Data analysis methods in optometry Part 8: Principal components analysis (PCA) and factor analysis (FA)

PCA/FA is a method of analyzing complex data sets in which there are no clearly defined X or Y variables. It has multiple uses including the study of the pattern of variation between individual entities such as patients with particular disorders and the detailed study of descriptive variables. In most applications, variables are related to a smaller number of ‘factors’ or PCs that account for the maximum variance in the data and hence, may explain important trends among the variables. An increasingly important application of the method is in the ‘validation’ of questionnaires that attempt to relate subjective aspects of a patients experience with more objective measures of vision.

Factor analysis and discriminant validity:a brief review of some practical issues

Growth in availability and ability of modern statistical software has resulted in greater numbers of research techniques being applied across the marketing discipline. However, with such advances come concerns that techniques may be misinterpreted by researchers. This issue is critical since misinterpretation could cause erroneous findings. This paper investigates some assumptions regarding: 1) the assessment of discriminant validity; and 2) what confirmatory factor analysis accomplishes. Examples that address these points are presented, and some procedural remedies are suggested based upon the literature. This paper is, therefore, primarily concerned with the development of measurement theory and practice. If advances in theory development are not based upon sound methodological practice, we as researchers could be basing our work upon shaky foundations.

Statnote 11: the two-factor analysis of variance

Experiments combining different groups or factors are a powerful method of investigation in applied microbiology. ANOVA enables not only the effect of individual factors to be estimated but also their interactions; information which cannot be obtained readily when factors are investigated separately. In addition, combining different treatments or factors in a single experiment is more efficient and often reduces the number of replications required to estimate treatment effects adequately. Because of the treatment combinations used in a factorial experiment, the degrees of freedom (DF) of the error term in the ANOVA is a more important indicator of the ‘power’ of the experiment than simply the number of replicates. A good method is to ensure, where possible, that sufficient replication is present to achieve 15 DF for each error term of the ANOVA. Finally, in a factorial experiment, it is important to define the design of the experiment in detail because this determines the appropriate type of ANOVA. We will discuss some of the common variations of factorial ANOVA in future statnotes. If there is doubt about which ANOVA to use, the researcher should seek advice from a statistician with experience of research in applied microbiology.

Cross-cultural confirmatory factor analysis of the CES-D in Spanish and Mexican dementia caregivers

The Center for Epidemiologic Studies-Depression Scale (CES-D) is the most frequently used scale for measuring depressive symptomatology in caregiving research. The aim of this study is to test its construct structure and measurement equivalence between caregivers from two Spanish-speaking countries. Face-to-face interviews were carried out with 595 female dementia caregivers from Madrid, Spain, and from Coahuila, Mexico. The structure of the CES-D was analyzed using exploratory and confirmatory factor analysis (EFA and CFA, respectively). Measurement invariance across samples was analyzed comparing a baseline model with a more restrictive model. Significant differences between means were found for 7 items. The results of the EFA clearly supported a four-factor solution. The CFA for the whole sample with the four factors revealed high and statistically significant loading coefficients for all items (except item number 4). When equality constraints were imposed to test for the invariance between countries, the change in chi-square was significant, indicating that complete invariance could not be assumed. Significant between-countries differences were found for three of the four latent factor mean scores. Although the results provide general support for the original four-factor structure, caution should be exercised on reporting comparisons of depression scores between Spanish-speaking countries.

Conclusion : on multiple levels of analysis, context, contingency and capital

In this concluding chapter, we bring together the threads and reflections on the chapters contained in this text and show how they relate to multi-level issues. The book has focused on the world of Human Resource Management (HRM) and the systems and practices it must put in place to foster innovation. Many of the contributions argue that in order to bring innovation about, organisations have to think carefully about the way in which they will integrate what is, in practice, organisationally relevant — but socially distributed — knowledge. They need to build a series of knowledge-intensive activities and networks, both within their own boundaries and across other important external inter-relationships. In so doing, they help to co-ordinate important information structures. They have, in effect, to find ways of enabling people to collaborate with each other at lower cost, by reducing both the costs of their co-ordination and the levels of unproductive search activity. They have to engineer these behaviours by reducing the risks for people that might be associated with incorrect ideas and help individuals, teams and business units to advance incomplete ideas that are so often difficult to codify. In short, a range of intangible assets must flow more rapidly throughout the organisation and an appropriate balance must be found between the rewards and incentives associated with creativity, novelty and innovation, versus the risks that innovation may also bring.

