Talk:Structural equation modeling

Goodness of Fit Section

When describing Chi Square, RMSEA, etc., it would be helpful to cite sources to say that if it is over .1, for example, the model does not fit well. It would be helpful to have the seminal work that is always cited in academic articles. —Preceding unsigned comment added by 72.253.236.36 (talk) 09:35, 25 February 2011 (UTC)

History

Much of this article is paraphrased from Westland, J.C. (2015) Structural Equation Modeling: From Paths to Networks, New York: Springer, 2015, though in its brevity, the article omits some significant information, and purposely distorts other information.J.christopherwestland (talk) 17:01, 28 November 2015 (UTC)

I have modified the content to accurately reflect both the history presented in Westland, J.C. (2015) as well as to correct misstatements concerning sample size. I have put in references to Westland (2010) that describe sample size calculations for these methods.J.christopherwestland (talk) 17:01, 28 November 2015 (UTC)

Today, SEM is a general term used to describe either of two computer based statistical fit software packages: PLS-PA and LISREL/AMOS software. Sewall Wright developed Path Analysis, which is different; Trygve Haavelmo helped develop systems of regression approaches, and derided PLS-PA and LISREL as unscientific. Simon and Pearl made no contributions to these software packages.

I just finished a graduate course devoted to SEM and had to read several textbooks and over 50 articles about the technique - you are right that SEM is a general term, but authors repeatedly describe it as a class of statistical techniques, not specifically "software packages," although this article does need to keep info about the software; it is important. Also, I think MPLUS needs to be mentioned in the family of statistical software that can be used. I'll plan to work on the overall definition based on my course readings, and may break out a section on software. Sharp-shinned.hawk (talk) 13:53, 24 January 2015 (UTC)

I'm happy to listen to any discussion of 'statistical algorithms' but I have studied the code (I was one of Claes Fornell's students, and had access to the original PLS-PA and LISREL source code) and there is no consistency in the application of any algorithms. The software depends on iterative search and varies from package to package. J.christopherwestland (talk) 17:01, 28 November 2015 (UTC)

SEM vs Graphical Models, Bayesian networks

Could someone please explain the difference?

SEM is a graphical model where the distributions are all assumed to be normal (usually, though there are extensions). Conditional independence (or local dependence) is expressed by the path diagram/effect matrix. More explanation is in the J. Pearl book (2000). —Preceding unsigned comment added by 80.174.156.18 (talk) 14:57, 21 September 2008 (UTC)

This assertion is silly ... SEM is a general term for different algorithms. Wright's 'path model' graphics are used to describe the models, but each has it's own algorithms. Bayesian networks are a different animal altogether -- there is a reasonably good article in wikipedia on that alreadyJ.christopherwestland (talk) 17:04, 28 November 2015 (UTC)

contributions

Hi, I think the article is excellent and am disturbed by the clean-up indicators at the beginning that make it appear that there are serious problems. Nothing is perfect; however I was greatly informed by the clear writing, which I felt was not too technical and perfectly understandable. --Littleelf (talk) 11:31, 31 March 2008 (UTC)

The article definitely needs cleanup and fact checking. Many of the assertions are untrue or inconsistent with each other, and the overall presentation is sloppy. The relationship of these approaches to the broader body of statistical methods (including links to other Wikipedia articles) needs to be added before this article is fully acceptable. J.christopherwestland (talk) 17:13, 28 November 2015 (UTC)

Hi, I would like to invite authors to contribute articles to this entry. Some things that can be included are:

1. Lesson plans for teaching SEM 2. List of good books on SEM 3. Key, "must read" articles

Why would a Wikipedia article contain 'lesson plans' and 'good books' and 'must read' articles ... these are all highly subjective and the suggestion seems to me to be an abuse of WikipediaJ.christopherwestland (talk) 17:13, 28 November 2015 (UTC)

Hello,

It seems that emphasizing the difference between linear multiple regression and SEM is important for this article. Does anybody know more about this? To be frankly, I cannot see any urgent needs to state that y_i is determined by y_0,1,2, ..., n (except for i) in SEM. What can be the situations that can be described better with SEM than with linear regression?

