Mathematical psychology.

Mathematical psychology is not, per se, a distinct branch of psychology. Indeed, mathematical psychologists can be found in any area of psychology. Rather, mathematical psychology characterizes the approach that mathematical psychologists take in their substantive domains. Mathematical psychologists are concerned primarily with developing theories and models of behavior that permit quantitative prediction of behavioral change under varying experimental conditions. There are as many mathematical approaches within psychology as there are substantive psychological domains. As with most theorists of any variety, the mathematical psychologist will typically start by considering the psychological phenomena and underlying structures or processes that she wishes to model. A mathematical model or theory (and we do not distinguish between them here) is a set of mathematical structures, including a set of linkage statements. These statements relate variables, equations, and so on with components of the psychological process of interest and possibly also aspects of the stimuli or environment. Regardless of the domain, then, the first step in a mathematical approach is to quantify the variables, both independent and dependent, measured to study a psychological process. Quantification permits variables to be represented as parameters in a mathematical equation or statistical expression, the goal and defining feature of the mathematical psychology enterprise. Mathematical psychologists, then, construct mathematical and statistical models of the processes they study. Some domains, such as vision, learning and memory, and judgment and decision making, which frequently measure easily quantifiable performance variables like accuracy and response time, exhibit a greater penetration of mathematical reasoning and a higher proportion of mathematical psychologists than other domains. Processes such as the behavior of individual neurons, information flow through visual pathways, evidence accumulation in decision making, and language production or development have all been subjected to a great deal of mathematical modeling. However, even problems like the dynamics of mental illness, problems falling in the domains of social or clinical psychology, have benefited from a mathematical modeling approach (e.g., see the special issue on modeling in clinical science in the Journal of Mathematical Psychology [Townsend & Neufeld, 2010]). The power of the mathematical approach arises when unrealized implications of particular model structures become obvious after the mathematical representation of the model has been written down. By contrast, although verbal models might possess logical structure, the inability to interpret concepts in a mathematical fashion means that we cannot derive their logical implications. The ability to make such derivations for mathematical representations leads to better testability of theories, improved experimental designs targeting specific model predictions, and better data analyses—such analyses frequently being rooted in the statistical properties of the model variables.