Operating on Functions with Variable Domains

The sum, difference, product and quotient of two functions with different domains are usually defined only on their common domain. This paper extends these definitions so that the sum and other operations are essentially defined anywhere that at least one of the components is defined. This idea is applied to propositions and events, expressed as indicator functions, to define conditional propositions and conditional events as three-valued indicator functions that are undefined when their condition is false. Extended operations of and, or, not and conditioning are then defined on these conditional events with variable conditions. The probabilities of the disjunction (or) and of the conjunction (and) of two conditionals are expressed in terms of the conditional probabilities of the component conditionals. In a special case, these are shown to be weighted averages of the component conditional probabilities where the weights are the relative probabilities of the various conditions. Next, conditional random variables are defined to be random variables X whose domain has been restricted by a condition on a second random variable Y. The extended sum, difference, product and conditioning operations on functions are then applied to these conditional random variables. The expectation of a random variable and the conditional expectation of a conditional random variable are recounted. Theorem 1 generalizes the standard result that the conditional expectation of the sum of two conditional random variables with disjoint and exhaustive conditions is a weighted sum of the conditional expectations of the component conditional random variables. Because of the extended operations, the theorem is true for arbitrary conditions. Theorem 2 gives a formula for the expectation of the product of two conditional random variables. After the definition of independence of two random variables is extended to accommodate the extended operations, it is applied to the formula of Theorem 2 to simplify the expectation of a product of conditional random variables. Two examples end the paper. The first concerns a work force of n workers of different output levels and work shifts. The second example involves two radars with overlapping surveillance regions and different detection error rates. One radar's error rate is assumed to be sensitive to fog and the other radar's error rate is assumed to be sensitive to air traffic density. The combined error rate over the combined surveillance region given heavy fog and moderate air traffic is computed.     

An analysis of generalized contextual control of conditional discriminations

This research asked whether performance engendered by contextual control procedures would generalize to novel matching-to-sample stimulus arrangements. Two studies were conducted with young adult participants. In Study 1, participants first were trained to perform the contextually controlled conditional discrimination, X-AB, where the sample-comparison relations A1B1 and A2B2 were reinforced in the presence of contextual stimulus X1, but the relations A1B2 and A2B1 were reinforced in the presence of X2. Then, a new conditional discrimination, CD, was established via an unreinforced-conditional-selection procedure. Next, participants were tested for X-CD contextual control performance. Participants selected the originally established CD relations in the presence of X1, but the opposite relations in the presence of X2. Next, an additional conditional relation, EF, was established. Then, participants received trials consisting of entirely novel contextual stimuli, Z1 and Z2, and EF samples and comparisons. Selections were consistent with contextual control; that is, participants selected the originally established EF relations in the presence of one of the novel contextual Z stimuli, but selected the opposite EF relations in the presence of the other contextual Z stimulus. Study 2 systematically replicated these results with naive participants and demonstrated the necessity of first establishing a conditional discrimination prior to tests for generalized contextual control. The findings are discussed in terms of unreinforced conditional selection, stimulus classes, and new ways in which contextual control performances can emerge.     

Conjoint control of performance in conditional discriminations by successive and simultaneous stimuli.

In a conditional discrimination, reinforcement of pigeons' responses to pairs of simultaneously presented wavelength stimuli depended on the orientation of white lines superimposed on the wavelengths. Over different conditions in Experiment 1, three wavelength differences were combined with two differences between successively presented line orientations. Measures of stimulus discriminability increased with increases in the difference between both orientation and wavelength stimuli. Conditional-discrimination performance was thus conjointly determined by stimulus disparity in the successive and simultaneous discriminations. In Experiment 2, ratios of rates of reinforcement contingent upon the two categories of correct responses were varied over several conditions for difficult and easy discriminations. Ratios of responses to wavelength pairs were sensitive to variations in the reinforcement ratio to a greater extent for the more difficult orientation discrimination than for the easier orientation discrimination. Performance in the conditional discrimination was therefore determined by the interacting effects of stimulus disparity and the relative rates of reinforcement contingent upon the two correct choices. It was concluded that the effect of temporally distant reinforcement on behavior in a prevailing schedule component is attenuated to an extent that depends on similarity of stimuli that delineate the successive components.     

Stimulus class formation and concept learning: establishment of within- and between-set generalization and transitive relationships via conditional discrimination procedures

Three students with moderate mental retardation were taught a complex stimulus class with a two-choice conditional discrimination procedure applied across eight 10-member stimulus sets. Each set was composed of five age-appropriate and five age-inappropriate examples of clothing, accessories, and leisure items (e.g., a Walkman radio). Discrimination training was programmed serially across each set, and generalization probes were conducted concurrently among all sets. Generalization probes consisted of unreinforced conditional matching trials with comparison items being drawn from (a) the set undergoing training (within-set probes), (b) sets not undergoing training (between-set probes), and (c) both sample and comparison items from different sets (transitive stimulus control probes). Results indicate that within-set generalization, between-set generalization, and transitive stimulus relations controlled responding by all 3 students for items that had been contingently associated with reinforcement. However, items that gained control of responding through within-set and between-set generalization alone (i.e., not acquired through contingent reinforcement) remained at baseline levels during transitive stimulus control probes. Results are discussed in terms of a taxonomy of multiple sources of stimulus control that underlie socially defined and maintained stimulus classes.     

