Procrustes Rotation Research Articles

A study: Semi-confirmatory factor analysis and reliability of the symptom assessment-45 questionnaire in Peruvian population

The study of psychopathological variables constitutes a valuable resource in the field of mental health, but even more so when there are evaluation support tools that facilitate its exploration. In addition, when these tools are valid and reliable, they provide optimal data, which guides towards a better intervention response, which is extremely necessary in the face of the reality that exists in terms of mental health needs. Therefore, this work seeks to analyze the evidence of the construct validity of SA-45, convergent validity, and discriminant validity. In addition, to find the reliability of the Symptom Assessment-45 Questionnaire (SA-45). The SA-45, and other measures (LOT-4, DASS-21) were applied to 1637 subjects from different regions of Peru, aged 18 to 65. Then, a semi-confirmatory factor analysis was performed by means of exploratory structural equation modeling (ESEM) and an objective matrix with Procrustes rotation to identify the hypothesized dimensions. All nine dimensions hypothesized by previous studies were found to have favorable factor congruence coefficients (FCC-total >.85), adequate factor saturations (λ >.30), acceptable to high internal consistency alpha, omega, and ordinal alpha coefficients (>.70), and satisfactory goodness-of-fit (RMSEA = 0.004, RMSR = 0.021, FCC = 0.99, GFI = 1.000). Likewise, the validity in relation to other instruments such as the DASS-21 and the LOT-R reports Pearson correlation coefficients with medium and large effect sizes (r =.30 and r =.50) in their dimensions. The SA-45 is a theoretically and psychometrically supported instrument for use in young and adult populations.

Factor analysis and reliability of the Assessment of Spirituality and Religious Sentiments (ASPIRES) Scale in an Iranian sample

ABSTRACT This study investigated the factor structure and reliability of the Assessment of Spirituality and Religious Sentiments (ASPIRES) Scale in an Iranian sample. A total of 409 subjects (213 females and 196 males) between the ages of 14 and 88 years (Mean = 34.37) were selected. Scores were submitted to a principal components analysis using an oblimin rotation. The resulting pattern loadings were then subjected to an orthogonal procrustean rotation that used US normative data as the target matrix. The resulting factor loadings were then compared to the US normative loadings and significant convergence was obtained, suggesting that the Iranian sample understood the underlying spiritual dimension in a psychologically similar manner as the US samples. Alpha reliability values were consistent with those obtained from US norms. The results suggest that the ASPIRES Scale generalized well and is a potentially useful instrument for use in an Iranian context.

Investigating paired comparisons after principal component analysis

Principal component analysis (PCA) is often used to explore sensory and consumer test data about products on multicollinear sensory attributes. In this paper, we propose an approach for investigating paired comparisons between products and their uncertainties in the principal components. We use the truncated total bootstrap (TTB) procedure to simulate virtual panels from the original data set. The virtual-panel results are then used to investigate the uncertainty of the paired comparisons. To justify the proposal, we provide an appendix that contains theoretical contributions. We introduce a new (“crossdiff”) matrix operator for obtaining paired comparisons and a procedure called crossdiff-unfolding. The notation and procedure facilitate proofs that help to justify investigating paired comparisons in the same space as the products. Specifically, we show that the principal components that optimally extract variance from the objects (e.g. products in the original coordinate system of sensory variables) also optimally extract variance from all paired comparisons. Also, we demonstrate that the same Procrustes rotation matrix that optimally superimposes a score matrix onto a target score matrix also optimally superimposes a matrix of crossdiff-unfolded paired difference scores onto a target matrix of crossdiff-unfolded paired difference scores. We account for mutual dependencies in TTB-derived paired difference scores, then visualize these results along with contours of their kernel-estimated densities. We apply the proposed methods to two data sets, one from quantitative sensory descriptive analysis from a trained panel and one from check-all-that-apply (CATA) data from a consumer cluster. Practical advantages of the methods for practitioners are described. PCA is used widely in sensory and consumer science to communicate results, and the proposed tools can strengthen interpretability.

