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single data correlation between two variables is insufficient to provide estimates of a reciprocal pair of modeled effects between those variables. The correlation might be accounted for by one of the reciprocal effects being stronger than the other effect, or the other effect being stronger than the one, or by effects of equal magnitude. Underidentified effect estimates can be rendered identified by introducing additional model and/or data constraints. For example, reciprocal effects can be rendered identified by constraining one effect estimate to be double, triple, or equivalent to, the other effect estimate, but the resultant estimates will only be trustworthy if the additional model constraint corresponds to the world's structure. Data on a third variable that directly causes only one of a pair of reciprocally causally connected variables can also assist identification. Constraining a third variable to not directly cause one of the reciprocally-causal variables breaks the symmetry otherwise plaguing the reciprocal effect estimates because that third variable must be more strongly correlated with the variable it causes directly than with the variable at the "other" end of the reciprocal which it impacts only indirectly. Notice that this again presumes the properness of the model's causal specification â 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. Theoretical demands for null/zero effects provide helpful constraints assisting estimation, though theories often fail to clearly report which effects are allegedly nonexistent.
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coefficients the modification indices report as likely to produce substantial improvements in fit. 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. 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. Consequently, such models remain misspecified despite the closer fit provided by additional coefficients. 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. MacCallum (1986) demonstrated that "even under favorable conditions, models arising from specification serchers must be viewed with caution." 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.
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initial measurement step providing scales or factor-scores which are to be used later in a path-structured model. This stepwise approach seems obvious but actually confronts severe underlying deficiencies. The segmentation into steps interferes with thorough checking of whether the scales or factor-scores validly represent the indicators, and/or validly report on latent level effects. A structural equation model simultaneously incorporating both the measurement and latent-level structures not only checks whether the latent factors appropriately coordinates the indicators, it also checks whether that same latent simultaneously appropriately coordinates each latentâs indictors with the indicators of theorized causes and/or consequences of that latent. If a latent is unable to do both these styles of coordination, the validity of that latent is questioned, and a scale or factor-scores purporting to measure that latent is questioned. The disagreements swirled around respect for, or disrespect of, evidence challenging the validity of postulated latent factors. The simmering, sometimes boiling, discussions resulted in a special issue of the journal
Structural Equation Modeling focused on a target article by Hayduk and Glaser followed by several comments and a rejoinder, all made freely available, thanks to the efforts of George Marcoulides.
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which fit the data identically well, have been called equivalent models. 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. If there is a perfect 1.0 correlation between X and Y and we model this as X causes Y, there will be perfect fit and zero residual error. But the model may not match the world because Y may actually cause X, or both X and Y may be responding to a common cause Z, or the world may contain a mixture of these effects (e.g. like a common cause plus an effect of Y on X), or other causal structures. The perfect fit does not tell us the model's structure corresponds to the world's structure, and this in turn implies that getting closer to perfect fit does not necessarily correspond to getting closer to the world's structure â maybe it does, maybe it doesn't. This makes it incorrect for a researcher to claim that even perfect model fit implies the model is correctly causally specified. For even moderately complex models, precisely equivalently-fitting models are rare. Models almost-fitting the data, according to any index, unavoidably introduce additional potentially-important yet unknown model misspecifications. These models constitute a greater research impediment.
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interpretation possibilities. For example, a common cause contributes to the covariance or correlation between two effected variables, because if the value of the cause goes up, the values of both effects should also go up (assuming positive effects) even if we do not know the full story underlying each cause. (A correlation is the covariance between two variables that have both been standardized to have variance 1.0). Another interpretive contribution might be made by expressing how two causal variables can both explain variance in a dependent variable, as well as how covariance between two such causes can increase or decrease explained variance in the dependent variable. That is, interpretation may involve explaining how a pattern of effects and covariances can contribute to decreasing a dependent variableâs variance. Understanding causal implications implicitly connects to understanding âcontrollingâ, and potentially explaining why some variables, but not others, should be controlled. As models become more complex these fundamental components can combine in non-intuitive ways, such as explaining how there can be no correlation (zero covariance) between two variables despite the variables being connected by a direct non-zero causal effect.
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adjustment for all the other operative/modeled causal mechanisms. Indirect effects are interpreted similarly, with the magnitude of a specific indirect effect equaling the product of the series of direct effects comprising that indirect effect. The units involved are the real scales of observed variablesâ values, and the assigned scale values for latent variables. A specified/fixed 1.0 effect of a latent on a specific indicator coordinates that indicatorâs scale with the latent variableâs scale. The presumption that the remainder of the model remains constant or unchanging may require discounting indirect effects that might, in the real world, be simultaneously prompted by a real unit increase. And the unit increase itself might be inconsistent with what is possible in the real world because there may be no known way to change the causal variableâs value. If a model adjusts for measurement errors, the adjustment permits interpreting latent-level effects as referring to variations in true scores.
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and the power to detect the misspecification. Researchers seeking to learn from their modeling (including potentially learning their model requires adjustment or replacement) will strive for as large a sample size as permitted by funding and by their assessment of likely population-based causal heterogeneity/homogeneity. If the available N is huge, modeling sub-sets of cases can control for variables that might otherwise disrupt causal homogeneity. Researchers fearing they might have to report their modelâs deficiencies are torn between wanting a larger N to provide sufficient power to detect structural coefficients of interest, while avoiding the power capable of signaling model-data inconsistency. The huge variation in model structures and data characteristics suggests adequate sample sizes might be usefully located by considering other researchersâ experiences (both good and bad) with models of comparable size and complexity that have been estimated with similar data.
227:. The endogenous latent variables are the true-score variables postulated as receiving effects from at least one other modeled variable. Each endogenous variable is modeled as the dependent variable in a regression-style equation. The exogenous latent variables are background variables postulated as causing one or more of the endogenous variables and are modeled like the predictor variables in regression-style equations. Causal connections among the exogenous variables are not explicitly modeled but are usually acknowledged by modeling the exogenous variables as freely correlating with one another. The model may include intervening variables â variables receiving effects from some variables but also sending effects to other variables. As in regression, each endogenous variable is assigned a residual or error variable encapsulating the effects of unavailable and usually unknown causes. Each latent variable, whether
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that each latent variable's values must fall somewhere along the observable scale possessed by one of the indicators. The 1.0 effect connecting a latent to an indicator specifies that each real unit increase or decrease in the latent variable's value results in a corresponding unit increase or decrease in the indicator's value. It is hoped a good indicator has been chosen for each latent, but the 1.0 values do not signal perfect measurement because this model also postulates that there are other unspecified entities causally impacting the observed indicator measurements, thereby introducing measurement error. This model postulates that separate measurement errors influence each of the two indicators of latent intelligence, and each indicator of latent achievement. The unlabeled arrow pointing to academic performance acknowledges that things other than intelligence can also influence academic performance.
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merely the modelâs estimated coefficients. Whether a model fits the data, and/or how a model came to fit the data, are paramount for interpretation. Data fit obtained by exploring, or by following successive modification indices, does not guarantee the model is wrong but raises serious doubts because these approaches are prone to incorrectly modeling data features. For example, exploring to see how many factors are required preempts finding the data are not factor structured, especially if the factor model has been âpersuadedâ to fit via inclusion of measurement error covariances. Dataâs ability to speak against a postulated model is progressively eroded with each unwarranted inclusion of a âmodification index suggestedâ effect or error covariance. It becomes exceedingly difficult to recover a proper model if the initial/base model contains several misspecifications.
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of SEM's economic branch led to procedural and terminological differences, though deep mathematical and statistical connections remain. The economic version of SEM can be seen in SEMNET discussions of endogeneity, and in the heat produced as Judea Pearl's approach to causality via directed acyclic graphs (DAG's) rubs against economic approaches to modeling. Discussions comparing and contrasting various SEM approaches are available but disciplinary differences in data structures and the concerns motivating economic models make reunion unlikely. Pearl extended SEM from linear to nonparametric models, and proposed causal and counterfactual interpretations of the equations. Nonparametric SEMs permit estimating total, direct and indirect effects without making any commitment to linearity of effects or assumptions about the distributions of the error terms.
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contribute to testing the original modelâs structure because the few new and focused effect coefficients must work in coordination with the modelâs original direct and indirect effects to coordinate the new indicators with the original indicators. If the original modelâs structure was problematic, the sparse new causal connections will be insufficient to coordinate the new indicators with the original indicators, thereby signaling the inappropriateness of the original modelâs coefficients through model-data inconsistency. The correlational constraints grounded in null/zero effect coefficients, and coefficients assigned fixed nonzero values, contribute to both model testing and coefficient estimation, and hence deserve acknowledgment as the scaffolding supporting the estimates and their interpretation.
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publishing models intentionally burying evidence of model-data inconsistency under an MDI (a mound of distracting indices). There seems no general justification for why a researcher should "accept" a causally wrong model, rather than attempting to correct detected misspecifications. And some portions of the literature seems not to have noticed that "accepting a model" (on the basis of "satisfying" an index value) suffers from an intensified version of the criticism applied to "acceptance" of a null-hypothesis. Introductory statistics texts usually recommend replacing the term "accept" with "failed to reject the null hypothesis" to acknowledge the possibility of Type II error. A Type III error arises from "accepting" a model hypothesis when the current data are sufficient to reject the model.
153:, and closed form algebraic calculations, as iterative solution search techniques were limited in the days before computers. The convergence of two of these developmental streams (factor analysis from psychology, and path analysis from sociology via Duncan) produced the current core of SEM. One of several programs Karl Jöreskog developed at Educational Testing Services, LISREL 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). 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.
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leads to concern for, and controversy over, the minimum number of indicators required to support a latent variable in a structural equation model. Researchers tied to factor tradition can be persuaded to reduce the number of indicators to three per latent variable, but three or even two indicators may still be inconsistent with a proposed underlying factor common cause. Hayduk and
Littvay (2012) discussed how to think about, defend, and adjust for measurement error, when using only a single indicator for each modeled latent variable. Single indicators have been used effectively in SE models for a long time, but controversy remains only as far away as a reviewer who has considered measurement from only the factor analytic perspective.
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the estimate-value maximizing the match with the data, or minimizing the differences from the data. 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 â whether the data are the variables' covariances/correlations, or the cases' actual values on the indicator variables. 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.
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precisely what makes this happens remains unspecified because a single effect coefficient does not contain sub-components available for integration into a structured story of how that effect arises. A more fine-grained SE model incorporating variables intervening between the cause and effect would be required to provide features constituting a story about how any one effect functions. Until such a model arrives each estimated direct effect retains a tinge of the unknown, thereby invoking the essence of a theory. A parallel essential unknownness would accompany each estimated coefficient in even the more fine-grained model, so the sense of fundamental mystery is never fully eradicated from SE models.
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values like the 1.0 values in Figure 2 that provide a scales for the latent variables, or values of 0.0 which assert causal disconnections such as the assertion of no-direct-effects (no arrows) pointing from
Academic Achievement to any of the four scales in Figure 1. SEM programs provide estimates and tests of the free coefficients, while the fixed coefficients contribute importantly to testing the overall model structure. Various kinds of constraints between coefficients can also be used. 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.
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inconsistency? Or, what weight should be given to indexes which show close or not-so-close data fit for some models? Or, should we be especially lenient toward, and ârewardâ, parsimonious models that are inconsistent with the data? Or, given that the RMSEA condones disregarding some real ill fit for each model degree of freedom, doesnât that mean that people testing models with null-hypotheses of non-zero RMSEA are doing deficient model testing? Considerable variation in statistical sophistication is required to cogently address such questions, though responses will likely center on the non-technical matter of whether or not researchers are required to report and respect evidence.
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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. The estimates in data-inconsistent models do not necessarily become "obviously wrong" by becoming statistically strange, or wrongly signed according to theory. 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. Failing models remain interpretable, but only as interpretations that conflict with available evidence.
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free model coefficients took on the estimated values. 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. 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).
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model mis-specification underpins more recent concern for addressing âendogeneityâ â a style of model mis-specification that interferes with estimation due to lack of independence of error/residual variables. In general, the controversy over the causal nature of structural equation models, including factor-models, has also been declining. Stan Mulaik, a factor-analysis stalwart, has acknowledged the causal basis of factor models. The comments by Bollen and Pearl regarding myths about causality in the context of SEM reinforced the centrality of causal thinking in the context of SEM.
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example, makes it easy to overlook that a researcher might begin with one terrible model and one atrocious model, and end by retaining the structurally terrible model because some index reports it as better fitting than the atrocious model. It is unfortunate that even otherwise strong SEM texts like Kline (2016) remain disturbingly weak in their presentation of model testing. Overall, the contributions that can be made by structural equation modeling depend on careful and detailed model assessment, even if a failing model happens to be the best available.
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coefficients to be estimated, the number of modeled variables, and Monte Carlo simulations addressing specific model coefficients. Sample size recommendations based on the ratio of the number of indicators to latents are factor oriented and do not apply to models employing single indicators having fixed nonzero measurement error variances. Overall, for moderate sized models without statistically difficult-to-estimate coefficients, the required sample sizes (Nâs) seem roughly comparable to the Nâs required for a regression employing all the indicators.
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carry effects from background variables through intervening variables to the downstream dependent variables. SEM interpretations encourage understanding how multiple worldly causal pathways can work in coordination, or independently, or even counteract one another. Direct effects may be counteracted (or reinforced) by indirect effects, or have their correlational implications counteracted (or reinforced) by the effects of common causes. The meaning and interpretation of specific estimates should be contextualized in the full model.
235:"simplifying" model specification via diagrams or by using equations permitting user-selected variable names, re-convert the user's model into some standard matrix-algebra form in the background. The "simplifications" are achieved by implicitly introducing default program "assumptions" about model features with which users supposedly need not concern themselves. Unfortunately, these default assumptions easily obscure model components that leave unrecognized issues lurking within the model's structure, and underlying matrices.
