In this thesis, the effect of different configurations of missing values on block designs, row-column designs, and diallel cross designs is investigated. The average variance of all pairwise treatment comparisons has been used as a measure of robustness by the majority of researchers. The maximum variance of comparisons is computed numerically, or developed theoretically, in this thesis for most patterns of missing data. The reduced normal equations can be solved with a suitable choice of generalised inverse, and formulae for the individual variances of pairwise treatment differences can also be derived. The effect of missing values on block designs, in particular randomised and balanced incomplete block designs is studied. It is shown that designs with a small number of treatments and a small number of blocks are severely affected by the loss of one, two or three observations. Larger designs are not as seriously affected when the average variance is considered, but there are a small number of pairwise treatment comparisons that suffer a large loss of efficiency. Row-column designs have also been investigated for similar patterns of missing data. The lack of orthogonality introduced by the loss of data in many situations complicates the analysis and derivation of general expressions of the variances. The loss of efficiency for small Latin square designs is substantial after the removal of only one or two units. Constructing a design with multiple squares is shown to reduce the impact of the missing data. Youden square designs also suffer a similar loss of information after the loss of a few observations, and it is also shown that the structure of the design affects the distributions of efficiencies for a given number of missing values.