Please see Westland (2015) which describes how SEM path analysis models (with latent variables) can be operationalized in systems of regression equations J.christopherwestland (talk) 17:13, 28 November 2015 (UTC)

I'll be working on this page at my leisure time, if any, as this is my major research topic. What would be the best way to incorporate path diagrams into Wikipedia? I can do matrices with MathML, or whatever interface Wikipedia has for math, but the path diagrams would obviously have to go as graphics files.

Stas Kolenikov

Hi

The 'historical' introduction right now is only focused on econometrics. But it might be more clarifying to state that SEM is a combination of factor analysis (Thurstone) and path analysis (Wright). This should ring some bells for most people coming to this page. As far as I know, Joreskog, not Pearl, was the first to formally combine the two in 1970 [[1]].

It was Hermann Wold, in the development of computational methods for PCA and canonical correlations in the 1930s, who originated these extensions of Wright's path analysis J.christopherwestland (talk) 17:13, 28 November 2015 (UTC)

best, daniel (sorry for not logging in, next time I will) —Preceding unsigned comment added by 80.174.156.18 (talk) 14:45, 21 September 2008 (UTC)

I too was surprised to see no mention of Joreskog, Sorbom, Muthen or other key figures in SEM. 131.96.47.13 (talk) 14:40, 13 October 2009 (UTC)

Most of the historical material was paraphrased from Westland (2015), and was done sloppily (thus the omission of key figures) J.christopherwestland (talk) 17:13, 28 November 2015 (UTC)

"SEM has several important advantages ..."

"SEM allows for multiple dependent variable, whereas OLS regressions allows only a single dependent variable." I would say no: SEM is a model where as OLS is an estimation technique. OLS can be used for SEM.

"SEM allows simultaneous tests of multiple groups". I do not understand this: OLS can be used for multivariate regression analysis... In this type of model/analysis there are several response/dependent variables.

"SEM accounts for measurement error, whereas OLS regression assumes perfect measurement." In multivariate regression analysis you have Y=XB+U, and the Y is assumed to be confounded with measurement error (the U's).

- fnielsen 11:24, 10 March 2006 (UTC)

To fnielsen: Most of the notes you made are SEM jargon.

OLS cannot be used for SEM because of the measurement error. Or rather the class of SEMs to which OLS is applicable is very narrow (recursive/trinagluar models where all variables are observed; those are not very interesting).

Multiple groups means that you might have different parameters for different subpopulations. In OLS, you would model that through interactions. In SEM, however, there are way more parameters than just slopes, so you may have say the same loadings, but different measurement error variances between males and females. Multiple group comparisons are then based on nested hypotheses where you allow some of the model parameters vary between groups.

OLS assumes X's are fixed. Typically, that's too much a luxury to assume with SEM where most of the observed variables are truly random variables and contain measurement error.

GLM means generalized linear model to most statisticians; using it for general linear model is rather awkward, I'd say.

- Stas Kolenikov 16 March 2006

SAS is the main perpetrator of this misunderstanding (PROC GLM). Btyner 15:15, 16 April 2006 (UTC)

Category

Should this page have another category? I thought I remembered reading that a page should have at least two categories but cannot find the policy at the moment. Any suggestions? --Kenneth M Burke 01:39, 2 August 2007 (UTC)

POV

The introduction I think might have some problems with point of view, i.e. that SEM is confirmatory rather than exploratory. Not to mention, I think it is plagiarized from another website. Not being an expert and where parts of the introduction does cite sources, I did not put a POV template on the page. This being said, maybe a more textbook language common to mathematics and research methods would be better than the "confirmatory" and "exploratory," which sounds like its from a bad sci-fi movie that didn't do its homework. Just casual thoughts and suggestions. --Kenneth M Burke 01:32, 20 August 2007 (UTC)

Well, after doing my own homework, that's what they call it. But, the amount of information that one can find on both exploratory and confirmatory SEM I think certainly shows problems with POV in the article. --Kenneth M Burke 01:43, 20 August 2007 (UTC)

I've come to strongly believe that the information on SEM as primarily confirmatory is biased. Even the most basic of introductory texts note that it is not exclusively confirmatory (Kline, 2005, p. 9, New York: Guilford Press, Principles and practice of Structural Equation Modeling). We need remember that SEM is not a single method, that there are many means and methods for SEM. --Kenneth M Burke 18:00, 28 October 2007 (UTC)

SEM- Censored data, categorical data, and Likert scale

Hi,
Though not an expert in SEM, I have developed an interest in the subject and The SEM softwares. I will recommend that,That the article include more information on censured Data and Categorical data.