Statnote 27: Principal components analysis (PCA)

In Statnotes 24 and 25, multiple linear regression, a statistical method that examines the relationship between a single dependent variable (Y) and two or more independent variables (X), was described. The principle objective of such an analysis was to determine which of the X variables had a significant influence on Y and to construct an equation that predicts Y from the X variables. ‘Principal components analysis’ (PCA) and ‘factor analysis’ (FA) are also methods of examining the relationships between different variables but they differ from multiple regression in that no distinction is made between the dependent and independent variables, all variables being essentially treated the same. Originally, PCA and FA were regarded as distinct methods but in recent times they have been combined into a single analysis, PCA often being the first stage of a FA. The basic objective of a PCA/FA is to examine the relationships between the variables or the ‘structure’ of the variables and to determine whether these relationships can be explained by a smaller number of ‘factors’. This statnote describes the use of PCA/FA in the analysis of the differences between the DNA profiles of different MRSA strains introduced in Statnote 26.

What's in a broken promissory obligation? Developing and testing a multiple component form measure of psychological contract breach

While the literature has suggested the possibility of breach being composed of multiple facets, no previous study has investigated this possibility empirically. This study examined the factor structure of typical component forms in order to develop a multiple component form measure of breach. Two studies were conducted. In study 1 (N = 420) multi-item measures based on causal indicators representing promissory obligations were developed for the five potential component forms (delay, magnitude, type/form, inequity and reciprocal imbalance). Exploratory factor analysis showed that the five components loaded onto one higher order factor, namely psychological contract breach suggesting that breach is composed of different aspects rather than types of breach. Confirmatory factor analysis provided further evidence for the proposed model. In addition, the model achieved high construct reliability and showed good construct, convergent, discriminant and predictive validity. Study 2 data (N = 189), used to validate study 1 results, compared the multiple-component measure with an established multiple item measure of breach (rather than a single item as in study 1) and also tested for discriminant validity with an established multiple item measure of violation. Findings replicated those in study 1. The findings have important implications for considering alternative, more comprehensive and elaborate ways of assessing breach.

Probabilistic principal component analysis

Principal component analysis (PCA) is a ubiquitous technique for data analysis and processing, but one which is not based upon a probability model. In this paper we demonstrate how the principal axes of a set of observed data vectors may be determined through maximum-likelihood estimation of parameters in a latent variable model closely related to factor analysis. We consider the properties of the associated likelihood function, giving an EM algorithm for estimating the principal subspace iteratively, and discuss the advantages conveyed by the definition of a probability density function for PCA.

Probabilistic principal component analysis

Principal component analysis (PCA) is a ubiquitous technique for data analysis and processing, but one which is not based upon a probability model. In this paper we demonstrate how the principal axes of a set of observed data vectors may be determined through maximum-likelihood estimation of parameters in a latent variable model closely related to factor analysis. We consider the properties of the associated likelihood function, giving an EM algorithm for estimating the principal subspace iteratively, and discuss the advantages conveyed by the definition of a probability density function for PCA.