Two-Stage Procrustes Rotation with Sparse Target Matrix and Least Squares Criterion with Regularization and Generalized Weighting

In factor analysis, a factor loading matrix is often rotated to a simple target matrix for its simplicity. For the purpose, Procrustes rotation minimizes the discrepancy between the target and rotated loadings using two types of approximation: 1) approximate the zeros in the target by the non-zeros in the loadings, and 2) approximate the non-zeros in the target by the non-zeros in the loadings. The central issue of Procrustes rotation considered in the article is that it equally treats the two types of approximation, while the former is more important for simplifying the loading matrix. Furthermore, a well-known issue of Simplimax is the computational inefficiency in estimating the sparse target matrix, which yields a considerable number of local minima. The research proposes a new rotation procedure that consists of the following two stages. The first stage estimates sparse target matrix with lesser computational cost by regularization technique. In the second stage, a loading matrix is rotated to the target, emphasizing on the approximation of non-zeros to zeros in the target by least squares criterion with generalized weighing that is newly proposed by the study. The simulation study and real data examples revealed that the proposed method surely simplifies loading matrices.

Assessments made easier: examining the use of a rating-based questionnaire to capture behavioral data in rehabilitant orangutans (Pongo pygmaeus morio)

Rehabilitation and release are commonly used for confiscated, surrendered, and rescued primates. To improve release efficacy it is important to generate accurate behavioral profiles of release candidates. Research on primates traditionally uses observer ratings to measure individual differences. This method is easily implemented, but its validity has been questioned. We evaluated whether observer ratings reflect behavioral data indicating forest adaptation in 18 free-ranging rehabilitant orangutans (Pongo pygmaeus morio). In 2017, we used a species-specific questionnaire to measure how often orangutans engaged in behaviors linked to living successfully in the wild (e.g., nest building) and the extent to which they express personality traits that may influence forest adaptation. We collected 11 months of observational data on 17 of the orangutans concurrently to validate the questionnaire items, and collected further questionnaire data for 16 of the individuals in 2019. We used regularized exploratory factor analysis (REFA) and parallel analysis to condense the ratings and determine that two factors could be reliably extracted. We conducted another REFA using the observational data, and calculated factor congruence coefficients following procrustean rotation. The first of the two factors represented forest skills and human aversion, and was congruent with observational data. The second factor reflected boldness, sociability, and exploration, and was not congruent with observational data. Ratings correlated significantly with observations for all five questionnaire items reflecting adaptation to forest life, and for three of seven items reflecting personality traits. We conclude that ratings can be a valid approach to obtain individual-based behavioral information reflecting forest adaptation in free-ranging rehabilitant orangutans, and may be particularly useful in summarizing behaviors relevant to forest adaptation that are otherwise challenging to gather in primates.

Discriminability and uncertainty in principal component analysis (PCA) of temporal check-all-that-apply (TCATA) data

Temporal check-all-that-apply (TCATA) data can be summarized and explored using principal component analysis (PCA). Here we analyze TCATA data on Syrah wines obtained from a trained sensory panel. We evaluate new and existing methods to explore the uncertainty in the PCA scores. To do so, we use the bootstrap procedure to obtain many virtual panels from the real panel’s data. Virtual-panel PCA scores are obtained using two methods. The first method, called the partial bootstrap (PB), obtains virtual-panel scores from regression. The second method, called the truncated total bootstrap (TTB), applies PCA to the virtual-panel results to obtain scores, which are truncated and superimposed on the real-panel scores by Procrustes rotation. We use the virtual scores from each method to investigate uncertainty in the real-panel PCA scores visually and numerically. To understand the uncertainty of the scores, we obtain confidence ellipses (CEs) and their areas, as well as confidence intervals (CIs) and their widths. Next, to determine whether PCA scores for different samples are well separated, we propose a procedure for approximating the standard errors of sample differences and correcting for multiple comparisons. We propose a discriminability index, and show that it can enhance the interpretability of PCA results. We incorporate graphical features into our PCA biplots to visualize discriminability. We did not find a large difference between the PB and TTB methods for understanding the uncertainty and discriminability in PCA scores. Although the TCATA data that we analyzed have a special structure, the methodological approaches presented here can be readily adapted to other applications of PCA.