The main contention is when do we consider a Likert scale to be continous? Can we Indexes in SEM? Wadson12 13:58, 9 November 2007 (UTC)

advantages and disadvantages of using SEM

i'm a college student now and i'm doing my final project. It is a reseach which is using SEm. Can anybody tell me what is the advantages and disadvantages of using SEM? How about comparison of SEM and another multivariat methods? thx before..

kind regard

donny

Monday, 3 November 2008

SEM is quite advanced. In comparison to PLS, SEM can caluculate specific goodness-of-fit indices. That is, you know how good your model fits the data. The main disadvantage of SEM is that you need a quite large sample (typically much larger than 300). PLS works fine with samples <200. And you must be aware that SEM wouldn't work correctly if you specify a model with formative rather than reflective measurement scales. In practice, his is often neglected, which often leads to nonsense findings. PLS can estimate both formative and reflective models. But in general, SEM is a fine method if you want to estimate models which build on multi-item measurement scales. 85.179.138.248 (talk) 05:45, 8 January 2012 (UTC)

independent variables in structurla mnodels

I am not sure if the independent cvariable in a structutural model could be an observable variable (no construct). I am very grateful if anyone clarify me that point Thanks a lot —Preceding unsigned comment added by 83.40.89.47 (talk) 11:56, 31 December 2008 (UTC)

TETRAD: External links embedded in the text

I am disturbed by the reference to the TETRAD project, which contains a link to a web site outside of wiki. Shouldn't external links go to the bottom of the article? And if each one of us inserted free advertising for our own project into wiki articles, what would become of wiki? 193.255.135.1 (talk) 13:17, 7 April 2010 (UTC)

Sample size section

"Complexities which increase information demands in structural model estimation increase with the number of potential combinations of latent variables; while the information supplied for estimation increases with the number of measured parameters times the number of observations in the sample size – both are non-linear. Sample size in SEM can be computed through two methods: the first as a function of the ratio of indicator variables to latent variables, and the second as a function of minimum effect, power and significance. Software and methods for computing both have been developed by Westland (2010)."

There is a long history addressing the issues of sample size in SEM. I have edited out what I feel are personal, unfounded and unscientific assertions from an anonymous poster. Indeed this quote is from my research: both the ECRA article and my Springer book "Structural Equation Models: From Path to Network Analysis". (JC Westland)

History is missing Joreskog

A piece on SEM without mentioning Karl Jöreskog? Very very strange!

As much as I respect Pearl's work, he cannot in my opinion be seen as "the person to formalize SEM" as this article claims. His work certainly has bearing on SEM but many went before him... — Preceding unsigned comment added by 137.56.58.180 (talk) 13:58, 14 January 2013 (UTC)

No Equations?

The Stuctural Equation Modelling page has no equations. So what exactly ARE these "structural equations" then? Can we see them/write them down? Or does the work "equation" in SEM not mean equation in the normal mathematical sense? — Preceding unsigned comment added by 31.221.13.140 (talk) 09:14, 13 March 2013 (UTC)

SEM-specific software

The section should contain notable entries only, and no links to their own websites. --Ronz (talk) 03:13, 8 April 2013 (UTC)

Please show me this in the WP guidelines.

The WP:Notability criterion applies to the main topic of an article, not to WP:references. Referencing an official web page that supports a claim is usually fine. The software R is extremely notible as it is widely used in statistics and data analysis. A reader of this article is reasonably likely to be an R user and would be interested in the various packages. Those packages are provided by various people, not including me.

Perhaps there needs to be a new page R packages to guide readers to the many contributed packages and task views. Tayste (edits) 19:11, 9 April 2013 (UTC)

I didn't mention WP:N, so let's not get off track.

WP:SOAP and WP:NPOV.

Trying to make it a stand-alone article would make it a WP:CFORK. --Ronz (talk) 20:37, 9 April 2013 (UTC)

Please show me which WP guideline states that the "section should contain notable entries only". It is your use of the word notable in this context that I am querying. Tayste (edits) 00:18, 10 April 2013 (UTC)

And which WP guideline forbids "links to their own websites", especially when I formatted them as references rather than direct links? Tayste (edits) 00:22, 10 April 2013 (UTC)

I didn't claim the policies contain those exact quotes, so let's not get off track.