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might have done to attain a structurally-improved understanding of the disciplineâs substance. The discipline ends up paying a real costs for index-based displacement of evidence of model misspecification. 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.
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the AIC (Akaike
Information Criterion) can be found in most SEM introductions. For each measure of fit, a decision as to what represents a good-enough fit between the model and the data reflects the researcher's modeling objective (perhaps challenging someone else's model, or improving measurement); whether or not the model is to be claimed as having been "tested"; and whether the researcher is comfortable "disregarding" evidence of the index-documented degree of ill fit.
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substantive context for the measured variables permitted clear causal, not merely predictive, understandings. O. D. Duncan introduced SEM to the social sciences in his 1975 book and SEM blossomed in the late 1970's and 1980's when increasing computing power permitted practical model estimation. In 1987 Hayduk provided the first book-length introduction to structural equation modeling with latent variables, and this was soon followed by Bollen's popular text (1989).
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dichotomous variables create more estimation difficulties than exogenous dichotomous variables). Most SEM programs provide several options for what is to be maximized or minimized to obtain estimates the model's coefficients. The choices often include maximum likelihood estimation (MLE), full information maximum likelihood (FIML), ordinary least squares (OLS), weighted least squares (WLS), diagonally weighted least squares (DWLS), and two stage least squares.
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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. 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.
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observed variables that would be observed if the estimated model effects actually controlled the observed variables' values. The "fit" of a model reports match or mismatch between the model-implied relationships (often covariances) and the corresponding observed relationships among the variables. Large and significant differences between the data and the model's implications signal problems. The probability accompanying a
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tended to defend fit-indexing rather than fit-testing. These discussions led to a target article in
Personality and Individual Differences by Paul Barrett who said: âIn fact, I would now recommend banning ALL such indices from ever appearing in any paper as indicative of model âacceptabilityâ or âdegree of misfitâ.â (page 821). Barrettâs article was also accompanied by commentary from both perspectives.
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428:. The fallaciousness of their claim that close-fit should be treated as good enough was demonstrated by Hayduk, Pazkerka-Robinson, Cummings, Levers and Beres who demonstrated a fitting model for Browne, et al.'s own data by incorporating an experimental feature Browne, et al. overlooked. The fault was not in the math of the indices or in the over-sensitivity of
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causal connections linking the postulated latent variables to variables that can be observed and whose values are available in some data set. Variations among the styles of latent causal connections, variations among the observed variables measuring the latent variables, and variations in the statistical estimation strategies result in the SEM toolkit including
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replicate data. Replication helps detect issues such as data mistakes (made by different research groups), but is especially weak at detecting misspecifications after exploratory model modification â as when confirmatory factor analysis (CFA) is applied to a random second-half of data following exploratory factor analysis (EFA) of first-half data.
70:) is a diverse set of methods used by scientists doing both observational and experimental research. SEM is used mostly in the social and behavioral sciences but it is also used in epidemiology, business, and other fields. A definition of SEM is difficult without reference to technical language, but a good starting place is the name itself.
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Viewing factor analysis as a data-reduction technique deemphasizes testing, which contrasts with path analytic appreciation for testing postulated causal connections â where the test result might signal model misspecification. The friction between factor analytic and path analytic traditions continue to surface in the literature.
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latent IQ variance is fixed at 1 to provide scale to the model. Figure 1 depicts measurement errors influencing each indicator of latent intelligence and each indicator of latent achievement. Neither the indicators nor the measurement errors of the indicators are modeled as influencing the latent variables.
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These discussions fueled disagreement over whether or not structural equation models should be tested for consistency with the data, and model testing became the next focus of discussions. Scholars having path-modeling histories tended to defend careful model testing while those with factor-histories
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or out-of-sample predictive power, change the interpretation criteria by diminishing concern for whether or not the modelâs coefficients have worldly counterparts. The fundamental features differentiating the five PLS modeling perspectives discussed by Rigdon, Sarstedt and Ringle point to differences
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Careful interpretation of both failing and fitting models can provide research advancement. To be dependable, the model should investigate academically informative causal structures, fit applicable data with understandable estimates, and not include vacuous coefficients. Dependable fitting models are
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test). Beyond-chance model-data inconsistency challenges both the coefficient estimates and the model's capacity for adjudicating the model's structure, irrespective of whether the inconsistency originates in problematic data, inappropriate statistical estimation, or incorrect model specification.
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One common problem is that a coefficient's estimated value may be underidentified because it is insufficiently constrained by the model and data. No unique best-estimate exists unless the model and data together sufficiently constrain or restrict a coefficient's value. For example, the magnitude of a
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There is a limit to how many coefficients can be estimated in a model. If there are fewer data points than the number of estimated coefficients, the resulting model is said to be "unidentified" and no coefficient estimates can be obtained. Reciprocal effect, and other causal loops, may also interfere
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A great advantage of SEM is that all of these measurements and tests occur simultaneously in one statistical estimation procedure, where all the model coefficients are calculated using all information from the observed variables. This means the estimates are more accurate than if a researcher were to
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An additional controversy that touched the fringes of the previous controversies awaits ignition. Factor models and theory-embedded factor structures having multiple indicators tend to fail, and dropping weak indicators tends to reduce the model-data inconsistency. Reducing the number of indicators
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The multiple ways of conceptualizing PLS models complicate interpretation of PLS models. Many of the above comments are applicable if a PLS modeler adopts a realist perspective by striving to ensure their modeled indicators combine in a way that matches some existing but unavailable latent variable.
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Causal interpretations of SE models are the clearest and most understandable but those interpretations will be fallacious/wrong if the modelâs structure does not correspond to the worldâs causal structure. Consequently, interpretation should address the overall status and structure of the model, not
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evidence of model-data inconsistency was too statistically solid to be dislodged or discarded, but people could at least be provided a way to distract from the "disturbing" evidence. Career-profits can still be accrued by developing additional indices, reporting investigations of index behavior, and
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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. endogenous
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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. Some, but not all, results are obtained without the "inconvenience"
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Traces of the historical convergence of the factor analytic and path analytic traditions persist as the distinction between the measurement and structural portions of models; and as continuing disagreements over model testing, and whether measurement should precede or accompany structural estimates.
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Though declining, traces of these controversies are scattered throughout the SEM literature, and you can easily incite disagreement by asking: What should be done with models that are significantly inconsistent with the data? Or by asking: Does model simplicity override respect for evidence of data
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A briefer controversy focused on competing models. Comparing competing models can be very helpful but there are fundamental issues that cannot be resolved by creating two models and retaining the better fitting model. The statistical sophistication of presentations like Levy and
Hancock (2007), for
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The caution appearing in the Model
Assessment section warrants repeat. Interpretation should be possible whether a model is or is not consistent with the data. The estimates report how the world would appear to someone believing the model â even if that belief is unfounded because the model happens
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SEM interpretations depart most radically from regression interpretations when a network of causal coefficients connects the latent variables because regressions do not contain estimates of indirect effects. SEM interpretations should convey the consequences of the patterns of indirect effects that
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Direct-effect estimates are interpreted in parallel to the interpretation of coefficients in regression equations but with causal commitment. Each unit increase in a causal variableâs value is viewed as producing a change of the estimated magnitude in the dependent variableâs value given control or
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The following table provides references documenting these, and other, features for some common indices: the RMSEA (Root Mean Square Error of
Approximation), SRMR (Standardized Root Mean Squared Residual), CFI (Confirmatory Fit Index), and the TLI (the Tucker-Lewis Index). Additional indices such as
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whether the researcher knowingly agrees to disregard evidence pointing to the kinds of misspecifications on which the index criteria were based. (If the index criterion is based on simulating a missing factor loading or two, using that criterion acknowledges the researcher's willingness to accept a
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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. Models with different causal structures
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Structural equation models attempt to mirror the worldly forces operative for causally homogeneous cases â namely cases enmeshed in the same worldly causal structures but whose values on the causes differ and who therefore possess different values on the outcome variables. Causal homogeneity can be
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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. The prolonged separation
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connect to one another. Structural equation models often contain postulated causal connections among some latent variables (variables thought to exist but which can't be directly observed). Additional causal connections link those latent variables to observed variables whose values appear in a data
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Figure 2. An example structural equation model before estimation. Similar to Figure 1 but without standardized values and fewer items. Because intelligence and academic performance are merely imagined or theory-postulated variables, their precise scale values are unknown, though the model specifies
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Figure 1. An example structural equation model after estimation. Latent variables are sometimes indicated with ovals while observed variables are shown in rectangles. Residuals and variances are sometimes drawn as double-headed arrows (shown here) or single arrows and a circle (as in Figure 2). The
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Structural equation modeling is fraught with controversies. Researchers from the factor analytic tradition commonly attempt to reduce sets of multiple indicators to fewer, more manageable, scales or factor-scores for later use in path-structured models. This constitutes a stepwise process with the
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Adding new latent variables entering or exiting the original model at a few clear causal locations/variables contributes to detecting model misspecifications which could otherwise ruin coefficient interpretations. The correlations between the new latentâs indicators and all the original indicators
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The larger the sample size, the greater the likelihood of including cases that are not causally homogeneous. Consequently, increasing N to improve the likelihood of being able to report a desired coefficient as statistically significant, simultaneously increases the risk of model misspecification,
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structural equation modeling program, "Why have you then added GFI?" to your LISREL program, Joreskog replied "Well, users threaten us saying they would stop using LISREL if it always produces such large chi-squares. So we had to invent something to make people happy. GFI serves that purpose." The
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probability reports it would be unlikely for the current data to have arisen if the current model structure constituted the real population causal forces â with the remaining differences attributed to random sampling variations. Browne, McCallum, Kim, Andersen, and Glaser presented a factor model
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A stronger effect connecting two latent variables implies the indicators of those latents should be more strongly correlated. Hence, a reasonable estimate of a latent's effect will be whatever value best matches the correlations between the indicators of the corresponding latent variables â namely
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Model coefficients fixed at zero, 1.0, or other values, do not require estimation because they already have specified values. 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
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at some value. The free coefficients may be postulated effects the researcher wishes to test, background correlations among the exogenous variables, or the variances of the residual or error variables providing additional variations in the endogenous latent variables. The fixed coefficients may be
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The LISREL program assigned Greek names to the elements in a set of matrices to keep track of the various model components. These names became relatively standard notation, though the notation has been extended and altered to accommodate a variety of statistical considerations. Texts and programs
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must be understood to mean "a model that conveys causal assumptions", not necessarily a model that produces validated causal conclusionsâmaybe it does maybe it does not. Collecting data at multiple time points and using an experimental or quasi-experimental design can help rule out certain rival
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Whether or not researchers are committed to seeking the worldâs structure is a fundamental concern. Displacing test evidence of model-data inconsistency by hiding it behind index claims of acceptable-fit, introduces the discipline-wide cost of diverting attention away from whatever the discipline
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Replication is unlikely to detect misspecified models which inappropriately-fit the 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
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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
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The controversy over model testing declined as clear reporting of significant model-data inconsistency becomes mandatory. Scientists do not get to ignore, or fail to report, evidence just because they do not like what the evidence reports. The requirement of attending to evidence pointing toward
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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. Researchers confronting data-inconsistent models can easily free
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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. The equations were estimated like ordinary regression equations but the
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The statistical insignificance of an effect estimate indicates the estimate could rather easily arise as a random sampling variation around a null/zero effect, so interpreting the estimate as a real effect becomes equivocal. As in regression, the proportion of each dependent variableâs variance
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SE model interpretation should connect specific model causal segments to their variance and covariance implications. A single direct effect reports that the variance in the independent variable produces a specific amount of variation in the dependent variableâs values, but the causal details of
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but the postulated structuring can also be presented using diagrams containing arrows as in
Figures 1 and 2. The causal structures imply that specific patterns should appear among the values of the observed variables. This makes it possible to use the connections between the observed variables'
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This logical weakness renders all fit indices "unhelpful" whenever a structural equation model is significantly inconsistent with the data, but several forces continue to propagate fit-index use. For example, Dag Sorbom reported that when someone asked Karl Joreskog, the developer of the first
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Researchers agree samples should be large enough to provide stable coefficient estimates and reasonable testing power but there is no general consensus regarding specific required sample sizes, or even how to determine appropriate sample sizes. Recommendations have been based on the number of
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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. criteria based on simulation of omitted factor loadings may not be appropriate for misspecification resulting from failure to include
378:
Research claiming to test or "investigate" a theory requires attending to beyond-chance model-data inconsistency. Estimation adjusts the model's free coefficients to provide the best possible fit to the data. The output from SEM programs includes a matrix reporting the relationships among the
872:
Even if each modeled effect is unknown beyond the identity of the variables involved and the estimated magnitude of the effect, the structures linking multiple modeled effects provide opportunities to express how things function to coordinate the observed variables â thereby providing useful
116:
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. Because a postulated model such as Figure 1 may not correspond to the worldly forces
525:
and whether the researcher has considered both alpha (Type I) and beta (Type II) errors in making their index-based decisions (E.g. if the model is significantly data-inconsistent, the "tolerable" amount of inconsistency is likely to differ in the context of medical, business, social and
896:
Interpretations become progressively more complex for models containing interactions, nonlinearities, multiple groups, multiple levels, and categorical variables. Effects touching causal loops, reciprocal effects, or correlated residuals also require slightly revised interpretations.
2075:
Satorra, A.; and Bentler, P. M. (1994) âCorrections to test statistics and standard errors in covariance structure analysisâ. In A. von Eye and C. C. Clogg (Eds.), Latent variables analysis: Applications for developmental research (pp. 399â419). Thousand Oaks, CA:
539:
A fundamental test of fit used in the calculation of many other fit measures. It is a function of the discrepancy between the observed covariance matrix and the model-implied covariance matrix. Chi-square increases with sample size only if the model is detectably
432:
testing. The fault was in Browne, MacCallum, and the other authors forgetting, neglecting, or overlooking, that the amount of ill fit cannot be trusted to correspond to the nature, location, or seriousness of problems in a model's specification.