Development of a near activity visual questionnaire to assess accommodating intraocular lenses

Purpose: To develop a questionnaire that subjectively assesses near visual function in patients with 'accommodating' intraocular lenses (IOLs). Methods: A literature search of existing vision-related quality-of-life instruments identified all questions relating to near visual tasks. Questions were combined if repeated in multiple instruments. Further relevant questions were added and item interpretation confirmed through multidisciplinary consultation and focus groups. A preliminary 19-item questionnaire was presented to 22 subjects at their 4-week visit post first eye phacoemulsification with 'accommodative' IOL implantation, and again 6 and 12 weeks post-operatively. Rasch Analysis, Frequency of Endorsement, and tests of normality (skew and kurtosis) were used to reduce the instrument. Cronbach's alpha and test-retest reliability (intraclass correlation coefficient, ICC) were determined for the final questionnaire. Construct validity was obtained by Pearson's product moment correlation (PPMC) of questionnaire scores to reading acuity (RA) and to Critical Print Size (CPS) reading speed. Criterion validity was obtained by receiver operating characteristic (ROC) curve analysis and dimensionality of the questionnaire was assessed by factor analysis. Results: Rasch Analysis eliminated nine items due to poor fit statistics. The final items have good separation (2.55), internal consistency (Cronbach's α = 0.97) and test-retest reliability (ICC = 0.66). PPMC of questionnaire scores with RA was 0.33, and with CPS reading speed was 0.08. Area under the ROC curve was 0.88 and Factor Analysis revealed one principal factor. Conclusion: The pilot data indicates the questionnaire to be internally consistent, reliable and a valid instrument that could be useful for assessing near visual function in patients with 'accommodating' IOLS. The questionnaire will now be expanded to include other types of presbyopic correction. © 2007 British Contact Lens Association.

Antecedents and consequences of new product development and international new product rollout timeliness:an examination of mediating and moderating influences

The study examined the relationships between antecedents, timeliness in NPD and INPR, and consequences. A conceptual framework was tested using 232 new products from South Korean firms. The hypothesized relationships among the constructs in the model were evaluated by multiple regression and hierarchal regression analyses using SPSS 12 as well as by structural equation modelling (SEM) using SIMPLIS LISREL. In addition, confirmatory factor analysis (CFA) was carried out using SIMPLIS LISREL. In the direct relationships, cross-functional linkages and marketing synergy exhibited a statistically significant effect on NPD timeliness. The results also supported the influences of the HQ-subsidiary/agent relationship and NPD timeliness on INPR timeliness as well as INPR timeliness on performance. In the mediating effect tests, marketing proficiency significantly accounts for the relationships between cross-functional linkages and NPD timeliness, between marketing synergy and NPD timeliness, and between the HQ-subsidiary/agent relationship and INPR timeliness. Technical proficiency also mediates the effect of the HQ-subsidiary/agent relationship on INPR timeliness. The influence of NPD timeliness on new product performance in target markets is attributed to INPR timeliness. As for the results of the external environmentals and standardization influences, competitive intensity moderates the relationship between NPD timeliness and new product performance. Technology change also moderates the relationship between cross-functional linkages and NPD timeliness and between timeliness in NPD and INPR and performance. Standardization has a moderating role on the relationship between NPD timeliness and INPR timeliness. This study presents the answers to research questions which concern what factors are predictors of criterion variables, how antecedents influence timeliness in NPD and INPR and when the direct relationships in the INPR process are strengthened.

Statnote 28:Canonical Variate Analysis (CVA)

The pattern of correlation between two sets of variables can be tested using canonical variate analysis (CVA). CVA, like principal components analysis (PCA) and factor analysis (FA) (Statnote 27, Hilton & Armstrong, 2011b), is a multivariate analysis Essentially, as in PCA/FA, the objective is to determine whether the correlations between two sets of variables can be explained by a smaller number of ‘axes of correlation’ or ‘canonical roots’.

The influence of the socio-educational reading environment in an Arab University upon English reading performance

