structural equation modeling


  • Hence model assessments consider: • whether the data contain reasonable measurements of appropriate variables, • whether the modeled case are causally homogeneous, (It makes
    no sense to estimate one model if the data cases reflect two or more different causal networks.)

  • Because a postulated model such as Figure 1 may not correspond to the worldly forces controlling the observed data measurements, the programs also provide model tests and
    diagnostic clues suggesting which indicators, or which model components, might introduce inconsistency between the model and observed data.

  • [4] The boundary between what is and is not a structural equation model is not always clear but SE models often contain postulated causal connections among a set of latent
    variables (variables thought to exist but which can’t be directly observed, like an attitude, intelligence or mental illness) and causal connections linking the postulated latent variables to variables that can be observed and whose values
    are available in some data set.

  • Fitting yet worldly-inconsistent models are especially likely to arise if a researcher committed to a particular model (for example a factor model having a desired number
    of factors) gets an initially-failing model to fit by inserting measurement error covariances “suggested” by modification indices.

  • The model’s implications for what the data should look like for a specific set of coefficient values depends on: a) the coefficients’ locations in the model (e.g.

  • Model specification[edit] Building or specifying a model requires attending to: • the set of variables to be employed, • what is known about the variables, • what is presumed
    or hypothesized about the variables’ causal connections and disconnections, • what the researcher seeks to learn from the modeling, • and the cases for which values of the variables will be available (kids?

  • A small χ2 probability reports it would be unlikely for the current data to have arisen if the current model structure constituted the real population causal forces – with
    the remaining differences attributed to random sampling variations.

  • This makes it possible to use the connections between the observed variables’ values to estimate the magnitudes of the postulated effects, and to test whether or not the observed
    data are consistent with the requirements of the hypothesized causal structures.

  • A modification index is an estimate of how much a model’s fit to the data would “improve” (but not necessarily how much the model’s structure would improve) if a specific
    currently-fixed model coefficient were freed for estimation.

  • Ordinary least squares estimates are the coefficient values that minimize the squared differences between the data and what the data would look like if the model was correctly
    specified, namely if all the model’s estimated features correspond to real worldly features.

  • A small χ2 probability reports it would be unlikely for the current data to have arisen if the modeled structure constituted the real population causal forces – with the remaining
    differences attributed to random sampling variations.

  • The output from SEM programs includes a matrix reporting the relationships among the observed variables that would be observed if the estimated model effects actually controlled
    the observed variables’ values.

  • [5][6][7][8][9] SEM researchers use computer programs to estimate the strength and sign of the coefficients corresponding to the modeled structural connections, for example
    the numbers connected to the arrows in Figure 1.

  • Numerous fit indices quantify how closely a model fits the data but all fit indices suffer from the logical difficulty that the size or amount of ill fit is not trustably
    coordinated with the severity or nature of the issues producing the data inconsistency.

  • Estimated values for free model coefficients are obtained by maximizing fit to, or minimizing difference from, the data relative to what the data’s features would be if the
    free model coefficients took on the estimated values.

  • “[31] Model misspecification may sometimes be corrected by insertion of coefficients suggested by the modification indices, but many more corrective possibilities are raised
    by employing a few indicators of similar-yet-importantly-different latent variables.

  • The probability accompanying a χ2 (chi-squared) test is the probability that the data could arise by random sampling variations if the estimated model constituted the real
    underlying population forces.

  • If a model remains inconsistent with the data despite selecting optimal coefficient estimates, an honest research response reports and attends to this evidence (often a significant
    model χ2 test).

  • For example, the magnitude of a single data correlation between two variables is insufficient to provide estimates of a reciprocal pair of modeled effects between those variables.

  • [33] The probability accompanying a χ2 test is the probability that the data could arise by random sampling variations if the current model, with its optimal estimates, constituted
    the real underlying population forces.

  • [10] A great advantage of SEM is that all of these measurements and tests occur simultaneously in one statistical estimation procedure, where all the model coefficient are
    calculated using all information from the observed variables.

  • Researchers confronting data-inconsistent models can easily free coefficients the modification indices report as likely to produce substantial improvements in fit.

  • [27] Such models are data-fit-equivalent though not causally equivalent, so at least one of the so-called equivalent models must be inconsistent with the world’s structure.

  • The factor-structured portion of the model incorporated measurement errors which permitted measurement-error-adjustment, though not necessarily error-free estimation, of effects
    connecting different postulated latent variables.

  • [27] One common problem is that a coefficient’s estimated value may be underidentified because it is insufficiently constrained by the model and data.

  • [32] The considerations relevant to using fit indices include checking: 1. whether data concerns have been addressed (to ensure data mistakes are not driving model-data inconsistency);
    2. whether criterion values for the index have been investigated for models structured like the researcher’s model (e.g.

  • (The estimation process minimizes the differences between the model and data but important and informative differences may remain.)

  • which variables are connected/disconnected), b) the nature of the connections between the variables (covariances or effects; with effects often assumed to be linear), c) the
    nature of the error or residual variables (often assumed to be independent of, or causally-disconnected from, many variables), and d) the measurement scales appropriate for the variables (interval level measurement is often assumed).

  • With maximum likelihood estimation, the numerical values of all the free model coefficients are individually adjusted (progressively increased or decreased from initial start
    values) until they maximize the likelihood of observing the sample data – whether the data are the variables’ covariances/correlations, or the cases’ actual values on the indicator variables.

  • The original model may contain causal misspecifications such as incorrectly directed effects, or incorrect assumptions about unavailable variables, and such problems cannot
    be corrected by adding coefficients to the current model.

  • [25] SEM analyses are popular in the social sciences because computer programs make it possible to estimate complicated causal structures, but the complexity of the models
    introduces substantial variability in the quality of the results.