I use the word "notable" to indicate two things: First, if an entry has it's own article, and that article meets WP:N, then it would likely to be fine. The entries' articles also provide a proper location for any official website that might exist per WP:ELOFFICIAL.

Second, if an entry can be shown to be worth noting with sources that are both independent and reliable that might not be enough to justify it's own article but demonstrates that it would be due weight to include in this article, then that would probably work as well.

Finally, we only add links in the body of an article as references per WP:EL. --Ronz (talk) 04:19, 10 April 2013 (UTC)

While you're on this hobby horse, you might like to stop by other pages with similar sections, e.g. Principal component analysis, Student's t-test, Generalized estimating equation... Tayste (edits) 00:30, 10 April 2013 (UTC)

Thanks for pointing them out. --Ronz (talk) 04:19, 10 April 2013 (UTC)

Planning to work on this page

Hi everyone, I just finished a grad course on SEM and plan to try to improve several aspects of this currently C-class article - I've got lots of citations handy from my course readings, so I figure it's a good opportunity to "give back." The "sample size" section and the fit indices seem to need citations in particular. Let me know if you were also planning to work on this page in the next few weeks! Happy to collaborate. Sharp-shinned.hawk (talk) 15:47, 24 January 2015 (UTC)

Draft for suggested revisions to get input from SEMNET

For the journal, see Structural Equation Modeling (journal).

An example structural equation model. Latent variables are drawn as circles. Manifest or measured variables are shown as squares. Residuals and variances are drawn as double headed arrows into an object. Note latent IQ variable fixed at 1 to provide scale to the model.

Structural equation modeling (SEM) includes a diverse set of mathematical models, computer algorithms, and statistical methods that fit networks of constructs to data. SEM includes confirmatory factor analysis, confirmatory composite analysis, path analysis, partial least squares path modeling, and latent growth modeling. The concept should not be confused with the related concept of structural models in econometrics, nor with structural models in economics. Structural equation models are often used to assess unobservable 'latent' constructs. They often invoke a measurement model that defines latent variables using one or more observed variables, and a structural model that imputes relationships between latent variables. The links between constructs of a structural equation model may be estimated with independent regression equations or through more involved approaches such as those employed in LISREL.

Use of SEM is commonly justified in the social sciences because of its ability to impute relationships between unobserved constructs (latent variables) and observable variables. To provide a simple example, the concept of human intelligence cannot be measured directly as one could measure height or weight. Instead, psychologists develop a hypothesis of intelligence and write measurement instruments with items (questions) designed to measure intelligence according to their hypothesis. They would then use SEM to test their hypothesis using data gathered from people who took their intelligence test. With SEM, "intelligence" would be the latent variable and the test items would be the observed variables.

A simplistic model suggesting that intelligence (as measured by four questions) can predict academic performance (as measured by SAT, ACT, and high school GPA) is shown above (top right). In SEM diagrams, latent variables are commonly shown as ovals and observed variables as rectangles. The diagram above shows how error (e) influences each intelligence question and the SAT, ACT, and GPA scores, but does not influence the latent variables. SEM provides numerical estimates for each of the parameters (arrows) in the model to indicate the strength of the relationships. Thus, in addition to testing the overall theory, SEM therefore allows the researcher to diagnose which observed variables are good indicators of the latent variables.

Various methods in structural equation modeling have been used in the sciences, business, and other fields. Criticism of SEM methods often addresses pitfalls in mathematical formulation, weak external validity of some accepted models and philosophical bias inherent to the standard procedures.

History

Structural equation modeling, as the term is currently used in sociology, psychology, and other social sciences evolved from the earlier methods in genetic path modeling of Sewall Wright. Their modern forms came about with computer-intensive implementations in the 1960s and 1970s. SEM evolved in three different streams: (1) systems of equation regression methods developed mainly at the Cowles Commission; (2) iterative maximum likelihood algorithms for path analysis developed mainly by Karl Gustav Jöreskog at the Educational Testing Service and subsequently at Uppsala University; and (3) iterative canonical correlation fit algorithms for path analysis also developed at Uppsala University by Hermann Wold. Much of this development occurred at a time that automated computing was offering substantial upgrades over the existing calculator and analogue computing methods available, themselves products of the proliferation of office equipment innovations in the late 20th century. The 2015 text Structural Equation Modeling: From Paths to Networks provides a history of the methods.

