回归混合模型：方法进展与软件实现

近来以个体为分析对象的方法日益受到研究者的重视, 其中潜类别和潜剖面模型最为流行。研究者在潜类别和潜剖面模型建模时往往需要进一步探讨协变量与潜分组之间的关系(即带有协变量的潜类别模型)。例如, 哪些变量预测个体类别归属, 以及个体的类别归属对结果变量的预测。本文对近年来研究者提出的各种方法进行了回顾和比较。包括当结果变量是分类变量的LTB法; 当结果变量是连续变量时的BCH和稳健三步法。在此基础上, 文章为应用研究者提供了Mplus软件示例, 并在最后对当前研究存在的问题和未来研究趋势进行了简要评价。     

Repeated,within‐session resurgence

Resurgence is a reliable, transient effect that only occasionally is replicated more than once within a single experiment or subject. In the present experiments, within‐session resurgence was generated repeatedly by dividing individual sessions into three phases (Training, Alternative‐Reinforcement, and Resurgence‐Test). In Experiments 1 and 2, resurgence reliably occurred in most of the 22‐30 daily sessions when responding was reinforced on, respectively, fixed‐ and variable‐interval schedules. Resurgence magnitude and duration did decrease across replications for some subjects, but not for others. To examine the utility of the procedure in studying the effects of an independent variable on resurgence, in Experiment 3 the effects of rich and lean baseline and alternative reinforcement rates on resurgence were compared. The target response was eliminated more rapidly, resurgence occurred more often, and usually was greater following rich alternative reinforcement rates. Resurgence was of greater magnitude when the baseline reinforcement rate was relatively lean compared to the alternative reinforcement rate. These experiments provide a reliable method for generating resurgence within individual sessions, instead of across multiple‐session conditions, that can be repeated over many successive sessions.     

Nonlinear Indicator-Level Moderation in Latent Variable Models

Linear, nonlinear, and nonparametric moderated latent variable models have been developed to investigate possible interaction effects between a latent variable and an external continuous moderator on the observed indicators in the latent variable model. Most moderation models have focused on moderators that vary across persons but not across the indicators (e.g., moderators like age and socioeconomic status). However, in many applications, the values of the moderator may vary both across persons and across indicators (e.g., moderators like response times and confidence ratings). Indicator-level moderation models are available for categorical moderators and linear interaction effects. However, these approaches require respectively categorization of the continuous moderator and the assumption of linearity of the interaction effect. In this article, parametric nonlinear and nonparametric indicator-level moderation methods are developed. In a simulation study, we demonstrate the viability of these methods. In addition, the methods are applied to a real data set pertaining to arithmetic ability.     

Unsignaled delay of reinforcement, relative time, and resistance to change

Two experiments with pigeons examined the effects of unsignaled, nonresetting delays of reinforcement on responding maintained by different reinforcement rates. In Experiment 1, 3-s unsignaled delays were introduced into each component of a multiple variable-interval (VI) 15-s VI 90-s VI 540-s schedule. When considered as a proportion of the preceding immediate reinforcement baseline, responding was decreased similarly for the three multiple-schedule components in both the first six and last six sessions of exposure to the delay. In addition, the relation between response rates and reinforcement rates was altered such that both parameters of the single-response version of the matching law (i.e., k and Re) were decreased. Experiment 2 examined the effects of unsignaled delays ranging from 0.5 s to 8.0 s on responding maintained by a multiple VI 20-s VI 120-s schedule of reinforcement. Response rates in both components increased with brief unsignaled delays and decreased with longer delays. As in Experiment 1, response rates as a proportion of baseline were affected similarly for the two components in both the first six and last six sessions of exposure to the delay. Unlike delays imposed between two stimulus events, the effects of delays between responses and reinforcers do not appear to be attenuated when the average time between reinforcers is longer. In addition, the disruptions produced by unsignaled delays appear to be inconsistent with the general finding that responding maintained by higher rates of reinforcement is less resistant to change.     

Selection of variables in exploratory factor analysis: An empirical comparison of a stepwise and traditional approach

The purpose of this study was to investigate and compare the performance of a stepwise variable selection algorithm to traditional exploratory factor analysis. The Monte Carlo study included six factors in the design; the number of common factors; the number of variables explained by the common factors; the magnitude of factor loadings; the number of variables not explained by the common factors; the type of anomaly evidenced by the poorly explained variables; and sample size. The performance of the methods was evaluated in terms of selection and pattern accuracy, and bias and root mean squared error of the structure coefficients. Results indicate that the stepwise algorithm was generally ineffective at excluding anomalous variables from the factor model. The poor selection accuracy of the stepwise approach suggests that it should be avoided.     

Bouts of responding on variable-interval schedules: effects of deprivation level

Rats obtained food pellets on a variable-interval schedule of reinforcement by nose poking a lighted key. After training to establish baseline performance (with the mean variable interval set at either 60, 120, or 240 s), the rats were given free access to food during the hour just before their daily session. This satiation operation reduced the rate of key poking. Analysis of the interresponse time distributions (log survivor plots) indicated that key poking occurred in bouts. Prefeeding lengthened the pauses between bouts, shortened the length of bouts (less reliably), and had a relatively small decremental effect on the response rate within bouts. That deprivation level affects mainly between-bout pauses has been reported previously with fixed-ratio schedules. Thus, when the focus is on bouts, the performances maintained by variable-interval schedules and fixed-ratio schedules are similarly affected by deprivation.     

Evidence against a constant-difference effect in concurrent-chains schedules

Savastano and Fantino (1996) reported that in concurrent-chains schedules, initial-link choice proportions remained constant as terminal-link durations increased as long as the subtractive difference between the two terminal-link schedules remained constant. Two experiments with pigeons were conducted to examine this constant-difference effect. Both experiments used equal variable-interval schedules as initial links. The terminal links were fixed delays to reinforcement in Experiment 1 and variable delays to reinforcement in Experiment 2. The durations of the terminal links were varied across conditions, but the difference between pairs of terminal links was always 10 s. In both experiments, preference for the shorter terminal link became less extreme as terminal-link durations increased, so a constant-difference effect was not found. It is argued, however, that this choice situation does not provide clear evidence for or against delay-reduction theory versus other theories of choice.     

Preratio pausing: effects of an alternative reinforcer on fixed- and variable-ratio responding

Seven rats responding under fixed-ratio or variable-ratio schedules of food reinforcement had continuous access to a drinking tube inserted into the operant chamber. Under different conditions they could drink either tap water or one of two saccharin solutions. In a baseline condition, the drinking bottle was empty. Preratio pausing was observed with both schedules, more so with the fixed-ratio than the variable-ratio schedule, and increasing the concentration of the saccharin solution increased the duration of pausing. Comparisons with baseline performances revealed that the additional pausing was largely, but not entirely, spent drinking. The results support the view that pausing under ratio schedules is a consequence of competition between the scheduled reinforcer and alternative reinforcers that also are available within the experimental environment.     

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.