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Factors Influencing Four Rules For Determining The Number Of Components To Retain
Abstract
The performance of four rules for determining the number of components to retain (Kaiser's eigenvalue greater than unity, Cattell's SCREE, Bartlett's test, and Velicer's MAP) was investigated across four systematically varied factors (sample size, number of variables, number of components, and component saturation). Ten sample correlation matrices were generated from each of 48 known population correlation matrices representing the combinations of conditions. The performance of the SCREE and MAP rules was generally the best across all situations. Bartlett's test was generally adequate except when the number of variables was close to the sample size. Kaiser's rule tended to severely overestimate the number of components.
... In reality, a finite sample size tends to cause the first eigenvalue to be computed as greater than 1 and subsequent eigenvalues to be computed as less than 1 due to sampling error and least squares bias (Horn, 1965). PA overcomes the effect of sampling error and is thus a sample-based alternative to the population-based K1 criterion (Carraher & Buckley, 1991;Zwick & Velicer, 1986).PA is based on the premise that real data with a valid underlying factor structure should have larger eigenvalues than parallel components derived from random data with the same sample size and number of variables (Green et al., 2018). By generating random data with the same number of observations (n) and variables (v), the correlation matrix for the data sets is used to extract the eigenvalues. ...
... By comparing the eigenvalues of the real and generated data, the number of factors is determined by the point where the last real eigenvalue is higher than the generated eigenvalue. Zwick and Velicer (1986) examined five commonly used methods for determining dimensionality across conditions as defined by the number of uncorrelated factors, sample size, number of variables per factor, loading size, and factorial complexity. They found that PA was the most accurate approach, but its performance was affected by sample size, factor loading, and number of variables. ...
The primary goal of exploratory factor analysis (EFA) is to determine the number of factors and their structure. Thus, the decision on the number of factors to retain is crucial. Nevertheless, researchers frequently overlook the precision of factor retention techniques and opt for unreliable methodologies instead. The objective of this study is to compare the efficiency of utilizing root mean square error of approximation (RMSEA) and parallel analysis (PA) methods for retaining factors in exploratory factor analysis (EFA). Two methods for comparing RMSEA, namely root deterioration per restriction (RDR) and RMSEA difference test, are employed for nested models. Although researchers use RMSEA to compare two different models, no studies have compared RMSEA and RDR methods. Thus, this study examined three different methods for factor retention. Monte-Carlo simulations were utilized to evaluate the accuracy of RDR compared to RMSEA difference testing and PA. The simulations show that RDR performs better than RMSEA difference testing and PA when the number of variables per factor is low. However, as the number of variables per factor increases, PA becomes more effective. This study provides guidance to researchers using EFA to select factor retention methods that suit different conditions.
... Table 3 reports the results of the factor analysis, displaying the correlation between the four measures of earnings myopia with one underlying factor correlated with all four measures. Following Horn (1965) and Zwick and Velicer (1986), we perform a parallel analysis to confirm whether this factor should be retained, presented in Fig. 2. This analysis compares the eigenvalues obtained using our dataset to what would be expected from a corresponding set of random data. ...
... However, eigenvalues from factor analysis represent the variation explained by the underlying factor and do not necessarily average 1.0. Additionally, researchers have increasingly criticized the Kaiser criterion as a method for factor extraction (Bandalos and Boehm-Kaufman 2009), with Zwick and Velicer (1986) finding that parallel analysis is the most accurate method for factor retention. As such, we use the parallel analysis approach for factor retention. ...
We examine the role of private equity in alleviating earnings myopia induced by public markets. We first construct a measure of earnings myopia and show that this measure varies as predicted with determinants and effects of myopia. Then we show that public firms exhibiting earnings myopia realize an increased likelihood of takeover by private equity buyers. Cross-sectional analyses indicate that this relation is strongest when costs of earnings myopia are likely higher. Following private equity takeovers, firms exhibiting greater measures of earnings myopia realize improvements to R&D investment and productivity. The results add to the understanding of the role of private equity in identifying and alleviating earnings myopia within U.S. capital markets. This is important given the increasing size of private equity assets under management. Takeover premiums paid for myopic firms suggest a cost of earnings myopia at approximately 6.9% of firm value.
... In order to avoid subjective criteria when deciding the number of factors, several methods have been developed: Horn's parallel analysis, Velicer's map test, Kaiser criteria, Cattell scree plot or variance explained criteria [61][62][63][64]. We proceed to compute the scree plot so as to obtain the number of main principal components and hence potential latent factors. ...
Several approaches to cognition and intelligence research rely on statistics-based models testing, namely factor analysis. In the present work we exploit the emerging dynamical systems perspective putting the focus on the role of the network topology underlying the relationships between cognitive processes. We go through a couple of models of distinct cognitive phenomena and yet find the conditions for them to be mathematically equivalent. We find a non-trivial attractor of the system that corresponds to the exact definition of a well-known network centrality and hence stress the interplay between the dynamics and the underlying network connectivity, showing that both of the two are relevant. The connectivity structure between cognitive processes is not known but yet it is not any. Regardless of the network considered, it is always possible to recover a positive manifold of correlations. However, we show that different network topologies lead to different plausible statistical models concerning correlations structure, ranging from one to multiple factors models and richer correlation structures.
