Optimal matching analysis in career research: A review and some best-practice recommendations

Optimal matching is a method for the analysis of sequential data. It allows researchers to detect patterns in career sequences or occupational trajectories. We first give a brief introduction to the method and review the existing career literature that employs optimal matching. To examine which data properties are required for optimal matching analysis, we conducted Monte Carlo simulations of career sequences with varying parameters for sequence length, sample size and missing items. We find that sequence length is the relevant factor for correct results, while sample size does not substantially affect result quality. Another important finding is that sequences with up to 30% elements missing can be used for optimal matching analysis. We also show which settings for the optimal matching procedure deliver the best results.     

Pigeons may not remember the stimuli that reinforced their recent behavior

In two experiments the conditioned reinforcing and delayed discriminative stimulus functions of stimuli that signal delays to reinforcement were studied. Pigeons' pecks to a center key produced delayed-matching-to-sample trials according to a variable-interval 60-s (or 30-s in 1 pigeon) schedule (Experiment 1) or a multiple variable-interval 20-s variable-interval 120-s schedule (Experiment 2). The trials consisted of a 2-s illumination of one of two sample key colors followed by delays ranging across phases from 0.1 to 27.0 s followed in turn by the presentation of matching and nonmatching comparison stimuli on the side keys. Pecks to the key color that matched the sample were reinforced with 4-s access to grain. Under some conditions of Experiment 1, pecks to nonmatching comparison stimuli produced a 4-s blackout and the start of the next interval. Under other conditions of Experiment 1 and each condition of Experiment 2, pecks to nonmatching stimuli had no effect and trials ended only when pigeons pecked the other, matching stimulus and received food. The functions relating pretrial response rates to delays differed markedly from those relating matching-to-sample accuracy to delays. Specifically, response rates remained relatively high until the longest delays (15.0 to 27.0 s) were arranged, at which point they fell to low levels. Matching accuracy was high at short delays, but fell to chance at delays between 3.0 and 9.0 s. In Experiment 2, both matching accuracy and response rates remained high over a wider range of delays in the variable-interval 120-s component relative to the variable-interval 20-s component. The difference in matching accuracy between the components was not due to an increased tendency in the variable-interval 20-s component toward proactive interference following short intervals. Thus, under these experimental conditions the conditioned reinforcing and the delayed discriminative functions of the sample stimulus depended on the same variables (delay and variable-interval value), but were nevertheless dissociated.     

An application of the matching law to evaluate the allocation of two- and three-point shots by college basketball players

We applied the matching equation to evaluate the allocation of two- and three-point shots by male and female college basketball players from a large Division 1 university. The matching law predicts that the proportion of shots taken from three-point range should match the proportional reinforcement rate produced by such shots. Thus, we compared the proportion of three-point shots taken relative to all shots to the proportion of three-point shots scored relative to all shots scored. However, the matching equation was adjusted to account for the greater reinforcer magnitude of the three-point basket (i.e., 1.5 times greater than the two-point basket reinforcer magnitude). For players with substantial playing time, results showed that the overall distribution of two- and three-point shots was predicted by the matching equation. Game-by-game shot distribution was variable, but the cumulative proportion of shots taken from three-point range as the season progressed was predicted almost perfectly on a player-by-player basis for both male and female basketball players.     

Molecular contingencies: reinforcement probability

Pigeons obtained food by responding in a discrete-trials two-choice probability-learning experiment involving temporal stimuli. A given response alternative, a left- or right-key peck, had 11 associated reinforcement probabilities within each session. Reinforcement probability for a choice was an increasing or a decreasing function of the time interval immediately preceding the choice. The 11 equiprobable temporal stimuli ranged from 1 to 11 sec in 1-sec classes. Preference tended to deviate from probability matching in the direction of maximizing; i.e., the percentage of choices of the preferred response alternative tended to exceed the probability of reinforcement for that alternative. This result was qualitatively consistent with probability-learning experiments using visual stimuli. The result is consistent with a molecular analysis of operant behavior and poses a difficulty for molar theories holding that local variations in reinforcement probability may safely be disregarded in the analysis of behavior maintained by operant paradigms.     