669:
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
2526:
Leitgöb, Heinz; Seddig, Daniel; Asparouhov, Tihomir; Behr, Dorothée; Davidov, Eldad; De Roover, Kim; Jak, Suzanne; Meitinger, Katharina; Menold, Natalja; Muthén, Bengt; Rudnev, Maksim; Schmidt, Peter; van de Schoot, Rens (February 2023).
496:
whether criterion values for the index have been investigated for models structured like the researcher's model (e.g. index criterion based on factor structured models are only appropriate if the researcher's model actually is factor
2203:
Hayduk, L. A.; Cummings, G.; Stratkotter, R.; Nimmo, M.; Grugoryev, K.; Dosman, D.; Gillespie, M.; Pazderka-Robinson, H. (2003) âPearlâs D-separation: One more step into causal thinking.â Structural Equation Modeling. 10 (2):
391:
probability reports it would be unlikely for the current data to have arisen if the modeled structure constituted the real population causal forces â with the remaining differences attributed to random sampling variations.
2301:
Hayduk, L.A.; Cummings, G.; Boadu, K.; Pazderka-Robinson, H.; Boulianne, S. (2007) âTesting! testing! one, two, three â Testing the theory in structural equation models!â Personality and Individual Differences. 42 (5):
2085:
Sorbom, D. "xxxxx" in Cudeck, R; du Toit R.; Sorbom, D. (editors) (2001) Structural Equation Modeling: Present and Future: Festschrift in Honor of Karl Joreskog. Scientific Software International: Lincolnwood,
411:"Accepting" failing models as "close enough" is also not a reasonable alternative. A cautionary instance was provided by Browne, MacCallum, Kim, Anderson, and Glaser who addressed the mathematics behind why the
2255:
Hayduk, L.A.; Stratkotter, R.; Rovers, M.W. (1997) âSexual Orientation and the Willingness of Catholic Seminary Students to Conform to Church Teachings.â Journal for the Scientific Study of Religion. 36 (3):
889:
to be wrong. Interpretation should acknowledge that the model coefficients may or may not correspond to âparametersâ â because the modelâs coefficients may not have corresponding worldly structural features.
460:
increase with N provides the good-news of increasing statistical power to detect model misspecification (namely power to detect Type II error). Some kinds of important misspecifications cannot be detected by
2466:
Zyphur, Michael J.; Allison, Paul D.; Tay, Louis; Voelkle, Manuel C.; Preacher, Kristopher J.; Zhang, Zhen; Hamaker, Ellen L.; Shamsollahi, Ali; Pierides, Dean C.; Koval, Peter; Diener, Ed (October 2020).
231:, is thought of as containing the cases' true-scores on that variable, and these true-scores causally contribute valid/genuine variations into one or more of the observed/reported indicator variables.
5483:
2213:
Hayduk, L.A. (2006) âBlocked-Error-R2: A conceptually improved definition of the proportion of explained variance in models containing loops or correlated residuals.â Quality and Quantity. 40: 629-649.
1606:
Hayduk, L.; Glaser, D.N. (2000) "Doing the Four-Step, Right-2-3, Wrong-2-3: A Brief Reply to Mulaik and Millsap; Bollen; Bentler; and Herting and Costner". Structural Equation Modeling. 7 (1): 111-123.
368:, (Negative variances, and correlations exceeding 1.0 or -1.0, are impossible. Statistically possible estimates that are inconsistent with theory may also challenge theory, and our understanding.)
1663:
Imbens, G.W. (2020). "Potential outcome and directed acyclic graph approaches to causality: Relevance for empirical practice in economics". Journal of Economic Literature. 58 (4): 11-20-1179.
419:
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. A small
612:
395:
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
1548:
Jöreskog, Karl; Gruvaeus, Gunnar T.; van Thillo, Marielle. (1970) ACOVS: A General Computer Program for Analysis of Covariance Structures. Princeton, N.J.; Educational Testing Services.
511:
whether satisfying criterion values on pairs of indices are required (e.g. Hu and Bentler report that some common indices function inappropriately unless they are assessed together.);
2110:
Hu, L.; Bentler, P.M. (1999) "Cutoff criteria for fit indices in covariance structure analysis: Conventional criteria versus new alternatives." Structural Equation Modeling. 6: 1-55.
89:
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.
2158:
Steiger, J. H.; and Lind, J. (1980) "Statistically Based Tests for the Number of Common Factors." Paper presented at the annual meeting of the Psychometric Society, Iowa City.
5476:
2185:
Herting, R.H.; Costner, H.L. (2000) âAnother perspective on âThe proper number of factorsâ and the appropriate number of steps.â Structural Equation Modeling. 7 (1): 92-110.
1582:
Jöreskog, Karl; Sorbom, Dag. (1976) LISREL III: Estimation of Linear Structural Equation Systems by Maximum Likelihood Methods. Chicago: National Educational Resources, Inc.
362:, (Substantive assessments may be devastated: by violating assumptions, by using an inappropriate estimator, and/or by encountering non-convergence of iterative estimators.)
2390:
2320:
Levy, R.; Hancock, G.R. (2007) âA framework of statistical tests for comparing mean and covariance structure models.â Multivariate Behavioral Research. 42(1): 33-66.
2194:
Hayduk, L. (1987) Structural Equation Modeling with LISREL: Essentials and Advances, page 20. Baltimore, Johns Hopkins University Press. ISBN 0-8018-3478-3 Page 20
885:
should be used if the dependent variable is involved in reciprocal or looped effects, or if it has an error variable correlated with any predictorâs error variable.
387:) test is the probability that the data could arise by random sampling variations if the estimated model constituted the real underlying population forces. A small
1957:
Browne, M.W.; MacCallum, R.C.; Kim, C.T.; Andersen, B.L.; Glaser, R. (2002) "When fit indices and residuals are incompatible." Psychological Methods. 7: 403-421.
1779:
Rigdon, E. (1995). "A necessary and sufficient identification rule for structural models estimated in practice." Multivariate Behavioral Research. 30 (3): 359-383.
5545:
5469:
4542:
1260:
5047:
5197:
4821:
1594:
Hayduk, L.; Glaser, D.N. (2000) "Jiving the Four-Step, Waltzing Around Factor Analysis, and Other Serious Fun". Structural Equation Modeling. 7 (1): 1-35.
2934:
5554:
3462:
2715:
Hu, Li-tze; Bentler, Peter M (1999). "Cutoff criteria for fit indexes in covariance structure analysis: Conventional criteria versus new alternatives".
918:
hypotheses but even a randomized experiments cannot fully rule out threats to causal claims. No research design can fully guarantee causal structures.
202:
facilitated by case selection, or by segregating cases in a multi-group model. A model's specification is not complete until the researcher specifies:
165:
of understanding experimental design, statistical control, the consequences of sample size, and other features contributing to good research design.
5559:
4595:
5034:
2167:
Steiger, J. H. (1990) "Structural Model Evaluation and Modification: An Interval Estimation Approach". Multivariate Behavioral Research 25:173-180.
1284:
Hayduk, L. (1987) Structural Equation Modeling with LISREL: Essentials and Advances. Baltimore, Johns Hopkins University Press. ISBN 0-8018-3478-3
5992:
2785:
2835:
Quintana, Stephen M.; Maxwell, Scott E. (1999). "Implications of Recent Developments in Structural Equation Modeling for Counseling Psychology".
2237:
Entwisle, D.R.; Hayduk, L.A.; Reilly, T.W. (1982) Early Schooling: Cognitive and Affective Outcomes. Baltimore: Johns Hopkins University Press.
39:
3070:
3035:
2774:
2605:
2529:"Measurement invariance in the social sciences: Historical development, methodological challenges, state of the art, and future perspectives"
1695:
1560:"LISREL: A General Computer Program for Estimating a Linear Structural Equation System Involving Multiple Indicators of Unmeasured Variables"
1507:
Wolfle, L.M. (1999) "Sewall Wright on the method of path coefficients: An annotated bibliography" Structural Equation Modeling: 6(3):280-291.
1336:
1236:
1193:
1160:
1127:
913:
Caution should be taken when making claims of causality even when experiments or time-ordered investigations have been undertaken. The term
5887:
5527:
3457:
3157:
1413:"An overview of structural equation modeling: Its beginnings, historical development, usefulness and controversies in the social sciences"
456:
increases with N that itself is a sign that something is detectably problematic. And second, for models that are detectably misspecified,
6078:
4061:
3209:
1064:
106:
3107:
5867:
5517:
2755:
243:
228:
217:
212:
and which coefficients will be given fixed/unchanging values (e.g. to provide measurement scales for latent variables as in Figure 2).
1765:
Kline, Rex. (2016) Principles and Practice of Structural Equation Modeling (4th ed). New York, Guilford Press. ISBN 978-1-4625-2334-4
374:. (The estimation process minimizes the differences between the model and data but important and informative differences may remain.)
5787:
4844:
4736:
3013:
2957:
2943:
2912:
1303:
415:
test can have (though it does not always have) considerable power to detect model misspecification. The probability accompanying a
2066:
Barrett, P. (2007). "Structural equation modeling: Adjudging model fit." Personality and Individual Differences. 42 (5): 815-824.
647:
Fit index where a value of zero indicates the best fit. Guidelines for determining a "close fit" using RMSEA are highly contested.
197:
and the cases for which values of the variables will be available (kids? workers? companies? countries? cells? accidents? cults?).
5593:
5449:
5022:
4896:
1069:
3123:
3045:
Lewis-Beck, Michael; Bryman, Alan E.; Bryman, Emeritus Professor Alan; Liao, Tim Futing (2004). "Structural Equation Modeling".
465:, so any amount of ill fit beyond what might be reasonably produced by random variations warrants report and consideration. The
6068:
5080:
4741:
4486:
3857:
3447:
2406:"Exploratory Structural Equation Modeling: An Integration of the Best Features of Exploratory and Confirmatory Factor Analysis"
98:
4071:
356:, (Models are unpersuasive if they omit features required by a theory, or contain coefficients inconsistent with that theory.)
5829:
5131:
4343:
4150:
4039:
3997:
2176:
Browne, M.W.; Cudeck, R. (1992) "Alternate ways of assessing model fit." Sociological Methods and Research. 21(2): 230-258.
652:
641:
3236:
901:
rarer than failing models or models inappropriately bludgeoned into fitting, but appropriately-fitting models are possible.
508:
whether the latest, not outdated, index criteria are being used (because the criteria for some indices tightened over time);
337:
Model assessment depends on the theory, the data, the model, and the estimation strategy. Hence model assessments consider:
5374:
4333:
3129:
150:
4383:
2225:
Millsap, R.E. (2007) âStructural equation modeling made difficult.â Personality and Individual Differences. 42: 875-881.
1824:"Describing qualitative research undertaken with randomised controlled trials in grant proposals: A documentary analysis"
658:
The SRMR is a popular absolute fit indicator. Hu and Bentler (1999) suggested .08 or smaller as a guideline for good fit.
6015:
5915:
5905:
5824:
5769:
4925:
4859:
4849:
4718:
4590:
4557:
4338:
4168:
2586:"Multilevel structural equation modeling for intensive longitudinal data: A practical guide for personality researchers"
2246:
Hayduk, L.A. (1994). âPersonal space: Understanding the simplex model.â Journal of Nonverbal Behavior., 18 (3): 245-260.
1075:
545:
142:
94:
4994:
4295:
1801:
Hayduk, L. (1996) LISREL Issues, Debates, and Strategies. Baltimore, Johns Hopkins University Press. ISBN 0-8018-5336-2
6042:
5857:
5269:
5070:
4049:
3718:
3182:
2871:
Bagozzi, Richard P; Yi, Youjae (2011). "Specification, evaluation, and interpretation of structural equation models".
1523:
Duncan, Otis Dudley. (1975). Introduction to Structural Equation Models. New York: Academic Press. ISBN 0-12-224150-9.
5154:
5121:
2332:"Review essay on Rex B. Kline's Principles and Practice of Structural Equation Modeling: Encouraging a fifth edition"
2311:
Mulaik, S.A. (2009) Foundations of Factor Analysis (second edition). Chapman and Hall/CRC. Boca Raton, pages 130-131.
1013:
Multiple group modelling with or without constraints between groups (genders, cultures, test forms, languages, etc.)
1217:
Pearl, J. (2009). Causality: Models, Reasoning, and Inference. Second edition. New York: Cambridge University Press.
6073:
5537:
5126:
4869:
4628:
4534:
4514:
4422:
4133:
3951:
3434:
3306:
255:
102:
4300:
4066:
3924:
1909:"Should researchers use single indicators, best indicators, or multiple indicators in structural equation models?"
556:
137:
Different yet mathematically related modeling approaches developed in psychology, sociology, and economics. Early
6063:
5882:
5709:
5673:
5642:
4886:
4654:
4375:
4229:
4158:
4078:
3936:
3917:
3625:
3346:
2904:
1970:
Hayduk, Leslie A.; Pazderka-Robinson, Hannah; Cummings, Greta G.; Levers, Merry-Jo D.; Beres, Melanie A. (2005).
1714:
Borsboom, Denny; Mellenbergh, Gideon J.; Van Heerden, Jaap (2003). "The theoretical status of latent variables".
5862:
4999:
3081:
5740:
5704:
5632:
5522:
5504:
5369:
5136:
4684:
4649:
4613:
4398:
3840:
3749:
3708:
3620:
3311:
3150:
1026:
436:
Many researchers tried to justify switching to fit-indices, rather than testing their models, by claiming that
4406:
4390:
2920:& Bonett, D.G. (1980), "Significance tests and goodness of fit in the analysis of covariance structures",
493:
whether data concerns have been addressed (to ensure data mistakes are not driving model-data inconsistency);
250:
showing the causal connections between the latent variables and the indicators. Exploratory and confirmatory
5603:
5278:
4891:
4831:
4768:
4128:
3990:
3980:
3830:
3744:
1058:
350:, (It makes no sense to estimate one model if the data cases reflect two or more different causal networks.)