A rapid method for quantifying small‐scale vegetation patch structure to complement conventional quadrat surveys

AbstractAimsVegetation sampling typically involves the use of quadrats to estimate species cover abundance. Such surveys do not generally record small‐scale vegetation patch structure at sub‐quadrat scales. Here we test a simple method to quantify patch structure that complements conventional techniques. We compare the two methods, and analyse metrics derived from small‐scale patch surveys with environment/management data.LocationNorthumberland, United Kingdom.MethodsWe recorded cover abundance of all species in an upland moor with 1‐m2 quadrats. These were divided into 100 ‘sub‐quadrats’, 10 cm × 10 cm, and the dominant and subdominant species identified. Patch metrics (number, area and shape) for individual species recorded as dominant or subdominant in the subquadrat survey were analysed using multi‐variate generalised linear models with environmental and management data. Sub‐quadrat data were also aggregated for each quadrat, to create species composition data. The two sets of compositional data, from whole‐quadrat and sub‐quadrat aggregations, were compared via Procrustes rotation of ordination scores.ResultsPatch number, area and shape for dominant and subdominant species were all significantly affected by soil pH, soil water content, slope and elevation. Effects of proximity to sheep tracks and drainage ditches were less consistent amongst species. Ordinations of vegetation data from conventional and sub‐quadrats were similar, with significant Procrustes R2 of 67% and 70% for dominant and subdominant species respectively.ConclusionsSub‐quadrat surveys can easily be used to complement existing whole‐quadrat surveys at little cost in time or resources. Their patch metrics can provide additional insights into the environmental and management drivers that may affect the growth of individual plants or clumps, potentially in relation to plant traits, and thus alter the overall community composition. The methods we describe can readily be adapted to other sizes of quadrats and sub‐quadrats in a wide range of vegetation communities.

Combinatorial projection pursuit analysis for exploring multivariate chemical data

Kurtosis-based projection pursuit analysis (kPPA) has demonstrated the ability to visualize multivariate data in a way that complements other exploratory data analysis tools, such as principal components analysis (PCA). It is especially useful for partitioning binary data sets (2k classes) with a balanced design. Since kPPA is not a variance-based method, it can often provide unsupervised class separation where other methods fail. However, when multiple classifications are possible (e.g. by gender, age, disease state, etc.), the projection provided by kPPA (corresponding to the global minimum kurtosis) will not necessarily be the one of greatest interest to the researcher. Fortunately, the optimization algorithm for kPPA allows for interrogation of projections obtained from numerous local minima. This strategy provides the basis of a new method described here, referred to as combinatorial projection pursuit analysis (CombPPA) because it presents alternative combinations of class separation. The method is truly exploratory in that it allows the landscape of interesting projections to be more fully probed. The approach uses Procrustes rotation to map local minima among the kPPA solutions, whereupon the researcher can visualize different projections. To demonstrate the new method, the clustering of grape juice samples using visible spectroscopy is presented as a model problem. This problem is well-suited to this type of study because there are eight classes of samples symmetrically partitioned into two classes by type (organic/non-organic) or four classes by brand. Results presented show the different combinations of projections that can be obtained, including the desired partitions. In addition, this work describes new enhancements to the kPPA algorithm that improve the orthogonality of solutions obtained.

