Psychology and Aging 1998, VbT. 13, No. 3,501-518 Copyright 1998 by the A A Molar Entropy Model of Age Differences in Spatial Memory Philip A. Allen, Miron Kaufman, Albert F. Smith, and Ruth E. Propper Cleveland State University Two very-short-term-memory, spatial scanning aging experiments were conducted involving a graph- ics character as a target stimulus. On the probe portion of a trial, the stimulus was presented in the same position as it was on the target portion of the trial (i.e., a same trial) 50% of the time. However, on the remaining 50% of the trials, the probe stimulus was shifted (or transposed) 1, 2, or 3 positions to the right or left of the original presentation (target) position. In Experiment 1, exposure duration was manipulated. In Experiment 2, the number of potential target display positions was manipulated. For both experiments, older adults showed larger transposition distance effects than younger adults for errors. In the past (e.g., P. A. Allen, 1990, 1991), this effect has been attributed to higher levels of internal noise (entropy) in older than younger adults. This research provides converging operations to this contention by using statistical physics methods to rigorously compute the entropy in a molar neural network across age groups. After successfully fitting the statistical mechanics model to the data, the model is proved to have external validity by fitting a simplified version of it to an earlier spatial memory aging experiment reported by P. R. Bruce and J. F. Herman (1986). The results of both traditional reaction time and error rate analyses, as well as the entropy modeling analyses, indicated that older adults exhibited higher levels of entropy than did the younger adults and that this effect appeared to be generalized across processing stage. In this article, we develop and test a molar model of age differences in spatial memory. In particular, we develop a model that accounts for age differences in spatial episodic memory by assuming that information processing occurs in a neural network and can be predicted by the "computational temperature" (Smolensky, 1986) of this network. Using statistical mechanics methods that are discussed in more detail later, we use the error data from three experiments to estimate the computational temperature of younger and older adults. This general neural network approach to examining age dif- ferences in information processing is not novel to our research. For example, Salthouse (1988) simulated an associative network to predict age differences in picture completion. Also, MacKay and Burke (1990) proposed a theoretical parallel distributed processing network to explain age differences in learning new information. We wanted to extend the earlier simulation work of Salthouse (1988) and the theoretical work of MacKay and Burke (1990) to develop both a mathematical theory and a psychological theory of age differences in spatial memory. By the term mathematical theory we mean a set of interrelated Philip A. Allen, Albert F. Smith, and Ruth E. Propper, Department of Psychology, Cleveland State University; Miron Kaufman, Department of Physics, Cleveland State University. Ruth E. Propper is now at Harvard Medical School and the Massachusetts Mental Health Center, Boston. This research was supported by the National Institute on Aging and National Institutes of Health Grant AG09282. We are grateful to Sanda Kaufman for her expert statistical advice on how to estimate the fit of empirical data points to theoretical models. We also thank Tim Weber and David Chancy for technical assistance. Correspondence concerning this article should be addressed to Philip A. Allen, Department of Psychology, Cleveland State University, Euclid Avenue at East 24th Street, Cleveland, Ohio 44115. Electronic mail may be sent to p.allen@popmail.csuohio.edu. definitions, theorems, and analytical techniques (Smolensky, 1986). By the term psychological theory we mean an integrated body of knowledge (including theoretical constructs and an accumulation of empirical results) about a certain aspect of human behavior—in this case, age differences in spatial memory. Our mathematical model is an application of the statistical physics framework of Boltzmann (1898/1966) and Gibbs (1961; see Hinton & Sejnowski, 1986; Smolensky, 1986) to cognitive science. Our work is related in spirit to the applications of the statistical physics framework in psychology promoted by parallel distributed processing research on harmony theory (Smolensky, 1986) and Boltzmann machines (Hinton & Sej- nowski, 1986). Harmony theory is a statistical physics model of information processing, according to which optimal neural processing (i.e., zero entropy) occurs when the computational temperature is lowest. As Smolensky (1986) noted, harmony theory has parallels in physics and psychology. Specifically, the relationship between harmony (the complement of entropy) and probability is mathematically analogous to the relationship be- tween (minus) energy and probability in statistical physics. In- deed, in statistical physics, this relationship is called the "Gibbs law" or "Boltzman law." Instead of minimizing entropy, however (as was the goal of Smolensky's, 1986, harmony theory), we were interested in determining what happens to an information-processing system when entropy is high in one group (older adults) compared with another (younger adults). Consequently, in this research we used quantitative methods developed for statistical physics to estimate the temperature of a physical system to derive measures of the computational temperature of a cognitive (information- processing) system. This method also allowed us to predict how increased age would affect this cognitive system. A major 501