confirmatory factor analysis

In confirmatory factor analysis, researchers are typically interested in studying the degree to which responses on a p x 1 vector of observable random variables can be used to assign a value to one or more unobserved variable(s) $\backslash xi$. The investigation is largely accomplished by estimating and evaluating the loading of each item used to tap aspects of the unobserved latent variable. That is, y[i] is the vector of observed responses predicted by the unobserved latent variable $\backslash xi$, which is defined as:

$Y=\backslash Lambda\backslash xi+\backslash epsilon$,

where $Y$ is the p x 1 vector of observed random variables, $\backslash xi$ are the unobserved latent variables and $\backslash Lambda$ is a p x k matrix with k equal to the number of latent variables.{{Cite journal|title = Confirmatory Factor Analysis of Ordinal Variables With Misspecified Models|journal = Structural Equation Modeling|date = 2010-07-13|issn = 1070-5511|pages = 392–423|volume = 17|issue = 3|doi = 10.1080/10705511.2010.489003|first1 = Fan|last1 = Yang-Wallentin|first2 = Karl G.|last2 = Jöreskog|first3 = Hao|last3 = Luo|s2cid = 122941470}} Since, $Y$ are imperfect measures of $\backslash xi$, the model also consists of error, $\backslash epsilon$. Estimates in the maximum likelihood (ML) case generated by iteratively minimizing the fit function,

$F\_\{\backslash mathrm\{ML\}\}=\backslash ln|\backslash Lambda\backslash Omega\backslash Lambda\{\text{'}\}+I-\backslash operatorname\{diag\}(\backslash Lambda\backslash Omega\backslash Lambda\{\text{'}\})|+\backslash operatorname\{tr\}(R(\backslash Lambda\backslash Omega\backslash Lambda\{\text{'}\}+I-\backslash operatorname\{diag\}(\backslash Lambda\backslash Omega\backslash Lambda\{\text{'}\}))^\{-1\})-\backslash ln(R)-p$

where $\backslash Lambda\backslash Omega\backslash Lambda\{\text{'}\}+I-\backslash operatorname\{diag\}(\backslash Lambda\backslash Omega\backslash Lambda\{\text{'}\})$ is the variance-covariance matrix implied by the proposed factor analysis model and $R$ is the observed variance-covariance matrix. That is, values are found for free model parameters that minimize the difference between the model-implied variance-covariance matrix and observed variance-covariance matrix.

Although numerous algorithms have been used to estimate CFA models, maximum likelihood (ML) remains the primary estimation procedure.{{Cite journal|last1=Flora|first1=David B.|last2=Curran|first2=Patrick J.|title=An Empirical Evaluation of Alternative Methods of Estimation for Confirmatory Factor Analysis With Ordinal Data.|journal=Psychological Methods|volume=9|issue=4|pages=466–491|doi=10.1037/1082-989x.9.4.466|pmid=15598100|pmc=3153362|year=2004}} That being said, CFA models are often applied to data conditions that deviate from the normal theory requirements for valid ML estimation. For example, social scientists often estimate CFA models with non-normal data and indicators scaled using discrete ordered categories.{{Cite journal|last1=Millsap|first1=Roger E.|last2=Yun-Tein|first2=Jenn|date=2004-07-01|title=Assessing Factorial Invariance in Ordered-Categorical Measures|journal=Multivariate Behavioral Research|volume=39|issue=3|pages=479–515|doi=10.1207/S15327906MBR3903_4|issn=0027-3171|doi-access=free}} Accordingly, alternative algorithms have been developed that attend to the diverse data conditions applied researchers encounter. The alternative estimators have been characterized into two general type: (1) robust and (2) limited information estimator.{{Cite journal|last=Bandalos|first=Deborah L.|date=2014-01-02|title=Relative Performance of Categorical Diagonally Weighted Least Squares and Robust Maximum Likelihood Estimation|journal=Structural Equation Modeling|volume=21|issue=1|pages=102–116|doi=10.1080/10705511.2014.859510|s2cid=123259681|issn=1070-5511}}

