## Optimal Experimental Design in the Presence of Nested Factors

A common occurrence in practical design of experiments is that one factor, called a nested factor, can only be varied for some but not all the levels of a categorical factor, called a branching factor. In this case, it is possible, but inefficient, to proceed by performing two experiments. One experiment would be run at the level(s) of the branching factor that allow for varying the second, nested, factor. The other experiment would only include the other level(s) of the branching factor. It is preferable to perform one experiment that allows for assessing the effects of both factors. Clearly, the effect of the nested factor then is conditional on the levels of the branching factor for which it can be varied. For example, consider an experiment comparing the performance of two machines where one machine has a switch that is missing for the other machine. The investigator wants to compare the two machines but also wants to understand the effect of flipping the switch. The main effect of the switch is conditional on the machine. This article describes several example situations involving branching factors and nested factors. We provide a model that is sensible for each situation, present a general method for constructing appropriate models, and show how to generate optimal designs given these models.