The effect of logarithmic transformation on estimating the parameters of the generalized matching law

This study examined stimulus class membership established via stimulus-reinforcer relations. Mentally retarded subjects learned conditional discriminations with four two-member sets of visual stimuli (A, B, C, and D). On arbitrary-matching trials, they selected comparison stimuli B1 and B2 conditionally upon samples A1 and A2, respectively, and C1 and C2 conditionally upon B1 and B2, respectively. On identity-matching trials, they selected all stimuli as comparisons conditionally upon identical stimuli as samples. Throughout training, correct selections of A1, B1, C1, and D1 were followed by one reinforcer, R1, and those of A2, B2, C2, and D2 were followed by another, R2. Subsequent tests documented the formation of two four-member stimulus classes, A1-B1-C1-D1 and A2-B2-C2-D2. The class membership of the A, B, and C stimuli could have been based on equivalence relations that resulted from the arbitrary-matching training. D1 and D2 had never appeared on arbitrary-matching trials, however. Their class membership must have been based on relations with R1 and R2, respectively. Results thus confirm a previous finding that stimulus classes can be expanded via stimulus-reinforcer relations. They also define more precisely the potential nature of those classes and the conditions under which class membership can be established.     

An incomplete data approach to the analysis of covariance structures

In this paper, linear structural equation models with latent variables are considered. It is shown how many common models arise from incomplete observation of a relatively simple system. Subclasses of models with conditional independence interpretations are also discussed. Using an incomplete data point of view, the relationships between the incomplete and complete data likelihoods, assuming normality, are highlighted. For computing maximum likelihood estimates, the EM algorithm and alternatives are surveyed. For the alternative algorithms, simplified expressions for computing function values and derivatives are given. Likelihood ratio tests based on complete and incomplete data are related, and an example on using their relationship to improve the fit of a model is given.This research forms part of the author's doctoral thesis and was supported by a Commonwealth Postgraduate Research Award. The author also wishes to acknowledge the support of CSIRO during the preparation of this paper and the referees' comments which led to substantial improvements.     

Six-member stimulus classes generated by conditional-discrimination procedures

In conditional-discrimination procedures with three sets of stimuli, A, B, and C, three stimuli per set (A1A2A3, B1B2B3, and C1C2C3), subjects (children and adults) learned to select Set-B and Set-C comparisons conditionally upon Set-A samples (A1B1, A1C1, A2B2, A2C2, A3B3, A3C3). If the conditional-discrimination procedures also generated equivalence relations, three 3-member stimulus classes would be demonstrable, A1B1C1, A2B2C2, and A3B3C3. In addition to these three sets, the present experiments used three other sets of stimuli--D, E, and F. The subjects learned to select Set-E and Set-F comparisons conditionally upon Set-D samples (D1E1, D1F1, D2E2, D2F2, D3E3, D3F3). This established a second group of three 3-member stimulus classes, D1E1F1, D2E2F2, and D3E3F3. In all, two groups of three 3-member classes were established by teaching subjects 12 conditional discriminations. The two groups of 3-member classes were then combined (successfully for 5 of 8 subjects) into a single group of three 6-member classes by teaching the subjects three more conditional relations (E1C1, E2C2, and E3C3). With three other children, enlarging the classes one member at a time also produced 6-member classes. As a consequence of class formation, 60 untrained conditional relations emerged from 15 that had been explicitly taught. Six of the subjects also proved capable of naming the stimuli consistently in accord with their class membership, but two subjects demonstrated class formation even in the absence of consistent naming.     

Pluralism, scientific knowledge, and the fallacy of overriding values

The paper examines one implication of pluralism, the view that all values are conditional and none are overriding. This implication is that since scientific knowledge is one of the conditional values, there are circumstances in which the pursuit of even the most basic scientific knowledge is legitimately curtailed. These circumstances occur when the pursuit of scientific knowledge conflicts with moral and political values which, in that context, are more important than it. The argument focuses on the case for and against space exploration in search of intelligent extraterrestrial life. The widely held supposition that search for pure scientific knowledge cannot be reasonably curtailed is identified as the fallacy of overriding values.     

Conditional covariance structure of generalized compensatory multidimensional items

Some nonparametric dimensionality assessment procedures, such as DIMTEST and DETECT, use nonparametric estimates of item pair conditional covariances given an appropriately chosen subtest score as their basic building blocks. Such conditional covariances given some subtest score can be regarded as an approximation to the conditional covariances given an appropriately chosen unidimensional latent composite, where the composite is oriented in the multidimensional test space direction in which the subtest score measures best. In this paper, the structure and properties of such item pair conditional covariances given a unidimensional latent composite are thoroughly investigated, assuming a semiparametric IRT modeling framework called a generalized compensatory model. It is shown that such conditional covariances are highly informative about the multidimensionality structure of a test. The theory developed here is very useful in establishing properties of dimensionality assessment procedures, current and yet to be developed, that are based upon estimating such conditional covariances.In particular, the new theory is used to justify the DIMTEST procedure. Because of the importance of conditional covariance estimation, a new bias reducing approach is presented. A byproduct of likely independent importance beyond the study of conditional covariances is a rigorous score information based definition of an item's and a score's direction of best measurement in the multidimensional test space.This paper is based on a chapter of the first author's doctoral dissertation, written at the University of Illinois and supervised by the second author. Part of this research has been presented at the annual meeting of the National Council on Measurement in Education, San Francisco, April 1995.The authors would like to thank Jeff Douglas, Xuming He and Ming-mei Wang for their comments and suggestions. The research of the first author was partially supported by an ETS/GREB Psychometric Fellowship, and by Educational Testing Service Research Allocation Project 884-01. The research of the second author was partially supported by NSF grant DMS 97-04474.     

Necessary versus Probable Cause

One finds in the systems of natural languages some explicit means of elaborating not only upon the directness of the causal relationship believed to exist between two events X and Y (i.e. some means of specifying just how inevitably event X gives or gave rise to event Y), but also some manner of indicating just who or what is understood to be the primary instigator of the caused event. The goal of the present paper is to explore these notions in detail and arrive at a formal, logic-based means of capturing them.