DNA metabarcoding and morphological methods show complementary patterns in the metacommunity organization of lentic epiphytic diatoms

Diatoms are important organisms in freshwater ecosystems due to their position as primary producers and therefore, analyzing their assemblages provides relevant information on ecosystem functioning. Diatoms have historically been identified based on morphological traits, which is time-consuming and requires well-trained specialists. Nevertheless, DNA barcoding offers an alternative approach to overcome some limitations of the morphological method. Here, we assess if both approaches are comparable methods to study patterns and mechanisms (including environmental filtering and dispersal limitation) of epiphytic diatom metacommunities using a comprehensive dataset from 22 Mediterranean ponds at different taxonomic resolutions. We used a fragment of rbcL barcode gene combined with High-Throughput Sequencing to infer diatom community composition. The overall degree of correspondence between both approaches was assessed by Procrustean rotation analysis and Procrustean randomization tests, whereas the role of local environmental variables and geographical distances was studied using a comprehensive combination of BIOENV, Mantel tests and distance-based redundancy analysis. Our results showed a relatively poor correspondence in the compositional variation of diatom metacommunity between both approaches. We speculate that the incompleteness of the reference database and the bioinformatics processing are the biases most likely affecting the molecular approach, whereas the limited counting effort and the presence of cryptic species are presumably the major biases related with the morphological method. On the other hand, variation in diatom community composition detected with both approaches was strongly related to the environmental template, which may be related with the narrow community-environment relationships in diatoms. Nevertheless, we found no significant relationship between compositional variation and geographical distances. Overall, our work shows the complementary nature of both approaches and highlights the importance of DNA metabarcoding to address empirical research questions of community ecology in freshwaters, especially once the reference databases include most genotypes of occurring taxa and bioinformatics biases are overcome.

Bootstrap confidence intervals for principal covariates regression.

Principal covariate regression (PCOVR) is a method for regressing a set of criterion variables with respect to a set of predictor variables when the latter are many in number and/or collinear. This is done by extracting a limited number of components that simultaneously synthesize the predictor variables and predict the criterion ones. So far, no procedure has been offered for estimating statistical uncertainties of the obtained PCOVR parameter estimates. The present paper shows how this goal can be achieved, conditionally on the model specification, by means of the bootstrap approach. Four strategies for estimating bootstrap confidence intervals are derived and their statistical behaviour in terms of coverage is assessed by means of a simulation experiment. Such strategies are distinguished by the use of the varimax and quartimin procedures and by the use of Procrustes rotations of bootstrap solutions towards the sample solution. In general, the four strategies showed appropriate statistical behaviour, with coverage tending to the desired level for increasing sample sizes. The main exception involved strategies based on the quartimin procedure in cases characterized by complex underlying structures of the components. The appropriateness of the statistical behaviour was higher when the proper number of components were extracted.

DNA metabarcoding and morphological methods show complementary patterns in the metacommunity organization of lentic epiphytic diatoms

Diatoms are important organisms in freshwater ecosystems due to their position as primary producers and therefore, analyzing their communities provides relevant information on ecosystem functioning. Diatoms have historically been identified based on morphological traits, which is time-consuming and requires well-trained specialists. Nevertheless, DNA barcoding offers an alternative approach to overcome some limitations of the morphological method. Here, we assess if both approaches are comparable methods to study patterns and mechanisms (including environmental filtering and dispersal limitation) of epiphytic diatom metacommunities using a comprehensive dataset from 22 Mediterranean ponds at different taxonomic resolutions. We used a fragment of rbcL barcode gene combined with High-Throughput Sequencing to infer diatom community composition. The overall degree of correspondence between both approaches was assessed by Procrustean rotation analysis and Procrustean randomization tests, whereas the role of local environmental variables and geographical distances was studied using a comprehensive combination of BIOENV, Mantel tests and distance-based redundancy analysis. Our results showed a relatively poor correspondence in the compositional variation of diatom metacommunity between both approaches. We speculate that the incompleteness of the reference database and the bioinformatics processing are the biases most likely affecting the molecular approach, whereas the limited counting effort and the presence of cryptic species are presumably the major biases related with the morphological method. On the other hand, variation in diatom community composition detected with both approaches was strongly related to the environmental template, which may be related with the narrow ecological niche and the strong preferences for particular substrata of some diatom species. Nevertheless, we found no significant relationship between compositional variation and geographical distances at regional extent. Overall, our work highlights the importance of DNA metabarcoding to address empirical research questions of community ecology in freshwaters, especially once the reference databases include most genotypes of occurring taxa and bioinformatics biases are overcome.

