S-multiplicativity of a stochastic matrix and applications to visual identification

A stochastic matrix is called S-multiplicative if the conditional probabilities in a specific subset S of its cells can be factorized into the product of two functions depending on the rows and columns, respectively. The concept of S-multiplicativity is related to Goodman's concept of quasi-independence and generalizes Falmagne's concept of a multiplicative confusion matrix, where S consists of the cells lying outside the main diagonal ($\text{D}$-multiplicativity). In the present study, the notion of S-multiplicativity is applied to the analysis of visual identification performance. The confusion matrices obtained from three identification experiments are tested for predictions of S-multiplicativity derived from the multicomponent theory of perception. If the stimuli do not overlap, i.e., different stimuli have no features in common, then the theory predicts a $\text{D}$-multiplicative confusion matrix. Alternatively, in the case of overlapping stimuli the theory predicts S-multiplicativity with respect to a more severely restricted subset S. These predictions are confirmed by the data.     

The relationship between Eysenck's major personality dimensions and simultaneous and sequential processing in children

The present study reexamined Eysenck's hypothesis that intelligence and personality are uncorrelated. Intelligence was measured with the Kaufman Assessment Battery for Children, yielding scores for simultaneous and sequential processing and achievement, and the Woodcock-Johnson Brief Scale. Personality was assessed with the Junior Eysenck Personality Questionnaire. Ss were 105 children, ages 8$\text{1}\text{2}$10$\text{1}\text{2}$yr. Although a process as well as a product-oriented intelligence test was employed, the correlational and discriminant analyses supported Eysenck's original hypothesis.     

Decision processes in children's recognition memory

Kindergarteners and third graders were given a continuous recognition memory task involving two-digit numbers. In addition, a rating scale consisting of photographs of various facial expressions was used to obtain confidence judgments from the Ss. Conventional analyses as well as signal detection analyses of the data revealed the following results: (a) the overall performance of the third graders was superior to that of the kindergarteners; (b) memory strength decreased as the number of intervening items increased; (c) there was no difference in the forgetting rates of the two grade levels; (d) the third graders exhibited a more liberal response bias than the kindergarteners; (e) both the hit rate (probability of correctly labeling an old stimulus as old) and the false-alarm rate (probability of incorrectly labeling a new stimulus as old) increased across blocks of items; (f) the increases in the hit rate and the false-alarm rate over blocks were due to a change in criterion from a relatively conservative level to a more lenient one; (g) the lower the S's level of confidence in judging an item as old, the lower was the probability of that item actually being old; (h) the third graders were better than the kindergarteners at gauging the accuracy of their recognition responses. It was concluded that with respect to recognition memory, chidren as young as $\text{5}\text{1}\text{2}$ years old are capable, to some extent, of monitoring their own memory states.     

Developmental changes in conditional reasoning: Linguistic or logical?

This study investigated reasoning with abstract conditional sentences as a function of age. Subjects from third-grade to eleventh-grade were required to evaluate the conclusion of several conditional arguments. The results confirmed the previously established finding that performance improves with age, particularly between 11 and 15 years. This finding could be interpreted to indicate that individuals become more logical as they get older. However, another possible interpretation is that the meaning of a conditional sentence like pthenq for naive Ss may not always be given by the truth function pq true, $\text{p}\text{q}$ false, $\text{p}\text{q}$ true, and $\text{p}\text{q}$ true. Further analysis suggested that at 9 years individuals treat the connective in the sentence, ifpthenq, as if it were either a conjunctive or a biconditional, that the conjunctive meaning disappears with increasing age and after 13 years is gradually superceded by the conditional.     

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.     

The modification of racial bias in preschool children

In previous studies using the Preschool Racial Attitude Measure (PRAM) and other measurement procedures, preschool children have displayed a tendency to evaluate light-skinned (Euro-American) persons more positively than dark-skinned (Afro-American) persons. Both Euro- and Afro-American children demonstrate this pro-Euro/anti-Afro ($\text{E+}\text{A?}$) bias, though it is less pronounced among Afro children.Two experiments designed to modify this racial bias are reported. The first study involved 39 Euro- and Afro-American preschool children and employed operant learning principles to modify $\text{E+}\text{A?}$ bias via a teaching machine procedure which provided reinforcement for pro-Afro/anti-Euro responses. Following training, the children showed a reduction in $\text{E+}\text{A?}$ bias on a test procedure not associated with the training. Follow-up testing of 30 of the children approximately 1 yr after the posttest revealed a tendency for the pro-Euro bias to be partially reestablished. The second study involved 70 Euro-American kindergarten children and was concerned with the modification of $\text{E+}\text{A?}$ bias via a classroom curriculum procedure and the possible influence of the race of the teacher associated with the curriculum. The experimental curriculum, designed to develop positive associations to dark-skinned persons and to the colors black and brown, was shown to have no effect on $\text{E+}\text{A?}$ bias, nor was race of teacher a variable of significance. Possible explantations for the findings of both experiments were discussed.     