Loose and confusing terminology has been used to obscure weaknesses in the methods. In particular, PLS-PA (the Lohmoller algorithm) has been conflated with partial least squares regression PLSR, which is a substitute for ordinary least squares regression and has nothing to do with path analysis. PLS-PA has been falsely promoted as a method that works with small datasets when other estimation approaches fail. Westland (2010) decisively showed this not to be true and developed an algorithm for sample sizes in SEM. Since the 1970s, the 'small sample size' assertion has been known to be false (see for example Dhrymes, 1972, 1974; Dhrymes & Erlat, 1972; Dhrymes et al., 1972; Gupta, 1969; Sobel, 1982).

Both LISREL and PLS-PA were conceived as iterative computer algorithms, with an emphasis from the start on creating an accessible graphical and data entry interface and extension of Wright's (1921) path analysis. Early Cowles Commission work on simultaneous equations estimation centered on Koopman and Hood's (1953) algorithms from the economics of transportation and optimal routing, with maximum likelihood estimation, and closed form algebraic calculations, as iterative solution search techniques were limited in the days before computers. Anderson and Rubin (1949, 1950) developed the limited information maximum likelihood estimator for the parameters of a single structural equation, which indirectly included the two-stage least squares estimator and its asymptotic distribution (Anderson, 2005) and Farebrother (1999). Two-stage least squares was originally proposed as a method of estimating the parameters of a single structural equation in a system of linear simultaneous equations, being introduced by Theil (1953a, 1953b, 1961) and more or less independently by Basmann (1957) and Sargan (1958). Anderson's limited information maximum likelihood estimation was eventually implemented in a computer search algorithm, where it competed with other iterative SEM algorithms. Of these, two-stage least squares was by far the most widely used method in the 1960s and the early 1970s.

Systems of regression equation approaches were developed at the Cowles Commission from the 1950s on, extending the transportation modeling of Tjalling Koopmans. Sewall Wright and other statisticians attempted to promote path analysis methods at Cowles (then at the University of Chicago). University of Chicago statisticians identified many faults with path analysis applications to the social sciences; faults which did not pose significant problems for identifying gene transmission in Wright's context, but which made path methods such as PLS-PA and LISREL problematic in the social sciences. Freedman (1987) summarized these objections in path analyses: "failure to distinguish among causal assumptions, statistical implications, and policy claims has been one of the main reasons for the suspicion and confusion surrounding quantitative methods in the social sciences" (see also Wold's (1987) response). Wright's path analysis never gained a large following among U.S. econometricians, but was successful in influencing Hermann Wold and his student Karl Jöreskog. Jöreskog's student Claes Fornell promoted LISREL in the US.

Advances in computers made it simple for novices to apply structural equation methods in the computer-intensive analysis of large datasets in complex, unstructured problems. The most popular solution techniques fall into three classes of algorithms: (1) ordinary least squares algorithms applied independently to each path, such as applied in the so-called PLS path analysis packages which estimate with OLS; (2) covariance analysis algorithms evolving from seminal work by Wold and his student Karl Jöreskog implemented in LISREL, AMOS, and EQS; and (3) simultaneous equations regression algorithms developed at the Cowles Commission by Tjalling Koopmans.

Pearl has extended SEM from linear to nonparametric models, and proposed causal and counterfactual interpretations of the equations. For example, excluding a variable Z from the arguments of an equation asserts that the dependent variable is independent of interventions on the excluded variable, once we hold constant the remaining arguments. Nonparametric SEMs permit the estimation of total, direct and indirect effects without making any commitment to the form of the equations or to the distributions of the error terms. This extends mediation analysis to systems involving categorical variables in the presence of nonlinear interactions. Bollen and Pearl survey the history of the causal interpretation of SEM and why it has become a source of confusions and controversies.

SEM path analysis methods are popular in the social sciences because of their accessibility; packaged computer programs allow researchers to obtain results without the inconvenience of understanding experimental design and control, effect and sample sizes, and numerous other factors that are part of good research design. Supporters say that this reflects a holistic, and less blatantly causal, interpretation of many real world phenomena – especially in psychology and social interaction – than may be adopted in the natural sciences; detractors suggest that many flawed conclusions have been drawn because of this lack of experimental control.

