Defficiency computes the D- or Ds-efficiency measure for some design with respect to some reference design.Defficiency(des, ref, sNames = NULL, names = NULL)Defficiency returns a single numeric.sNames and names default to the corresponding argument which was used to find the reference design.D efficiency is defined as $$\left(\frac{\left|M(\xi,\bar{\theta})\right|}{\left|M(\xi^{*},\bar{\theta})\right|}\right)^{1/n}$$ and Ds efficiency as $$\left(\frac{\left|M_{11}(\xi,\bar{\theta})-M_{12}(\xi,\bar{\theta})M_{22}^{-1}(\xi,\bar{\theta})M_{12}^{T}(\xi,\bar{\theta})\right|}{\left|M_{11}(\xi^{*},\bar{\theta})-M_{12}(\xi^{*},\bar{\theta})M_{22}^{-1}(\xi^{*},\bar{\theta})M_{12}^{T}(\xi^{*},\bar{\theta})\right|}\right)^{1/s}$$
where $M_{11}$ is the submatrix corresponding to the parameters in sNames, $M_{22}$ is the submatrix corresponding to the parameters in names which are not in sNames, and $M_{12}$ is defined as the resulting off diagonal submatrix.
FedorovWynn, update_reference, update.desigh