Compounds,confounds, and models in developmental information processing: A reply to Trabasso and Foellinger

A point by point examination of Trabasso and Foellinger's paper shows their criticisms of my 1970 work to be based on errors of fact (regarding the data I reported) and errors of method (regarding proper procedures for model evaluation). Factual errors are refuted by summarizing crucial data reported in 1970 but ignored by my critics. Errors of method are refuted by contrasting my $\text{CSVI}\text{BE}$ model and data with Trabasso and Foellinger's model and data in the light of the scientific epistemology of model evaluation. The issues of “general” versus “local” models, their “empirical scope,” “number of empirical parameters,” and “simplifying assumptions” are examined. Five ways of evaluating models are distinguished, and in all five my 1970 model is shown to be superior to that of Trabasso and Foellinger. The 1970 data (in light of the controls built into the CSVI task) and new unreported data exhibiting developmental step functions confirm the model of a developmental growth of M (mental energy, working memory), which occurs concurrently with and independently from the growth of executive/control structures. Executive growth alone cannot explain the obtained results.     

Classification of concatenation measurement structures according to scale type

A relational structure is said to be of scale type (M,N) iff M is the largest degree of homogeneity and N the least degree of uniqueness (Narens, 1981a, Narens, 1981b) of its automorphism group.Roberts (in Proceedings of the first Hoboken Symposium on graph theory, New York: Wiley, 1984; in Proceedings of the fifth international conference on graph theory and its applications, New York: Wiley, 1984) has shown that such a structure on the reals is either ordinal or M is less than the order of at least one defining relation (Theorem 1.2). A scheme for characterizing N is outlined in Theorem 1.3. The remainder of the paper studies the scale type of concatenation structures 〈X, ?, ° 〉, where ? is a total ordering and ° is a monotonic operation. Section 2 establishes that for concatenation structures with M>0 and N<∞ the only scale types are (1,1), (1,2), and (2,2), and the structures for the last two are always idempotent. Section 3 is concerned with such structures on the real numbers (i.e., candidates for representations), and it uses general results of Narens for real relational structures of scale type (M, M) (Theorem 3.1) and of Alper (Journal of Mathematical Psychology, 1985, 29, 73–81) for scale type (1, 2) (Theorem 3.2). For M>0, concatenation structures are all isomorphic to numerical ones for which the operation can be written $\text{x°y = yf(}\text{x}\text{y}\text{)}$, where f is strictly increasing and $\text{f(x)}\text{x}$ is strictly decreasing (unit structures). The equation f(x^?)=f(x)^? is satisfied for all x as follows: for and only for ? = 1 in the (1,1) case; for and only for ?=kⁿ, k > 0 fixed, and n ranging over the integers, in the (1, 2) case; and for all ?>0 in the (2, 2) case (Theorems 3.9, 3.12, and 3.13). Section 4 examines relations between concatenation catenation and conjoint structures, including the operation induced on one component by the ordering of a conjoint structure and the concept of an operation on one component being distributive in a conjoint structure. The results, which are mainly of interest in proving other results, are mostly formulated in terms of the set of right translations of the induced operation. In Section 5 we consider the existence of representations of concatenation structures. The case of positive ones was dealt with earlier (Narens & Luce (Journal of Pure & Applied Algebra27, 1983, 197–233). For idempotent ones, closure, density, solvability, and Archimedean are shown to be sufficient (Theorem 5.1). The rest of the section is concerned with incomplete results having to do with the representation of cases with M>0. A variety of special conditions, many suggested by the conjoint equivalent of a concatenation structure, are studied in Section 6. The major result (Theorem 6.4) is that most of these concepts are equivalent to bisymmetry for idempotent structures that are closed, dense, solvable, and Dedekind complete. This result is important in Section 7, which is devoted to a general theory of scale type (2, 2) for the utility of gambles. The representation is a generalization of the usual SEU model which embodies a distinctly bounded form of rationality; by the results of Section 6 it reduces to the fully rational SEU model when rationality is extended beyond the simplest equivalences. Theorem 7.3 establishes that under plausible smoothness conditions, the ratio scale case does not introduce anything different from the (2, 2) case. It is shown that this theory is closely related to, but somewhat more general, than Kahneman and Tversky's (Econometrica47, 1979, 263–291) prospect theory.     