Towards a common metric for assessing heroin-dependent patient satisfaction with medications: Testing methadone and buprenorphine-naloxone

BackgroundPatient satisfaction with methadone or buprenorphine-naloxone can be multidimensionally and specifically assessed by using, respectively, the Scale to Assess Satisfaction with Medications for Addiction Treatment–Methadone for Heroin addiction (SASMAT-METHER) or the SASMAT–Buprenorphine-Naloxone for Heroin addiction (SASMAT-BUNHER). The factor structures of the SASMAT-METHER and SASMAT-BUNHER show substantial commonalities. The objective of the present study is to evaluate the replicability of the SASMAT-METHER factor structure using data from the SASMAT-BUNHER development study in order to obtain an instrument that can be used to compare patient satisfaction with methadone vs. buprenorphine-naloxone. MethodSecondary analysis of SASMAT-BUNHER data provided by 205 participants in the original validation study of that scale (Pérez de los Cobos et al., 2018). Using the SASMAT-METHER component solution (17 items, 3 factors) as the target structure, a principal component analysis was performed on the data set comprised of the corresponding 17 SASMAT-BUNHER items using an oblique semi-specified Procrustean rotation. Additionally, Tucker congruence coefficients were computed to examine the correspondence between the two solutions. ResultThe factor structures of SASMAT-METHER and the 17-item version of the SASMAT-BUNHER can be considered equal given that the overall Tucker’s congruence coefficient of factorial similarity was 0.972, with individual component congruencies ranging from 0.960 to 0.995. ConclusionsThe SASMAT-METHER component solution can serve as a single common tool to compare methadone vs. buprenorphine-naloxone in terms of patient satisfaction. This finding supports the feasibility of using a common metric to specifically assess satisfaction with medications to treat heroin dependence.

Elevated CO2 Affects the Soil Organic Carbon Fractions and Their Relation to Soil Microbial Properties in the Rhizosphere of Robinia pseudoacacia L. Seedlings in Cd-Contaminated Soils

As the global climates change, elevated CO2 and soil contamination by heavy metal co-occur in natural ecosystems, which are anticipated to affect soil organic carbon fractions (SOC) and their relation to soil microbial activities, but this issue has not been extensively examined. We investigated the response of SOC and their relation with soil microorganisms and enzyme activities in rhizosphere soils of Robinia pseudoacacia L. seedlings to elevated CO2 plus cadmium (Cd) contamination. We found that elevated CO2 significantly (p < 0.05) stimulated total organic carbon (TOC) (8.6%), dissolved organic carbon (DOC) (32.6%), microbial biomass carbon (MBC) (13.5%), bacteria (11.6%), fungi (20.9%), actinomycetes (15.3%), urease (20.1%), dehydrogenase (15.8%), invertase (11.1%), and β-glucosidase (11.9%), and DOC, MBC, bacteria, actinomycetes, urease, and invertase presented smaller growth trend in the range of 500–700 μmol mol−1 CO2 than in the range of 385–500 μmol mol−1 CO2. Cd decreased DOC (30.1%), MBC (24.9%), bacteria (21.5%), actinomycetes (15.9%), and enzyme activities. Elevated CO2 offsets the negative effect of Cd on SOC and microbial activities (except for TOC and L-asparaginase). Procrustes rotation test was used to determine the drivers (elevated CO2, Cd, and CO2 + Cd) of the relation between SOC and microbial activities, revealing the correlations between SOC, soil microorganisms, and enzyme activities were higher under elevated CO2 than under elevated CO2 + Cd. Our results suggest elevated CO2 could stimulate soil fertility and microecological cycle in the rhizosphere microenvironment exposed to heavy metal by affecting the relationship between SOC and soil microbial properties.

