same numbers of observations were taken from each treatment group or in each submost subcell. When these numbers are the same, the data are referred to as balanced data; in contrast, when the numbers of observations in the cells are not all equal, the data are known as unbalanced data. In general, it is desirable to have equal numbers of observations in each subclass since the experiments with unbalanced data are much more complex and difficult to analyze and interpret than the ones with balanced data. However, in many practical situations, it is not always possible to have equal numbers of observations for the treatments or groups. Even if an experiment is well-thought-out and planned to be balanced, it may run into problems during execution due to circumstances beyond the control of the experimenter; for example, missing values or deletion of faulty observations may result in different sample sizes in different groups or cells. In many cases, the data may arise through a sample survey where the numbers of observations per group cannot be predetermined, or through an experiment designed to yield balanced data but which actually may result in unbalanced data because some plants or animals may die, patients may drop out or be taken out of the study. For example, in many clinical investigations involving a follow-up, patients may decide to discontinue their participation, they may withdraw due to side effects, they may die, or they are simply lost to follow-up.