Extensions of an existing circuit theory for current perpendicular to plane magnetoresistance and current-driven torque in noncollinear magnetic-multilayer pillar devices are presented. Our expressions for monodomain critical-current threshold ${J}_{c}$ and giant magnetoresistance $\mathrm{\ensuremath{\Delta}}R$ are firstly derived in terms of assumed spin-channel resistances for each of the two ferromagnets. Spinflips are thus neglected. We find a class of closed linear relationships connecting ${J}_{c}^{\ensuremath{-}1}$ and $\mathrm{\ensuremath{\Delta}}R$. We then derive more general expressions for these quantities which take into account spin-flip relaxation. In this case, we assume analytically calculable linear $2\ifmmode\times\else\texttimes\fi{}2$ current-voltage matrices for the separate two-channel ferromagnets. These expressions again lead to a class of closed linear relationships connecting ${J}_{c}^{\ensuremath{-}1}$ and $\mathrm{\ensuremath{\Delta}}R$. The latter generalization gives a simple theoretical framework to take into account bulk and interfacial spin flip and more complicated multilayer structures often used in experiments.