Slow and fast chess performance across three expert levels

ObjectivesSlow and fast thinking are crucial for human decision making in several domains of human activity including sports. These cognitive processes are remarkable in the intellectually demanding sport of chess. Slow and fast thinking underlie chess performance. However, the relative influence of each process has elicited controversial findings. Moreover, individual differences in chess skill are likely to moderate the integration of both processes.DesignThe simultaneous change over six time points in slow and fast chess performance was analyzed with a cross-domain latent curve model (LCM).MethodArchival data from an extensive group of chess players (n = 32,173) were included in these analyses at untitled, intermediate, and advanced levels of expertise. Intercept and slope latent factors of growth were specified and correlated for both processes.ResultsThere were remarkable differences in the change in slow and fast performance regarding the three expert levels, and in the concurrent interrelationship of both processes. The interdependence between both processes was more robust for the advanced than for the untitled and intermediate players.ConclusionsThese findings suggest that a better integration of slow and fast performance is produced at higher levels of expertise.     

Improving visual acuity in a myopic child: Assessing compliance and effectiveness

Improvements in visual acuity following vision training were evaluated for an 11$\text{1}\text{2}$-yr-old myopic male. Initial increases in the distance at which the S could discriminate letters were found. However, performance began to deteriorate as training progressed. A negative-reinforcement procedure was employed in order to rule out motivational factors potentially related to this decreased performance. Using a changing criterion within an ABCBC reversal design, the distance at which the S correctly discriminated letters increased by more than $\text{4}\text{1}\text{2}$ times and was clearly related to the reinforcement procedure.     

Perception in chess

This paper develops a technique for isolating and studying the perceptual structures that chess players perceive. Three chess players of varying strength — from master to novice — were confronted with two tasks: (1) A perception task, where the player reproduces a chess position in plain view, and (2) de Groot's (1965) short-term recall task, where the player reproduces a chess position after viewing it for 5 sec. The successive glances at the position in the perceptual task and long pauses in the memory task were used to segment the structures in the reconstruction protocol. The size and nature of these structures were then analyzed as a function of chess skill.     

Personality and leisure activities: an illustration with chess players

The present study investigated the relationship between personality and involvement in a leisure activity: chess playing. The participants comprised three groups of highly competitive chess players, moderately competitive chess players, and a comparison group of non-players (n = 20 each). The results showed that of the six personality characteristics under investigation all chess players differed from the comparison group in terms of unconventional thinking and orderliness. In addition, highly competitive players differed from non-players in being also significantly more suspicious. The three groups did not differ significantly on neuroticism, aggressive tendency, and hostility. Implications concerning future studies of the relationship between personality and involvement in competitive leisure activities are discussed.     

A complete database of international chess players and chess performance ratings for varied longitudinal studies

Chess is an oft-used study domain in psychology and artificial intelligence because it is well defined, its performance rating systems allow easy identification of experts and their development, and chess playing is a complex intellectual task. However, usable computerized chess data have been very limited. The present article has two aims. The first is to highlight the methodological value of chess data and how researchers can use them to address questions in quite different areas. The second is to present a computerized database of all international chess players and official performance ratings beginning from the inaugural 1970 international rating list. The database has millions of records and gives complete longitudinal official performance data for over 60,000 players from 1970 to the present. Like a time series of population censuses, these data can be used for many different research and teaching purposes. Three quite different studies, conducted by the author using the database, are described.     

Positive-difference structures and bilinear utility functions

Let (M₁, f), (M₂, g) be mixture sets and let ? be a binary preference relation on M₁ × M₂. By using the concept of positive-difference structures, necessary and sufficient conditions are given for the existence of a real-valued utility function u on M₁ × M₂ which represents ? and possesses the bilinearity property

$\text{u(?(α, x}{}_1\text{,x}{}_2\text{),g(β, y}{}_1\text{, y}{}_2\text{))}\text{=}\text{αu(x}{}_1\text{, g(βy}{}_1\text{, y}{}_2\text{))}\text{+}\text{(1 ? α) u(x}{}_2\text{, g(β, y}{}_1\text{, y}{}_2\text{))}\text{=}\text{βu(?(α,x}{}_1\text{, x}{}_2\text{),y}{}_1\text{)}\text{+}\text{(1 ? β) u(?(α,x}{}_1\text{, x}{}_2\text{),y}{}_2\text{)}$

, for all α, β ∈ [0, 1], all x₁, x₂ ∈ M₁ and all y₁, y₂ ∈ M₂. Moreover, uniqueness up to positive linear transformations can be proved for those utility functions. Finally an outline is given of applications of these results in expected utility theory.     

