Exploratory factor analysis: Difference between revisions

Revision as of 15:03, 30 April 2021

Revision as of 08:29, 25 May 2021

Line 10:

===Maximum likelihood (ML)===

The maximum likelihood method has many advantages in that it allows researchers to compute of a wide range of indexes of the [[goodness of fit]] of the model, it allows researchers to test the [[statistical significance]] of factor loadings, calculate correlations among factors and compute [[confidence interval]]s for these parameters.<ref>{{cite journal | last1 = Cudeck | first1 = R. | last2 = O'Dell | first2 = L. L. | year = 1994 | title = Applications of standard error estimates in unrestricted factor analysis: Significance tests for factor loadings and correlations | journal = Psychological Bulletin | volume = 115 | issue = 3| pages = 475–487 | doi = 10.1037/0033-2909.115.3.475 | pmid = 8016288 }}</ref> ML is the best choice when data are normally distributed because “it allows for the computation of a wide range of indexes of the goodness of fit of the model [and] permits statistical significance testing of factor loadings and correlations among factors and the computation of confidence intervals”.<ref name =Fabrigar/>

===Principal axis factoring (PAF)===

Line 38:

===Cattell's (1966) scree plot===

{{Main|Scree plot}}

Compute the eigenvalues for the correlation matrix and plot the values from largest to smallest. Examine the graph to determine the last substantial drop in the magnitude of eigenvalues. The number of plotted points before the last drop is the number of factors to include in the model.<ref name="Cattell, R. B. 1966"/> This method has been criticized because of its subjective nature (i.e., there is no clear objective definition of what constitutes a substantial drop).<ref>{{cite journal | last1 = Kaiser | first1 = H. F. | year = 1970 | title = A second generation little jiffy | journal = Psychometrika | volume = 35 | issue = 4 | pages = 401–415 | doi = 10.1007/bf02291817 }}</ref> As this procedure is subjective, Courtney (2013) does not recommend it.<ref name="pareonline.net"/>

===Revelle and Rocklin (1979) very simple structure===

Line 48:

There are different methods that can be used to assess model fit:<ref name =Fabrigar/>

*'''Likelihood ratio statistic:'''<ref>Lawley, D. N. (1940). The estimation of factor loadings by the method of maximumlikelihood. Proceedings of the Royal Society ofedinborough, 60A, 64-82.</ref> Used to test the null hypothesis that a model has perfect model fit. It should be applied to models with an increasing number of factors until the result is nonsignificant, indicating that the model is not rejected as good model fit of the population. This statistic should be used with a large sample size and normally distributed data. There are some drawbacks to the likelihood ratio test. First, when there is a large sample size, even small discrepancies between the model and the data result in model rejection.<ref name =Humphreys/><ref>{{cite journal | last1 = Hakstian | first1 = A. R. | last2 = Rogers | first2 = W. T. | last3 = Cattell | first3 = R. B. | year = 1982 | title = The behavior of number-offactors rules with simulated data | journal = Multivariate Behavioral Research | volume = 17 | issue = 2| pages = 193–219 | doi = 10.1207/s15327906mbr1702_3 | pmid = 26810948 }}</ref><ref>{{cite journal|last=Harris|first=M. L.|author2=Harris, C. W.|title=A Factor Analytic Interpretation Strategy|journal=Educational and Psychological Measurement|date=1 October 1971|volume=31|issue=3|pages=589–606|doi=10.1177/001316447103100301}}</ref> When there is a small sample size, even large discrepancies between the model and data may not be significant, which leads to underfactoring.<ref name =Humphreys/> Another disadvantage of the likelihood ratio test is that the null hypothesis of perfect fit is an unrealistic standard.<ref name=Maccallum>{{cite journal | last1 = Maccallum | first1 = R. C. | year = 1990 | title = The need for alternative measures of fit in covariance structure modeling | journal = Multivariate Behavioral Research | volume = 25 | issue = 2| pages = 157–162 | doi=10.1207/s15327906mbr2502_2| pmid = 26794477 }}</ref><ref name=Browne>{{cite journal | last1 = Browne | first1 = M. W. | last2 = Cudeck | first2 = R. | year = 1992 | title = Alternative ways of assessing model fit | journal = Sociological Methods and Research | volume = 21 | issue = 2 | pages = 230–258 | doi = 10.1177/0049124192021002005 }}</ref>

*'''Root mean square error of approximation (RMSEA) fit index:''' RMSEA is an estimate of the discrepancy between the model and the data per degree of freedom for the model. Values less that .05 constitute good fit, values between 0.05 and 0.08 constitute acceptable fit, a values between 0.08 and 0.10 constitute marginal fit and values greater than 0.10 indicate poor fit .<ref name =Browne/><ref>Steiger, J. H. (1989). EzPATH: A supplementary module for SYSTAT andsygraph. Evanston, IL: SYSTAT</ref> An advantage of the RMSEA fit index is that it provides confidence intervals which allow researchers to compare a series of models with varying numbers of factors.

