Balanced and unbalanced designs

Balanced designs have the same number of replicate observations in each sample. Thus a one-factor model Y = A+ε will be balanced if sample sizes all take the same value n at each of the a levels of factor A. Balanced designs are generally straightforward to analyse because factors are completely independent of each other and the total sum of squares (SS) can be partitioned completely among the various terms in the model. The SS explained by each term is simply the improvement in the residual SS as that term is added to the model. These are often termed ‘sequential SS’ or ‘Type I SS’.

Designs become unbalanced when some sampling units are lost, destroyed or cannot be measured, or when practicalities mean that it is easier to sample some populations than others. For nested models, imbalance may result from unequal nesting as well as unequal sample sizes. Thus a nested model Y = B′(A)+ε will be balanced only if each of the a levels of factor A has b levels of factor B′, and each of the ba level of B′ has n replicate observations. For factorial models, an imbalance means that some combinations of treatments have more observations than others. An extreme case of unbalanced data arises in factorial designs where there are no observations for one or more combinations of treatments, resulting in missing samples and a substantially more complicated analysis.