When ML is implemented with data that deviates away from the assumptions of normal theory, CFA models may produce biased parameter estimates and misleading conclusions.{{Cite journal|last=Li|first=Cheng-Hsien|date=2015-07-15|title=Confirmatory factor analysis with ordinal data: Comparing robust maximum likelihood and diagonally weighted least squares|journal=Behavior Research Methods|language=en|volume=48|issue=3|pages=936–949|doi=10.3758/s13428-015-0619-7|pmid=26174714|issn=1554-3528|doi-access=free}} Robust estimation typically attempts to correct the problem by adjusting the normal theory model χ² and standard errors. For example, Satorra and Bentler (1994) recommended using ML estimation in the usual way and subsequently dividing the model χ² by a measure of the degree of multivariate kurtosis.{{Cite journal|last1=Bryant|first1=Fred B.|last2=Satorra|first2=Albert|date=2012-07-20|title=Principles and Practice of Scaled Difference Chi-Square Testing|journal=Structural Equation Modeling|volume=19|issue=3|pages=372–398|doi=10.1080/10705511.2012.687671|issn=1070-5511|hdl=10230/46110|s2cid=53601390|hdl-access=free}} An added advantage of robust ML estimators is their availability in common SEM software (e.g., LAVAAN).{{Cite journal|title=lavaan: An R Package for Structural Equation Modeling {{!}} Rosseel {{!}} Journal of Statistical Software|journal=Journal of Statistical Software|volume=48|issue=2|doi=10.18637/jss.v048.i02|year=2012|last1=Rosseel|first1=Yves|doi-access=free}}

Unfortunately, robust ML estimators can become untenable under common data conditions. In particular, when indicators are scaled using few response categories (e.g., disagree, neutral, agree) robust ML estimators tend to perform poorly. Limited information estimators, such as weighted least squares (WLS), are likely a better choice when manifest indicators take on an ordinal form.{{Cite journal|last1=Rhemtulla|first1=Mijke|last2=Brosseau-Liard|first2=Patricia É.|last3=Savalei|first3=Victoria|title=When can categorical variables be treated as continuous? A comparison of robust continuous and categorical SEM estimation methods under suboptimal conditions.|journal=Psychological Methods|volume=17|issue=3|pages=354–373|doi=10.1037/a0029315|pmid=22799625|year=2012}} Broadly, limited information estimators attend to the ordinal indicators by using polychoric correlations to fit CFA models.{{Cite journal|last1=Yang-Wallentin|first1=Fan|last2=Jöreskog|first2=Karl G.|last3=Luo|first3=Hao|date=2010-07-13|title=Confirmatory Factor Analysis of Ordinal Variables With Misspecified Models|journal=Structural Equation Modeling|volume=17|issue=3|pages=392–423|doi=10.1080/10705511.2010.489003|s2cid=122941470|issn=1070-5511}} Polychoric correlations capture the covariance between two latent variables when only their categorized form is observed, which is achieved largely through the estimation of threshold parameters.{{Cite journal|last=Olsson|first=Ulf|title=Maximum likelihood estimation of the polychoric correlation coefficient|journal=Psychometrika|language=en|volume=44|issue=4|pages=443–460|doi=10.1007/BF02296207|issn=0033-3123|year=1979|s2cid=119716465}}

{{main article|Exploratory factor analysis}}

Both exploratory factor analysis (EFA) and confirmatory factor analysis (CFA) are employed to understand shared variance of measured variables that is believed to be attributable to a factor or latent construct. Despite this similarity, however, EFA and CFA are conceptually and statistically distinct analyses.

The goal of EFA is to identify factors based on data and to maximize the amount of variance explained.{{cite journal |last=Suhr |first=D. D. |year=2006 |title=Exploratory or confirmatory factor analysis? |journal=Statistics and Data Analysis |volume=31 |access-date=April 20, 2012 |url=https://support.sas.com/resources/papers/proceedings/proceedings/sugi31/200-31.pdf }} The researcher is not required to have any specific hypotheses about how many factors will emerge, and what items or variables these factors will comprise. If these hypotheses exist, they are not incorporated into and do not affect the results of the statistical analyses. By contrast, CFA evaluates a priori hypotheses and is largely driven by theory. CFA analyses require the researcher to hypothesize, in advance, the number of factors, whether or not these factors are correlated, and which items/measures load onto and reflect which factors.Thompson, B. (2004). Exploratory and confirmatory factor analysis: Understanding concepts and applications. Washington, DC, US: American Psychological Association. As such, in contrast to exploratory factor analysis, where all loadings are free to vary, CFA allows for the explicit constraint of certain loadings to be zero.

