A fully crossed and randomized within-subjects design is a type of experimental design used in research to investigate the effects of one or more independent variables on a dependent variable. In this type of design, each participant is exposed to all levels of the independent variable(s) in a random order.
A fully crossed design means that each level of each independent variable is paired with every level of every other independent variable. For example, if we have two independent variables, A and B, with two levels each, a fully crossed design would involve four conditions: A1B1, A1B2, A2B1, and A2B2. In a fully crossed design, each participant is randomly assigned to one of the conditions and completes the experiment in that condition before moving on to the next condition.
A randomized within-subjects design means that the order in which the conditions are presented to each participant is randomized. For example, if we have four conditions in a fully crossed design, the order in which each participant completes the conditions might be randomized so that some participants start with condition A1B1, while others start with condition A2B2.
The advantage of a fully crossed and randomized within-subjects design is that it allows researchers to control for individual differences between participants, since each participant serves as their own control. This type of design can also increase statistical power, as it allows for more precise estimation of the effects of the independent variables.
However, fully crossed and randomized within-subjects designs also have some limitations. These designs may be subject to order effects, where the order in which the conditions are presented may affect the dependent variable. Researchers can try to minimize these effects through counterbalancing, where the order of conditions is varied across participants in a systematic way. Another limitation is that fully crossed and randomized within-subjects designs may be more prone to attrition or dropout, as participants may become fatigued or lose interest over the course of the experiment.