A solution to the weighted procrustes problem in which the transformation is in agreement with the loss function

This paper provides a generalization of the Procrustes problem in which the errors are weighted from the right, or the left, or both. The solution is achieved by having the orthogonality constraint on the transformation be in agreement with the norm of the least squares criterion. This general principle is discussed and illustrated by the mathematics of the weighted orthogonal Procrustes problem.     

How a Group Goal May Reduce Social Matching in Group Performance: Shifts in Standards for Determining a Fair Contribution of Effort

Abstract

The authors investigated whether the presence of a specific group goal would reduce social matching (i.e., matching one's own performance to the performance expected from others) by serving as an alternative standard. As predicted, when there was no specific goal, the participants matched their own performance to the performance expected from other group members. When there was a specific group goal, the women no longer engaged in social matching, although that effect did not emerge among the men. Instead, the women's mean personal performance was close to the performance level representing an equal share of the group goal. Moreover, the participants' perceptions of a fair contribution mediated the performance of the men and the women, both in the presence and in the absence of a goal.     

The other-race effect does not rely on memory: Evidence from a matching task

Viewers are typically better at remembering faces from their own race than from other races; however, it is not yet established whether this effect is due to memorial or perceptual processes. In this study, UK and Egyptian viewers were given a simultaneous face-matching task, in which the target faces were presented upright or upside down. As with previous research using face memory tasks, participants were worse at matching other-race faces than own-race faces and showed a stronger face inversion effect for own-race faces. However, subjects' performance on own and other-race faces was highly correlated. These data provide strong evidence that difficulty in perceptual encoding of unfamiliar faces contributes substantially to the other-race effect and that accounts based entirely on memory cannot capture the full data. Implications for forensic settings are also discussed.     

Response and time allocation in concurrent second-order schedules

Six pigeons were trained on two-key concurrent variable-interval schedules in which the required response was the completion of a fixed number of key pecks. When the required number of pecks was equal on the two keys, response- and time-allocation ratios under-matched obtained reinforcement rate ratios. A similar result was found when the required number of pecks was unequal, except that performance, measured in response terms, was biased to the shorter required number of pecks and was less sensitive to reinforcement-rate changes. No such differences were found in the data on time spent responding. When the variable-interval schedules were kept constant and the required numbers of pecks were systematically varied, response ratios changed inversely with the ratio of the required number of pecks, but time-allocation ratios varied directly with the same independent variable. Thus, on response measures, pigeons “prefer” the schedule with the smaller peck requirement, but on time measures they “prefer” the schedule with the larger peck requirement. This finding is inconsistent with a commonsense notion of choice, which sees response and time-allocation measures as equivalent.     

THE EFFECT OF MODELLING ON DRINKING RATE

Three male college seniors were asked to drink beer at their normal rate in a simulated tavern setting. Each was paired with a confederate, also a male college senior, in an ABACA single subject design. In the baseline conditions, the confederate matched the drinking rate of the subject. Baseline and all subsequent conditions were continued in 1-hr sessions until a stable drinking rate was achieved. In Condition B, the confederate drank either one third more or one third less than the subject's baseline rate. In Condition C, the direction was reversed. All three subjects closely matched the confederate's drinking rate, whether high or low. All subjects reported they were unaware of the true purpose of the study.     

Choice: Some quantitative relations

Six pigeons responded in fifty-six conditions on a concurrent-chains procedure. Conditions included several with equal initial links and unequal terminal links, several with unequal initial links and equal terminal links, and several with both unequal initial and terminal links. Although the delay-reduction hypothesis accounted well for choice when the initial links were equal (mean deviation of .04), it fit the data poorly when the initial links were unequal (mean deviation of .18). A modification of the delay-reduction hypothesis, replacing the rates of reinforcement with the square roots of these rates, fit the data better than either the unmodified delay-reduction equation or Killeen's (1982) model. The modified delay-reduction equation was also consistent with data from prior studies using concurrent chains. The absolute rates of responding in each terminal link were well described by the same hyperbola (Herrnstein, 1970) that describes response rates on simple interval schedules.