Direction in the directed network models of SEM arises from presumed cause-effect assumptions made about reality. Social interactions and artifacts are often epiphenomena – secondary phenomena that are difficult to directly link to causal factors. An example of a physiological epiphenomenon is, for example, time to complete a 100-meter sprint. A person may be able to improve their sprint speed from 12 seconds to 11 seconds, but it will be difficult to attribute that improvement to any direct causal factors, like diet, attitude, weather, etc. The 1 second improvement in sprint time is an epiphenomenon – the holistic product of interaction of many individual factors.

General approach to SEM

Although each technique in the SEM family is different, the following aspects are common to many SEM methods.

Model specification

Two main components of models are distinguished in SEM: the structural model showing potential causal dependencies between endogenous and exogenous variables, and the measurement model showing the relations between latent variables and their indicators. Exploratory and confirmatory factor analysis models, for example, contain only the measurement part, while path diagrams can be viewed as SEMs that contain only the structural part.

In specifying pathways in a model, the modeler can posit two types of relationships: (1) free pathways, in which hypothesized causal (in fact counterfactual) relationships between variables are tested, and therefore are left 'free' to vary, and (2) relationships between variables that already have an estimated relationship, usually based on previous studies, which are 'fixed' in the model.

A modeler will often specify a set of theoretically plausible models in order to assess whether the model proposed is the best of the set of possible models. Not only must the modeler account for the theoretical reasons for building the model as it is, but the modeler must also take into account the number of data points and the number of parameters that the model must estimate to identify the model. An identified model is a model where a specific parameter value uniquely identifies the model (recursive definition), and no other equivalent formulation can be given by a different parameter value. A data point is a variable with observed scores, like a variable containing the scores on a question or the number of times respondents buy a car. The parameter is the value of interest, which might be a regression coefficient between the exogenous and the endogenous variable or the factor loading (regression coefficient between an indicator and its factor). If there are fewer data points than the number of estimated parameters, the resulting model is "unidentified", since there are too few reference points to account for all the variance in the model. The solution is to constrain one of the paths to zero, which means that it is no longer part of the model.

Estimation of free parameters

Parameter estimation is done by comparing the actual covariance matrices representing the relationships between variables and the estimated covariance matrices of the best fitting model. This is obtained through numerical maximization via expectation–maximization of a fit criterion as provided by maximum likelihood estimation, quasi-maximum likelihood estimation, weighted least squares or asymptotically distribution-free methods. This is often accomplished by using a specialized SEM analysis program, of which several exist.

Assessment of model and model fit

Having estimated a model, analysts will want to interpret the model. Estimated paths may be tabulated and/or presented graphically as a path model. The impact of variables is assessed using path tracing rules (see path analysis).

It is important to examine the "fit" of an estimated model to determine how well it models the data. This is a basic task in SEM modeling, forming the basis for accepting or rejecting models and, more usually, accepting one competing model over another. The output of SEM programs includes matrices of the estimated relationships between variables in the model. Assessment of fit essentially calculates how similar the predicted data are to matrices containing the relationships in the actual data.

Formal statistical tests and fit indices have been developed for these purposes. Individual parameters of the model can also be examined within the estimated model in order to see how well the proposed model fits the driving theory. Most, though not all, estimation methods make such tests of the model possible.

Of course as in all statistical hypothesis tests, SEM model tests are based on the assumption that the correct and complete relevant data have been modeled. In the SEM literature, discussion of fit has led to a variety of different recommendations on the precise application of the various fit indices and hypothesis tests.

There are differing approaches to assessing fit. Traditional approaches to modeling start from a null hypothesis, rewarding more parsimonious models (i.e. those with fewer free parameters), to others such as AIC that focus on how little the fitted values deviate from a saturated model (i.e. how well they reproduce the measured values), taking into account the number of free parameters used. Because different measures of fit capture different elements of the fit of the model, it is appropriate to report a selection of different fit measures. Guidelines (i.e., "cutoff scores") for interpreting fit measures, including the ones listed below, are the subject of much debate among SEM researchers.