Accounting for the dimensionality of the dependence in analyses of contingency tables obtained with Check-All-That-Apply and Free-Comment

Check-All-That-Apply (CATA) and Free-Comment (FC) provide a so-called contingency table containing citation counts of words or descriptors (columns) by products (rows). This table is most often analysed using correspondence analysis (CA). CA aims at decomposing dependence between products and descriptors into axes of maximal and decreasing dependencies, which is reasonable if the dependence has been previously established by a chi-square test. However, the p-value of this test is not valid when the observations are not independent or when the contingency table contains too many low expected citation rates. In addition, rejecting independence with a chi-square test only means that at least the first CA axis captures some dependence. This paper presents a test to determine the number of axes of the CA that capture significant dependence and proposes a Monte-Carlo approach to compute valid p-values for this test. The variability in the products’ coordinates in the CA space is often evaluated by means of a total bootstrap procedure. The paper proposes to rely on this test to determine the number of axes to consider for the Procrustes rotations of such a procedure. Finally, to investigate which words are cited more often for each product, the paper proposes performing Fisher’s exact tests per cell on the derived contingency table obtained by reversing the CA computations on the axes capturing significant dependence. The benefits of accounting for the dimensionality of the dependence in the analyses are demonstrated on real CATA data.

A three-step procedure for analysis of circumplex models: An example of narcissism located within the circumplex of personality metatraits

Circumplex models are widely utilized in the field of personality and individual differences research. Although within the literature one could find many suggestions on how to analyze such models, none of them are comprehensive enough. Within the current paper we propose a three-step procedure, which will systematize and standardize the analysis of circumplex models and the localization of external variables within such circumplex models. First, we propose to verify the underlying circumplex structure through structural equation modeling. Second, we propose to test the possibility to locate external variables in the space of the empirical circumplex model through the investigation of their structural summary profiles. Finally, we propose to test the congruence between empirical locations and theoretical predictions within the circumplex structure through the Procrustes rotation. The three-step procedure is described using the example of narcissism embedded within the Circumplex of Personality Metatraits model. This paper is supplemented by pedagogical tutorials assisting other researchers in applying the three-step procedure to their own data.

Anthropogenic and climate change impacts on lake-water chemistry over the past 20 years, Upper Midwest, United States

ABSTRACT Water data collected from 33 lakes across the Upper Midwest in two sampling regimes spanning ~ 20 years were used to detect changes in water chemistry. Significant increases in chloride, total dissolved solids, and pH, and decreases in magnesium were observed between the two datasets. Rates of change per year were greatest for Cl (0.088 mg L−1 yr−1) and TDS (1.88 mg L−1 yr−1), with road salt application and an increase in impervious surfaces related to urbanization being the suggested causes. The decrease in atmospheric sulphate deposition over the time period was a likely driver for higher pH. Principal component analyses of 10 water chemistry variables for each sampling period were compared using Procrustes rotation, with sample scores from 2013 representing the change in water chemistry (ΔWC). Correlations between mean annual temperature (r = 0.545, p = 0.001) and total annual precipitation (r = −0.672, p = 0.000) with ΔWC warranted the development of a regression model that explained 50% of the variance in the ΔWC (adjusted r2 = 0.50, p < 0.001). The two datasets, collected 20 years apart, reveal trends similar to those observed in other studies.