319:
5039:
4976:
5956:
5782:
5647:
5637:
5588:
5316:
5246:
4731:
4618:
3615:
3512:
3419:
3298:
3197:
2922:
2405:
996:
985:
424:
they viewed as acceptable despite the model being significantly inconsistent with their data according to
173:
The following considerations apply to the construction and assessment of many structural equation models.
110:
5437:
4315:
6037:
5997:
5961:
5946:
5897:
5841:
5668:
5341:
5283:
5226:
5052:
4945:
4854:
4580:
4464:
4323:
4205:
4197:
4012:
3908:
3886:
3845:
3810:
3777:
3723:
3698:
3653:
3592:
3552:
3354:
3177:
2384:
910:
in PLS modelersâ objectives, and corresponding differences in model features warranting interpretation.
663:
35:
6027:
5420:
4310:
2976:
Haavelmo, Trygve (January 1943). "The Statistical Implications of a System of Simultaneous Equations".
2816:
1678:
Bollen, Kenneth A.; Pearl, Judea (2013). "Eight Myths About Causality and Structural Equation Models".
2012:
Note the correction of .922 to .992, and the correction of .944 to .994 in the Hayduk, et al. Table 1.
1360:
Curran, Patrick J. (2003-10-01). "Have Multilevel Models Been Structural Equation Models All Along?".
54:
45:
6002:
5941:
5928:
5877:
5777:
5699:
5678:
5652:
5264:
4839:
4788:
4764:
4726:
4644:
4623:
4575:
4454:
4432:
4401:
4187:
4138:
4056:
4029:
3985:
3941:
3703:
3479:
3359:
2627:
974:
129:
Structural equation modeling (SEM) began differentiating itself from correlation and regression when
2585:
2584:
Sadikaj, Gentiana; Wright, Aidan G.C.; Dunkley, David M.; Zuroff, David C.; Moskowitz, D.S. (2021),
329:
of sections of text to one or more sub-topic articles which are then summarized in the main article.
6020:
5951:
5836:
5803:
5755:
5745:
5724:
5719:
5598:
5580:
5565:
5496:
5411:
5336:
5259:
4940:
4704:
4697:
4659:
4567:
4547:
4519:
4252:
4118:
4113:
4103:
4095:
3913:
3874:
3764:
3754:
3663:
3442:
3398:
3316:
3241:
3143:
1539:
Bollen, K. (1989). Structural Equations with Latent Variables. New York, Wiley. ISBN 0-471-01171-1.
980:
518:
value, degrees of freedom, and probability will be available for models reporting indices based on
326:
146:
4986:
1471:
Wright, Sewall. (1921) "Correlation and causation". Journal of Agricultural Research. 20: 557-585.
6032:
5936:
5925:
5750:
5425:
5236:
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4811:
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2888:
2852:
2694:
2566:
2528:
2508:
1646:
1393:
1254:
1037:
Structural equation modeling programs differ widely in their capabilities and user requirements.
2027:
Hayduk, Leslie (2014). "Seeing Perfectly Fitting Factor Models That Are Causally Misspecified".
1972:"Structural equation model testing and the quality of natural killer cell activity measurements"
1615:
Westland, J.C. (2015). Structural Equation Modeling: From Paths to Networks. New York, Springer.
5987:
5714:
5624:
5615:
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5301:
5293:
5113:
5104:
5029:
4960:
4816:
4801:
4776:
4664:
4605:
4471:
4459:
4085:
4002:
3946:
3869:
3713:
3635:
3414:
3288:
3066:
3031:
3009:
2953:
2939:
2929:
2908:
2808:
2770:
2751:
2686:
2647:
2601:
2558:
2500:
2433:
2425:
2364:
2003:
1940:
1855:
1731:
1691:
1638:
1442:
1385:
1377:
1342:
1332:
1309:
1299:
1242:
1232:
1189:
1156:
1123:
1007:
625:
444:
probability decreases) with increasing sample size (N). There are two mistakes in discounting
138:
3023:
1822:
Drabble, Sarah J.; o'Cathain, Alicia; Thomas, Kate J.; Rudolph, Anne; Hewison, Jenny (2014).
17:
5683:
5550:
5356:
5311:
5075:
5062:
4955:
4930:
4864:
4796:
4674:
4282:
4175:
4108:
4021:
3968:
3787:
3658:
3452:
3251:
3218:
3119:
The causal interpretation of structural equations (or SEM survival kit) by Judea Pearl 2000.
3058:
3050:
3019:
2985:
2917:
2880:
2844:
2800:
2732:
2724:
2678:
2639:
2593:
2548:
2540:
2490:
2480:
2450:
2417:
2372:
2343:
2282:
2036:
1993:
1983:
1930:
1920:
1882:
1845:
1835:
1723:
1683:
1490:
1432:
1424:
1369:
1181:
1148:
1115:
1087:
384:
191:
what is presumed or hypothesized about the variables' causal connections and disconnections,
2404:
Marsh, Herbert W.; Morin, Alexandre J.S.; Parker, Philip D.; Kaur, Gurvinder (2014-03-28).
30:
This article is about the general structural modeling. For the use of structural models in
5966:
5872:
5808:
5273:
5017:
4879:
4806:
4481:
4355:
4328:
4305:
4274:
3901:
3896:
3850:
3580:
3231:
2421:
2271:"On Comparing Results from CB-SEM and PLS-SEM: Five Perspectives and Five Recommendations"
1052:
534:
469:
model test, possibly adjusted, is the strongest available structural equation model test.
251:
5813:
2363:
Alex James Ing, Alvaro Andrades, Marco Raffaele Cosenza, Jan Oliver Korbel (2024-06-13).
551:
An index of relative model fit: The preferred model is the one with the lowest AIC value.
5461:
5910:
5222:
5217:
3680:
3610:
3256:
2597:
1935:
1908:
1850:
1823:
1437:
1412:
3024:"Non-linear structural equation models: The Kenny-Judd model with interaction effects"
2950:
Structural Equation Modeling with AMOS - Basic Concepts, Applications, and Programming
1998:
1971:
6057:
5982:
5512:
5492:
5379:
5346:
5209:
5170:
4981:
4950:
4414:
4368:
3973:
3675:
3502:
3266:
3261:
2892:
2856:
2570:
2544:
2512:
1873:
MacCallum, Robert (1986). "Specification searches in covariance structure modeling".
1002:
130:
3532:
2698:
5321:
5254:
5231:
5146:
4476:
3772:
3670:
3605:
3547:
3469:
3424:
2968:
2804:
1397:
1046:
31:
2643:
1559:
3132:, a collection of previously used multi-item scales to measure constructs for SEM
2682:
1687:
5570:
5364:
5326:
5009:
4910:
4772:
4585:
4552:
4044:
3961:
3956:
3600:
3557:
3537:
3517:
3507:
3276:
2287:
2270:
1886:
1727:
3054:
2666:
2365:"Integrating Multi-Modal Cancer Data Using Deep Latent Variable Path Modelling"
1373:
1185:
1152:
1119:
1061: â Simultaneous observation and analysis of more than one outcome variable
206:
which effects and/or correlations/covariances are to be included and estimated,
4210:
3690:
3390:
3321:
3271:
3246:
3166:
2884:
2728:
2376:
1494:
1428:
1081:
633:
74:
2848:
2690:
2651:
2504:
2485:
2468:
2429:
2040:
1925:
1642:
1627:"The Cowles Commission's Contributions to Econometrics at Chicago, 1939-1955"
1381:
1346:
1246:
354:
whether the model appropriately represents the theory or features of interest
4363:
4215:
3835:
3630:
3542:
3527:
3522:
3487:
2454:
1840:
1313:
621:
78:
3118:
2812:
2562:
2469:"From Data to Causes I: Building A General Cross-Lagged Panel Model (GCLM)"
2437:
2007:
1944:
1859:
1735:
1446:
1389:
209:
which effects and other coefficients are forbidden or presumed unnecessary,
2737:
1988:
3879:
3497:
3374:
3369:
3364:
3336:
3028:
Advanced structural equation modeling: Concepts, issues, and applications
342:
whether the data contain reasonable measurements of appropriate variables
84:
2786:"Applications of Structural Equation Modeling in Psychological Research"
2495:
254:
models, for example, focus on the causal measurement connections, while
5384:
5085:
2997:
2553:
1650:
1626:
372:
the remaining consistency, or inconsistency, between the model and data
3112:
3108:
Structural equation modeling page under David Garson's StatNotes, NCSU
3062:
2667:"A Review of Eight Software Packages for Structural Equation Modeling"
2628:"SEMDS: An R Package for Structural Equation Multidimensional Scaling"
2348:
2331:
1084: â A network consisting of links or arcs between nodes or factors
5306:
4287:
4261:
4241:
3492:
3283:
2989:
489:
The considerations relevant to using fit indices include checking:
3126:: journal articles and book chapters on structural equation models
2269:
Rigdon, Edward E.; Sarstedt, Marko; Ringle, Christian M. (2017).
145:
estimation centered on Koopman and Hood's (1953) algorithms from
3226:
3006:
Structural Equation Modeling: Concepts, Issues, and Applications
1682:. Handbooks of Sociology and Social Research. pp. 301â328.
5465:
5195:
4762:
4509:
3808:
3578:
3195:
3139:
258:
more closely correspond to SEMs latent structural connections.
27:
Form of causal modeling that fit networks of constructs to data
877:
explained by variations in the modeled causes are provided by
302:
73:
SEM involves a model representing how various aspects of some
3135:
3026:. In Marcoulides, George A.; Schumacker, Randall E. (eds.).
568:
565:
562:
238:
Two main components of models are distinguished in SEM: the
3124:
Structural Equation Modeling Reference List by Jason Newsom
3080:
Schermelleh-Engel, K.; Moosbrugger, H.; MĂŒller, H. (2003),
905:
Non-causal PLS models, such as those focusing primarily on
2142:
2140:
1481:
Wright, Sewall (1934). "The Method of Path Coefficients".
1178:
Encyclopedia of Educational Leadership and Administration
3047:
The SAGE Encyclopedia of Social Science Research Methods
3030:. Thousand Oaks, CA: Sage Publications. pp. 57â88.
2748:
Structural Equation Modeling: Foundations and Extensions
1329:
Structural equation modeling: foundations and extensions
2767:
Principles and Practice of Structural Equation Modeling
1229:
Principles and practice of structural equation modeling
1092:
Pages displaying short descriptions of redirect targets
951:
Extensions, modeling alternatives, and statistical kin
530:
Some of the more commonly used fit statistics include
261:
Modelers specify each coefficient in a model as being
181:
Building or specifying a model requires attending to:
1010:, hierarchical models (e.g. people nested in groups)
559:
194:
what the researcher seeks to learn from the modeling,
5048:
Autoregressive conditional heteroskedasticity (ARCH)
1558:
Jöreskog, Karl Gustav; van Thillo, Mariella (1972).
5975:
5924:
5896:
5850:
5796:
5768:
5733:
5692:
5661:
5623:
5612:
5579:
5536:
5503:
5355:
5292:
5245:
5208:
5163:
5145:
5112:
5103:
5061:
5008:
4969:
4918:
4909:
4830:
4787:
4717:
4683:
4637:
4604:
4566:
4533:
4445:
4354:
4273:
4228:
4196:
4149:
4094:
4020:
4011:
3821:
3763:
3737:
3689:
3644:
3591:
3478:
3433:
3407:
3389:
3345:
3297:
3217:
3208:
360:
whether the estimates are statistically justifiable
3113:Issues and Opinion on Structural Equation Modeling
3082:"Evaluating the fit of structural equation models"
2590:The Handbook of Personality Dynamics and Processes
606:
514:whether a model test is, or is not, available. (A
82:set. The causal connections are represented using
2964:Structural equation models in the social sciences
2626:Vera, José Fernando; Mair, Patrick (2019-09-03).
1049: â Conceptual model in philosophy of science
348:whether the modeled case are causally homogeneous
792:References indicating two-index or paired-index
4596:Multivariate adaptive regression splines (MARS)
1680:Handbook of Causal Analysis for Social Research
1459:
1110:Salkind, Neil J. (2007). "Intelligence Tests".
366:the substantive reasonableness of the estimates
318:may benefit from being shortened by the use of
105:, multi-group modeling, longitudinal modeling,
242:showing potential causal dependencies between
5477:
3151:
121:calculate each part of the model separately.
8:
2389:: CS1 maint: multiple names: authors list (
2221:
2219:
2062:
2060:
2058:
2056:
2054:
2052:
2050:
2022:
2020:
2018:
1965:
1963:
1907:Hayduk, Leslie A.; Littvay, Levente (2012).
1817:
1815:
1813:
1811:
1809:
1807:
607:{\displaystyle {\mathit {AIC}}=2k-2\ln(L)\,}
2873:Journal of the Academy of Marketing Science
1797:
1795:
1793:
1791:
1789:
1787:
1785:
1775:
1773:
1771:
1535:
1533:
1531:
1529:
1519:
1517:
1515:
1513:
452:does not increase with increasing N, so if
216:The latent level of a model is composed of
5620:
5484:
5470:
5462:
5205:
5192:
5109:
4915:
4784:
4759:
4530:
4506:
4234:
4017:
3818:
3805:
3588:
3575:
3214:
3205:
3192:
3158:
3144:
3136:
2935:Structural Equations with Latent Variables
2901:Latent Variable Models and Factor Analysis
1761:
1759:
1757:
1755:
1753:
1751:
1749:
1747:
1745:
1296:Structural equations with latent variables
1280:
1278:
1276:
1274:
1272:
1270:
1259:: CS1 maint: location missing publisher (
1112:Encyclopedia of Measurement and Statistics
2903:Kendall's Library of Statistics, vol. 7,
2899:Bartholomew, D. J., and Knott, M. (1999)
2784:MacCallum, Robert; Austin, James (2000).
