Exploratory analysis of contingency tables by loglinear formulation and generalizations of correspondence analysis

Goodman's (1979, 1981, 1985) loglinear formulation for bi-way contingency tables is extended to tables with or without missing cells and is used for exploratory purposes. A similar formulation is done for three-way tables and generalizations of correspondence analysis are deduced. A generalized version of Goodman's algorithm, based on Newton's elementary unidimensional method is used to estimate the scores in all cases.This research was partially supported by National Science and Engineering Research Council of Canada, Grant No. A8724. The author is grateful to the reviewers and the editor for helpful comments.     

The analysis of three-way contingency tables by three-mode association models

TheRC(M) association model (Goodman, 1979, 1985, 1986, 1991) is useful for analyzing the relationship between the variables of a 2-way cross-classification. The models presented here are generalizations of theRC(M) association model for 3-way tables. The family of models proposed here, 3-mode association models, use Tucker's 3-mode components model (Tucker, 1964, 1966; Kroonenberg, 1983) to represent either the three factor interaction or the combined effects of two and three factor interactions. An example from a study in developmental psychology (Kramer & Gottman, 1992) is provided to illustrate the usefulness of the proposed models.I thank Stanley Wasserman, Laurie Kramer, Ulf Böckenholt, Larwence Hubert, Jeffrey Tanaka, and five anonymous reviewers for valuable comments.     

Exact conditional tests for cross-classifications: Approximation of attained significance levels

A procedure is proposed for approximating attained significance levels of exact conditional tests. The procedure utilizes a sampling from the null distribution of tables having the same marginal frequencies as the observed table. Application of the approximation through a computer subroutine yields precise approximations for practically any table dimensions and sample size.     

Multivariate normal integrals and contingency tables with ordered categories

Ak-dimensional multivariate normal distribution is made discrete by partitioning thek-dimensional Euclidean space with rectangular grids. The collection of probability integrals over the partitioned cubes is ak-dimensional contingency table with ordered categories. It is shown that loglinear model with main effects plus two-way interactions provides an accurate approximation for thek-dimensional table. The complete multivariate normal integral table is computed via the iterative proportional fitting algorithm from bivariate normal integral tables. This approach imposes no restriction on the correlation matrix. Comparisons with other numerical integration algorithms are reported. The approximation suggests association models for discretized multivariate normal distributions and contingency tables with ordered categories.The contingency-table approach occurred to me while I was collaborating with Paul Holland of the Educational Testing Service in 1985 on bivariate dependence functions. Holland maintains a belief that the continuous can learn from the discrete. This work is a reassertion of his claim.This research was sponsored by the National Science Council, Republic of China.     

The analysis of group contingency data

The data analyses utilized in group contingency projects are reviewed. Previous studies are cited to emphasize advantages of nonconsolidated ("individual") over consolidated analyses. Several procedures are described that enable applied researchers to incorporate nonconsolidated data analyses in group contingency studies.     

Correspondence analysis of incomplete contingency tables

Correspondence analysis can be described as a technique which decomposes the departure from independence in a two-way contingency table. In this paper a form of correspondence analysis is proposed which decomposes the departure from the quasi-independence model. This form seems to be a good alternative to ordinary correspondence analysis in cases where the use of the latter is either impossible or not recommended, for example, in case of missing data or structural zeros. It is shown that Nora's reconstitution of order zero, a procedure well-known in the French literature, is formally identical to our correspondence analysis of incomplete tables. Therefore, reconstitution of order zero can also be interpreted as providing a decomposition of the residuals from the quasi-independence model. Furthermore, correspondence analysis of incomplete tables can be performed using existing programs for ordinary correspondence analysis.     

Some exact conditional tests of independence forR ×C cross-classification tables

Exact conditional tests of independence in cross-classification tables are formulated based on the ² statistic and statistics with stronger operational interpretations, such as some nominal and ordinal measures of association. Guidelines for the table dimensions and sample sizes for which the tests are economically implemented on a computer are given. Some selected sample sizes and marginal distributions are used in a numerical comparison between the significance levels of the approximate and exact conditional tests based on the ² statistic.The authors are grateful for the suggestions of the referees and for computer funding provided by the Northeast Regional Data Center at the University of Florida.     

