Asymptotic standard errors of irt observed-score equating methods

A method of the IRT observed-score equating using chain equating through a third test without equating coefficients is presented with the assumption of the three-parameter logistic model. The asymptotic standard errors of the equated scores by this method are obtained using the results given by M. Liou and P.E. Cheng. The asymptotic standard errors of the IRT observed-score equating method using a synthetic examinee group with equating coefficients, which is a currently used method, are also provided. Numerical examples show that the standard errors by these observed-score equating methods are similar to those by the corresponding true score equating methods except in the range of low scores.The author is indebted to Michael J. Kolen for access to the real data used in this article and anonymous reviewers for their corrections and suggestions on this work.     

Choice in a variable environment: effects of unequal reinforcer distributions

Six pigeons were trained in a procedure in which sessions included seven unsignaled components, each offering two pecking keys, and each providing a potentially different reinforcer ratio between the two keys. Across conditions, various combinations of reinforcer ratios and reinforcer-magnitude ratios were used to create unequal reinforcer distributions between the two alternatives when averaged across a session. The results extended previous research using the same basic procedure that had included only reinforcer distributions symmetrical around 1:1. Data analyses suggested that the variables controlling choice operated at a number of levels: First, individual reinforcers had local effects on choice; second, sequences of successive reinforcers obtained at the same alternative (continuations) had cumulative effects; and, third, when these sequences themselves occurred with greater frequency, their effects further cumulated. A reinforcer obtained at the other alternative following a sequence of continuations (a discontinuation) had a large effect and apparently reset choice to levels approximating the sessional reinforcer ratio.     

Concurrent-schedule performance in transition: changeover delays and signaled reinforcer ratios

Six pigeons were trained in experimental sessions that arranged six or seven components with various concurrent-schedule reinforcer ratios associated with each. The order of the components was determined randomly without replacement. Components lasted until the pigeons had received 10 reinforcers, and were separated by 10-s blackout periods. The component reinforcer ratios arranged in most conditions were 27:1, 9:1, 3:1, 1:1, 1:3, 1:9 and 1:27; in others, there were only six components, three of 27:1 and three of 1:27. In some conditions, each reinforcement ratio was signaled by a different red-yellow flash frequency, with the frequency perfectly correlated with the reinforcer ratio. Additionally, a changeover delay was arranged in some conditions, and no changeover delay in others. When component reinforcer ratios were signaled, sensitivity to reinforcement values increased from around 0.40 before the first reinforcer in a component to around 0.80 before the 10th reinforcer. When reinforcer ratios were not signaled, sensitivities typically increased from zero to around 0.40. Sensitivity to reinforcement was around 0.20 lower in no-changeover-delay conditions than in changeover-delay conditions, but increased in the former after exposure to changeover delays. Local analyses showed that preference was extreme towards the reinforced alternative for the first 25 s after reinforcement in changeover-delay conditions regardless of whether components were signaled or not. In no-changeover-delay conditions, preference following reinforcers was either absent, or, following exposure to changeover delays, small. Reinforcers have both local and long-term effects on preference. The former, but not the latter, is strongly affected by the presence of a changeover delay. Stimulus control may be more closely associated with longer-term, more molar, reinforcer effects.     

Instructions as discriminative stimuli

Four undergraduates were exposed to a fixed-ratio schedule under an instruction to respond slowly and to a differential-reinforcement-of-low-rate 5-s schedule under an instruction to respond rapidly. Following this, a fixed-interval schedule was in effect under those same two sets of instructions. For 3 of 4 subjects, response rates were higher with the instruction to respond slowly than with the instruction to respond rapidly during the fixed-interval schedule. For the remaining subject, low-rate responding with the instruction to respond rapidly continued during the first 17 reinforcements of the fixed-interval schedule. Such control by instructions was not observed for other subjects exposed only to a fixed-interval schedule, with or without instructions. The results demonstrate that the effect of instructions can be altered by contingencies and suggest that instructions can function as discriminative stimuli.     