2736:
2552:
2494:
2484:
2347:
2286:
2146:
2029:Educational and Psychological Measurement
1997:
1987:
1934:
1924:
1849:
1839:
1709:
1707:
1569:. ETS-RB-72-56 – via US Government.
1436:
1213:
1211:
1209:
1207:
1205:
968:Exploratory Structural Equation Modeling
603:
561:
560:
558:
448:on this basis. First, for proper models,
244:endogenous and exogenous latent variables
113:and hierarchical or multilevel modeling.
2264:
2262:
2233:
2231:
1602:
1600:
1590:
1588:
707:Standardized Root Mean Squared Residual
704:Root Mean Square Error of Approximation
679:
505:model missing a factor loading or two.);
53:
44:
5993:Numerical smoothing and differentiation
2106:
2104:
2102:
2100:
2098:
2096:
2094:
2092:
1902:
1900:
1898:
1896:
1673:
1671:
1669:
1578:
1576:
1102:
653:Standardized Root Mean Squared Residual
642:Root Mean Square Error of Approximation
5122:KaplanâMeier estimator (product limit)
2382:
1567:Research Bulletin: Office of Education
1252:
40:Structural Equation Modeling (journal)
2422:10.1146/annurev-clinpsy-032813-153700
2131:
2119:
1483:The Annals of Mathematical Statistics
777:References proposing revised/changed,
278:Estimation of free model coefficients
7:
5528:Iteratively reweighted least squares
5432:
5132:Accelerated failure time (AFT) model
2410:Annual Review of Clinical Psychology
823:References recommending against use
185:the set of variables to be employed,
5444:
4727:Analysis of variance (ANOVA, anova)
1331:(2nd ed.). Los Angeles: SAGE.
1065:Partial least squares path modeling
780:disagreements over critical values
107:partial least squares path modeling
5546:Pearson product-moment correlation
4822:CochranâMantelâHaenszel statistics
3448:Pearson product-moment correlation
2598:10.1016/b978-0-12-813995-0.00033-9
959:Categorical intervening variables
839:Sample size, power, and estimation
188:what is known about the variables,
25:
3089:Methods of Psychological Research
1078: â Type of statistical model
6026:
5443:
5431:
5419:
5406:
5405:
2545:10.1016/j.ssresearch.2022.102805
1976:BMC Medical Research Methodology
1913:BMC Medical Research Methodology
1828:BMC Medical Research Methodology
1362:Multivariate Behavioral Research
1176:"Structural Equation Modeling".
1143:"Structural Equation Modeling".
1070:Partial least squares regression
1022:Structural Equation Model Trees
1016:Multi-method multi-trait models
956:Categorical dependent variables
307:
169:General steps and considerations
5081:Least-squares spectral analysis
2473:Organizational Research Methods
501:appropriate control variables);
99:confirmatory composite analysis
4062:Mean-unbiased minimum-variance
2805:10.1146/annurev.psych.51.1.201
2592:, Elsevier, pp. 855â885,
2336:Canadian Studies in Population
1631:Journal of Economic Literature
632:is the maximized value of the
600:
594:
325:Summary style may involve the
1:
5375:Geographic information system
4591:Simultaneous equations models
3130:Handbook of Management Scales
2644:10.1080/10705511.2018.1561292
151:maximum likelihood estimation
18:Structural equation modelling
6016:Regression analysis category
5906:Response surface methodology
4558:Coefficient of determination
4169:Uniformly most powerful test
2769:(Third ed.). Guilford.
2717:Structural Equation Modeling
2683:10.1080/00031305.2012.708641
2665:Narayanan, A. (2012-05-01).
2632:Structural Equation Modeling
1688:10.1007/978-94-007-6094-3_15
1145:Encyclopedia of Epidemiology
1076:Simultaneous equations model
546:Akaike information criterion
95:confirmatory factor analysis
64:Structural equation modeling
5888:FrischâWaughâLovell theorem
5858:Mean and predicted response
5127:Proportional hazards models
5071:Spectral density estimation
5053:Vector autoregression (VAR)
4487:Maximum posterior estimator
3719:Randomized controlled trial
2837:The Counseling Psychologist
2793:Annual Review of Psychology
2288:10.15358/0344-1369-2017-3-4
1887:10.1037/0033-2909.100.1.107
1728:10.1037/0033-295X.110.2.203
1460:MacCallum & Austin 2000
1294:Bollen, Kenneth A. (1989).
1055: â Probabilistic model
922:Controversies and movements
881:, though the Blocked-Error
6095:
6079:Structural equation models
5538:Correlation and dependence
4887:Multivariate distributions
3307:Average absolute deviation
3055:10.4135/9781412950589.n979
2962:Goldberger, A. S. (1972).
1374:10.1207/s15327906mbr3804_5
1231:(4th ed.). New York.
1186:10.4135/9781412939584.n544
1153:10.4135/9781412953948.n443
1120:10.4135/9781412952644.n220
1072: â Statistical method
149:and optimal routing, with
29:
6011:
5883:Minimum mean-square error
5770:Decomposition of variance
5674:Growth curve (statistics)
5643:Generalized least squares
5401:
5204:
5191:
4875:Structural equation model
4783:
4758:
4529:
4505:
4237:
4211:Score/Lagrange multiplier
3817:
3804:
3626:Sample size determination
3587:
3574:
3204:
3191:
3173:
2885:10.1007/s11747-011-0278-x
2729:10.1080/10705519909540118
2671:The American Statistician
2377:10.1101/2024.06.13.598616
1429:10.1007/s11135-017-0469-8
1090: â Statistical model
1019:Random intercepts models
526:psychological contexts.).
5741:Generalized linear model
5633:Simple linear regression
5523:Non-linear least squares
5505:Computational statistics
5370:Environmental statistics
4892:Elliptical distributions
4685:Generalized linear model
4614:Simple linear regression
4384:HodgesâLehmann estimator
3841:Probability distribution
3750:Stochastic approximation
3312:Coefficient of variation
2905:Edward Arnold Publishers
2849:10.1177/0011000099274002
2486:10.1177/1094428119847278
2041:10.1177/0013164414527449
1926:10.1186/1471-2288-12-159
1625:Christ, Carl F. (1994).
1027:Multidimensional scaling
5030:Cross-correlation (XCF)
4638:Non-standard predictors
4072:LehmannâScheffĂ© theorem
3745:Adaptive clinical trial
3004:Hoyle, R H (ed) (1995)
2533:Social Science Research
2455:10.3389/psyg.2019.01139
2330:Hayduk, Leslie (2018).
1841:10.1186/1471-2288-14-24
1495:10.1214/aoms/1177732676
1059:Multivariate statistics
710:Confirmatory Fit Index
682:Features of Fit Indices
229:exogenous or endogenous
38:. For the journal, see
6069:Latent variable models
6033:Mathematics portal
5957:Orthogonal polynomials
5783:Analysis of covariance
5648:Weighted least squares
5638:Ordinary least squares
5589:Ordinary least squares
5426:Mathematics portal
5247:Engineering statistics
5155:NelsonâAalen estimator
4732:Analysis of covariance
4619:Ordinary least squares
4543:Pearson product-moment
3947:Statistical functional
3858:Empirical distribution
3691:Controlled experiments
3420:Frequency distribution
3198:Descriptive statistics
2923:Psychological Bulletin
2750:(2nd ed.). SAGE.
1875:Psychological Bulletin
1417:Quality & Quantity
1327:Kaplan, David (2009).
1227:Kline, Rex B. (2016).
997:Measurement invariance
986:Latent growth modeling
971:Fusion validity models
794:criteria are required
608:
440:increases (and hence
143:simultaneous equations
111:latent growth modeling
60:
51:
5998:System identification
5962:Chebyshev polynomials
5947:Numerical integration
5898:Design of experiments
5842:Regression validation
5669:Polynomial regression
5594:Partial least squares
5342:Population statistics
5284:System identification
5018:Autocorrelation (ACF)
4946:Exponential smoothing
4860:Discriminant analysis
4855:Canonical correlation
4719:Partition of variance
4581:Regression validation
4425:(JonckheereâTerpstra)
4324:Likelihood-ratio test
4013:Frequentist inference
3925:Locationâscale family
3846:Sampling distribution
3811:Statistical inference
3778:Cross-sectional study
3765:Observational studies
3724:Randomized experiment
3553:Stem-and-leaf display
3355:Central limit theorem
2147:Hu & Bentler 1999
1989:10.1186/1471-2288-5-1
1411:Tarka, Piotr (2017).
664:Comparative Fit Index
609:
57:
48:
36:Structural estimation
6003:Moving least squares
5942:Approximation theory
5878:Studentized residual
5868:Errors and residuals
5863:GaussâMarkov theorem
5778:Analysis of variance
5700:Nonlinear regression
5679:Segmented regression
5653:General linear model
5571:Confounding variable
5518:Linear least squares
5265:Probabilistic design
4850:Principal components
4693:Exponential families
4645:Nonlinear regression
4624:General linear model
4586:Mixed effects models
4576:Errors and residuals
4553:Confounding variable
4455:Bayesian probability
4433:Van der Waerden test
4423:Ordered alternative
4188:Multiple comparisons
4067:RaoâBlackwellization
4030:Estimating equations
3986:Statistical distance
3704:Factorial experiment
3237:Arithmetic-Geometric
3115:, SEM in IS Research
3022:; Yang, Fan (1996).
2948:Byrne, B. M. (2001)
1716:Psychological Review
1025:Structural Equation
993:Longitudinal models
975:Item response theory
965:Deep Path Modelling
765:for critical values
747:for critical values
557:
265:to be estimated, or
6021:Statistics category
5952:Gaussian quadrature
5837:Model specification
5804:Stepwise regression
5662:Predictor structure
5599:Total least squares
5581:Regression analysis
5566:Partial correlation
5497:regression analysis
5337:Official statistics
5260:Methods engineering
4941:Seasonal adjustment
4709:Poisson regressions
4629:Bayesian regression
4568:Regression analysis
4548:Partial correlation
4520:Regression analysis
4119:Prediction interval
4114:Likelihood interval
4104:Confidence interval
4096:Interval estimation
4057:Unbiased estimators
3875:Model specification
3755:Up-and-down designs
3443:Partial correlation
3399:Index of dispersion
3317:Interquartile range
2765:Kline, Rex (2011).
2746:Kaplan, D. (2008).
1298:. New York: Wiley.
981:Latent class models
684:
177:Model specification
147:transport economics
6038:Statistics outline
5937:Numerical analysis
5357:Spatial statistics
5237:Medical statistics
5137:First hitting time
5091:Whittle likelihood
4742:Degrees of freedom
4737:Multivariate ANOVA
4670:Heteroscedasticity
4482:Bayesian estimator
4447:Bayesian inference
4296:KolmogorovâSmirnov
4181:Randomization test
4151:Testing hypotheses
4124:Tolerance interval
4035:Maximum likelihood
3930:Exponential family
3863:Density estimation
3823:Statistical theory
3783:Natural experiment
3729:Scientific control
3646:Survey methodology
3332:Standard deviation
2822:on 28 January 2015
680:
604:
61:
52:
6074:Regression models
6051:
6050:
6043:Statistics topics
5988:Calibration curve
5797:Model exploration
5764:
5763:
5734:Non-normal errors
5625:Linear regression
5616:statistical model
5459:
5458:
5397:
5396:
5393:
5392:
5332:National accounts
5302:Actuarial science
5294:Social statistics
5187:
5186:
5183:
5182:
5179:
5178:
5114:Survival function
5099:
5098:
4961:Granger causality
4802:Contingency table
4777:Survival analysis
4754:
4753:
4750:
4749:
4606:Linear regression
4501:
4500:
4497:
4496:
4472:Credible interval
4441:
4440:
4224:
4223:
4040:Method of moments
3909:Parametric family
3870:Statistical model
3800:
3799:
3796:
3795:
3714:Random assignment
3636:Statistical power
3570:
3569:
3566:
3565:
3415:Contingency table
3385:
3384:
3252:Generalized/power
3072:978-0-7619-2363-3
3037:978-1-317-84380-1
3020:Jöreskog, Karl G.
2776:978-1-60623-876-9
2607:978-0-12-813995-0
2349:10.25336/csp29397
1697:978-94-007-6093-6
1338:978-1-4129-1624-0
1238:978-1-4625-2334-4
1195:978-0-7619-3087-7
1162:978-1-4129-2816-8
1129:978-1-4129-1611-0
1008:Multilevel models
836:
835:
763:proposed wording
745:proposed wording
731:Basic References
718:RMSEA = sq-root((
626:statistical model
620:is the number of
335:
334:
274:with estimation.
