Cloud vertical structure influences the fluxes of precipitation and radiation throughout the atmosphere. This structure is not predicted in large-scale models but is instead applied in the form of overlap assumptions. In their current guise, overlap assumptions apply to the presence or absence of clouds, and new data sets have led to the development of empirical formulations described by exponential decay from maximum to random overlap over a characteristic length scale. At the same time, cloud parameterizations in many large-scale models have been moving toward assumed PDF schemes that predict the distribution of total water within each grid cell, which will require overlap assumptions that may be applied to cells with specified internal variability. This paper uses a month-long cloud-resolving model simulation of continental convection to develop overlap assumptions for use with assumed PDF cloud schemes in large-scale models. An observing system simulation experiment shows that overlap assumptions derived from millimeter-wavelength cloud radar observations can be strongly affected by the presence of precipitation and convective clouds and, to a lesser degree, by limited sampling and reliance on the frozen turbulence assumption. Current representations of overlap can be extended with good accuracy to treat the rank correlation of total water in each grid cell, which provides a natural way to treat vertical structure in assumed PDF cloud schemes. The scale length that describes an exponential fit to the rank correlation of total water depends on the state of the atmosphere: convection is associated with greater vertical coherence (longer scale lengths), while wind shear decreases vertical coherence (shorter scale lengths). The new overlap assumptions are evaluated using cloud physical properties, microphysical process rates, and top-of-atmosphere radiative fluxes. These quantities can be reproduced very well when the exact cloud structure is replaced with its statistical equivalent and somewhat less well when the time mean vertical structure is imposed. Overlap formulations that treat total water can also be used to determine the variability in clear-air relative humidity, which might be used by convection and aerosol parameterizations.