EFA is often considered to be more appropriate than CFA in the early stages of scale development because CFA does not show how well the items load on the non-hypothesized factors.{{cite journal |last=Kelloway |first=E. K. |year=1995 |title=Structural equation modelling in perspective |journal=Journal of Organizational Behavior |volume=16 |issue=3 |pages=215–224 |doi=10.1002/job.4030160304 }} Another strong argument for the initial use of EFA, is that the misspecification of the number of factors at an early stage of scale development will typically not be detected by confirmatory factor analysis. At later stages of scale development, confirmatory techniques may provide more information by the explicit contrast of competing factor structures.

EFA is sometimes reported in research when CFA would be a better statistical approach.{{cite journal |last=Levine |first=T. R. |year=2005 |title=Confirmatory factor analysis and scale validation in communication research |journal=Communication Research Reports |volume=22 |issue=4 |pages=335–338 |doi=10.1080/00036810500317730 |s2cid=145125871 }} It has been argued that CFA can be restrictive and inappropriate when used in an exploratory fashion.{{cite journal |last=Browne |first=M. W. |year=2001 |title=An overview of analytic rotation in exploratory factor analysis |journal=Multivariate Behavioral Research |volume=36 |issue=1 |pages=111–150 |doi=10.1207/S15327906MBR3601_05 |s2cid=9598774 }} However, the idea that CFA is solely a “confirmatory” analysis may sometimes be misleading, as modification indices used in CFA are somewhat exploratory in nature. Modification indices show the improvement in model fit if a particular coefficient were to become unconstrained.{{cite book |last=Gatignon |first=H. |year=2010 |chapter=Confirmatory Factor Analysis |title=Statistical Analysis of Management Data |pages=59–122 |publisher=Springer |doi=10.1007/978-1-4419-1270-1_4 |isbn=978-1-4419-1269-5 }} Likewise, EFA and CFA do not have to be mutually exclusive analyses; EFA has been argued to be a reasonable follow up to a poor-fitting CFA model.{{cite journal |last=Schmitt |first=T. A. |year=2011 |title=Current methodological considerations in exploratory and confirmatory factor analysis |journal=Journal of Psychoeducational Assessment |volume=29 |issue=4 |pages=304–321 |doi=10.1177/0734282911406653 |s2cid=4490758 }}

Structural equation modeling software is typically used for performing confirmatory factor analysis. LISREL,[http://luna.cas.usf.edu/~mbrannic/files/pmet/cfa.htm CFA with LISREL] {{webarchive|url=https://web.archive.org/web/20090528095559/http://luna.cas.usf.edu/~mbrannic/files/pmet/cfa.htm |date=2009-05-28 }} EQS,Byrne, B. M. (2006). Structural equation modeling with EQS: Basic concepts, application, and programming. New Jersey: Lawrence Elbaum Associates. AMOS,[http://www.indiana.edu/~statmath/stat/all/cfa/cfa3.html CFA using AMOS] Mplus[http://www.statmodel.com Mplus homepage] and LAVAAN package in R{{Cite web | url=http://lavaan.ugent.be/ | title=The lavaan Project}} are popular software programs. There is also the Python package {{proper name|semopy 2}}.{{cite arXiv |last1=Meshcheryakov |first1=Georgy |last2=Igolkina |first2=Anna A. |last3=Samsonova |first3=Maria G. |date=2021-06-09 |title=semopy 2: A Structural Equation Modeling Package with Random Effects in Python |class=stat.AP |eprint=2106.01140 }} CFA is also frequently used as a first step to assess the proposed measurement model in a structural equation model. Many of the rules of interpretation regarding assessment of model fit and model modification in structural equation modeling apply equally to CFA. CFA is distinguished from structural equation modeling by the fact that in CFA, there are no directed arrows between latent factors. In other words, while in CFA factors are not presumed to directly cause one another, SEM often does specify particular factors and variables to be causal in nature. In the context of SEM, the CFA is often called 'the measurement model', while the relations between the latent variables (with directed arrows) are called 'the structural model'.

