On scaling models applied to data from several groups

Statistical methods are presented to facilitate a more complete analysis of results obtained when a scaling model is applied to data from two or more groups. These methods can be used to (a) compare the corresponding estimated latent distributions obtained using the scaling model applied to the different groups, (b) compare the corresponding estimated item reliabilities (or item response error rates) for the different groups, and (c) test whether the scaling model applied to the several groups can be replaced by a more parsimonious scaling model that includes various homogeneity constraints (i.e., constraints that describe which parameters in the model are the same for the several groups). Various kinds of scaling models are considered here in the multiple-group context.Support for this research was provided in part by the National Science Foundation, to Clogg by Grant No. SES-7823759 and to Goodman by Grant No. SES-8303838. Clogg and Goodman were Fellows at the Center for Advanced Study in the Behavioral Sciences when part of the research was done, with financial support provided in part by National Science Foundation grant BNS-8011494 to the Center. The authors are indebted to Mark P. Becker and James W. Shockey for helpful comments.     

The integration of multidimensional scaling and multivariate analysis with optimal transformations

The recent history of multidimensional data analysis suggests two distinct traditions that have developed along quite different lines. In multidimensional scaling (MDS), the available data typically describe the relationships among a set of objects in terms of similarity/dissimilarity (or (pseudo-)distances). In multivariate analysis (MVA), data usually result from observation on a collection of variables over a common set of objects. This paper starts from a very general multidimensional scaling task, defined on distances between objects derived from one or more sets of multivariate data. Particular special cases of the general problem, following familiar notions from MVA, will be discussed that encompass a variety of analysis techniques, including the possible use of optimal variable transformation. Throughout, it will be noted how certain data analysis approaches are equivalent to familiar MVA solutions when particular problem specifications are combined with particular distance approximations.This research was supported by the Royal Netherlands Academy of Arts and Sciences (KNAW). An earlier version of this paper was written during a stay at McGill University in Montréal; this visit was supported by a travel grant from the Netherlands Organization for Scientific Research (NWO). I am grateful to Jim Ramsay and Willem Heiser for their encouragement and helpful suggestions, and to the Editor and referees for their constructive comments.     

Procrastination by pigeons: preference for larger, more delayed work requirements.

In three experiments, pigeons chose between alternatives that required the completion of a small ratio schedule early in the trial or a larger ratio schedule later in the trial. Completion of the ratio requirement did not lead to an immediate reinforcer, but simply allowed the events of the trial to continue. In Experiment 1, the ratio requirements interrupted periods in which food was delivered on a variable-time schedule. In Experiments 2 and 3, each ratio requirement was preceded and followed by a delay, and only one reinforcer was delivered, at the end of each trial. Two of the experiments used an adjusting-ratio procedure in which the ratio requirement was increased and decreased over trials so as to estimate an indifference point--a ratio size at which the two alternatives were chosen about equally often. These experiments found clear evidence for "procrastination"--the choice of a larger but more delayed response requirement. In some cases, subjects chose the more delayed ratio schedule even when it was larger than the more immediate alternative by a factor of four or more. The results suggest that as the delay to the start of a ratio requirement is increased, it has progressively less effect on choice behavior, in much the same way that delaying a positive reinforcer reduces it effect on choice.     

Polytomous IRT models and monotone likelihood ratio of the total score

In a broad class of item response theory (IRT) models for dichotomous items the unweighted total score has monotone likelihood ratio (MLR) in the latent trait. In this study, it is shown that for polytomous items MLR holds for the partial credit model and a trivial generalization of this model. MLR does not necessarily hold if the slopes of the item step response functions vary over items, item steps, or both. MLR holds neither for Samejima's graded response model, nor for nonparametric versions of these three polytomous models. These results are surprising in the context of Grayson's and Huynh's results on MLR for nonparametric dichotomous IRT models, and suggest that establishing stochastic ordering properties for nonparametric polytomous IRT models will be much harder.Hemker's research was supported by the Netherlands Research Council, Grant 575-67-034. Junker's research was supported in part by the National Institutes of Health, Grant CA54852, and by the National Science Foundation, Grant DMS-94.04438.     

On the robustness of maximum likelihood scaling for violations of the error model

A Monte Carlo study was conducted to investigate the robustness of the assumed error distribution in maximum likelihood estimation models for multidimensional scaling. Data sets generated according to the lognormal, the normal, and the rectangular distribution were analysed with the log-normal error model in Ramsay's MULTISCALE program package. The results show that violations of the assumed error distribution have virtually no effect on the estimated distance parameters. In a comparison among several dimensionality tests, the corrected version of thex ² test, as proposed by Ramsay, yielded the best results, and turned out to be quite robust against violations of the error model.     