Some of the more commonly used measures of fit include:

• Chi-squared
  • A fundamental measure of fit used in the calculation of many other fit measures. Conceptually it is a function of the sample size and the difference between the observed covariance matrix and the model covariance matrix.
• Akaike information criterion (AIC)
  • A test of relative model fit: The preferred model is the one with the lowest AIC value.
  • ${\mathit {AIC}}=2k-2\ln(L)\,$
  • where k is the number of parameters in the statistical model, and L is the maximized value of the likelihood of the model.
• Root Mean Square Error of Approximation (RMSEA)
  • Fit index where a value of zero indicates the best fit. While the guideline for determining a "close fit" using RMSEA is highly contested, most researchers concur that an RMSEA of .1 or more indicates poor fit.
• Standardized Root Mean Residual (SRMR)
  • The SRMR is a popular absolute fit indicator. Hu and Bentler (1999) suggested .08 or smaller as a guideline for good fit. Kline (2011) suggested .1 or smaller as a guideline for good fit.
• Comparative Fit Index (CFI)
  • In examining baseline comparisons, the CFI depends in large part on the average size of the correlations in the data. If the average correlation between variables is not high, then the CFI will not be very high. A CFI value of .95 or higher is desirable.

For each measure of fit, a decision as to what represents a good-enough fit between the model and the data must reflect other contextual factors such as sample size, the ratio of indicators to factors, and the overall complexity of the model. For example, very large samples make the Chi-squared test overly sensitive and more likely to indicate a lack of model-data fit.

Model modification

The model may need to be modified in order to improve the fit, thereby estimating the most likely relationships between variables. Many programs provide modification indices which may guide minor modifications. Modification indices report the change in χ² that result from freeing fixed parameters: usually, therefore adding a path to a model which is currently set to zero. Modifications that improve model fit may be flagged as potential changes that can be made to the model. Modifications to a model, especially the structural model, are changes to the theory claimed to be true. Modifications therefore must make sense in terms of the theory being tested, or be acknowledged as limitations of that theory. Changes to measurement model are effectively claims that the items/data are impure indicators of the latent variables specified by theory.

Models should not be led by MI, as Maccallum (1986) demonstrated: "even under favorable conditions, models arising from specification searches must be viewed with caution."

Sample size and power

While researchers agree that large sample sizes are required to provide sufficient statistical power and precise estimates using SEM, there is no general consensus on the appropriate method for determining adequate sample size. Generally, the considerations for determining sample size include the number of observations per parameter, the number of observations required for fit indexes to perform adequately, and the number of observations per degree of freedom. Researchers have proposed guidelines based on simulation studies, professional experience, and mathematical formulas.

Sample size requirements to achieve a particular significance and power in SEM hypothesis testing are similar for the same model when any of the three algorithms (PLS-PA, LISREL or systems of regression equations) are used for testing.

Interpretation and communication

The set of models are then interpreted so that claims about the constructs can be made, based on the best fitting model.

Caution should always be taken when making claims of causality even when experimentation or time-ordered studies have been done. The term causal model must be understood to mean "a model that conveys causal assumptions", not necessarily a model that produces validated causal conclusions. Collecting data at multiple time points and using an experimental or quasi-experimental design can help rule out certain rival hypotheses but even a randomized experiment cannot rule out all such threats to causal inference. Good fit by a model consistent with one causal hypothesis invariably entails equally good fit by another model consistent with an opposing causal hypothesis. No research design, no matter how clever, can help distinguish such rival hypotheses, save for interventional experiments.

As in any science, subsequent replication and perhaps modification will proceed from the initial finding.

Advanced uses

• Measurement invariance
• Multiple group modelling: This is a technique allowing joint estimation of multiple models, each with different sub-groups. Applications include behavior genetics, and analysis of differences between groups (e.g., gender, cultures, test forms written in different languages, etc.).
• Latent growth modeling
• Hierarchical/multilevel models; item response theory models
• Mixture model (latent class) SEM
• Alternative estimation and testing techniques
• Robust inference
• Survey sampling analyses
• Multi-method multi-trait models
• Structural Equation Model Trees

SEM-specific software

Several software packages exist for fitting structural equation models. LISREL was the first such software, initially released in the 1970s.

There are also several packages for the R open source statistical environment. The OpenMx R package provides an open source and enhanced version of the Mx application.

Scholars consider it good practice to report which software package and version was used for SEM analysis because they have different capabilities and may use slightly different methods to perform similarly-named techniques.