248:measurement model
139:Cowles Commission
16:(Redirected from
6086:
6064:Graphical models
6031:
6030:
5788:Multivariate AOV
5684:Local regression
5621:
5613:Regression as a
5604:Ridge regression
5551:Rank correlation
5486:
5479:
5472:
5463:
5447:
5446:
5435:
5434:
5424:
5423:
5409:
5408:
5312:Crime statistics
5206:
5193:
5110:
5076:Fourier analysis
5063:Frequency domain
5043:
4990:
4956:Structural break
4916:
4865:Cluster analysis
4812:Log-linear model
4785:
4760:
4701:
4675:Homoscedasticity
4531:
4507:
4426:
4418:
4410:
4409:(KruskalâWallis)
4394:
4379:
4334:Cross validation
4319:
4301:AndersonâDarling
4248:
4235:
4206:Likelihood-ratio
4198:Parametric tests
4176:Permutation test
4159:1- & 2-tails
4050:Minimum distance
4022:Point estimation
4018:
3969:Optimal decision
3920:
3819:
3806:
3788:Quasi-experiment
3738:Adaptive designs
3589:
3576:
3453:Rank correlation
3215:
3206:
3193:
3160:
3153:
3146:
3137:
3096:
3086:
3076:
3041:
3001:
2896:
2860:
2831:
2829:
2827:
2821:
2815:. Archived from
2790:
2780:
2761:
2742:
2740:
2703:
2702:
2662:
2656:
2655:
2623:
2617:
2616:
2615:
2614:
2581:
2575:
2574:
2556:
2523:
2517:
2516:
2498:
2488:
2463:
2457:
2448:
2442:
2441:
2401:
2395:
2394:
2388:
2380:
2360:
2354:
2353:
2351:
2327:
2321:
2318:
2312:
2309:
2303:
2299:
2293:
2292:
2290:
2266:
2257:
2253:
2247:
2244:
2238:
2235:
2226:
2223:
2214:
2211:
2205:
2201:
2195:
2192:
2186:
2183:
2177:
2174:
2168:
2165:
2159:
2156:
2150:
2144:
2135:
2129:
2123:
2117:
2111:
2108:
2087:
2083:
2077:
2073:
2067:
2064:
2045:
2044:
2024:
2013:
2011:
2001:
1991:
1967:
1958:
1955:
1949:
1948:
1938:
1928:
1904:
1891:
1890:
1870:
1864:
1863:
1853:
1843:
1819:
1802:
1799:
1780:
1777:
1766:
1763:
1740:
1739:
1711:
1702:
1701:
1675:
1664:
1661:
1655:
1654:
1622:
1616:
1613:
1607:
1604:
1595:
1592:
1583:
1580:
1571:
1570:
1564:
1555:
1549:
1546:
1540:
1537:
1524:
1521:
1508:
1505:
1499:
1498:
1478:
1472:
1469:
1463:
1457:
1451:
1450:
1440:
1408:
1402:
1401:
1357:
1351:
1350:
1324:
1318:
1317:
1291:
1285:
1282:
1265:
1264:
1258:
1250:
1224:
1218:
1215:
1200:
1199:
1173:
1167:
1166:
1140:
1134:
1133:
1107:
1093:
1088:Bayesian Network
809:
761:NON-Factor Model
721:
685:
613:
611:
610:
605:
572:
571:
521:
517:
480:
468:
464:
459:
455:
451:
447:
443:
439:
431:
427:
422:
418:
414:
398:
390:
382:
330:
311:
310:
303:
299:Model assessment
240:structural model
21:
6094:
6093:
6089:
6088:
6087:
6085:
6084:
6083:
6054:
6053:
6052:
6047:
6025:
6007:
5971:
5967:Chebyshev nodes
5920:
5916:Bayesian design
5892:
5873:Goodness of fit
5846:
5819:
5809:Model selection
5792:
5760:
5729:
5688:
5657:
5614:
5608:
5575:
5532:
5499:
5490:
5460:
5455:
5418:
5389:
5351:
5288:
5274:quality control
5241:
5223:Clinical trials
5200:
5175:
5159:
5147:Hazard function
5141:
5095:
5057:
5041:
5004:
5000:BreuschâGodfrey
4988:
4965:
4905:
4880:Factor analysis
4826:
4807:Graphical model
4779:
4746:
4713:
4699:
4679:
4633:
4600:
4562:
4525:
4524:
4493:
4437:
4424:
4416:
4408:
4392:
4377:
4356:Rank statistics
4350:
4329:Model selection
4317:
4275:Goodness of fit
4269:
4246:
4220:
4192:
4145:
4090:
4079:Median unbiased
4007:
3918:
3851:Order statistic
3813:
3792:
3759:
3733:
3685:
3640:
3583:
3581:Data collection
3562:
3474:
3429:
3403:
3381:
3341:
3293:
3210:Continuous data
3200:
3187:
3169:
3164:
3104:
3084:
3079:
3073:
3044:
3038:
3018:
2990:10.2307/1905714
2975:
2870:
2867:
2865:Further reading
2834:
2825:
2823:
2819:
2788:
2783:
2777:
2764:
2758:
2745:
2714:
2711:
2706:
2664:
2663:
2659:
2625:
2624:
2620:
2612:
2610:
2608:
2583:
2582:
2578:
2525:
2524:
2520:
2465:
2464:
2460:
2449:
2445:
2403:
2402:
2398:
2381:
2362:
2361:
2357:
2329:
2328:
2324:
2319:
2315:
2310:
2306:
2300:
2296:
2268:
2267:
2260:
2254:
2250:
2245:
2241:
2236:
2229:
2224:
2217:
2212:
2208:
2202:
2198:
2193:
2189:
2184:
2180:
2175:
2171:
2166:
2162:
2157:
2153:
2145:
2138:
2130:
2126:
2118:
2114:
2109:
2090:
2084:
2080:
2074:
2070:
2065:
2048:
2026:
2025:
2016:
1969:
1968:
1961:
1956:
1952:
1906:
1905:
1894:
1872:
1871:
1867:
1821:
1820:
1805:
1800:
1783:
1778:
1769:
1764:
1743:
1713:
1712:
1705:
1698:
1677:
1676:
1667:
1662:
1658:
1624:
1623:
1619:
1614:
1610:
1605:
1598:
1593:
1586:
1581:
1574:
1562:
1557:
1556:
1552:
1547:
1543:
1538:
1527:
1522:
1511:
1506:
1502:
1480:
1479:
1475:
1470:
1466:
1458:
1454:
1410:
1409:
1405:
1359:
1358:
1354:
1339:
1326:
1325:
1321:
1306:
1293:
1292:
1288:
1283:
1268:
1251:
1239:
1226:
1225:
1221:
1216:
1203:
1196:
1175:
1174:
1170:
1163:
1142:
1141:
1137:
1130:
1109:
1108:
1104:
1100:
1091:
1053:Graphical model
1043:
1035:
990:Link functions
953:
924:
854:
841:
807:
806:Index based on
778:
722:- d)/(d(N-1)))
719:
555:
554:
519:
515:
478:
466:
462:
457:
453:
449:
445:
441:
437:
429:
425:
420:
416:
412:
396:
388:
380:
331:
324:
312:
308:
301:
280:
252:factor analysis
179:
171:
127:
77:are thought to
43:
28:
23:
22:
15:
12:
11:
5:
6092:
6090:
6082:
6081:
6076:
6071:
6066:
6056:
6055:
6049:
6048:
6046:
6045:
6040:
6035:
6023:
6018:
6012:
6009:
6008:
6006:
6005:
6000:
5995:
5990:
5985:
5979:
5977:
5973:
5972:
5970:
5969:
5964:
5959:
5954:
5949:
5944:
5939:
5933:
5931:
5922:
5921:
5919:
5918:
5913:
5911:Optimal design
5908:
5902:
5900:
5894:
5893:
5891:
5890:
5885:
5880:
5875:
5870:
5865:
5860:
5854:
5852:
5848:
5847:
5845:
5844:
5839:
5834:
5833:
5832:
5827:
5822:
5817:
5806:
5800:
5798:
5794:
5793:
5791:
5790:
5785:
5780:
5774:
5772:
5766:
5765:
5762:
5761:
5759:
5758:
5753:
5748:
5743:
5737:
5735:
5731:
5730:
5728:
5727:
5722:
5717:
5712:
5710:Semiparametric
5707:
5702:
5696:
5694:
5690:
5689:
5687:
5686:
5681:
5676:
5671:
5665:
5663:
5659:
5658:
5656:
5655:
5650:
5645:
5640:
5635:
5629:
5627:
5618:
5610:
5609:
5607:
5606:
5601:
5596:
5591:
5585:
5583:
5577:
5576:
5574:
5573:
5568:
5563:
5557:
5555:Spearman's rho
5548:
5542:
5540:
5534:
5533:
5531:
5530:
5525:
5520:
5515:
5509:
5507:
5501:
5500:
5491:
5489:
5488:
5481:
5474:
5466:
5457:
5456:
5454:
5453:
5441:
5429:
5415:
5402:
5399:
5398:
5395:
5394:
5391:
5390:
5388:
5387:
5382:
5377:
5372:
5367:
5361:
5359:
5353:
5352:
5350:
5349:
5344:
5339:
5334:
5329:
5324:
5319:
5314:
5309:
5304:
5298:
5296:
5290:
5289:
5287:
5286:
5281:
5276:
5267:
5262:
5257:
5251:
5249:
5243:
5242:
5240:
5239:
5234:
5229:
5220:
5218:Bioinformatics
5214:
5212:
5202:
5201:
5196:
5189:
5188:
5185:
5184:
5181:
5180:
5177:
5176:
5174:
5173:
5167:
5165:
5161:
5160:
5158:
5157:
5151:
5149:
5143:
5142:
5140:
5139:
5134:
5129:
5124:
5118:
5116:
5107:
5101:
5100:
5097:
5096:
5094:
5093:
5088:
5083:
5078:
5073:
5067:
5065:
5059:
5058:
5056:
5055:
5050:
5045:
5037:
5032:
5027:
5026:
5025:
5023:partial (PACF)
5014:
5012:
5006:
5005:
5003:
5002:
4997:
4992:
4984:
4979:
4973:
4971:
4970:Specific tests
4967:
4966:
4964:
4963:
4958:
4953:
4948:
4943:
4938:
4933:
4928:
4922:
4920:
4913:
4907:
4906:
4904:
4903:
4902:
4901:
4900:
4899:
4884:
4883:
4882:
4872:
4870:Classification
4867:
4862:
4857:
4852:
4847:
4842:
4836:
4834:
4828:
4827:
4825:
4824:
4819:
4817:McNemar's test
4814:
4809:
4804:
4799:
4793:
4791:
4781:
4780:
4763:
4756:
4755:
4752:
4751:
4748:
4747:
4745:
4744:
4739:
4734:
4729:
4723:
4721:
4715:
4714:
4712:
4711:
4695:
4689:
4687:
4681:
4680:
4678:
4677:
4672:
4667:
4662:
4657:
4655:Semiparametric
4652:
4647:
4641:
4639:
4635:
4634:
4632:
4631:
4626:
4621:
4616:
4610:
4608:
4602:
4601:
4599:
4598:
4593:
4588:
4583:
4578:
4572:
4570:
4564:
4563:
4561:
4560:
4555:
4550:
4545:
4539:
4537:
4527:
4526:
4523:
4522:
4517:
4511:
4510:
4503:
4502:
4499:
4498:
4495:
4494:
4492:
4491:
4490:
4489:
4479:
4474:
4469:
4468:
4467:
4462:
4451:
4449:
4443:
4442:
4439:
4438:
4436:
4435:
4430:
4429:
4428:
4420:
4412:
4396:
4393:(MannâWhitney)
4388:
4387:
4386:
4373:
4372:
4371:
4360:
4358:
4352:
4351:
4349:
4348:
4347:
4346:
4341:
4336:
4326:
4321:
4318:(ShapiroâWilk)
4313:
4308:
4303:
4298:
4293:
4285:
4279:
4277:
4271:
4270:
4268:
4267:
4259:
4250:
4238:
4232:
4230:Specific tests
4226:
4225:
4222:
4221:
4219:
4218:
4213:
4208:
4202:
4200:
4194:
4193:
4191:
4190:
4185:
4184:
4183:
4173:
4172:
4171:
4161:
4155:
4153:
4147:
4146:
4144:
4143:
4142:
4141:
4136:
4126:
4121:
4116:
4111:
4106:
4100:
4098:
4092:
4091:
4089:
4088:
4083:
4082:
4081:
4076:
4075:
4074:
4069:
4054:
4053:
4052:
4047:
4042:
4037:
4026:
4024:
4015:
4009:
4008:
4006:
4005:
4000:
3995:
3994:
3993:
3983:
3978:
3977:
3976:
3966:
3965:
3964:
3959:
3954:
3944:
3939:
3934:
3933:
3932:
3927:
3922:
3906:
3905:
3904:
3899:
3894:
3884:
3883:
3882:
3877:
3867:
3866:
3865:
3855:
3854:
3853:
3843:
3838:
3833:
3827:
3825:
3815:
3814:
3809:
3802:
3801:
3798:
3797:
3794:
3793:
3791:
3790:
3785:
3780:
3775:
3769:
3767:
3761:
3760:
3758:
3757:
3752:
3747:
3741:
3739:
3735:
3734:
3732:
3731:
3726:
3721:
3716:
3711:
3706:
3701:
3695:
3693:
3687:
3686:
3684:
3683:
3681:Standard error
3678:
3673:
3668:
3667:
3666:
3661:
3650:
3648:
3642:
3641:
3639:
3638:
3633:
3628:
3623:
3618:
3613:
3611:Optimal design
3608:
3603:
3597:
3595:
3585:
3584:
3579:
3572:
3571:
3568:
3567:
3564:
3563:
3561:
3560:
3555:
3550:
3545:
3540:
3535:
3530:
3525:
3520:
3515:
3510:
3505:
3500:
3495:
3490:
3484:
3482:
3476:
3475:
3473:
3472:
3467:
3466:
3465:
3460:
3450:
3445:
3439:
3437:
3431:
3430:
3428:
3427:
3422:
3417:
3411:
3409:
3408:Summary tables
3405:
3404:
3402:
3401:
3395:
3393:
3387:
3386:
3383:
3382:
3380:
3379:
3378:
3377:
3372:
3367:
3357:
3351:
3349:
3343:
3342:
3340:
3339:
3334:
3329:
3324:
3319:
3314:
3309:
3303:
3301:
3295:
3294:
3292:
3291:
3286:
3281:
3280:
3279:
3274:
3269:
3264:
3259:
3254:
3249:
3244:
3242:Contraharmonic
3239:
3234:
3223:
3221:
3212:
3202:
3201:
3196:
3189:
3188:
3186:
3185:
3180:
3174:
3171:
3170:
3165:
3163:
3162:
3155:
3148:
3140:
3134:
3133:
3127:
3121:
3116:
3110:
3103:
3102:External links
3100:
3099:
3098:
3077:
3071:
3042:
3036:
3016:
3002:
2973:
2972:40, 979- 1001.