On the basis of early transitivity judgments

This research had two aims. The first was to test three explanations of performance on N-term series tasks by young children: the labeling model of B.DeBoysson-Bardies and K. O'Regan (1973), Nature (London), 246, 531–534, the sequential-contiguity model of L. Breslow (1981, Psychological Bulletin, 89, 325–351), and the ordered array or image model of C. A. Riley and T. Trabasso (1974, Journal of Experimental Child Psychology, 17, 187–202). In the first experiment, 5-year-old children were taught additional premises which would interfere with labeling and sequential-contiguity processes, but not with forming an ordered array. Reasoning performance was essentially comparable to previous results with the paradigm, thus supporting the ordered array model. The second aim was to reexamine children's ability to learn sets of premises which can be assembled into an ordered array, since there was reason to believe that previous studies had created false positives. In the second experiment, 3- to 7-year-old children were taught either overlapping (a > b, b > c, …) or nonoverlapping (a > b, c > d, …) premises. Overlapping premises can be integrated into an ordered array (a, b, c, d, e), but nonoverlapping premises cannot. However, the overlapping condition proved more difficult, and the success rate for preschoolers ($\text{3}\text{1}\text{2}\text{-}\text{to 4}\text{1}\text{2}\text{-year-olds}$) was of zero order. This raises doubts about their ability to learn a set of premises of the kind required for transitive inference. These doubts were strengthened by the third experiment which showed that when premises were not presented in serial order, preschool ($\text{3}\text{1}\text{2}\text{-}\text{to 4}\text{1}\text{2}\text{-year-old}$) children could not learn the premises of an N-term series task.     

Response mode and choice consistency: A test of unfolding theory

Coombs, Donnell, and Kirk (1978. Journal of Experimental Psychology, 4, 497–512), in a study of risk preferences, collected data using both pick $\text{1}\text{3}$ and reject $\text{1}\text{3}$ response modes. Although the preference orders derived from the two response modes were identical, the pick data contained a greater number of inconsistencies than the reject data. In the present study, predictions were derived from unfolding theory (Coombs, 1964. A theory of data. New York: Wiley) regarding the relative consistency of pick and reject response modes. An experiment performed as a test of these predictions supported the unfolding model suggesting that differences in inconsistency between response modes could be attributed to the fineness of the grid of working midpoints imposed upon the choice process by the response mode.     

An alternative scaling method for priorities in hierarchical structures

The early contributions of Saaty have spawned a multitude of applications of principal right (PR) eigenvector “scaling” of a dominance matrix [R]. Prior to Saaty's work (1977–1984) scaling of dominance matrices received little attention in multidimensional scaling, e.g., see Shepard (1972, pp. 26–27). This eigenvector method (EM) of scaling [R] yields u_i scores (weights) popularly used at each branching of the Analytic Hierarchy Process (AHP) technique that has been increasingly applied in multiple criterion analysis of utility, preference, probability, and performance. In this paper, it is proposed that an alternate least squares method (LSM) scaling technique yielding least squares optimal scores (weights) provides $\text{w}{}_{\mathrm{i}}{}^1$ values having a number of important advantages over u_i scores popularly utilized to date.     

Chess knowledge predicts chess memory even after controlling for chess experience: Evidence for the role of high-level processes

The expertise effect in memory for chess positions is one of the most robust effects in cognitive psychology. One explanation of this effect is that chess recall is based on the recognition of familiar patterns and that experts have learned more and larger patterns. Template theory and its instantiation as a computational model are based on this explanation. An alternative explanation is that the expertise effect is due, in part, to stronger players having better and more conceptual knowledge, with this knowledge facilitating memory performance. Our literature review supports the latter view. In our experiment, a sample of 79 chess players were given a test of memory for chess positions, a test of declarative chess knowledge, a test of fluid intelligence, and a questionnaire concerning the amount of time they had played nontournament chess and the amount of time they had studied chess. We determined the numbers of tournament games the players had played from chess databases. Chess knowledge correlated .67 with chess memory and accounted for 16% of the variance after controlling for chess experience. Fluid intelligence accounted for an additional 13% of the variance. These results support the conclusion that both high-level conceptual processing and low-level recognition of familiar patterns play important roles in memory for chess positions.     

