The problem of estimating factorial treatment contrasts in a multiple response setting is considered. In the multiple response problem considered here, several drugs may be administered singly or in combination with several other drugs to a number of subjects according to a cross-over design. A measurement is taken on each of the drugs administered in a given time period. The designs of interest for the multiple response problem are block designs with independent multiple responses and missing observations. It is shown that the lower bound for the average variance of a set of contrasts in the usual block design setting with a single measured response also provides a bound for a similar set of contrasts in the multiple response setting.