2960:
2946:
2927:
2926:, 88, 588â606.
2915:
2897:
2866:
2863:
2862:
2861:
2843:(4): 485â527.
2832:
2781:
2775:
2762:
2757:978-1412916240
2756:
2743:
2738:2027.42/139911
2710:
2707:
2705:
2704:
2677:(2): 129â138.
2657:
2638:(5): 803â818.
2618:
2606:
2576:
2518:
2479:(4): 651â687.
2458:
2443:
2396:
2355:
2322:
2313:
2304:
2294:
2258:
2248:
2239:
2227:
2215:
2206:
2196:
2187:
2178:
2169:
2160:
2151:
2136:
2134:, p. 206.
2124:
2122:, p. 205.
2112:
2088:
2078:
2068:
2046:
2035:(6): 905â926.
2014:
1959:
1950:
1892:
1865:
1803:
1781:
1767:
1741:
1722:(2): 203â219.
1703:
1696:
1665:
1656:
1617:
1608:
1596:
1584:
1572:
1550:
1541:
1525:
1509:
1500:
1489:(3): 161â215.
1473:
1464:
1462:, p. 209.
1452:
1403:
1368:(4): 529â569.
1352:
1337:
1319:
1304:
1286:
1266:
1237:
1219:
1201:
1194:
1168:
1161:
1135:
1128:
1101:
1099:
1096:
1095:
1094:
1085:
1079:
1073:
1067:
1062:
1056:
1050:
1042:
1039:
1034:
1031:
1030:
1029:
1023:
1020:
1017:
1014:
1011:
1005:
1000:
994:
991:
988:
983:
978:
972:
969:
966:
963:
960:
957:
952:
949:
923:
920:
853:
852:Interpretation
850:
840:
837:
834:
833:
831:
829:
827:
825:of this index
820:
819:
816:
813:
810:
803:
802:
800:
798:
796:
789:
788:
786:
784:
782:
774:
773:
771:
769:
767:
757:
756:
754:
752:
749:
739:
738:
736:
734:
732:
728:
727:
725:
723:
716:
712:
711:
708:
705:
702:
698:
697:
694:
691:
688:
674:
673:
672:
671:
661:
660:
659:
650:
649:
648:
639:
638:
637:
614:
602:
599:
596:
593:
590:
587:
584:
581:
578:
575:
570:
567:
564:
552:
543:
542:
541:
528:
527:
523:
512:
509:
506:
502:
498:
494:
376:
375:
369:
363:
357:
351:
345:
333:
332:
315:
313:
306:
300:
297:
279:
276:
214:
213:
210:
207:
199:
198:
195:
192:
189:
186:
178:
175:
170:
167:
126:
123:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
6091:
6080:
6077:
6075:
6072:
6070:
6067:
6065:
6062:
6061:
6059:
6044:
6041:
6039:
6036:
6034:
6029:
6024:
6022:
6019:
6017:
6014:
6013:
6010:
6004:
6001:
5999:
5996:
5994:
5991:
5989:
5986:
5984:
5983:Curve fitting
5981:
5980:
5978:
5974:
5968:
5965:
5963:
5960:
5958:
5955:
5953:
5950:
5948:
5945:
5943:
5940:
5938:
5935:
5934:
5932:
5930:
5929:approximation
5927:
5923:
5917:
5914:
5912:
5909:
5907:
5904:
5903:
5901:
5899:
5895:
5889:
5886:
5884:
5881:
5879:
5876:
5874:
5871:
5869:
5866:
5864:
5861:
5859:
5856:
5855:
5853:
5849:
5843:
5840:
5838:
5835:
5831:
5828:
5826:
5823:
5821:
5820:
5812:
5811:
5810:
5807:
5805:
5802:
5801:
5799:
5795:
5789:
5786:
5784:
5781:
5779:
5776:
5775:
5773:
5771:
5767:
5757:
5754:
5752:
5749:
5747:
5744:
5742:
5739:
5738:
5736:
5732:
5726:
5723:
5721:
5718:
5716:
5713:
5711:
5708:
5706:
5705:Nonparametric
5703:
5701:
5698:
5697:
5695:
5691:
5685:
5682:
5680:
5677:
5675:
5672:
5670:
5667:
5666:
5664:
5660:
5654:
5651:
5649:
5646:
5644:
5641:
5639:
5636:
5634:
5631:
5630:
5628:
5626:
5622:
5619:
5617:
5611:
5605:
5602:
5600:
5597:
5595:
5592:
5590:
5587:
5586:
5584:
5582:
5578:
5572:
5569:
5567:
5564:
5561:
5560:Kendall's tau
5558:
5556:
5552:
5549:
5547:
5544:
5543:
5541:
5539:
5535:
5529:
5526:
5524:
5521:
5519:
5516:
5514:
5513:Least squares
5511:
5510:
5508:
5506:
5502:
5498:
5494:
5493:Least squares
5487:
5482:
5480:
5475:
5473:
5468:
5467:
5464:
5452:
5451:
5442:
5440:
5439:
5430:
5428:
5427:
5422:
5416:
5414:
5413:
5404:
5403:
5400:
5386:
5383:
5381:
5380:Geostatistics
5378:
5376:
5373:
5371:
5368:
5366:
5363:
5362:
5360:
5358:
5354:
5348:
5347:Psychometrics
5345:
5343:
5340:
5338:
5335:
5333:
5330:
5328:
5325:
5323:
5320:
5318:
5315:
5313:
5310:
5308:
5305:
5303:
5300:
5299:
5297:
5295:
5291:
5285:
5282:
5280:
5277:
5275:
5271:
5268:
5266:
5263:
5261:
5258:
5256:
5253:
5252:
5250:
5248:
5244:
5238:
5235:
5233:
5230:
5228:
5224:
5221:
5219:
5216:
5215:
5213:
5211:
5210:Biostatistics
5207:
5203:
5199:
5194:
5190:
5172:
5171:Log-rank test
5169:
5168:
5166:
5162:
5156:
5153:
5152:
5150:
5148:
5144:
5138:
5135:
5133:
5130:
5128:
5125:
5123:
5120:
5119:
5117:
5115:
5111:
5108:
5106:
5102:
5092:
5089:
5087:
5084:
5082:
5079:
5077:
5074:
5072:
5069:
5068:
5066:
5064:
5060:
5054:
5051:
5049:
5046:
5044:
5042:(BoxâJenkins)
5038:
5036:
5033:
5031:
5028:
5024:
5021:
5020:
5019:
5016:
5015:
5013:
5011:
5007:
5001:
4998:
4996:
4995:DurbinâWatson
4993:
4991:
4985:
4983:
4980:
4978:
4977:DickeyâFuller
4975:
4974:
4972:
4968:
4962:
4959:
4957:
4954:
4952:
4951:Cointegration
4949:
4947:
4944:
4942:
4939:
4937:
4934:
4932:
4929:
4927:
4926:Decomposition
4924:
4923:
4921:
4917:
4914:
4912:
4908:
4898:
4895:
4894:
4893:
4890:
4889:
4888:
4885:
4881:
4878:
4877:
4876:
4873:
4871:
4868:
4866:
4863:
4861:
4858:
4856:
4853:
4851:
4848:
4846:
4843:
4841:
4838:
4837:
4835:
4833:
4829:
4823:
4820:
4818:
4815:
4813:
4810:
4808:
4805:
4803:
4800:
4798:
4797:Cohen's kappa
4795:
4794:
4792:
4790:
4786:
4782:
4778:
4774:
4770:
4766:
4761:
4757:
4743:
4740:
4738:
4735:
4733:
4730:
4728:
4725:
4724:
4722:
4720:
4716:
4710:
4706:
4702:
4696:
4694:
4691:
4690:
4688:
4686:
4682:
4676:
4673:
4671:
4668:
4666:
4663:
4661:
4658:
4656:
4653:
4651:
4650:Nonparametric
4648:
4646:
4643:
4642:
4640:
4636:
4630:
4627:
4625:
4622:
4620:
4617:
4615:
4612:
4611:
4609:
4607:
4603:
4597:
4594:
4592:
4589:
4587:
4584:
4582:
4579:
4577:
4574:
4573:
4571:
4569:
4565:
4559:
4556:
4554:
4551:
4549:
4546:
4544:
4541:
4540:
4538:
4536:
4532:
4528:
4521:
4518:
4516:
4513:
4512:
4508:
4504:
4488:
4485:
4484:
4483:
4480:
4478:
4475:
4473:
4470:
4466:
4463:
4461:
4458:
4457:
4456:
4453:
4452:
4450:
4448:
4444:
4434:
4431:
4427:
4421:
4419:
4413:
4411:
4405:
4404:
4403:
4400:
4399:Nonparametric
4397:
4395:
4389:
4385:
4382:
4381:
4380:
4374:
4370:
4369:Sample median
4367:
4366:
4365:
4362:
4361:
4359:
4357:
4353:
4345:
4342:
4340:
4337:
4335:
4332:
4331:
4330:
4327:
4325:
4322:
4320:
4314:
4312:
4309:
4307:
4304:
4302:
4299:
4297:
4294:
4292:
4290:
4286:
4284:
4281:
4280:
4278:
4276:
4272:
4266:
4264:
4260:
4258:
4256:
4251:
4249:
4244:
4240:
4239:
4236:
4233:
4231:
4227:
4217:
4214:
4212:
4209:
4207:
4204:
4203:
4201:
4199:
4195:
4189:
4186:
4182:
4179:
4178:
4177:
4174:
4170:
4167:
4166:
4165:
4162:
4160:
4157:
4156:
4154:
4152:
4148:
4140:
4137:
4135:
4132:
4131:
4130:
4127:
4125:
4122:
4120:
4117:
4115:
4112:
4110:
4107:
4105:
4102:
4101:
4099:
4097:
4093:
4087:
4084:
4080:
4077:
4073:
4070:
4068:
4065:
4064:
4063:
4060:
4059:
4058:
4055:
4051:
4048:
4046:
4043:
4041:
4038:
4036:
4033:
4032:
4031:
4028:
4027:
4025:
4023:
4019:
4016:
4014:
4010:
4004:
4001:
3999:
3996:
3992:
3989:
3988:
3987:
3984:
3982:
3979:
3975:
3974:loss function
3972:
3971:
3970:
3967:
3963:
3960:
3958:
3955:
3953:
3950:
3949:
3948:
3945:
3943:
3940:
3938:
3935:
3931:
3928:
3926:
3923:
3921:
3915:
3912:
3911:
3910:
3907:
3903:
3900:
3898:
3895:
3893:
3890:
3889:
3888:
3885:
3881:
3878:
3876:
3873:
3872:
3871:
3868:
3864:
3861:
3860:
3859:
3856:
3852:
3849:
3848:
3847:
3844:
3842:
3839:
3837:
3834:
3832:
3829:
3828:
3826:
3824:
3820:
3816:
3812:
3807:
3803:
3789:
3786:
3784:
3781:
3779:
3776:
3774:
3771:
3770:
3768:
3766:
3762:
3756:
3753:
3751:
3748:
3746:
3743:
3742:
3740:
3736:
3730:
3727:
3725:
3722:
3720:
3717:
3715:
3712:
3710:
3707:
3705:
3702:
3700:
3697:
3696:
3694:
3692:
3688:
3682:
3679:
3677:
3676:Questionnaire
3674:
3672:
3669:
3665:
3662:
3660:
3657:
3656:
3655:
3652:
3651:
3649:
3647:
3643:
3637:
3634:
3632:
3629:
3627:
3624:
3622:
3619:
3617:
3614:
3612:
3609:
3607:
3604:
3602:
3599:
3598:
3596:
3594:
3590:
3586:
3582:
3577:
3573:
3559:
3556:
3554:
3551:
3549:
3546:
3544:
3541:
3539:
3536:
3534:
3531:
3529:
3526:
3524:
3521:
3519:
3516:
3514:
3511:
3509:
3506:
3504:
3503:Control chart
3501:
3499:
3496:
3494:
3491:
3489:
3486:
3485:
3483:
3481:
3477:
3471:
3468:
3464:
3461:
3459:
3456:
3455:
3454:
3451:
3449:
3446:
3444:
3441:
3440:
3438:
3436:
3432:
3426:
3423:
3421:
3418:
3416:
3413:
3412:
3410:
3406:
3400:
3397:
3396:
3394:
3392:
3388:
3376:
3373:
3371:
3368:
3366:
3363:
3362:
3361:
3358:
3356:
3353:
3352:
3350:
3348:
3344:
3338:
3335:
3333:
3330:
3328:
3325:
3323:
3320:
3318:
3315:
3313:
3310:
3308:
3305:
3304:
3302:
3300:
3296:
3290:
3287:
3285:
3282:
3278:
3275:
3273:
3270:
3268:
3265:
3263:
3260:
3258:
3255:
3253:
3250:
3248:
3245:
3243:
3240:
3238:
3235:
3233:
3230:
3229:
3228:
3225:
3224:
3222:
3220:
3216:
3213:
3211:
3207:
3203:
3199:
3194:
3190:
3184:
3181:
3179:
3176:
3175:
3172:
3168:
3161:
3156:
3154:
3149:
3147:
3142:
3141:
3138:
3131:
3128:
3125:
3122:
3120:
3117:
3114:
3111:
3109:
3106:
3105:
3101:
3094:
3090:
3083:
3078:
3074:
3068:
3064:
3060:
3056:
3052:
3048:
3043:
3039:
3033:
3029:
3025:
3021:
3017:
3015:
3014:0-8039-5318-6
3011:
3007:
3003:
2999:
2995:
2991:
2987:
2983:
2979:
2974:
2971:
2970:
2965:
2961:
2959:
2958:0-8058-4104-0
2955:
2951:
2947:
2945:
2944:0-471-01171-1
2941:
2937:
2936:
2931:
2930:Bollen, K. A.