A locus problem involving the three-dimensional city-block metric

In connection with multidimensional scaling, representations have been considered of the form abDcd ← ?(f(a), f(b)) ≦ ?(f(c), f(d)), for all a, b, c, d ∈ A, where A is a nonvoid finite set, D is a four-place relation on A, f is a function from A into Euclidean n-space, $\text{R}{}^{\mathrm{n}}$, and ? is a metric in $\text{R}{}^{\mathrm{n}}$. For particular metrics there exist finite universal axiomatizations which are necessary and sufficient for the above representation. On the other hand, it is known that no such axiomatizations can be given for either the supremum metric or the ordinary Euclidean metric. Methods for showing this apply easily to the city-block metrics in $\text{R}{}^1$ and $\text{R}{}^2$. This article describes a computer-aided verification of a locus result which shows the impossibility of finite universal axiomatizability for the case of the city-block metric in $\text{R}{}^3$. The result was obtained by dealing with 21,780 cases, each of which involved a set of 10 equations in 12 unknowns along with a related set of inequalities.     

Chess databases as a research vehicle in psychology: Modeling large data

The game of chess has often been used for psychological investigations, particularly in cognitive science. The clear-cut rules and well-defined environment of chess provide a model for investigations of basic cognitive processes, such as perception, memory, and problem solving, while the precise rating system for the measurement of skill has enabled investigations of individual differences and expertise-related effects. In the present study, we focus on another appealing feature of chess—namely, the large archive databases associated with the game. The German national chess database presented in this study represents a fruitful ground for the investigation of multiple longitudinal research questions, since it collects the data of over 130,000 players and spans over 25 years. The German chess database collects the data of all players, including hobby players, and all tournaments played. This results in a rich and complete collection of the skill, age, and activity of the whole population of chess players in Germany. The database therefore complements the commonly used expertise approach in cognitive science by opening up new possibilities for the investigation of multiple factors that underlie expertise and skill acquisition. Since large datasets are not common in psychology, their introduction also raises the question of optimal and efficient statistical analysis. We offer the database for download and illustrate how it can be used by providing concrete examples and a step-by-step tutorial using different statistical analyses on a range of topics, including skill development over the lifetime, birth cohort effects, effects of activity and inactivity on skill, and gender differences.     

Similar threshold functions from contrasting neural signal models

Three neural signal models of increment threshold detection are compared. All assume that the criterion for threshold is the attainment of a critical, minimum neural signal (or difference between two neural signals), and that the signal due to a test flash of intensity λ in the absence of a background light is $\text{λ}\text{(λ + σ)}$ (where σ is the semi-saturation constant). The models differ in the manner in which a background light of intensity θ is assumed to affect the signal. One model (due to Alpern et al., 1970a, Alpern et al., 1970b, Alpern et al., 1970c) assumes that the test flash signal, $\text{λ}\text{(λ + σ)}$, is attenuated by the multiplicative factor $\text{θ}{}_{\mathrm{D}}\text{(θ + θ}{}_{\mathrm{D}}\text{)}$ (where θ_D is a constant interpreted as sensory noise); another model specifies that the test flash signal is simply reduced (by subtraction) by the amount $\text{θ}\text{(θ + K)}$ (K a constant). One main result of this paper is that in the absence of pigment bleaching, these two models imply indistinguishable increment threshold functions. Further, a necessary and sufficient condition for each model guaranteeing the absence of saturation with steady backgrounds is found to be empirically satisfied. A third model is considered where the background field is assumed both to contribute to the neural signal and simultaneously to attenuate it (via a gain change). These assumptions are closely related to theoretical accounts of color induction and color perception. Though this model needs further investigation, it appears to be in better accord with actual increment threshold data than the others.     