脑出血后智力障碍与局部脑血流量相关性研究

作者用长谷川智力测定量表法测定了６１例脑出血患者的智力,用ＳＰＥＣＴ定量测定法测定了双侧大脑半球各１５个脑区的局部血流量,并计算其比值。分析了脑出血、局部脑血流量、局部脑血流量比值与长谷川分数的关系。结果提示脑出血时,除由于病变直接破坏与智力有关的结构外,左半球（特别是左颞叶及左缘上回和角回）局部脑血流量下降,缘上回和角回、颞上回后部、颞中回后部、颞下回局部脑血流量比值失衡亦可能是造成智力下降的原因。     

Pausing under variable-ratio schedules: Interaction of reinforcer magnitude, variable-ratio size, and lowest ratio

Pigeons pecked a key under two-component multiple variable-ratio schedules that offered 8-s or 2-s access to grain. Postreinforcement pausing and the rates of responding following the pause (run rates) in each component were measured as a function of variable-ratio size and the size of the lowest ratio in the configuration of ratios comprising each schedule. In one group of subjects, variable-ratio size was varied while the size of the lowest ratio was held constant. In a second group, the size of the lowest ratio was varied while variable-ratio size was held constant. For all subjects, the mean duration of postreinforcement pausing increased in the 2-s component but not in the 8-s component. Postreinforcement pauses increased with increases in variable-ratio size (Group 1) and with increases in the lowest ratio (Group 2). In both groups, run rates were slightly higher in the 8-s component than in the 2-s component. Run rates decreased slightly as variable-ratio size increased, but were unaffected by increases in the size of the lowest ratio. These results suggest that variable-ratio size, the size of the lowest ratio, and reinforcer magnitude interact to determine the duration of postreinforcement pauses.     

Response suppression by visual stimuli paired with postsession d-amphetamine injections in the pigeon

Responding of pigeons, maintained under a fixed-interval 3-minute schedule of food presentation, was decreased on days that the color of the lights illuminating the food magazine was changed and d-amphetamine (1.0 mg/kg, i.m.) was injected after the session. Responding was not decreased by keylight color changes paired with postsession d-amphetamine or by postsession injections of saline. Administration of pentobarbital (3.0 to 5.6 mg/kg), but not d-amphetamine (.3 to 3.0 mg/kg), before the session increased rates of responding suppressed by drug-paired magazine lights. Responding maintained under a fixed-ratio 30-response schedule was not decreased when differently colored magazine lights were paired with a low (.3 mg/kg) postsession dose of d-amphetamine; with high (3.0 mg/kg) postsession doses, however, responding was completely suppressed after two pairings. The effects of pairing magazine stimuli with an intermediate (1.0 mg/kg) postsession dose of d-amphetamine depended upon the magnitude of prior postsession doses. After being paired with a low dose, stimuli paired with 1.0 mg/kg did not suppress responding. After being paired with a high dose, stimuli paired with 1.0 mg/kg completely suppressed responding. The suppression of food-maintained responding by stimuli paired with postsession drug administration depends upon both behavioral and pharmacological variables.     

Ratio responding as a function of concurrent avoidance schedules, yoked shocks, and ratio value

Fixed-ratio food-reinforced responding in rats was studied alone and with concurrent shock avoidance or with concurrent response-independent shocks matched to those that occurred in the avoidance condition. Under each condition, fixed-ratio size was increased over successive daily sessions. Fixed-ratio response rate generally passed through a maximum as a function of fixed-ratio size. Decreased fixed-ratio responding at values beyond the maximum occurred when (1) the time to complete a fixed ratio approximated the response-shock interval of the avoidance schedule, (2) the shock rate increased, and/or (3) the ratio requirements were so high that ratio strain occurred. Avoidance rates decreased slightly as fixed-ratio size increased.     

Probabilistic multidimensional scaling: Complete and incomplete data

Simple procedures are described for obtaining maximum likelihood estimates of the location and uncertainty parameters of the Hefner model. This model is a probabilistic, multidimensional scaling model, which assigns a multivariate normal distribution to each stimulus point. It is shown that for such a model, standard nonmetric and metric algorithms are not appropriate. A procedure is also described for constructing incomplete data sets, by taking into consideration the degree of familiarity the subject has for each stimulus. Maximum likelihood estimates are developed both for complete and incomplete data sets. This research was supported by National Science Grant No. SOC76-20517. The first author would especially like to express his gratitude to the Netherlands Institute for Advanced Study for its very substantial help with this research.