Human Sensitivity To Concurrent Schedules Of Reinforcement: Effects Of Observing Schedule-correlated Stimuli

The determinants of human sensitivity to concurrent variable-interval variable-interval schedules of reinforcement have been difficult to identify, in part because of procedural differences separating published experiments. This experiment investigated vigilance to stimuli correlated with concurrent schedules. Across phases, 3 college students were provided with either no schedule-correlated stimuli, an observing response that provided brief access to the stimuli, or a contingency that required the subject to identify the stimulus correlated with the source of each obtained reinforcer. Sensitivity, as quantified by the generalized matching equation, was low when no stimuli were available. When the stimuli were response contingent, 1 subject observed them, and her behavior became more sensitive to the distribution of reinforcers across the concurrent schedules. When the procedure required discrimination of the stimulus correlated with each reinforcer, the other 2 subjects also observed the stimuli, and their schedule sensitivity was increased as well. These results implicate procedural differences, rather than inherent behavioral differences, as the source of differences in sensitivity to schedules of reinforcement between humans and nonhumans.     

Relative versus absolute reinforcement effects: implications for preference assessments

We compared results obtained in two previous studies on reinforcer identification (Fisher et al., 1992; Pace, Ivancic, Edwards, Iwata, & Page, 1985) by combining methodologies from both studies. Eight individuals with mental retardation participated. During Phase 1, two preference assessments were conducted, one in which stimuli were presented singly (SS method) and one in which stimuli were presented in pairs (PS method). Based on these results, two types of stimuli were identified for each participant: High-preference (HP) stimuli were those selected on 75% or more trials during both preference assessments; low-preference (LP) stimuli were those selected on 100% of the SS trials but on 25% or fewer of the PS trials. During Phase 2, the reinforcing effects of HP and LP stimuli were evaluated in reversal designs under two test conditions: concurrent and single schedules of continuous reinforcement. Two response options were available under the concurrent-schedule condition: One response produced access to the HP stimulus; the other produced access to the LP stimulus. Only one response option was available under the single-schedule condition, and that response produced access only to the LP stimulus. Results indicated that 7 of the 8 participants consistently showed preference for the HP stimulus under the concurrent schedule. However, when only the LP stimulus was available during the single-schedule condition, response rates for 6 of the 7 participants were as high as those observed for the HP stimulus during the concurrent-schedule condition (1 participant showed no reinforcement effect). These results indicate that, although the concurrent-schedule procedure is well suited to the assessment of relative reinforcement effects (preference for one reinforcer over another), absolute reinforcement effects associated with a given stimulus may be best examined under single-schedule conditions.     

Effects of Interspersing Additional Brief Math Problems on Student Performance and Perception of Math Assignments: Getting Students to Prefer to Do More Work

Three experiments were conducted to assess the effects of the interspersing procedure on students' mathematics performance, perceptions of mathematics assignments, and preference for mathematics assignments when more work was added. In Experiment I, college students worked computation problems from two assignments. The control assignment contained 15 three-digit by two-digit problems (3 × 2) and the experimental assignments contained 18 3 × 2 problems and 6 additional interspersed one-digit by one-digit problems (1 × 1). Students then rated the assignments and chose one for homework. Results showed significantly more students preferred (i.e., chose it for homework) the experimental assignment with the additional target (i.e., 3 × 2) and interspersed problems. Furthermore, significantly more students rated this experiment as requiring less effort and being less difficult. Moreover, students' mathematics performance was not affected by the procedure. Experiment II strengthened the internal validity by showing that students did not prefer the experimental assignment because it contained more target problems. Experiment III showed that the interspersing procedure was not powerful enough to extend findings beyond the 20% level (i.e., 40 and 60% more conditions). Results showed how the interspersing procedure could be used to encourage students to choose to do more target computation tasks. Discussion focuses on the applied value of the interspersing procedure and plausible causal mechanisms that may account for current and past findings related to student choice and interspersing brief tasks.     