Extensions of Rasch's multiplicative poisson model

Consideration will be given to a model developed by Rasch that assumes scores observed on some types of attainment tests can be regarded as realizations of a Poisson process. The parameter of the Poisson distribution is assumed to be a product of two other parameters, one pertaining to the ability of the subject and a second pertaining to the difficulty of the test. Rasch's model is expanded by assuming a prior distribution, with fixed but unknown parameters, for the subject parameters. The test parameters are considered fixed. Secondly, it will be shown how additional between- and within-subjects factors can be incorporated. Methods for testing the fit and estimating the parameters of the model will be discussed, and illustrated by empirical examples.     

Analysis of contingency tables by ideal point discriminant analysis

Cross-classified data are frequently encountered in behavioral and social science research. The loglinear model and dual scaling (correspondence analysis) are two representative methods of analyzing such data. An alternative method, based on ideal point discriminant analysis (DA), is proposed for analysis of contingency tables, which in a certain sense encompasses the two existing methods. A variety of interesting structures can be imposed on rows and columns of the tables through manipulations of predictor variables and/or as direct constraints on model parameters. This, along with maximum likelihood estimation of the model parameters, allows interesting model comparisons. This is illustrated by the analysis of several data sets.Presented as the Presidential Address to the Psychometric Society's Annual and European Meetings, June, 1987. Preparation of this paper was supported by grant A6394 from the Natural Sciences and Engineering Research Council of Canada. Thanks are due to Chikio Hayashi of University of the Air in Japan for providing the ISM data, and to Jim Ramsay and Ivo Molenaar for their helpful comments on an earlier draft of this paper.     

Comparing item characteristic curves

Test items are often evaluated and compared by contrasting the shapes of their item characteristics curves (ICC's) or surfaces. The current paper develops and applies three general (i.e., nonparametric) comparisons of the shapes of two item characteristic surfaces: (i) proportional latent odds, (ii) uniform relative difficulty, and (iii) item sensitivity. Two items may be compared in these ways while making no assumption about the shapes of item characteristic surfaces for other items, and no assumption about the dimensionality of the latent variable. Also studied is a method for comparing the relative shapes of two item characteristic curves in two examinee populations.The author is grateful to Paul Holland, Robert Mislevy, Tue Tjur, Rebecca Zwick, the editor and reviewers for valuable comments on the subject of this paper, to Mari A. Pearlman for advice on the pairing of items in the examples, and to Dorothy Thayer for assistance with computing.     

College students' responding to and rating of contingency relations: The role of temporal contiguity

Two experiments investigated the role of temporal contiguity in college students' responding to and rating of contingency relations during operant conditioning. Schedules were devised that determined when but not whether appetitive or aversive events would occur. Subjects' reports concerning the schedules were obtained by means of a 200-point rating scale, anchored by the phrases “prevents the light from occurring” (−100) and “causes the light to occur” (+100). When tapping a telegraph key advanced the time of point gain, responding was maintained or increased and subjects gave positive ratings. When tapping a telegraph key advanced the time of point loss, subjects also gave positive ratings, but responding now decreased. When key tapping delayed the time of point gain, responding decreased and subjects gave negative ratings. When key tapping delayed the time of point loss, subjects also gave negative ratings, but responding now increased. These findings implicate response-outcome contiguity as an important contributor to causal perception and to reinforcement and punishment effects. Other accounts—such as those stressing the local probabilistic relation between response and outcome or the molar correlation between response rate and outcome rate—were seen to be less preferred interpretations of these and other results.     

Color preference in pigeons: stimulus intensity and reinforcement contingency effects in the avoidance of blue stimuli.