2928:
2925:
2924:
2919:
2918:Bentler, P.M.
2916:
2914:
2913:0-340-69243-X
2910:
2906:
2902:
2898:
2894:
2890:
2886:
2882:
2878:
2874:
2869:
2868:
2864:
2858:
2854:
2850:
2846:
2842:
2838:
2833:
2818:
2814:
2810:
2806:
2802:
2798:
2794:
2787:
2782:
2778:
2772:
2768:
2763:
2759:
2753:
2749:
2744:
2739:
2734:
2730:
2726:
2722:
2718:
2713:
2712:
2708:
2700:
2696:
2692:
2688:
2684:
2680:
2676:
2672:
2668:
2661:
2658:
2653:
2649:
2645:
2641:
2637:
2633:
2629:
2622:
2619:
2609:
2603:
2599:
2595:
2591:
2587:
2580:
2577:
2572:
2568:
2564:
2560:
2555:
2550:
2546:
2542:
2538:
2534:
2530:
2522:
2519:
2514:
2510:
2506:
2502:
2497:
2492:
2487:
2482:
2478:
2474:
2470:
2462:
2459:
2456:
2452:
2447:
2444:
2439:
2435:
2431:
2427:
2423:
2419:
2416:(1): 85â110.
2415:
2411:
2407:
2400:
2397:
2392:
2386:
2378:
2374:
2370:
2366:
2359:
2356:
2350:
2345:
2341:
2337:
2333:
2326:
2323:
2317:
2314:
2308:
2305:
2298:
2295:
2289:
2284:
2280:
2276:
2275:Marketing ZFP
2272:
2265:
2263:
2259:
2252:
2249:
2243:
2240:
2234:
2232:
2228:
2222:
2220:
2216:
2210:
2207:
2200:
2197:
2191:
2188:
2182:
2179:
2173:
2170:
2164:
2161:
2155:
2152:
2149:, p. 27.
2148:
2143:
2141:
2137:
2133:
2128:
2125:
2121:
2116:
2113:
2107:
2105:
2103:
2101:
2099:
2097:
2095:
2093:
2089:
2082:
2079:
2072:
2069:
2063:
2061:
2059:
2057:
2055:
2053:
2051:
2047:
2042:
2038:
2034:
2030:
2023:
2021:
2019:
2015:
2009:
2005:
2000:
1995:
1990:
1985:
1981:
1977:
1973:
1966:
1964:
1960:
1954:
1951:
1946:
1942:
1937:
1932:
1927:
1922:
1918:
1914:
1910:
1903:
1901:
1899:
1897:
1893:
1888:
1884:
1880:
1876:
1869:
1866:
1861:
1857:
1852:
1847:
1842:
1837:
1833:
1829:
1825:
1818:
1816:
1814:
1812:
1810:
1808:
1804:
1798:
1796:
1794:
1792:
1790:
1788:
1786:
1782:
1776:
1774:
1772:
1768:
1762:
1760:
1758:
1756:
1754:
1752:
1750:
1748:
1746:
1742:
1737:
1733:
1729:
1725:
1721:
1717:
1710:
1708:
1704:
1699:
1693:
1689:
1685:
1681:
1674:
1672:
1670:
1666:
1660:
1657:
1652:
1648:
1644:
1640:
1636:
1632:
1628:
1621:
1618:
1612:
1609:
1603:
1601:
1597:
1591:
1589:
1585:
1579:
1577:
1573:
1568:
1561:
1554:
1551:
1545:
1542:
1536:
1534:
1532:
1530:
1526:
1520:
1518:
1516:
1514:
1510:
1504:
1501:
1496:
1492:
1488:
1484:
1477:
1474:
1468:
1465:
1461:
1456:
1453:
1448:
1444:
1439:
1434:
1430:
1426:
1423:(1): 313â54.
1422:
1418:
1414:
1407:
1404:
1399:
1395:
1391:
1387:
1383:
1379:
1375:
1371:
1367:
1363:
1356:
1353:
1348:
1344:
1340:
1334:
1330:
1323:
1320:
1315:
1311:
1307:
1305:0-471-01171-1
1301:
1297:
1290:
1287:
1281:
1279:
1277:
1275:
1273:
1271:
1267:
1262:
1256:
1248:
1244:
1240:
1234:
1230:
1223:
1220:
1214:
1212:
1210:
1208:
1206:
1202:
1197:
1191:
1187:
1183:
1179:
1172:
1169:
1164:
1158:
1154:
1150:
1146:
1139:
1136:
1131:
1125:
1121:
1117:
1113:
1106:
1103:
1097:
1089:
1086:
1083:
1080:
1077:
1074:
1071:
1068:
1066:
1063:
1060:
1057:
1054:
1051:
1048:
1045:
1044:
1040:
1038:
1032:
1028:
1024:
1021:
1018:
1015:
1012:
1009:
1006:
1004:
1003:Mixture model
1001:
998:
995:
992:
989:
987:
984:
982:
979:
976:
973:
970:
967:
964:
961:
958:
955:
954:
950:
948:
944:
940:
936:
932:
928:
921:
919:
916:
911:
908:
902:
898:
894:
890:
886:
884:
880:
874:
870:
866:
862:
858:
851:
849:
845:
838:
832:
830:
828:
826:
822:
821:
817:
814:
811:
805:
804:
801:
799:
797:
795:
791:
790:
787:
785:
783:
781:
776:
775:
772:
770:
768:
766:
762:
759:
758:
755:
753:
751:.06 wording?
750:
748:
744:
741:
740:
737:
735:
733:
730:
729:
726:
724:
717:
714:
713:
709:
706:
703:
700:
699:
695:
692:
689:
687:
686:
683:
678:
668:
667:
665:
662:
657:
656:
654:
651:
646:
645:
643:
640:
636:of the model.
635:
631:
627:
623:
619:
615:
597:
591:
588:
585:
582:
579:
576:
573:
553:
550:
549:
547:
544:
540:misspecified.
538:
537:
536:
533:
532:
531:
524:
513:
510:
507:
503:
499:
495:
492:
491:
490:
487:
483:
474:
470:
434:
409:
405:
401:
393:
386:
373:
370:
367:
364:
361:
358:
355:
352:
349:
346:
343:
340:
339:
338:
328:
323:
321:
320:summary style
316:This article
314:
305:
304:
298:
296:
292:
288:
284:
277:
275:
271:
268:
264:
259:
257:
253:
249:
245:
241:
236:
232:
230:
226:
224:
220:
211:
208:
205:
204:
203:
196:
193:
190:
187:
184:
183:
182:
176:
174:
168:
166:
162:
158:
154:
152:
148:
144:
140:
135:
132:
131:Sewall Wright
124:
122:
118:
114:
112:
108:
104:
103:path analysis
100:
96:
90:
87:
86:
80:
76:
71:
69:
65:
56:
47:
41:
37:
33:
19:
5976:Applications
5815:
5693:Non-standard
5448:
5436:
5417:
5410:
5322:Econometrics
5272: /
5255:Chemometrics
5232:Epidemiology
5225: /
5198:Applications
5040:ARIMA model
4987:Q-statistic
4936:Stationarity
4874:
4832:Multivariate
4775: /
4771: /
4769:Multivariate
4767: /
4707: /
4703: /
4477:Bayes factor
4376:Signed rank
4288:
4262:
4254:
4242:
3937:Completeness
3773:Cohort study
3671:Opinion poll
3606:Missing data
3593:Study design
3548:Scatter plot
3470:Scatter plot
3463:Spearman's Ï
3425:Grouped data
3092:
3088:
3046:
3027:
3005:
2981:
2978:Econometrica
2977:
2969:Econometrica
2967:
2963:
2949:
2933:
2921:
2900:
2876:
2872:
2840:
2836:
2824:. Retrieved
2817:the original
2796:
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2766:
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2709:Bibliography
2674:
2670:
2660:
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2611:, retrieved
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2521:
2496:11343/247887
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743:Factor Model
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32:econometrics
5450:WikiProject
5365:Cartography
5327:Jurimetrics
5279:Reliability
5010:Time domain
4989:(LjungâBox)
4911:Time-series
4789:Categorical
4773:Time-series
4765:Categorical
4700:(Bernoulli)
4535:Correlation
4515:Correlation
4311:JarqueâBera
4283:Chi-squared
4045:M-estimator
3998:Asymptotics
3942:Sufficiency
3709:Interaction
3621:Replication
3601:Effect size
3558:Violin plot
3538:Radar chart
3518:Forest plot
3508:Correlogram
3458:Kendall's Ï
2984:(1): 1â12.
2879:(1): 8â34.
2799:: 201â226.
2554:1874/431763
2281:(3): 4â16.
1881:: 107â120.
701:Index Name
385:chi-squared
256:path models
6058:Categories
5851:Background
5814:Mallows's
5317:Demography
5035:ARMA model
4840:Regression
4417:(Friedman)
4378:(Wilcoxon)
4316:Normality
4306:Lilliefors
4253:Student's
4129:Resampling
4003:Robustness
3991:divergence
3981:Efficiency
3919:(monotone)
3914:Likelihood
3831:Population
3664:Stratified
3616:Population
3435:Dependence
3391:Count data
3322:Percentile
3299:Dispersion
3232:Arithmetic
3167:Statistics
3095:(2): 23â74
3063:2022/21973
2826:25 January
2613:2023-11-03
2539:: 102805.
2132:Kline 2011
2120:Kline 2011
1098:References
1082:Causal map
670:desirable.
634:likelihood
622:parameters
535:Chi-square
246:, and the
219:endogenous
75:phenomenon
5926:Numerical
4698:Logistic
4465:posterior
4391:Rank sum
4139:Jackknife
4134:Bootstrap
3952:Bootstrap
3887:Parameter
3836:Statistic
3631:Statistic
3543:Run chart
3528:Pie chart
3523:Histogram
3513:Fan chart
3488:Bar chart
3370:L-moments
3257:Geometric
2938:. Wiley,
2893:167896719
2857:145586057
2691:0003-1305
2652:1070-5511
2571:253343751
2513:181878548
2505:1094-4281
2430:1548-5943
1643:0022-0515
1382:0027-3171
1347:225852466
1255:cite book
1247:934184322
592:
583:−
327:splitting
225:variables
223:exogenous
85:equations
5756:Logistic
5746:Binomial
5725:Isotonic
5720:Quantile
5412:Category
5105:Survival
4982:Johansen
4705:Binomial
4660:Isotonic
4247:(normal)
3892:location
3699:Blocking
3654:Sampling
3533:QâQ plot
3498:Box plot
3480:Graphics
3375:Skewness
3365:Kurtosis
3337:Variance
3267:Heronian
3262:Harmonic
3008:. SAGE,
2932:(1989).
2813:10751970
2723:: 1â55.
2699:59460771
2563:36796989
2438:24313568
2256:455-467.
2204:289-311.
2008:15636638
1945:23088287
1860:24533771
1736:12747522
1447:29416184
1390:26777445
1314:18834634
1180:. 2006.
1147:. 2008.
1041:See also
1033:Software
962:Copulas
715:Formula
644:(RMSEA)
141:work on
79:causally
5751:Poisson
5438:Commons
5385:Kriging
5270:Process
5227:studies
5086:Wavelet
4919:General
4086:Plug-in
3880:L space
3659:Cluster
3360:Moments
3178:Outline
2998:1905714
2369:bioRxiv
2302:841-850
1936:3506474
1919:: 159.
1851:3937073
1651:2728422
1438:5794813
1398:7384127
999:models
977:models
655:(SRMR)
624:in the
125:History
5715:Robust
5307:Census
4897:Normal
4845:Manova
4665:Robust
4415:2-way
4407:1-way
4245:-test
3916:
3493:Biplot
3284:Median
3277:Lehmer
3219:Center
3069:
3034:
3012:
2996:
2956:
2952:.LEA,
2942:
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2891:
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1999:546216
1996:
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1933:
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1159:
1126:
690:RMSEA
666:(CFI)
628:, and
616:where
548:(AIC)
34:, see
4931:Trend
4460:prior
4402:anova
4291:-test
4265:-test
4257:-test
4164:Power
4109:Pivot
3902:shape
3897:scale
3347:Shape
3327:Range
3272:Heinz
3247:Cubic
3183:Index
3085:(PDF)
2994:JSTOR
2889:S2CID
2853:S2CID
2820:(PDF)
2789:(PDF)
2695:S2CID
2567:S2CID
2509:S2CID
2076:Sage.
1982:: 1.
1647:JSTOR
1563:(PDF)
1394:S2CID
693:SRMR
267:fixed
5495:and
5164:Test
4364:Sign
4216:Wald
3289:Mode
3227:Mean
3067:ISBN
3032:ISBN
3010:ISBN
2954:ISBN
2940:ISBN
2909:ISBN
2828:2015
2809:PMID
2771:ISBN
2752:ISBN
2687:ISSN
2648:ISSN
2602:ISBN
2559:PMID
2501:ISSN
2434:PMID
2426:ISSN
2391:link
2004:PMID
1941:PMID
1856:PMID
1732:PMID
1692:ISBN
1639:ISSN
1443:PMID
1386:PMID
1378:ISSN
1343:OCLC
1333:ISBN
1310:OCLC
1300:ISBN
1261:link
1243:OCLC
1233:ISBN
1190:ISBN
1157:ISBN
1124:ISBN
818:Yes
812:Yes
696:CFI
263:free
221:and
5830:BIC
5825:AIC
4344:BIC
4339:AIC
3059:hdl
3051:doi
2986:doi
2881:doi
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