Imputation of missing categorical data by maximizing internal consistency

This paper suggests a method to supplant missing categorical data by reasonable replacements. These replacements will maximize the consistency of the completed data as measured by Guttman's squared correlation ratio. The text outlines a solution of the optimization problem, describes relationships with the relevant psychometric theory, and studies some properties of the method in detail. The main result is that the average correlation should be at least 0.50 before the method becomes practical. At that point, the technique gives reasonable results up to 10–15% missing data.We thank Anneke Bloemhoff of NIPG-TNO for compiling and making the Dutch Life Style Survey data available to use, and Chantal Houée and Thérèse Bardaine, IUT, Vannes, France, exchange students under the COMETT program of the EC, for computational assistance. We also thank Donald Rubin, the Editors and several anonymous reviewers for constructive suggestions.     

Diathesis-stress models of psychopathology: A quantitative genetic perspective

When genetics is considered in diathesis-stress models of psychopathology, it is often assumed that heredity provides the diathesis and environmental factors are responsible for the stressor. We discuss two quantitative genetic models relevant to the diathesis-stress construct. One model focuses on genotype-environment interaction, which is the usual way in which genetic influence is represented in diathesis-stress models. A second model—genotype-environment correlation—provides an alternative that represents both genetic and environmental influences relevant to the development of psychopathology. Implications of these models for clinical research are discussed.     

Low-dimensional chaos maps learning in a model neuropil (olfactory bulb)

Quantification of a chaotic system can be made by calculating the correlation dimension (D2) of the data that the system generates (Packard et al., 1980). The D2 algorithm, however, requires stationarity of the generator, a feature that biological data rarely reflect (Mayer-Kress et al., 1988). So we developed the “point correlation dimension” (PD2), an algorithm that accurately tracks D2 in linked data of different dimensions (Carpeggiani et al., 1991). We now present a mathematical argument that, for stationary data, individual PD2s converge to D2 and we demonstrate that the algorithm rejects contributions made by bursts of noise. Data were obtained from the surface of the olfactory bulb of the conscious rabbit (64 electrodes, 640 Hz each, 1.3 sec epochs) before and after presentation of a novel or habituated odor. D2 could be calculated in only 1 of 10 novel-odor trials, whereas PD2 could be calculated in all. Both algorithms indicated that a novel odor evokes a spatially uniform dimensional increase. The PD2 uniquely exhibited the dimensional decreases that occur during inspiration and the gradients of mean dimension present during the nonstimulated control state. These control gradients remained unchanged without odor experience, but showed spatially specific PD2 increases following odor habituation. It is interpreted that, 1) the PD2 issensitive, accurate, and appropriate for dimensional assessment of biological data, 2) that during analysis of unfamiliar information a singleglobal process is transiently evoked in the neuropil, and 3) after experience multiplespatially specific processes tonically map the sites of learning. Grant Support: National Institutes of Health, HL 31164 and NS27745     

Least-squares approximation of an improper correlation matrix by a proper one

An algorithm is presented for the best least-squares fitting correlation matrix approximating a given missing value or improper correlation matrix. The proposed algorithm is based upon a solution for Mosier's oblique Procrustes rotation problem offered by ten Berge and Nevels. A necessary and sufficient condition is given for a solution to yield the unique global minimum of the least-squares function. Empirical verification of the condition indicates that the occurrence of non-optimal solutions with the proposed algorithm is very unlikely. A possible drawback of the optimal solution is that it is a singular matrix of necessity. In cases where singularity is undesirable, one may impose the additional nonsingularity constraint that the smallest eigenvalue of the solution be , where is an arbitrary small positive constant. Finally, it may be desirable to weight the squared errors of estimation differentially. A generalized solution is derived which satisfies the additional nonsingularity constraint and also allows for weighting. The generalized solution can readily be obtained from the standard unweighted singular solution by transforming the observed improper correlation matrix in a suitable way.     

Combinatorial data analysis: Association and partial association

A combinatorial data analysis strategy is reviewed that is designed to compare two arbitrary measures of proximity defined between the objects from some set. Based on a particular cross-product definition of correspondence between these two numerically specified notions of proximity (typically represented in the form of matrices), extensions are then pursued to indices of partial association that relate the observed pattern of correspondence between the first two proximity measures to a third. The attendant issues of index normalization and significance testing are discussed; the latter is approached through a simple randomization model implemented either through a Monte Carlo procedure or distributional approximations based on the first three moments. Applications of the original comparison strategy and its extensions to partial association may be developed for a variety of methodological and substantive tasks. Besides rank correlation, we emphasize the topics of spatial autocorrelation for one variable and spatial association between two and mention the connection to the usual randomization approach for one-way analysis-of-variance.     

Self-selection biases in correlational studies based on questionnaires

It is commonly held that even where questionnaire response is poor, correlational studies are affected only by loss of degrees of freedom or precision. We show that this supposition is not true. If the decision to respond is correlated with a substantive variable of interest, then regression or analysis of variance methods based upon the questionnaire results may be adversely affected by self-selection bias. Moreover such bias may arise even where response is 100%. The problem in both cases arises where selection information is passed to the score indirectly via the disturbance or individual effects, rather than entirely via the observable explanatory variables. We suggest tests for the ensuing self-selection bias and possible ways of handling the ensuing problems of inference.The author would like to thank members of the Psychology Department at UWA for references on the illustrative example, namely the effects of sex-stereotyping on achievements in mathematics; thanks are also due to the anonymous reviewers whose comments resulted in material improvements to the paper.     

A family of chance-corrected association coefficients for metric scales

A chance-corrected version of the family of association coefficients for metric scales proposed by Zegers and ten Berge is presented. It is shown that a matrix with chance-corrected coefficients between a number of variables is Gramian. The members of the chance-corrected family are shown to be partially ordered. The author is obliged to Jos ten Berge for helpful comments.     

A generalized family of coefficients of relational agreement for numerical scales

A family of coefficients of relational agreement for numerical scales is proposed. The theory is a generalization to multiple judges of the Zegers and ten Berge theory of association coefficients for two variables and is based on the premise that the choice of a coefficient depends on the scale type of the variables, defined by the class of admissible transformations. Coefficients of relational agreement that denote agreement with respect to empirically meaningful relationships are derived for absolute, ratio, interval, and additive scales. The proposed theory is compared to intraclass correlation, and it is shown that the coefficient of additivity is identical to one measure of intraclass correlation.The author thanks the Editor and anonymous reviewers for helpful suggestions.     

Interpreting canonical correlation analysis through biplots of structure correlations and weights

This paper extends the biplot technique to canonical correlation analysis and redundancy analysis. The plot of structure correlations is shown to the optimal for displaying the pairwise correlations between the variables of the one set and those of the second. The link between multivariate regression and canonical correlation analysis/redundancy analysis is exploited for producing an optimal biplot that displays a matrix of regression coefficients. This plot can be made from the canonical weights of the predictors and the structure correlations of the criterion variables. An example is used to show how the proposed biplots may be interpreted.     

New lower and upper bounds for communality in factor analysis

We derive several relationships between communalities and the eigenvalues for ap ×p correlation matrix under the usual factor analysis model. For suitable choices ofj, _j (), where _j () is thej-th largest eigenvalue of , provides either a lower or an upper bound to the communalities for some of the variables. We show that for at least one variable, 1 - _p () improves on the use of squared mulitiple correlation coefficient as a lower bound.This research was done while the second author was at Tokyo Institute of Technology.