Research into FL/EFL macro-reading (the effect of the broader context of reading) has been little explored in spite of its importance in the FL/EFL reading programmes. This study was designed to build on previous work by explaining in more depth the influence of the socio-educational reading environment in an Arab university (Al-Fateh University in Tripoli, Libya) - as reported by students, upon these students' reading ability in English and Arabic (particularly the former). Certain aspects of the lecturers' reading habits and attitudes and classroom operation were also investigated. Written cloze tests in English and Arabic and self-administered questionnaires were given to 125 preliminary-year undergraduates in three faculties of Al-Fateh University on the basis of their use of English as a medium of instruction (one representing the Arts' stream and two representing the Science stream). Twenty two lecturers were interviewed and observed by an inventory technique along with twenty other preliminary-year students. Factor analysis and standard multiple regression technique were among the statistical methods used to analyse the main data. The findings demonstrate a significant relationship between reading ability in English and the reading individual and environmental variables - as defined in the study. A combination of common and different series of such predictors were found accountable for the variation (43% for the first year English specialist; 48% for the combined Medicine student sample) in the English reading tests. Also found was a significant, though not very large, relationship between reading ability in Arabic and the reading environment. Non-statistical but objective analyses, based on the present data, also revealed an overall association between English reading performance and an important number of reading environmental variables - where many `poor' users of the reading environment (particularly the academic one) obtained low scores in the English cloze tests. Accepting the limitations of a single study, it is nevertheless clear that the reading environment at the University is in need of improvement and that students' use of it also requires better guidance and training in how to use it effectively. Suggestions are made for appropriate educational changes.

A survey based analysis of IT adoption and 3PLs' performance

Purpose: In today's competitive scenario, effective supply chain management is increasingly dependent on third-party logistics (3PL) companies' capabilities and performance. The dissemination of information technology (IT) has contributed to change the supply chain role of 3PL companies and IT is considered an important element influencing the performance of modern logistics companies. Therefore, the purpose of this paper is to explore the relationship between IT and 3PLs' performance, assuming that logistics capabilities play a mediating role in this relationship. Design/methodology/approach: Empirical evidence based on a questionnaire survey conducted on a sample of logistics service companies operating in the Italian market was used to test a conceptual resource-based view (RBV) framework linking IT adoption, logistics capabilities and firm performance. Factor analysis and ordinary least square (OLS) regression analysis have been used to test hypotheses. The focus of the paper is multidisciplinary in nature; management of information systems, strategy, logistics and supply chain management approaches have been combined in the analysis. Findings: The results indicate strong relationships among data gathering technologies, transactional capabilities and firm performance, in terms of both efficiency and effectiveness. Moreover, a positive correlation between enterprise information technologies and 3PL financial performance has been found. Originality/value: The paper successfully uses the concept of logistics capabilities as mediating factor between IT adoption and firm performance. Objective measures have been proposed for IT adoption and logistics capabilities. Direct and indirect relationships among variables have been successfully tested. © Emerald Group Publishing Limited.

Differences between younger and older adults in the structure of suicidal intent and its correlates

OBJECTIVE: To analyze differences in the variables associated with severity of suicidal intent and in the main factors associated with intent when comparing younger and older adults. DESIGN: Observational, descriptive cross-sectional study. SETTING: Four general hospitals in Madrid, Spain. PARTICIPANTS: Eight hundred seventy suicide attempts by 793 subjects split into two groups: 18-54 year olds and subjects older than 55 years. MEASUREMENTS: The authors tested the factorial latent structure of suicidal intent through multigroup confirmatory factor analysis for categorical outcomes and performed statistical tests of invariance across age groups using the DIFFTEST procedure. Then, they tested a multiple indicators-multiple causes (MIMIC) model including different covariates regressed on the latent factor "intent" and performed two separate MIMIC models for younger and older adults to test for differential patterns. RESULTS: Older adults had higher suicidal intent than younger adults (z = 2.63, p = 0.009). The final model for the whole sample showed a relationship of intent with previous attempts, support, mood disorder, personality disorder, substance-related disorder, and schizophrenia and other psychotic disorders. The model showed an adequate fit (chi²[12] = 22.23, p = 0.035; comparative fit index = 0.986; Tucker-Lewis index = 0.980; root mean square error of approximation = 0.031; weighted root mean square residual = 0.727). All covariates had significant weights in the younger group, but in the older group, only previous attempts and mood disorders were significantly related to intent severity. CONCLUSIONS: The pattern of variables associated with suicidal intent varies with age. Recognition, and treatment of geriatric depression may be the most effective measure to prevent suicidal behavior in older adults.