... The Barlett's Test of Sphericity compares the population correlation matrix to an identity matrix (Wilson & Martin, 1983). When the Bartlett's test p value is <.05, clusters of linked variables are significant (Zwick & Velicer, 1982;Watson, 2017). Our dataset confirms strong relationships between items (χ1(153)=1735.749, ...
Efficient and reliable bus services are essential components of a
sustainable urban transportation system. In densely populated countries like Bangladesh, where mobility challenges are exacerbated by rapid urbanization, understanding the existing bus service conditions and customer satisfaction levels becomes crucial. This study employs a comprehensive approach to assess
the current state of bus services and analyze customer perceptions in Chandra-Hemayetpur Route in Savar, Dhaka. Through a combination of quantitative surveys and qualitative interviews, 18 factors were examined including bus availability, comfort in bus, bus fare rate, delay time and accessibility.
Moreover, customer satisfaction was also measured trough principal component analysis with the regression model and identify areas for improvement in the bus service. The findings of this research highlight the critical issues within the four component system, such as bus condition, driver’s skills, delay time; comfort in bus; uncomfortable space in bus seat; contractor behavior and crime in the bus. The research found that a significant proportion of bus passengers, above 50%, expressed dissatisfaction or high levels of dissatisfaction with the 18 factors
assessed in the Chandra - Hemayetpur route segment of Savar. The total satisfaction level is 3.30, suggesting a proximity to dissatisfaction on the Likert
scale (1 to 5; where 4 = dissatisfied). The insights from this study provide valuable guidance for policymakers and transport authorities to enhance the quality of bus services, thereby fostering a more efficient and customer-oriented public transportation network. This research contributes to the broader discourse on satellite town to city connecting road in developing nations and underscores the significance of addressing public transportation challenges for sustainable urban development.
... There were six random eigenvalues from the parallel analysis [41][42][43] that were greater than 1.0. For the NAQ-IV, an initial four-factor solution was chosen for better grouping of items along content dimensions, as parallel analysis tends to overestimate the number of extracted factors [44]. ...
Introduction: The image of a nurse is a source of concern due to its impact on recruitment into the profession, political decisions about the profession, and how the image affects nursing practice. For these reasons, one of the long-term challenges is to assess and maintain a favorable public image that respects the utility and value of the nursing profession. Aim: This study aims to validate an instrument for assessing the image of the nurse as perceived by Italian citizens. Methods: A non�probabilistic sample of 564 people participated in the study between 2022 and 2023. Sociodemographic information of the Italian citizens was collected, and the instrument used to evaluate the perceived public image was the Nursing Attitudes Questionnaire (NAQ). The psychometric properties of the Italian version of the NAQ (NAQ-IV) were calculated using Cronbach’s alpha, item–total correlations,
skewness, and kurtosis. Factor analysis was performed using principal axis factoring and the varimax rotation method. Results: Factor analysis revealed a four-factor model explaining more than 60.52% of the variance, with the largest variance explained by the “Role and Professionalism” factor (34.08%).
The internal consistency calculation showed a Cronbach’s alpha of 0.89 for the scale and between 0.88 and 0.89 among the factors; all the items verified the item–total correlation and response variability criteria. Conclusions: The NAQ-IV could be a valid tool for assessing the perception of Italian citizens.
However, further studies are recommended to evaluate the reliability of the instrument, especially in the evolving professional profile and social health welfar
Introduction
Parental Reflective Functioning describes the parents’ ability to view their child as motivated by mental states. The Parental Reflective Functioning Questionnaire (PRFQ) represents an 18-item and three-factor self-report measure. Our goal was to conduct the first German validation study.
Method
In a community sample of 378 mothers of children aged 10.2–78.6 months, we used Confirmatory Factor Analysis with a cross-validation approach to assess model fit. Reliability was measured using Cronbach’s α and McDonald’s ω. Concurrent validity was assessed using correlations with relevant constructs.
Results
The three-factor structure of the original validation could be confirmed. The German model only needed minor modifications: two items had to be removed, and one error covariance was added. The resulting 16-item questionnaire with the three subscales “Pre-mentalizing”, “Interest and Curiosity about Mental States”, and “Certainty about Mental States” was successfully cross-validated (CFI = .94, TLI = .93, SRMR = .07, RMSEA = .04 (CI [.01, .06])). These factors were related in theoretically expected ways to parental attachment dimensions, emotional availability, parenting stress, and infant attachment status.
Conclusion
While reliability could still be improved, the German 16-item version of the PRFQ represents a valid measure of parental reflective functioning.
Background and Objective
Evidence of short‐term exposure to particulate matter with an aerodynamic diameter ≤1 μm (PM 1 ) on hospital admission for respiratory diseases (RDs) is limited. We aimed to estimate the associated risk of PM 1 on hospital admissions for RDs.
Methods
In this time‐stratified case‐crossover study, we assigned cases who had been admitted to hospital for RDs in Guangdong, China between 2016 and 2019. Exposure to PM 1 was assigned on the basis of the patient's residence for each case day and its control days. Conditional logistic regression models and distributed lag nonlinear models were used to quantify the association of PM 1 exposure with hospital admission for RDs at lag 0–1 days.