Pooling peripheral information: Averages versus extreme values

Suppose, as an idealization, that sensory intensity is coded in peripheral channels as identical Poisson pulse trains with intensity parameter a power function of signal intensity. Discrimination models based on either an average count computed over a fixed time or an average time computed for a fixed count per channel have difficulty in fitting the Weber function ($\text{ΔI}\text{I}$ versus I) if the free parameters are constrained to ranges determined from other experiments (magnitude estimation, reaction time). Here we study a different decision rule, namely, the most extreme observation in either the counting or timing mode. Our extremum-counting model, but not two timing ones, accounts very nicely for the Weber function. However, the ROC curves for these extremum models, which agree in shape with data of Green and Luce, yield estimates for the intensity parameter which are much larger than predicted by the power function growth used to calculate ΔI and about twice as large as those estimated from reaction time data collected in the same experiment.     

Restricted thresholds for interval orders: A case of nonaxiomatizability by a universal sentence

Let $\text{P}$_n be the class of all finite interval orders that can be interval-represented using no more than n interval lengths or threshold levels. Thus $\text{P}$₁ is the class of finite semiorders, and $\text{P}$, the union of the $\text{P}$_n, is the class of finite interval orders. While each of $\text{P}$₁ and $\text{P}$ is axiomatizable by a universal sentence in first-order logic, no $\text{P}$_n for $\text{n ≧ 2}$ is axiomatizable in the same sense.     

The effects of speed on skilled chess performance

Two types of mechanisms may underlie chess skill: fast mechanisms, such as recognition, and slow mechanisms, such as search through the space of possible moves and responses. Speed distinguishes these mechanisms, so I examined archival data on blitz chess (5 min for the whole game), in which the opportunities for search are greatly reduced. If variation in fast processes accounts for substantial variation in chess skill, performance in blitz chess should correlate highly with a player's overall skill. In addition, restricting search processes should tend to equalize skill difference between players, but this effect should decrease as overall skill level increases. Analyses of three samples of blitz chess tournaments supported both hypotheses. Search is undoubtedly important, but up to 81% of variance in chess skill (measured by rating) was accounted for by how players performed with less than 5% of the normal time available.     

Difference measurement and simple scalability with restricted solvability

Let A, B be two sets, with B ? A × A, and ≤ a binary relation on B. The problem analyzed here is that of the existence of a mapping u: A → R, satisfying:

$\text{(a,b) ? (}\text{a}\text{?}\text{,}\text{b}\text{?}\text{)}\text{iff}\text{∨∧ μ(b) ? μ(a) ? μ(}\text{b}\text{?}\text{) ? μ(}\text{a}\text{?}\text{)}$

whenever (a, b), (a′, b′) ∈ B. In earlier discussions of this problem, it is usually assumed that B is connected on A. Here, we only assume that B satisfies a certain convexity property. The resulting system provides an appropriate axiomatization of Fechner's scaling procedures. The independence of axioms is discussed. A more general representation is also analyzed:

$\text{(a,b) ? (}\text{a}\text{?}\text{,}\text{b}\text{?}\text{)}\text{iff}\text{∨∧ F[μ(b), μ(a)] ? F[μ}\text{b}\text{?}\text{]}$

, where F is strictly increasing in the first argument, and strictly decreasing in the second. Sufficient conditions are presented, and a proof of the representation theorem is given.     

Planning abilities and chess: A comparison of chess and non‐chess players on the Tower of London task

Playing chess requires problem‐solving capacities in order to search through the chess problem space in an effective manner. Chess should thus require planning abilities for calculating many moves ahead. Therefore, we asked whether chess players are better problem solvers than non‐chess players in a complex planning task. We compared planning performance between chess (N=25) and non‐chess players (N=25) using a standard psychometric planning task, the Tower of London (ToL) test. We also assessed fluid intelligence (Raven Test), as well as verbal and visuospatial working memory. As expected, chess players showed better planning performance than non‐chess players, an effect most strongly expressed in difficult problems. On the other hand, they showed longer planning and movement execution times, especially for incorrectly solved trials. No differences in fluid intelligence and verbal/visuospatial working memory were found between both groups. These findings indicate that better performance in chess players is associated with disproportionally longer solution times, although it remains to be investigated whether motivational or strategic differences account for this result.