{{see also|Regression validation|Statistical model validation}}

In CFA, several statistical tests are used to determine how well the model fits to the data. Note that a good fit between the model and the data does not mean that the model is “correct”, or even that it explains a large proportion of the covariance. A “good model fit” only indicates that the model is plausible.Schermelleh-Engel, K., Moosbrugger, H., & Müller, H. (2003). Evaluating the fit of structural equation models: Tests of significance and descriptive goodness-of-fit measures, Methods of Psychological Research Online, 8(2), 23-74 When reporting the results of a confirmatory factor analysis, one is urged to report: a) the proposed models, b) any modifications made, c) which measures identify each latent variable, d) correlations between latent variables, e) any other pertinent information, such as whether constraints are used.Jackson, D. L., Gillaspy, J. A., & Purc-Stephenson, R. (2009). Reporting practices in confirmatory factor analysis: An overview and some recommendations. Psychological Methods, 14(1), 6-23. With regard to selecting model fit statistics to report, one should not simply report the statistics that estimate the best fit, though this may be tempting. Though several varying opinions exist, Kline (2010) recommends reporting the chi-squared test, the root mean square error of approximation (RMSEA), the comparative fit index (CFI), and the standardised root mean square residual (SRMR).

Absolute fit indices determine how well the a priori model fits, or reproduces the data.McDonald, R. P., & Ho, M. H. R. (2002). Principles and practice in reporting statistical equation analyses. Psychological Methods, 7(1), 64-82 Absolute fit indices include, but are not limited to, the Chi-Squared test, RMSEA, GFI, AGFI, RMR, and SRMR.Hooper, D., Coughlan, J., & Mullen, M.R. (2008). Structural equation modelling: Guidelines for determining model fit. Journal of Business Research Methods, 6, 53–60

The chi-squared test indicates the difference between observed and expected covariance matrices. Values closer to zero indicate a better fit; smaller difference between expected and observed covariance matrices. Chi-squared statistics can also be used to directly compare the fit of nested models to the data. One difficulty with the chi-squared test of model fit, however, is that researchers may fail to reject an inappropriate model in small sample sizes and reject an appropriate model in large sample sizes. As a result, other measures of fit have been developed.

The root mean square error of approximation (RMSEA) avoids issues of sample size by analyzing the discrepancy between the hypothesized model, with optimally chosen parameter estimates, and the population covariance matrix. The RMSEA ranges from 0 to 1, with smaller values indicating better model fit. A value of .06 or less is indicative of acceptable model fit.{{cite journal|last1=Hu|first1=Li-tze|last2=Bentler|first2=Peter M.|title=Cutoff criteria for fit indexes in covariance structure analysis: Conventional criteria versus new alternatives|journal=Structural Equation Modeling|volume=6|issue=1|year=1999|pages=1–55|issn=1070-5511|doi=10.1080/10705519909540118|hdl=2027.42/139911|hdl-access=free}}{{cite book | last=Brown | first=Timothy | title=Confirmatory factor analysis for applied research | publisher=The Guilford Press | location=New York London | year=2015 | isbn=978-1-4625-1779-4 | page=72}}

The root mean square residual (RMR) and standardized root mean square residual (SRMR) are the square root of the discrepancy between the sample covariance matrix and the model covariance matrix. The RMR may be somewhat difficult to interpret, however, as its range is based on the scales of the indicators in the model (this becomes tricky when there are multiple indicators with varying scales; e.g., two questionnaires, one on a 0–10 scale, the other on a 1–3 scale). The standardized root mean square residual removes this difficulty in interpretation, and ranges from 0 to 1, with a value of .08 or less being indicative of an acceptable model.

The goodness of fit index (GFI) is a measure of fit between the hypothesized model and the observed covariance matrix. The adjusted goodness of fit index (AGFI) corrects the GFI, which is affected by the number of indicators of each latent variable. The GFI and AGFI range between 0 and 1, with a value of over .9 generally indicating acceptable model fit.Baumgartner, H., & Hombur, C. (1996). Applications of structural equation modeling in marketing and consumer research: A review. International Journal of Research in Marketing, 13, 139-161.