On the reciprocity of proximity relations

The degree of reciprocity of a proximity order is the proportion, P(1), of elements for which the closest neighbor relation is symmetric, and the R value of each element is its rank in the proximity order from its closest neighbor. Assuming a random sampling of points, we show that Euclidean n-spaces produce a very high degree of reciprocity, $\text{P(1) ≥}\text{1}\text{2}$, and correspondingly low R values, E(R) ≤ 2, for all n. The same bounds also apply to homogeneous graphs, in which the same number of edges meet at every node. Much less reciprocity and higher R values, however, can be attained in finite tree models and in the contrast model in which the “distance” between objects is a linear function of the numbers of their common and distinctive features.     

Effect of prior Pavlovian discrimination training upon learning an operant discrimination

The effect of Pavlovian discrimination training with two stimuli upon subsequent learning of an operant discrimination involving those stimuli was studied. After preliminary lever press training, the lever was removed and thirsty rats received noncontingent pairings between S₁ (a tone or a clicker) and water reinforcements, whereas S₂ (a clicker or a tone) occurred always without reinforcement. This procedure presumably established S₁ as a positive CS for respondent behavior, whereas S₂ was established as an inhibitory CS. Following this training, the lever was reintroduced and the rats were trained on an operant (lever pressing) discrimination involving S₁ and S₂. For the Consistent Ss, S₁ was the S^D and S₂ the S^Δ in the operant discrimination; for the Reversed Ss, S₂ served as S^D and S₁ as S^Δ. The Consistent Ss learned the operant discrimination significantly faster than did the Reversed Ss. The result emphasizes the importance of respondents, conditioned to S^D and S^Δ, which modulate operant performance to these stimuli.     

Training research in aging on the fluid ability of inductive reasoning

The modifiability of older adults' performance on Induction tasks was examined through cognitive training. The posttraining performance of 52 older adults ($\text{X}\text{age = 70.3}\text{years}$) was assessed with regard to a transfer-of-training paradigm and maintenance of training effects at three posttests (one week, one month, six months). A pattern of differential training transfer across the posttest battery of fluid and crystallized intelligence measures was predicted, with the largest training effects expected for near transfer (Induction) measures. The predicted pattern of training transfer was obtained at one-week and one-month posttests, with significant training effects to the nearest Induction measure. Large retest (practice) effects across posttests also occurred for training and control groups. This research contributes to the position that modifiability of intellectual performance through cognitive intervention extends across the adult life span.     

Propositional self-control in children

The scope of Dulany's (1968) Theory of Propositional Control was examined using 54 $\text{11}\text{1}\text{2}\text{-, 8}\text{1}\text{2}\text{-,}\text{and 4}\text{1}\text{2}\text{-year-olds}$. Significance of the reinforcer was manipulated orthogonally to value of the reinforcer. With a six attribute concept attainment task $\text{4}\text{1}\text{2}\text{-}\text{year-olds}$ did not learn in either of two significant ways: a performance change over blocks, nor the acquisition of a rule of reinforcement. Children of $\text{8}\text{1}\text{2}$ and $\text{11}\text{1}\text{2}$ learned in both of the above senses. In addition, for the latter subjects the central equations of the theory held with strength and subjects' intentions accounted for more variance in performance than did the experimental manipulations. In supplementary experiments using a simplified task some $\text{4}\text{1}\text{2}\text{-}\text{year-olds}$ learned. When only the $\text{4}\text{1}\text{2}\text{-}\text{year-olds}$ who learned to a criterion were examined the central equations of the theory were nearly as strong as for subjects at ages $\text{8}\text{1}\text{2}$ and $\text{11}\text{1}\text{2}$. There was no evidence for learning without awareness at any age level.     