Results
A total of 408, 658 hospital admissions for total RDs were recorded in the study period. Each 10 μg/m ³ increase in PM 1 was significantly associated with a 1.39% (95% confidence interval [CI]: 0.87%–1.91%), 1.97% (95% CI: 1.06%–2.87%) and 1.69% (95% CI: 0.67%–2.71%) increase in odds of hospital admissions for total RDs, chronic obstructive pulmonary disease (COPD) and pneumonia. The excess fraction of hospital admission for total RDs attributable to PM 1 exposure was 6.03%, while 6.59% for COPD and 7.48% for pneumonia. Besides, higher excess fractions were more pronounced for hospital admission of total RDs in older patients (>75 years).
Conclusion
Our results support that PM 1 is associated with increased risks of hospital admissions for RDs. It emphasizes the needs to pay attention to the effects of PM 1 on respiratory health, especially among elderly patients.
Theoretical problems with the factor analysis model have resulted in increased interest in component analysis as an alternative. It is therefore of interest to assess empirically some of the asserted differences between the methods. One of the positive features sometimes attributed to factor analysis, as opposed to component analysis, is superior stability of results under sampling from a population of variables. The comparative stability of three methods, Maximum Liklihood Factor Analysis, principal component analysis, and rescaled image analysis, is investigated. Random samples are drawn from a "population" of variables. The sample pattern is then compared with the corresponding population pattern. The results suggest that none of the three methods can be considered superior with respect to stability.
Theoretical problems with the factor analysis model, such as the factor indeterminacy issue, have resulted in increased interest in component analysis as an alternative. Indeed, component analysis alternatives are already in common use. It is, therefore, of interest to assess empirically some of the assumed differences between the methods. Of particular interest is the relation between component scores and "factor score estimates." The correlations between principal component scores, rescaled image scores, and three factor score estimates are investigated. Three pattern positions are considered: (1) the "canonical" position, (2) the varimax position, and (3) a position of "maximal" similarity which was achieved by procrustes rotation. The correlations between corresponding factors were near unity for both the canonical and "maximal" position. The correlations for the varimax position are slightly lower, generally in the .80 to .90 range. These results suggest there is little practical difference between principal component scores, rescaled image scores, and factor score estimates.
The inter-rater reliability of Cattell's scree and Linn's mean square ratio test of the number of factors was studied. Sample correlation matrices were generated from a population correlation matrix by means of standard Monte Carlo procedures such that there were 100 samples based on each of 3 sample sizes. Each matrix was factored and the scree test and the mean square ratio test were each applied by five raters. For both tests, the inter-rater reliabilities were very low. These results suggest that inexperienced factor analysts should be wary of these tests of the number of factors.
Given a population-covariance matrix, this program can generate any number of sample-covariance (or correlation) matrices, based on any sample sizes. If no population matrix is input, the program generates random correlation matrices.
In most well designed researches larger substantive factors show a break in size from those of a trivial lower degree of variance, which are due to error and other sources. The only short test for the number of such factors that has been repeatedly supported by plasmode results is the scree. The present experiment extends this empirical testing of the scree by taking 15 plasmodes, which vary in (1) number of variables, (2) of factors, (3) degree of obliquity, from orthogonal cases on, (4) presence of simple structure, (5) presence of error, extending to factoring random numbers, and (6) size of communality. The results support the scree and show it superior to the Kaiser-Guttman unity root criterion. Apart from the scree itself, a brief examination is made of the reliability of its use in the hands of trained and untrained judges. A second evaluation of the intrinsic validity of the scree is made by examining internal consistency of item and parcel factoring, of different sampling of variables and of change of population. The result again supports its capacity to locate the correct number of factors within narrow limits.
Then correlation matrices based upon real and random data with squared multiple correlations in the diagonals are factored, the hypothesis is that the point at which the curves of the latent roots cross indicates the number of common factors. Sampling studies confirm the hypothesis when the common factor model provides a good fit to the data. When small overlapping, nonrandom factors are introduced, the expected value of the number of common factors can still be the number of major factor in the population when the nonrandom "noise" is small compared to sampling error "noise." This criterion for the number of common factors, furthermore, is more accurate than the method of maximum likelihood.
It is demonstrated that Cattell's scree test and Bartlett's chi-square test for the number of factors are both based on the same rationale, so the former reflects statistical (subject sampling) variability and the latter usually involves psychometric (variable sampling) influences. If the alpha-level (implicit in the scree test) is set the same, the two tests should lead to the same conclusions. Analyses with some examples suggest that if the alpha-level for the Bartlett test is set (explicitly) in the neighborhood of .0003 for sample Ns of 100 to 150, the results from applications of this test will indicate approximately the same number of factors as estimated on the basis of a scree test determined on a much larger (N 600) sample. Used in this way, the Bartlett test may yield fairly good "population" estimates of the number of factors. Relationships between the Bartlett test, hence the scree test, and tests for a common factor model and for the significance of a correlation matrix are explicated.