Penalized optimal scaling for ordinal variables with an application to international classification of functioning core sets

Ordinal data occur frequently in the social sciences. When applying principal component analysis (PCA), however, those data are often treated as numeric, implying linear relationships between the variables at hand; alternatively, non-linear PCA is applied where the obtained quantifications are sometimes hard to interpret. Non-linear PCA for categorical data, also called optimal scoring/scaling, constructs new variables by assigning numerical values to categories such that the proportion of variance in those new variables that is explained by a predefined number of principal components (PCs) is maximized. We propose a penalized version of non-linear PCA for ordinal variables that is a smoothed intermediate between standard PCA on category labels and non-linear PCA as used so far. The new approach is by no means limited to monotonic effects and offers both better interpretability of the non-linear transformation of the category labels and better performance on validation data than unpenalized non-linear PCA and/or standard linear PCA. In particular, an application of penalized optimal scaling to ordinal data as given with the International Classification of Functioning, Disability and Health (ICF) is provided.     

Lifespan normative data (18–89 years) for Raven's Advanced Progressive Matrices Set I

Raven's Advanced Progressive Matrices (APM) Set I is a validated and brief test of fluid intelligence, ideal for use in busy clinical settings. However, there is a dearth of normative data allowing an accurate interpretation of APM scores. To address this, we present normative data from across the adult lifespan (18–89 years) for the APM Set I. Data are presented in five age cohorts (total N = 352), including two older adult cohorts (65–79 years and 80–89 years), which allows age-standardized assessment. We also present data from a validated measure of premorbid intellectual ability, which was absent from previous standardizations of longer forms of the APM. In line with previous findings, a striking age-related decline was noted, beginning relatively early in adulthood and most marked amongst lower-scoring individuals. Older adults did not demonstrate difficulty with specific test items or make an increased proportion of specific errors. Sex was not a significant predictor of performance. The data set is of particular use in the neuropsychological assessment of older adults, given the known susceptibility of fluid intelligence to both the effects of normal ageing and acquired brain injury in older age. The results are discussed in light of theories of neurological ageing.     

Psychometric properties and factor structure of the Fear Questionnaire phobia subscale items in two normative samples

Normative data for the Fear Questionnaire (Marks & Mathews, 1979), a popular self-report instrument measuring phobic concerns, were collected from both a community and a collegiate sample. The covariation and internal consistency of the blood-injury phobia (BI), social phobia (SO), and agoraphobia (AG) subscales were assessed in each sample, and the factor structure of the items from these three subscales was examined. Results indicated that, in general, community subjects reported more phobic concerns than did collegiate subjects, and females reported greater phobic concerns than did males. Results from confirmatory factor analyses suggest that the three factors of BI, SO, and AG did not emerge from either data set. Follow-up exploratory factor analyses did identify the general factors of blood-injury phobia and agoraphobia. Normative data from the present study are compared to those obtained previously, and directions for future research are provided.     

Social networks with two sets of actors

Traditional network research analyzes relational ties within a single group of actors: the models presented in this paper involve relational ties exist beteen two distinct sets of actors. Statistical models for traditional networks in which relations are measured within a group simplify when modeling unidirectional relations measured between groups. The traditional paradigm results in a one-mode socionatrix; the network paradigm considered in this paper results in a two-mode socionatrix; A statistical model is presented, illustrated on a sample data set, and compared to its traditional counterpart. Extensions are discussed, including those that model multivariate relations simultaneously, and those that allow for the inclustion of attributes of the individuals in the group.We thank the Editor and two anonymous reviewers for their helpful comments. We are also grateful to George Barnett for allowing us to analyze his data.     

Data Smoothing: Prediction of Human Behavior, Detection of Behavioral Patterns, and Monitoring Treatment Effectiveness in Single-Subject Behavioral Studies

Data-smoothing can be particularly useful in predicting human behavior, detecting behavioral patterns, and monitoring treatment effectiveness in highly variable single-subject behavioral experiments that cannot be determined by only visual inspection of their graphs. Using an example from the applied behavior analytic literature, the use of moving-average and exponential data-smoothing aided the detection of the unique behavioral patterns of a child with autism across different treatments. Furthermore, the utility of the data-smoothing procedures to monitor and control the effectiveness of an intervention is discussed. The ease of their calculations suggest use of data-smoothing by behavior analysts whenever the effects of particular interventions are questionable.     

