Is the 'simple carry-over' model useful?

It is argued that the models usually employed for carry-over effects in cross-over trials are not reasonable. The example of a non-linear dose response for a dose-finding trial arranged in a Williams square is developed to show that if carry-over is present to any appreciable degree the usual statistical models provide no guaranteed protection against its effects. It is concluded that the most reasonably defended assumption about carry-over effects is that no important carry-over has taken place and that, where this assumption cannot be defended, statistical models provide no satisfactory substitute for it.