Urban distribution centres in historical cities from the perspective of residents, retailers and carriers

The heavy flow of vehicles in historical cities leads to negative impacts for city logistics and for the preservation of its heritage. The use of an urban distribution centre (UDC) is a solution to minimise these impacts. This paper compares the operational, economic, social, cultural, and environmental aspects that influence the implementation of a UDC in a Brazilian historical city, considering the viewpoints of the agents: residents, retailers and carriers. Having a more complete picture of these aspects by the agents, this paper also sheds light on corporate social responsibility (CSR) in urban freight transport (UFT). A survey was developed with each agent. Factor analysis is used to find the factors for implementation of a UDC and the Structural Equivalence by Orthogonal Procrustes Rotation is used to compare the factors. The findings reveal different points of view for these agents. Residents attached greater importance to factors related to habitability and sustainability in the city. Retailers focused on economic environmental issues, while carriers emphasised operational and environmental aspects.

A hybrid algorithm for low-rank approximation of nonnegative matrix factorization

Nonnegative matrix factorization (NMF) is a recently developed method for data analysis. So far, most of known algorithms for NMF are based on alternating nonnegative least squares (ANLS) minimization of the squared Euclidean distance between the original data matrix and its low-rank approximation. In this paper, we first develop a new NMF algorithm, in which a Procrustes rotation and a nonnegative projection are alternately performed. The new algorithm converges very rapidly. Then, we propose a hybrid NMF (HNMF) algorithm that combines the new algorithm with the low-rank approximation based NMF (lraNMF) algorithm. Furthermore, we extend the HNMF algorithm to nonnegative Tucker decomposition (NTD), which leads to a hybrid NTD (HNTD) algorithm. The simulations verify that the HNMF algorithm performs well under various noise conditions, and HNTD has a comparable performance to the low-rank approximation based sequential NTD (lraSNTD) algorithm for sparse representation of tensor objects.

A Primer to (Cross-Cultural) Multi-Group Invariance Testing Possibilities in R.

Psychology has become less WEIRD in recent years, marking progress toward becoming a truly global psychology. However, this increase in cultural diversity is not matched by greater attention to cultural biases in research. A significant challenge in culture-comparative research in psychology is that any comparisons are open to possible item bias and non-invariance. Unfortunately, many psychologists are not aware of problems and their implications, and do not know how to best test for invariance in their data. We provide a general introduction to invariance testing and a tutorial of three major classes of techniques that can be easily implemented in the free software and statistical language R. Specifically, we describe (1) confirmatory and multi-group confirmatory factor analysis, with extension to exploratory structural equation modeling, and multi-group alignment; (2) iterative hybrid logistic regression as well as (3) exploratory factor analysis and principal component analysis with Procrustes rotation. We pay specific attention to effect size measures of item biases and differential item function. Code in R is provided in the main text and online (see https://osf.io/agr5e/), and more extended code and a general introduction to R are available in the Supplementary Materials.

Permutimin: Factor Rotation to Simple Structure with Permutation of Variables

Factor rotation is usually performed for a p-variables -factors loading matrix so that the resulting rotated matrix has a simple structure. This simple structure was originally defined by Thurstone (1947) by specifying how zero elements are arranged in the loading matrix. In this article, we propose a new rotation technique, which is directly based on Thurstone’s definition. It can give a p-variables -factors target matrix of zero and nonzero elements, which stands for the properties to be possessed by the rotated loading matrix. However, it is unknown how the rows of the target matrix are associated with those of the loading matrix. In the proposed procedure, a loading matrix is rotated simultaneously with a permutation of the rows of the target matrix, so that the rotated loading matrix is optimally matched to the permuted target matrix in a least squares sense. Its novel feature is the use of permutation, thus we call the technique Permutimin. Its algorithm is presented, with Thurstone’s definition of simple structure modified so as to specify the target matrix uniquely. Permutimin is illustrated with real data examples. Finally, we discuss the relationships between Permutimin and Procrustes rotation.