Millsap and Meredith (1988) have developed a generalization of principal components analysis for the simultaneous analysis of a number of variables observed in several populations or on several occasions. The algorithm they provide has some disadvantages. The present paper offers two alternating least squares algorithms for their method, suitable for small and large data sets, respectively. Lower and upper bounds are given for the loss function to be minimized in the Millsap and Meredith method. These can serve to indicate whether or not a global optimum for the simultaneous components analysis problem has been attained.

A number of methods for the analysis of three-way data are described and shown to be variants of principal components analysis (PCA) of the two-way supermatrix in which each two-way slice is “strung out” into a column vector. The methods are shown to form a hierarchy such that each method is a constrained variant of its predecessor. A strategy is suggested to determine which of the methods yields the most useful description of a given three-way data set.

The Maxbet method is a generalized principal components analysis of a data set, where the group structure of the variables is taken into account. Similarly, 3-block[12,13] partial Maxdiff method is a generalization of covariance analysis, where only the covariances between blocks (1, 2) and (1, 3) are taken into account. The aim of this paper is to give the global maximum for the 2-block Maxbet and 3-block[12,13] partial Maxdiff problems by picking the best solution from the complete solution set for the multivariate eigenvalue problem involved. To do this, we generalize the characteristic polynomial of a matrix to a system of two characteristic polynomials, and provide the complete solution set of the latter via Sylvester resultants. Examples are provided.

In this paper, the statistical significance of the contribution of variables to the principal components in principal components analysis (PCA) is assessed nonparametrically by the use of permutation tests. We compare a new strategy to a strategy used in previous research consisting of permuting the columns (variables) of a data matrix independently and concurrently, thus destroying the entire correlational structure of the data. This strategy is considered appropriate for assessing the significance of the PCA solution as a whole, but is not suitable for assessing the significance of the contribution of single variables. Alternatively, we propose a strategy involving permutation of one variable at a time, while keeping the other variables fixed. We compare the two approaches in a simulation study, considering proportions of Type I and Type II error. We use two corrections for multiple testing: the Bonferroni correction and controlling the False Discovery Rate (FDR). To assess the significance of the variance accounted for by the variables, permuting one variable at a time, combined with FDR correction, yields the most favorable results. This optimal strategy is applied to an empirical data set, and results are compared with bootstrap confidence intervals.

Multitrait-Multimethod (MTMM) matrices are often analyzed by means of confirmatory factor analysis (CFA). However, fitting MTMM models often leads to improper solutions, or non-convergence. In an attempt to overcome these problems, various alternative CFA models have been proposed, but with none of these the problem of finding improper solutions was solved completely. In the present paper, an approach is proposed where improper solutions are ruled out altogether and convergence is guaranteed. The approach is based on constrained variants of components analysis (CA). Besides the fact that these methods do not give improper solutions, they have the advantage that they provide component scores which can later on be used to relate the components to external variables. The new methods are illustrated by means of simulated data, as well as empirical data sets.

Procedures for oblique rotation of factors or principal components typically focus on rotating the pattern matrix such that it becomes optimally simple. An important oblique rotation method that does so is Harris and Kaiser's (1964) independent cluster (HKIC) rotation. In principal components analysis, a case can be made for interpreting the components on the basis of the component weights rather than on the basis of the pattern, so it seems desirable to rotate the components such that the weights rather than the pattern become optimally simple. In the present paper, it is shown that HKIC rotates the components such that both the pattern and the weights matrix become optimally simple. In addition, it is shown that the pattern resulting from HKIC rotation is columnwise proportional to the associated weights matrix, which implies that the interpretation of the components does not depend on whether it is based on the pattern or on the component weights matrix. It is also shown that the latter result only holds for HKIC rotation and slight modifications of it.

We offer an introduction to the five papers that make up this special section. These papers deal with a range of the methodological challenges that face researchers analyzing fMRI data—the spatial, multilevel, and longitudinal nature of the data, the sources of noise, and so on. The papers all provide analyses of data collected by a multi-site consortium, the Function Biomedical Informatics Research Network. Due to the sheer volume of data, univariate procedures are often applied, which leads to a multiple comparisons problem (since the data are necessarily multivariate). The papers in this section include interesting applications, such as a state-space model applied to these data, and conclude with a reflection on basic measurement problems in fMRI. All in all, they provide a good overview of the challenges that fMRI data present to the standard psychometric toolbox, but also to the opportunities they offer for new psychometric modeling.

In this paper we discuss the use of a recent dimension reduction technique called Locally Linear Embedding, introduced by Roweis and Saul, for performing an exploratory latent structure analysis. The coordinate variables from the locally linear embedding describing the manifold on which the data reside serve as the latent variable scores. We propose the use of semiparametric penalized spline methods for reconstruction of the manifold equations that approximate the data space. We also discuss a crossvalidation strategy that can guide in selecting an appropriate number of latent variables. Synthetic as well as real data sets are used to illustrate the proposed approach. A nonlinear latent structure representation of a data set also serves as a data visualization tool.