A marginalization model for the multidimensional unfolding analysis of ranking data

A marginalization model for the multidimensional unfolding analysis of ranking data is presented. A subject samples one of a number of random points that are multivariate normally distributed. The subject perceives the distances from the point to all the stimulus points fixed in the same multidimensional space. The distances are error perturbed in this perception process. He/she produces a ranking dependent on these error-perturbed distances. The marginal probability of a ranking is obtained according to this ranking model and by integrating out the subject (ideal point) parameters, assuming the above distribution. One advantage of the model is that the individual differences are captured using the posterior probabilities of subject points. Three sets of ranking data are analyzed by the model.     

Guidelines for training in the ethical conduct of scientific research

Historically, scientists in training have learned the rules of ethical conduct by the example of their advisors and other senior scientists and by practice. This paper is intended to serve as a guide for the beginning scientist to some fundamental principles of scientific research ethics. The paper focuses less on issues of outright dishonesty or fraud, and more on the positive aspects of ethical scientific behavior; in other words, what a scientist should do to maintain a high level of ethical conduct in research. There are a number of fairly specific rules, guidelines, or commonly accepted operating principles that have evolved for the ethical conduct of science. In order to discuss this code of ethics, this paper is divided into sections dealing with specific areas of scientific ethics. These areas are: data collection and storage, ownership of data, confidentiality, communication, authorship, collaboration, the peer review system, and rules of dealing with ethical complaints. Illustrative case histories are presented to provide examples of the type of ethical dispute or problem being discussed. If scientific trainees learn the accepted rules of behavior that govern the conduct of science, ethical problems that arise out of ignorance, misunderstanding, or poor communication can be avoided.     

A simplification of a result by zellini on the maximal rank of symmetric three-way arrays

Zellini (1979, Theorem 3.1) has shown how to decompose an arbitrary symmetric matrix of ordern ×n as a linear combination of 1/2n(n+1) fixed rank one matrices, thus constructing an explicit tensor basis for the set of symmetricn ×n matrices. Zellini's decomposition is based on properties of persymmetric matrices. In the present paper, a simplified tensor basis is given, by showing that a symmetric matrix can also be decomposed in terms of 1/2n(n+1) fixed binary matrices of rank one. The decomposition implies that ann ×n ×p array consisting ofp symmetricn ×n slabs has maximal rank 1/2n(n+1). Likewise, an unconstrained INDSCAL (symmetric CANDECOMP/PARAFAC) decomposition of such an array will yield a perfect fit in 1/2n(n+1) dimensions. When the fitting only pertains to the off-diagonal elements of the symmetric matrices, as is the case in a version of PARAFAC where communalities are involved, the maximal number of dimensions can be further reduced to 1/2n(n–1). However, when the saliences in INDSCAL are constrained to be nonnegative, the tensor basis result does not apply. In fact, it is shown that in this case the number of dimensions needed can be as large asp, the number of matrices analyzed.     

Decompositions and biplots in three-way correspondence analysis

In this paper correspondence analysis for three-way contingency tables is presented using three-way generalisations of the singular value decomposition. It is shown that in combination with Lancaster's (1951) additive decomposition of interactions in three-way tables, a detailed analysis is possible of the deviations from independence. Finally, biplots are shown to produce powerful graphical representations of the results from three-way correspondence analyses. An example from child development is used to illustrate the theoretical developments.     

Some uniqueness results for PARAFAC2

Whereas the unique axes properties of PARAFAC1 have been examined extensively, little is known about uniqueness properties for the PARAFAC2 model for covariance matrices. This paper is concerned with uniqueness in the rank two case of PARAFAC2. For this case, Harshman and Lundy have recently shown, subject to mild assumptions, that PARAFAC2 is unique when five (covariance) matrices are analyzed. In the present paper, this result is sharpened. PARAFAC2 is shown to be usually unique with four matrices. With three matrices it is not unique unless a certain additional assumption is introduced. If, for instance, the diagonal matrices of weights are constrained to be non-negative, three matrices are enough to have uniqueness in the rank two case of PARAFAC2. The authors are obliged to Richard Harshman for stimulating this research, and to the Associate Editor and reviewers for suggesting major improvements in the presentation.