See also

• Causal model
• Graphical model
• Multivariate statistics
• Partial least squares path modeling
• Partial least squares regression
• Simultaneous equations model
• Structural Equations with Latent Variables

Bibliography

• Hu, Li‐tze; Bentler, Peter M (1999). "Cutoff criteria for fit indexes in covariance structure analysis: Conventional criteria versus new alternatives". Structural Equation Modeling: A Multidisciplinary Journal. 6: 1–55. doi:10.1080/10705519909540118. hdl:2027.42/139911.
• Kaplan, D. (2008). Structural Equation Modeling: Foundations and Extensions (2nd ed.). SAGE. ISBN 978-1412916240.
• Kline, Rex (2011). Principles and Practice of Structural Equation Modeling (Third ed.). Guilford. ISBN 978-1-60623-876-9.
• MacCallum, Robert; Austin, James (2000). "Applications of Structural Equation Modeling in Psychological Research" (PDF). Annual Review of Psychology. 51: 201–226. doi:10.1146/annurev.psych.51.1.201. PMID 10751970. Retrieved 25 January 2015.
• Quintana, Stephen M.; Maxwell, Scott E. (1999). "Implications of Recent Developments in Structural Equation Modeling for Counseling Psychology". The Counseling Psychologist. 27 (4): 485–527. doi:10.1177/0011000099274002.

Further reading

• Bagozzi, Richard P; Yi, Youjae (2011). "Specification, evaluation, and interpretation of structural equation models". Journal of the Academy of Marketing Science. 40 (1): 8–34. doi:10.1007/s11747-011-0278-x.
• Bartholomew, D. J., and Knott, M. (1999) Latent Variable Models and Factor Analysis Kendall's Library of Statistics, vol. 7, Edward Arnold Publishers, ISBN 0-340-69243-X
• Bentler, P.M. & Bonett, D.G. (1980), "Significance tests and goodness of fit in the analysis of covariance structures", Psychological Bulletin, 88, 588-606.
• Bollen, K. A. (1989). Structural Equations with Latent Variables. Wiley, ISBN 0-471-01171-1
• Byrne, B. M. (2001) Structural Equation Modeling with AMOS - Basic Concepts, Applications, and Programming.LEA, ISBN 0-8058-4104-0
• Goldberger, A. S. (1972). Structural equation models in the social sciences. Econometrica 40, 979- 1001.
• Haavelmo, Trygve (January 1943). "The Statistical Implications of a System of Simultaneous Equations". Econometrica. 11 (1): 1–12. doi:10.2307/1905714. JSTOR 1905714.
• Hoyle, R H (ed) (1995) Structural Equation Modeling: Concepts, Issues, and Applications. SAGE, ISBN 0-8039-5318-6
• Jöreskog, Karl G.; Yang, Fan (1996). "Non-linear structural equation models: The Kenny-Judd model with interaction effects". In Marcoulides, George A.; Schumacker, Randall E. (eds.). Advanced structural equation modeling: Concepts, issues, and applications. Thousand Oaks, CA: Sage Publications. pp. 57–88. ISBN 978-1-317-84380-1.
• Lewis-Beck, Michael; Bryman, Alan E.; Bryman, Emeritus Professor Alan; Liao, Tim Futing (2004). "Structural Equation Modeling". The SAGE Encyclopedia of Social Science Research Methods. doi:10.4135/9781412950589.n979. hdl:2022/21973. ISBN 978-0-7619-2363-3.
• Schermelleh-Engel, K.; Moosbrugger, H.; Müller, H. (2003), "Evaluating the fit of structural equation models" (PDF), Methods of Psychological Research, 8 (2): 23–74.

External links

• Ed Rigdon's Structural Equation Modeling Page: people, software and sites
• Structural equation modeling page under David Garson's StatNotes, NCSU
• Issues and Opinion on Structural Equation Modeling, SEM in IS Research
• The causal interpretation of structural equations (or SEM survival kit) by Judea Pearl 2000.
• Structural Equation Modeling Reference List by Jason Newsom: journal articles and book chapters on structural equation models
• Handbook of Management Scales, a collection of previously used multi-item scales to measure constructs for SEM

Prof. Eric A. Youngstrom (talk) 23:17, 18 December 2020 (UTC)