Relative fit indices (also called “incremental fit indices”{{cite book |authorlink=Jeffrey S. Tanaka |last=Tanaka |first=J. S. |year=1993 |chapter=Multifaceted conceptions of fit in structure equation models |editor-first=K. A. |editor-last=Bollen |editor2-first=J. S. |editor2-last=Long |title=Testing structural equation models |pages=136–162 |location=Newbury Park, CA |publisher=Sage |isbn=0-8039-4506-X }} and “comparative fit indices”{{cite journal |last=Bentler |first=P. M. |authorlink=Peter M. Bentler |year=1990 |title=Comparative fit indexes in structural models |journal=Psychological Bulletin |volume=107 |issue=2 |pages=238–46 |doi=10.1037/0033-2909.107.2.238 |pmid=2320703 }}) compare the chi-square for the hypothesized model to one from a “null”, or “baseline” model. This null model almost always contains a model in which all of the variables are uncorrelated, and as a result, has a very large chi-square (indicating poor fit). Relative fit indices include the normed fit index and comparative fit index.

The normed fit index (NFI) analyzes the discrepancy between the chi-squared value of the hypothesized model and the chi-squared value of the null model.{{cite journal |last1=Bentler |first1=P. M. |last2=Bonett |first2=D. G. |year=1980 |title=Significance tests and goodness of fit in the analysis of covariance structures |journal=Psychological Bulletin |volume=88 |issue= 3|pages=588–606 |doi=10.1037/0033-2909.88.3.588 }} However, NFI tends to be negatively biased. The non-normed fit index (NNFI; also known as the Tucker–Lewis index, as it was built on an index formed by Tucker and Lewis, in 1973{{cite journal |last1=Tucker |first1=L. R. |authorlink=Ledyard Tucker |last2=Lewis |first2=C. |year=1973 |title=A reliability coefficient for maximum likelihood factor analysis |journal=Psychometrika |volume=38 |issue= |pages=1–10 |doi=10.1007/BF02291170 |s2cid=50680436 }}) resolves some of the issues of negative bias, though NNFI values may sometimes fall beyond the 0 to 1 range. Values for both the NFI and NNFI should range between 0 and 1, with a cutoff of .95 or greater indicating a good model fit.{{cite journal |last1=Hu |first1=L. |last2=Bentler |first2=P. M. |year=1999 |title=Cutoff criteria for fit indexes in covariance structure analysis: Conventional criteria versus new alternatives |journal=Structural Equation Modeling |volume=6 |issue=1 |pages=1–55 |doi=10.1080/10705519909540118 }}

The comparative fit index (CFI) analyzes the model fit by examining the discrepancy between the data and the hypothesized model, while adjusting for the issues of sample size inherent in the chi-squared test of model fit, and the normed fit index. CFI values range from 0 to 1, with larger values indicating better fit. Previously, a CFI value of .90 or larger was considered to indicate acceptable model fit. However, a 1999 study indicated that a value greater than .90 is needed to ensure that misspecified models are not deemed acceptable. Thus, a CFI value of .95 or higher is presently accepted as an indicator of good fit.

To estimate the parameters of a model, the model must be properly identified. That is, the number of estimated (unknown) parameters (q) must be less than or equal to the number of unique variances and covariances among the measured variables; p(p + 1)/2. This equation is known as the "t rule". If there is too little information available on which to base the parameter estimates, then the model is said to be underidentified, and model parameters cannot be estimated appropriately.{{cite journal |last1=Babyak |first1=M. A. |last2=Green |first2=S. B. |year=2010 |title=Confirmatory factor analysis: An introduction for psychosomatic medicine researchers |journal=Psychosomatic Medicine |volume=72 |issue=6 |pages=587–597 |doi=10.1097/PSY.0b013e3181de3f8a |pmid=20467001 |s2cid=23528566 |doi-access=free }}

• Factor analysis
• Exploratory factor analysis
• Latent variable model
• Measurement invariance

{{reflist}}

• Brown, T. A. (2006). Confirmatory factor analysis for applied research. New York: Guilford.
• DiStefano, C., & Hess, B. (2005). Using confirmatory factor analysis for construct validation: An empirical review. Journal of Psychoeducational Assessment, 23, 225-241.
• Harrington, D. (2009). Confirmatory factor analysis. New York: Oxford University Press.
• Maruyama, G. M. (1998). Basics of structural equation modeling. Thousand Oaks, CA: Sage.

• [http://www.indiana.edu/~statmath/stat/all/cfa/index.html Center for Statistical and Mathematical Computing at Indiana University]

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Category:Structural equation models

Category:Factor analysis