In a procedure intended to determine color preference in pigeons (which partially replicated Catania, Owens, & von Lossberg, 1983), two keys were illuminated by different colors drawn from a set of amber, red, green, or blue stimuli; this was followed by the presentation of grain when either of the two colors was pecked. The grain was illuminated alternately across trials with the colors presented on the keys. In Experiment 1 the intensity of the color stimuli used was not equalized, whereas in Experiment 2 the intensity of the colors was equalized. The low preference for blue found in Experiment 1, as measured by differential key pecking, was not found in Experiment 2. The discriminability of the intensity-equalized colors was confirmed in Experiment 2a, in which equal-intensity color discrimination problems were presented. In Experiment 3, as in Catania et al. (1983), a response-independent reinforcement schedule was used, but with intensity-equalized colors. In contrast to Experiment 2, very low preference for blue was found here and in Experiment 4, which used a within-subject procedure. These findings suggest that pigeon color preference may be a function of intensity, but all controlling variables have not as yet been identified.     

Age differences in instability, contingency, and level of self-esteem across the life span

We investigated age differences in instability, contingency, and level of self-esteem from age 13 to 72 years, using data from 1386 individuals who participated in a diary study over 25 days. Instability and contingency of self-esteem decreased from adolescence to old age, whereas level of self-esteem increased. Big Five personality traits predicted the level, but not the slope, of the trajectories of self-esteem characteristics. Age differences in self-esteem characteristics did not merely reflect age differences in instability and level of positive and negative affect. Finally, self-esteem characteristics showed a stable pattern of interrelations across the life span. Overall, the findings suggest that people’s self-esteem tends to become better adjusted—i.e., more stable, less contingent, and higher—across the life course.     

Effects of a pairing contingency on behavior in a three-person programmed environment

Four groups of three subjects resided for 10 days within a continuously programmed environment. Subjects followed a behavioral program of contingently scheduled activities that determined individual and social behaviors. A triadic condition was in effect when all three subjects were required to select simultaneous access to a group area before it became available for a social episode. A dyadic condition was in effect when access to a group area was permitted to any combination of two, and only two, subjects. The effects of these two conditions on individual and social behaviors were studied in reversal designs with several successive days devoted to each condition. Results showed that durations of social activities and synchronization of individual activities were greater during triadic conditions than during dyadic conditions. Under both conditions, wake-sleep cycles departed from a typical day-night rhythm for most subjects. Instances when subjects did not respond to each others' attempts to initiate conversations using the intercom were generally more frequent during dyadic than triadic conditions. Physical distance during triadic social episodes was found to be related to sociability levels during dyadic conditions.     

The effects of a cooperation contingency on behavior in a continuous three-person environment

Five groups of three subjects resided for 10 or 15 days within a continuously programmed environment. Subjects followed a programmatic arrangement of required and optional private and social activities that determined the individual and group baseline behaviors into which experimental operations were introduced and withdrawn. A cooperation condition was in effect when all three subjects were required to select simultaneous access to a group area before it became available for use. A noncooperation condition was in effect when access to a group area could be selected by individual subjects, without regard to the other subjects' selections. For all groups, the effects of these two conditions on individual and group behaviors were investigated in reversal designs where several successive days occurred under each condition. Groups 1, 4, and 5 had the noncooperation condition interposed between cooperation conditions. Groups 2 and 3 had the cooperation condition interposed between noncooperation conditions. Durations of triadic activities, per cent of time in triadic activities, intercom use, and intersubject program synchronization were greater during cooperation conditions than during noncooperation conditions. These data show that a cooperation contingency within the behavioral program affected both social behavior and the collateral individual behavior necessary to execute the cooperation response.     