  • index criterion based on factor structured models are only appropriate if the researcher’s model actually is factor structured); 3. whether the kinds of potential misspecifications
    in the current model correspond to the kinds of misspecifications on which the index criterion are based (e.g.

  • The frictions created by disagreements over the necessity of correcting model misspecifications will likely increase with increasing use of non-factor-structured models, and
    with use of fewer, more-precise, indicators of similar yet importantly-different latent variables.

  • criteria based on simulation of omitted factor loadings may not be appropriate for misspecification resulting from failure to include appropriate control variables); 4. whether
    the researcher knowingly agrees to disregard evidence pointing to the kinds of misspecifications on which the index criteria were based.

  • [20][21] Viewing factor analysis as a data-reduction technique deemphasizes testing, which contrasts with path analytic appreciation for testing postulated causal connections
    – where the test result might signal model misspecification.

  • The appropriate statistical feature to maximize or minimize to obtain estimates depends on the variables’ levels of measurement (estimation is generally easier with interval
    level measurements than with nominal or ordinal measures), and where a specific variable appears in the model (e.g.

  • Underidentified effect estimates can be rendered identified by introducing additional model and/or data constraints.

  • Coefficient estimates in data-inconsistent (“failing”) models are interpretable, as reports of how the world would appear to someone believing a model that conflicts with
    the available data.

  • If the replicate data is within random variations of the original data, the same incorrect coefficient placements that provided inappropriate-fit to the original data will
    likely also inappropriately-fit the replicate data.

  • For example, reciprocal effects can be rendered identified by constraining one effect estimate to be double, triple, or equivalent to, the other effect estimate,[29] but the
    resultant estimates will only be trustworthy if the additional model constraint corresponds to the world’s structure.

  • Some of the more commonly used fit statistics include • Chi-square o A fundamental test of fit used in the calculation of many other fit measures.

  • [35] Many researchers tried to justify switching to fit-indices, rather than testing their models, by claiming that χ2 increases (and hence χ2 probability decreases) with
    increasing sample size (N).

  • [11] History Structural equation modeling (SEM) began differentiating itself from correlation and regression when Sewall Wright provided explicit causal interpretations for
    a set of regression-style equations based on a solid understanding of the physical and physiological mechanisms producing direct and indirect effects among his observed variables.

  • This logical weakness renders all fit indices “unhelpful” whenever a structural equation model is significantly inconsistent with the data,[36] but several forces continue
    to propagate fit-index use.

  • Browne, McCallum, Kim, Andersen, and Glaser presented a factor model they viewed as acceptable despite the model being significantly inconsistent with their data according
    to χ2.

  • The model may include intervening variables – variables receiving effects from some variables but also sending effects to other variables.

  • This simultaneously introduces a substantial risk of moving from a causally-wrong-and-failing model to a causally-wrong-but-fitting model because improved data-fit does not
    provide assurance that the freed coefficients are substantively reasonable or world matching.

  • Statistically possible estimates that are inconsistent with theory may also challenge theory, and our understanding.)

  • • whether the model appropriately represents the theory or features of interest, (Models are unpersuasive if they omit features required by a theory, or contain coefficients
    inconsistent with that theory.)

  • Structural equation models often contain postulated causal connections among some latent variables (variables thought to exist but which can’t be directly observed).

  • Some, but not all, results are obtained without the “inconvenience” of understanding experimental design, statistical control, the consequences of sample size, and other features
    contributing to good research design.

  • Additional causal connections link those latent variables to observed variables whose values appear in a data set.

  • if the model is significantly data-inconsistent, the “tolerable” amount of inconsistency is likely to differ in the context of medical, business, social and psychological

  • SEM programs provide estimates and tests of the free coefficients, while the fixed coefficients contribute importantly to testing the overall model structure.

  • Replication helps detect issues such as data mistakes (made by different research groups), but is especially weak at detecting misspecifications after exploratory model modification
    – as when confirmatory factor analysis (CFA) is applied to a random second-half of data following exploratory factor analysis (EFA) of first-half data.

  • One of several programs Karl Jöreskog developed at Educational Testing Services, LISREL[17][18][19] embedded latent variables (which psychologists knew as the latent factors
    from factor analysis) within path-analysis-style equations (which sociologists inherited from Wright and Duncan).

  • A Type III error arises from “accepting” a model hypothesis when the current data are sufficient to reject the model.

  • Theoretical demands for null/zero effects provide helpful constraints assisting estimation, though theories often fail to clearly report which effects are allegedly nonexistent.

  • The estimates may even closely match a theory’s requirements but the remaining data inconsistency renders the match between the estimates and theory unable to provide succor.

  • A cautionary instance was provided by Browne, MacCallum, Kim, Anderson, and Glaser who addressed the mathematics behind why the χ2 test can have (though it does not always
    have) considerable power to detect model misspecification.

  • Wright’s path analysis influenced Hermann Wold, Wold’s student Karl Jöreskog, and Jöreskog’s student Claes Fornell, but SEM never gained a large following among U.S. econometricians,
    possibly due to fundamental differences in modeling objectives and typical data structures.

  • Criticisms of SEM methods hint at: disregard of available model tests, problems in the model’s specification, a tendency to accept models without considering external validity,
    and potential philosophical biases.

  • [27][6][16] The model specification depends on what is known from the literature, the researcher’s experience with the modeled indicator variables, and the features being
    investigated by using the specific model structure.

  • [28] Notice that this again presumes the properness of the model’s causal specification – namely that there really is a direct effect leading from the third variable to the
    variable at this end of the reciprocal effects and no direct effect on the variable at the “other end” of the reciprocally connected pair of variables.


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