Reporting contingencies of reinforcement in concurrent schedules

Five pigeons were trained on concurrent variable-interval schedules in which two intensities of yellow light served as discriminative stimuli in a switching-key procedure. A conditional discrimination involving a simultaneous choice between red and green keys followed every reinforcer obtained from both alternatives. A response to the red side key was occasionally reinforced if the prior reinforcer had been obtained from the bright alternative, and a response to the green side key was occasionally reinforced if the prior reinforcer had been obtained from the dim alternative. Measures of the discriminability between the concurrent-schedule alternatives were obtained by varying the reinforcer ratio for correct red and correct green responses across conditions in two parts. Part 1 arranged equal rates of reinforcement in the concurrent schedule, and Part 2 provided a 9:1 concurrent-schedule reinforcer ratio. Part 3 arranged a 1:9 reinforcer ratio in the conditional discrimination, and the concurrent-schedule reinforcer ratio was varied across conditions. Varying the conditional discrimination reinforcer ratio did not affect response allocation in the concurrent schedule, but varying the concurrent-schedule reinforcer ratio did affect conditional discrimination performance. These effects were incompatible with a contingency-discriminability model of concurrent-schedule performance (Davison & Jenkins, 1985), which implies a constant discriminability parameter that is independent of the obtained reinforcer ratio. However, a more detailed analysis of conditional discrimination performance showed that the discriminability between the concurrent-schedule alternatives decreased with time since changing over to an alternative. This effect, combined with aspects of the temporal distribution of reinforcers obtained in the concurrent schedules, qualitatively predicted the molar results and identified the conditions that operate whenever contingency discriminability remains constant.     

Matching under nonindependent variable-ratio schedules of drug reinforcement.

Response-contingent deliveries of oral pentobarbital maintained responding of 3 rhesus monkeys during daily 3-hr sessions. Deliveries of pentobarbital were arranged under nonindependent concurrent variable-ratio variable-ratio schedules. Responses to either schedule counted toward completion of both variable-ratio schedule requirements. This schedule is similar in some respects to conventional concurrent variable-interval variable-interval schedules, in which passage of time counts toward completion of the interval value on both schedules. Restricted nonindependent concurrent variable-ratio variable-ratio schedules were also studied. On that schedule, when a drug delivery was assigned to one spout, it had to be collected before responses on the opposite spout again counted toward completion of the schedule requirements. Relative reinforcer magnitude was varied by changing the drug concentration on one schedule while keeping the drug concentration constant on the other variable-ratio schedule. Under both types of concurrent variable-ratio schedules, the relative rate of responding corresponded to the relative drug intake. Unlike earlier studies of concurrent variable-interval variable-interval intravenous cocaine reinforcement, preference was proportionate to concentration, and exclusive preferences did not develop. The relationship between relative rate of responding and relative drug intake was well described by the generalized matching law.     

Contrast and undermatching with regular or irregular alternation of components

Behavioral contrast and response-ratio sensitivity to reinforcement were compared in multiple schedules in which components alternated strictly or according to a pseudorandom sequence. Average component durations in the two regimes were always 60 s, and order of presentation of component alternation regimes was counterbalanced across subjects. In Part 1, the reinforcer rate in one component was reduced from 60 per hour to zero, while that in the other component was unchanged. Positive behavioral contrast occurred in the constant component in that response rates increased, but neither the reliability nor the magnitude of contrast was affected by the manner in which components alternated. Part 2 was similar, except that a number of different reinforcer rates were used in the varied component. Neither contrast nor sensitivity of response ratios to changes in reinforcer ratios depended on the regime of component alternation. Thus, the predictability in time of future reinforcement conditions, which is a feature of regular multiple scheduling, does not appear to be a determinant of multiple-schedule performance.