The emergence of functional play in infants: Evidence for a major cognitive transition

Sixty-four infants, eight boys and eight girls at each of four ages, $\text{9}\text{1}\text{2}\text{, 11}\text{1}\text{2}\text{, 13}\text{1}\text{2}\text{,}\text{and}\text{15}\text{1}\text{2}$ months, were observed during a 15-minute free play session. Three types of play (stereotypical, relational, and functional) and the number of appropriate uses for specific realistic toys were recorded. Stereotypical play (87% of total active play at $\text{9}\text{1}\text{2}$ months) gave way to relational (35% at $\text{13}\text{1}\text{2}$ months) and functional play (52% at $\text{15}\text{1}\text{2}$ months) as the dominant activity. Both functional play and the number of different appropriate uses of toys were rare at $\text{9}\text{1}\text{2}$ months, appeared reliably at $\text{11}\text{1}\text{2}$ months and increased lineraly through $\text{15}\text{1}\text{2}$ months. A change in the quality of play from indiscriminate mouthing, waving, banging, and fingering of objects to the matching of appropriate uses for a large array of toys indicates that the child's manipulations become more object specific and functional over this 6-month age span. Among possible explanations for the linear increase in the frequency and diversity of appropriate toy uses is the suggestion that an important cognitive change occurs toward the end of the first year of life. A case study is presented to illustrate the application of the play procedure for both assessment and treatment.     

Two-generator semigroups of binary relations

We study upper bounds on the size of the semigroups generated two randomly chosen n × n Boolean matrices having exactly N one entries. In general, as $\text{N}\text{n}$ increases the semigroups tend to be smaller. If N is unrestricted, or is a function which is at least [$\text{((2 + ?)n}{}^3\text{log}\text{n)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$] the probability tends to one that the size of the semigroup is exactly 3. If N is a function which is at least [(r + 1 + ?)n log n] the probability tends to one that the size of the semigroup will be no more than $\text{2}{}^{\mathrm{<mtext>(</mtext><mtext>(n?1)</mtext><mtext>r</mtext><mtext>)+1</mtext>}}\text{? 1}$. However, if N is any function w(n) such that $\text{w(n)}\text{n}{}^2\text{→ 0}$ and w(n) > n, the average size of the resulting semigroups will be at least . This phenomenon is caused by some semigroups of extremely large size. It is thought that w(n) > n and $\text{w(n)}\text{n}{}^2\text{→ 0}$ hold for those matrices usually encountered in sociology and psychology.     

On unpredictability as a causal factor in “learned helplessness”

In two replications, two groups of dogs were exposed to a series of uncontrollable, electric shocks. For one group the shocks were preceded by a tone (i.e., Paired). For the other group the shocks were randomly related to the tones and hence unpredictable (i.e., Random). Each replication also included a third group; in the first it was exposed only to the series of tones (CS-only), while in the second, it was exposed only to a series of shocks (Shocks-only). Then, all dogs were required to learn a discriminative choice escape/avoidance task in which the required response was to lift the correct paw in the presence of each of two visual S^Ds to escape or avoid the shocks [($\text{S}{}_1{}^{\mathrm{D}}\text{?R}{}_1\text{)}\text{(S}{}_2{}^{\mathrm{D}}\text{?R}{}_2$)]. Dogs preexposed to random tones and shocks were least successful in learning the task relative to those groups which experienced either predicted shocks, only the tones, or only the shocks, which in turn did not differ from each other. These results permitted the inference that the proactive interference with choice behavior following random tone CSs and shocks was attributable to a learned irrelevance generalized with respect to CSs.     

Treatment of a selectively mute third-grade child

A third-grade girl who did not speak in school was treated by school personnel using principles of behavior modification. A step-by-step procedure was designed which first encouraged speech in a one-to-one setting with an adult. Gradually elements of the regular school situation were introduced into the setting. The final phrase of treatment involved moving the elements back into the regular classroom. At the end of the school year, $\text{2}\text{1}\text{2}$ months following termination of treatment, the subject was still speaking freely.     