Relatively little is known about how the diet of chronically undernourished children may impact cardiometabolic biomarkers. The objective of this exploratory study was to characterise relationships between dietary patterns and the cardiometabolic profile of 153 3–5-year-old Peruvian children with a high prevalence of chronic undernutrition. We collected monthly dietary recalls from children when they were 9–24 months old. At 3–5 years, additional dietary recalls were collected, and blood pressure, height, weight, subscapular skinfolds and fasting plasma glucose, insulin and lipid profiles were assessed. Nutrient intakes were expressed as average density per 100 kcals (i) from 9 to 24 months and (ii) at follow-up. The treelet transform and sparse reduced rank regress'ion (RRR) were used to summarize nutrient intake data. Linear regression models were then used to compare these factors to cardiometabolic outcomes and anthropometry. Linear regression models adjusting for subscapular skinfold-for-age Z-scores (SSFZ) were then used to test whether observed relationships were mediated by body composition. 26 % of children were stunted at 3–5 years old. Both treelet transform and sparse RRR-derived child dietary factors are related to protein intake and associated with total cholesterol and SSFZ. Associations between dietary factors and insulin were attenuated after adjusting for SSFZ, suggesting that body composition mediated these relationships. Dietary factors in early childhood, influenced by protein intake, are associated with cholesterol profiles, fasting glucose and body fat in a chronically undernourished population.

Reading difficulties are prevalent worldwide, including in economically developed countries, and are associated with low academic achievement and unemployment. Longitudinal studies have identified several early childhood predictors of reading ability, but studies frequently lack genotype data that would enable testing of predictors with heritable influences. The National Child Development Study (NCDS) is a UK birth cohort study containing direct reading skill variables at every data collection wave from age 7 years through to adulthood with a subsample (final n = 6431) for whom modern genotype data are available. It is one of the longest running UK cohort studies for which genotyped data are currently available and is a rich dataset with excellent potential for future phenotypic and gene-by-environment interaction studies in reading. Here, we carry out imputation of the genotype data to the Haplotype Reference Panel, an updated reference panel that offers greater imputation quality. Guiding phenotype choice, we report a principal components analysis of nine reading variables, yielding a composite measure of reading ability in the genotyped sample. We include recommendations for use of composite scores and the most reliable variables for use during childhood when conducting longitudinal, genetically sensitive analyses of reading ability.

Dietary pattern analysis is typically based on dimension reduction and summarises the diet with a small number of scores. We assess ‘joint and individual variance explained’ (JIVE) as a method for extracting dietary patterns from longitudinal data that highlights elements of the diet that are associated over time. The Auckland Birthweight Collaborative Study, in which participants completed an FFQ at ages 3·5 (n 549), 7 (n 591) and 11 (n 617), is used as an example. Data from each time point are projected onto the directions of shared variability produced by JIVE to yield dietary patterns and scores. We assess the ability of the scores to predict future BMI and blood pressure measurements of the participants and make a comparison with principal component analysis (PCA) performed separately at each time point. The diet could be summarised with three JIVE patterns. The patterns were interpretable, with the same interpretation across age groups: a vegetable and whole grain pattern, a sweets and meats pattern and a cereal v. sweet drinks pattern. The first two PCA-derived patterns were similar across age groups and similar to the first two JIVE patterns. The interpretation of the third PCA pattern changed across age groups. Scores produced by the two techniques were similarly effective in predicting future BMI and blood pressure. We conclude that when data from the same participants at multiple ages are available, JIVE provides an advantage over PCA by extracting patterns with a common interpretation across age groups.

Data on the combination of foods consumed simultaneously at specific eating occasions are scarce, primarily due to a lack of assessment tools. We applied a recently developed meal coding system to multiple-day dietary intake data for assessing its ability to estimate food and nutrient intakes and characterise meal-based dietary patterns in the Japanese context. A total of 242 Japanese adults completed sixteen non-consecutive-day weighed dietary records, including 14 734 eating occasions (3788 breakfasts, 3823 lunches, 3856 dinners and 3267 snacks). Common food group combinations were identified by meal type to identify a range of generic meals. Dietary intake was calculated on the basis of not only the standard food composition database but also the substituted generic meal database. In total, eighty generic meals (twenty-three breakfasts, twenty-one lunches, twenty-four dinners and twelve snacks) were identified. The Spearman correlation coefficients between food group intakes calculated based on the standard food composition database and the substituted generic meal database ranged from 0·26 to 0·85 (median 0·69). The corresponding correlations for nutrient intakes ranged from 0·17 to 0·82 (median 0·61). A total of eleven meal patterns were established using principal components analysis, and these accounted for 39·1 % of total meal variance. Considerable variation in patterns was seen in meal type inclusion and choice of staple foods (bread, rice and noodles) and drinks, and also in meal constituents. In conclusion, this study demonstrated the usefulness of a meal coding system for assessing habitual diet, providing a scientific basis towards the development of simple meal-based dietary assessment tools.