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.     

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.     

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.     

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 an interdependent group contingency on engagement in physical education

We examined whether a group contingency increased engagement during elementary school physical education sessions. The intervention employed procedures (explicit instruction, goal setting, and reinforcement) drawn from the first tier of classwide function‐related intervention teams (CW‐FIT; Wills et al., 2009). Results showed salutary increases in engagement that corresponded with the teacher's implementation.     

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.     

Base-rate neglect as a function of base rates in probabilistic contingency learning

When humans predict criterion events based on probabilistic predictors, they often lend excessive weight to the predictor and insufficient weight to the base rate of the criterion event. In an operant analysis, using a matching-to-sample paradigm, Goodie and Fantino (1996) showed that humans exhibit base-rate neglect when predictors are associated with criterion events through physical similarity. In partial replications of their studies, we demonstrated similar effects when the predictors resembled the criterion events in terms of similarly skewed base rates. Participants' predictions were biased toward the more (or less) frequent criterion event following the more (or less) frequent predictor. This finding adds to the growing evidence for pseudocontingencies (Fiedler & Freytag, 2004), a framework that stresses base-rate influences on contingency learning.     

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.     

Stimulus control topographies and tests of symmetry in pigeons

Pigeons were tested for symmetry after A-B training under conditions designed to avoid problems that may prevent its emergence, namely the change of stimulus location in testing relative to training and the lack of requisite discrimination training. In Experiment 1, samples appeared in two locations during baseline training to minimize the impact of stimulus location. Experiments 2 and 3 included multiple-location training along with additional identity and symbolic matching training, respectively, to explicitly train all of the simultaneous and successive stimulus discriminations required for testing. Experiment 4 provided reinforcement for symmetrical matching relations with some stimulus sets (with multiple-location training) prior to symmetry testing with different sets. In all experiments, pigeons showed no evidence of symmetry despite the fact that baseline (A-B) matching transferred to novel locations. Additional tests for reflexivity (Experiment 2) yielded similar outcomes. These results indicate that the change in stimulus location is not the sole reason that pigeons do not show symmetry and increase the plausibility of arguments that symmetry and other indexes of stimulus equivalence may be beyond the capabilities of the pigeon.     

Testing for symmetry in the conditional discriminations of language-trained chimpanzees

If subjects are taught to match Stimulus A to B and then, without further training, match B to A, they have passed a test of symmetry. It has been suggested that non-humans' lack of success on symmetry tests might be overcome by giving them a history of symmetry exemplar training, that is, by directly teaching a large number of conditional relations (e.g., AB, CD, EF,...) and also directly training the "reverse" of these relations (e.g., BA, DC, FE,...). The chimpanzee subjects of the present study, Sherman, Austin, and Lana, had already received extensive symmetry exemplar training as a result of attempts to teach a selection-based language system of lexigrams. The present study systematically subjected 2 of these chimps (Sherman and Lana), for the first time, to standard symmetry tests in controlled conditions. Both chimps failed the tests, even when their correct responses on test trials were reinforced. The findings do not support the exemplar training hypothesis, and cast doubt upon whether the chimps can pass tests of stimulus equivalence.