Reactions of neural, psychological, and social systems are rarely, if ever, independent of previous inputs and states. The potential for serial order carryover effects from one condition to the next in a sequence of experimental trials makes counterbalancing of condition order an essential part of experimental design. Here, a method is proposed for generating counterbalanced sequences for repeated-measures designs including those with multiple observations of each condition on one participant and self-adjacencies of conditions. Condition ordering is reframed as a graph theory problem. Experimental conditions are represented as vertices in a graph and directed edges between them represent temporal relationships between conditions. A counterbalanced trial order results from traversing an Euler circuit through such a graph in which each edge is traversed exactly once. This method can be generalized to counterbalance for higher-order serial order carryover effects as well as to create intentional serial order biases.Modern graph theory provides tools for finding other types of paths through such graph representations, providing a tool for generating experimental condition sequences with useful properties.
3Neural systems, human and otherwise, are characterized by a continuous stream of behavior in which the current state is dependent not only on current inputs but also on previous states and inputs. The history of experimental psychology is replete with examples. The speed with which a participant names a picture can depend on the preceding picture (e.g, Bartram, 1974;Biederman & Cooper, 1991a;Goldstein, 1958), a behavioral effect known as priming. Prolonged viewing of a visual stimulus can cause adaptation to its color, orientation, or other features and lead to subsequent aftereffects (e.g., Blakemore & Campbell, 1969;Blakemore & Nachmias, 1971;Bradley, Switkes, & De Valois, 1988). The hemodynamic response (during functional magnetic resonance imaging, FMRI) to a stimulus is attenuated when it has been preceded by a similar stimulus (Grill-Spector & Malach, 2001;Haushofer, Baker, Livingstone, & Kanwisher, 2008;Henson & Rugg, 2003;Vuilleumier, Henson, Driver, & Dolan, 2002). The brain's response to transcranial magnetic stimulation (TMS) also depends on previous visual stimulation (e.g., Pasley, Allen, & Freeman, 2009;Silvanto, Muggleton, & Walsh, 2008). Similar situations can arise in questionnaire assessments used in social psychology and clinical contexts. For instance, survey results can vary depending on the order of the questions (e.g., Faulkner & Cogan, 1990;Gama, Correia, & Lunet, 2009) and election results can be affected by which candidates precede which on the ballot (e.g. Miller & Krosnick, 1998). All of these are examples of serial order carryover effects, i.e. the response in one experimental trial in a sequence depends on the preceding trial(s) in that sequence. Because learning and adaptation are central to the functioning of living things, carryover effects are ubiquitous within all domains...