Line 72:

==Factor rotation==

Factor rotation is a commonly employed step in EFA, used to aide interpretation of factor matrixes.<ref name="Browne2001">{{cite journal |last1=Browne |first1=Michael W. |title=An Overview of Analytic Rotation in Exploratory Factor Analysis |journal=Multivariate Behavioral Research |date=January 2001 |volume=36 |issue=1 |pages=111–150 |doi=10.1207/S15327906MBR3601_05}}</ref><ref name="Sass2010">{{cite journal |last1=Sass |first1=Daniel A. |last2=Schmitt |first2=Thomas A. |title=A Comparative Investigation of Rotation Criteria Within Exploratory Factor Analysis |journal=Multivariate Behavioral Research |date=29 January 2010 |volume=45 |issue=1 |pages=73–103 |doi=10.1080/00273170903504810|pmid=26789085 }}</ref><ref name="Schmitt2011">{{cite journal |last1=Schmitt |first1=Thomas A. |last2=Sass |first2=Daniel A. |title=Rotation Criteria and Hypothesis Testing for Exploratory Factor Analysis: Implications for Factor Pattern Loadings and Interfactor Correlations |journal=Educational and Psychological Measurement |date=February 2011 |volume=71 |issue=1 |pages=95–113 |doi=10.1177/0013164410387348}}</ref> For any solution with two or more factors there are an infinite number of orientations of the factors that will explain the data equally well. Because there is no unique solution, a researcher must select a single solution from the infinite possibilities. The goal of factor rotation is to [[Rotation of axes|rotate]] factors in multidimensional space to arrive at a solution with best simple structure. There are two main types of factor rotation: [[Orthogonality|orthogonal]] and [[Angle#Types of angles|oblique]] rotation.

===Orthogonal rotation===

Line 79:

[[Varimax rotation]] is an orthogonal rotation of the factor axes to maximize the variance of the squared loadings of a factor (column) on all the variables (rows) in a factor matrix, which has the effect of differentiating the original variables by extracted factor. Each factor will tend to have either large or small loadings of any particular variable. A varimax solution yields results which make it as easy as possible to identify each variable with a single factor. This is the most common orthogonal rotation option.<ref name =Fabrigar/>

Quartimax rotation is an orthogonal rotation that maximizes the squared loadings for each variable rather than each factor. This minimizes the number of factors needed to explain each variable. This type of rotation often generates a general factor on which most variables are loaded to a high or medium degree.<ref name=Neuhaus>{{cite journal|last=Neuhaus|first=Jack O|author2=Wrigley, C.|title=The Quartimax Method|journal=British Journal of Statistical Psychology|date=1954|volume=7|issue=2|pages=81-9181–91|doi=10.1111/j.2044-8317.1954.tb00147.x}}</ref>

Equimax rotation is a compromise between varimax and quartimax criteria.

Line 92:

Common factor analysis software is capable of producing an unrotated solution. This refers to the result of a [[#Principal_axis_factoring_(PAF)|principal axis factoring]] with no further rotation. The so-called unrotated solution is in fact an orthogonal rotation that maximizes the variance of the first factors. The unrotated solution tends to give a general factor with loadings for most of the variables. This may be useful if many variables are correlated with each other, as revealed by one or a few dominating [[eigenvalue|eigenvalues]] on a [[scree plot]].

The usefulness of an unrotated solution was emphasized by a [[meta analysis]] of studies of cultural differences. This revealed that many published studies of cultural differences have given similar factor analysis results, but rotated differently. Factor rotation has obscured the similarity between the results of different studies and the existence of a strong general factor, while the unrotated solutions were much more similar.<ref name="Fog2020">{{cite journal|last=Fog|first=A. |title=A Test of the Reproducibility of the Clustering of Cultural Variables |journal=Cross-Cultural Research |year=2020 |volume=55 |pages=29–57 |doi=10.1177/1069397120956948}}</ref><ref>{{Cite journal|title=Examining Factors in 2015 TIMSS Australian Grade 4 Student Questionnaire Regarding Attitudes Towards Science Using Exploratory Factor Analysis (EFA)|url=https://twasp.info/journal/gi93583P/examining-factors-in-2015-timss-australian-grade-4-student-questionnaire-regarding-attitudes-towards-science-using-exploratory-factor-analysis-efa|journal=North American Academic Research|volume=3}}</ref>

==Factor interpretation==