The role of the human extrastriate visual cortex in mirror symmetry discrimination: a TMS-adaptation study

The human visual system is able to efficiently extract symmetry information from the visual environment. Prior neuroimaging evidence has revealed symmetry-preferring neuronal representations in the dorsolateral extrastriate visual cortex; the objective of the present study was to investigate the necessity of these representations in symmetry discrimination. This was accomplished by the use of state-dependent transcranial magnetic stimulation, which combines the fine resolution of adaptation paradigms with the assessment of causality. Subjects were presented with adapters and targets consisting of dot configurations that could be symmetric along either the vertical or horizontal axis (or they could be non-symmetric), and they were asked to perform a symmetry discrimination task on the targets while fixating the center of the screen. TMS was applied during the delay between the adapter and the test stimulus over one of four different sites: Left or Right V1/V2, or left or right dorsolateral extrastriate cortex (DLO). TMS over both Left and Right DLO reduced the adaptation effect in detecting vertical and horizontal symmetry, although the Left DLO effect on horizontal symmetry and the Right DLO effect on both vertical and horizontal symmetry were present only when considering subjects who showed a behavioral adaptation effect in the baseline No-TMS condition. Application of TMS over the Left or Right V1/V2 did not modulate the adaptation effect. Overall, these data suggest that both the Left and Right DLO contain neuronal representations tuned to mirror symmetry which play a causal role in symmetry discrimination.     

Testing the conditional independence and monotonicity assumptions of item response theory

When item characteristic curves are nondecreasing functions of a latent variable, the conditional or local independence of item responses given the latent variable implies nonnegative conditional covariances between all monotone increasing functions of a set of item responses given any function of the remaining item responses. This general result provides a basis for testing the conditional independence assumption without first specifying a parametric form for the nondecreasing item characteristic curves. The proposed tests are simple, have known asymptotic null distributions, and possess certain optimal properties. In an example, the conditional independence hypothesis is rejected for all possible forms of monotone item characteristic curves.The author acknowledges Paul W. Holland for valuable conversations on the subject of this paper; Henry Braun and Fred Lord for comments at a presentation on this subject which led to improvements in the paper; Carl H. Haag for permission to use the data in §4; Bruce Kaplan for assistance with computing; and two referees for helpful suggestions. Requests for reprints should be sent to Paul R. Rosenbaum     

The role of the magazine-response contingency on signal-directed responding in pigeons

Three experiments used a procedure for directly delivering water into the thirsty pigeon's mouth to explore the role that the instrumental magazine contingency plays in autoshaped responding. Magazine training was accomplished by requiring birds to contact a “magazine” key when it was illuminated, in order to obtain intraoral injections of water. Other subjects received water injections independent of a magazine-response contingency. Subsequently, magazine-trained subjects showed a transient enhancement in responding to the illumination of another “signal” key, whether that stimulus was presented alone or paired with water delivery. The overall level of maintained autoshaped responding was little influenced by the instrumental magazine contingency, although the within-trial time course of signal-directed responding was affected. When illumination of the signal key preceded access to water only when the signal was not contacted, pigeons directed many responses toward the signal key. However, there was no evidence that the instrumental magazine contingency enhanced responding on this omission schedule of reinforcement. These results thus confirm a small, but measurable contribution of the magazine-response contingency to signal-directed responding; they fail to support the conclusion that the instrumental magazine contingency greatly affects the outcomes of autoshaping studies.     

The role of contingencies and "principles of behavioral variation" in pigeons' pecking

Staddon and Simmelhag's proposal that behavior is produced by “principles of behavioral variation” instead of contingencies of reinforcement was tested in two experiments. In the first experiment pigeons were exposed to either a fixed-interval schedule of response-contingent reinforcement, an autoshaping schedule of stimulus-contingent reinforcement, or a fixed-time schedule of noncontingent reinforcement. Pigeons exposed to contingent reinforcement came to peck more rapidly than those exposed to noncontingent reinforcement. Staddon and Simmelhag's “principles of behavioral variation” included the proposal that patterns (interim and terminal) were a function of momentary probability of reinforcement. In the second experiment pigeons were exposed to either a fixed-time or a random-time schedule of noncontingent reinforcement. Pecking showed a constant frequency of occurrence over postfood time on the random-time schedule. Most behavior showed patterns on the fixed-time schedule that differed in overall shape (i.e., interim versus terminal) from those shown on the random-time schedule. It was concluded that both the momentary probability of reinforcement and postfood time can affect patterning.