On the scales of measurement

Let $\text{X}$ = 〈X, ≧, R₁, R₂…〉 be a relational structure, 〈$\text{X, ≧}$〉 be a Dedekind complete, totally ordered set, and n be a nonnegative integer. $\text{X}$ is said to satisfy n-point homogeneity if and only if for each x₁,…, x_n, y₁,…, y_n such that x₁ ? x₂ ? … ? x_n and y₁ ? y₂ … ? y_n, there exists an automorphism α of $\text{X}$ such that α(x₁) = y_i. $\text{X}$ is said to satisfy n-point uniqueness if and only if for all automorphisms β and γ of $\text{X}$, if β and γ agree at n distinct points of $\text{X}$, then β and γ are identical. It is shown that if $\text{X}$ satisfies n-point homogeneity and n-point uniqueness, then n ≦ 2, and for the case n = 1, $\text{X}$ is ratio scalable, and for the case n = 2, interval scalable. This result is very general and may in part provide an explanation of why so few scale types have arisen in science. The cases of 0-point homogeneity and infinite point homogeneity are also discussed.     

Pavlovian processes do mediate control of the “advance” response strategy

Two parallel experiments used pigeons in a transfer of control design to determine the basis of the transfer of the use of the “advance strategy” across successive discriminations (W. K. Honig & H. Lindsay, Learning and Motivation, 1975, 6, 157–178). Pigeons were first trained under instrumental contingencies to use an “advance” response to maximize their exposure to S^D and minimize their exposure to S^Δ. Then orthogonal stimuli (key lights in Experiment 1 and diffuse illuminations in Experiment 2) were independently established through separate discriminative Pavlovian procedures as CS+ and CS? or as CS^o's. Later in special tests, it was demonstrated that the associative values of the Pavlovian CS's were a major factor governing the use of the advance strategy: only pigeons exposed to $\text{CS+}\text{CS?}$ used the advance response to regulate their exposure to either CS. Although these experiments showed that the necessary and sufficient conditions for utilization of the advance strategy are the Pavlovian associative values of the discriminative stimuli, opportunity for differential “operant” responding during the discriminative stimuli seems to be a contributor to the optimal use of the advance response across successive discriminations.     

Development of relational learning: Effects of instruction and delay of transfer

Preschool children were trained on a color relevant oddity problem by one of three methods: increased salience of the oddity relationship, instruction on the solution rule, or a combination of increased salience and rule instruction. The youngest Ss solved the training problem when the solution rule was provided but not under the salience procedure. The two older groups solved in all conditions and errors decreased with age. The generality of the solution was determined by performance on a form relevant transfer task. The youngest Ss made fewer errors on the transfer problem following rule instruction training than following training in the combined procedure. The intermediate age group made fewer errors when the transfer task was administered one week after training than one minute after training. The transfer problem was easily solved by the oldest Ss and performance was independent of training procedure and amount of delay. These results were interpreted as reflecting developmental differences in relational concept learning.     

Cultural differences on WISC-R verbal items

The item difficulty patterns of four groups of nonreferred, average children— Anglos, Blacks, Chicanos and Bermudians—were compared on each of the verbal subtests of the Wechsler Intelligence Scale for Children-Revised. The analysis was done for $\text{7}\text{1}\text{2}$ years old and $\text{10}\text{1}\text{2}$ year olds separately. Although this procedure identified a small number of items differentially difficult for one group of children or another, the item difficulty curves for the four groups were remarkably parallel, considering the differences in the children's cultures.     

Decision making in a case of mixed uncertainty: A normative model

We consider a case of uncertainty which is frequently met in various fields, e.g., in parametric statistics: Events {θ}, θ ∈ ∵, are members of family $\text{E}$ on which the decision maker possesses no information at all; however, conditionally on the realization of {θ}, he is able to affix probabilities to all members of another family of events, $\text{F}$. We assume that the decision maker: (1) has a rational behavior under complete ignorance, for decisions whose results only depend on events of $\text{E}$; (2) with {θ} known, maximizes his conditional expected utility for decisions whose results only depend on events of $\text{F}$; (3) has (unconditional) preferences which are consistent with his conditional ones. These assumptions are shown to be sufficient to ensure an approximate representation of the decision maker's preference by a real-valued function W which has the form $\text{W(f) = v[}\text{Inf}{}_{\mathrm{\theta \in \because }}\text{E}{}_{\mathrm{\theta }}\text{(u°f),}\text{Sup}{}_{\mathrm{\theta \in \because }}\text{E}{}_{\mathrm{\theta }}\text{(u°f)]}$, where u and v, respectively, characterize the decision maker's attitudes toward risk and toward complete ignorance.