Structural Equation Modeling

Overview Structural equation modeling (SEM) grows out of and serves purposes similar to multiple regression, but in a more powerful way which takes into account the modeling of interactions, nonlinearities, correlated independents, measurement error, correlated error terms, multiple latent independents each measured by multiple indicators, and one or more latent dependents also each with multiple indicators. SEM may be used as a more powerful alternative to multiple regression, path analysis, factor analysis, time series analysis, and analysis of covariance. That is, these procedures may be seen as special cases of SEM, or, to put it another way, SEM is an extension of the general linear model (GLM) of which multiple regression is a part. Advantages of SEM compared to multiple regression include more flexible assumptions (particularly allowing interpretation even in the face of multicollinearity), use of confirmatory factor analysis to reduce measurement error by having multiple indicators per latent variable, the attraction of SEM's graphical modeling interface, the desirability of testing models overall rather than coefficients individually, the ability to test models with multiple dependents, the ability to model mediating variables rather than be restricted to an additive model (in OLS regression the dependent is a function of the Var1 effect plus the Var2 effect plus the Var3 effect, etc.), the ability to model error terms, the ability to test coefficients across multiple between-subjects groups, and ability to handle difficult data (time series with autocorrelated error, non-normal data, incomplete data). Moreover, where regression is highly susceptible to error of interpretation by misspecification, the SEM strategy of comparing alternative models to assess relative model fit makes it more robust. SEM is usually viewed as a confirmatory rather than exploratory procedure, using one of three approaches: Strictly confirmatory approach: A model is tested using SEM goodness-of-fit tests to determine if the pattern of variances and covariances in the data is consistent with a structural (path) model specified by the researcher. However as other unexamined models may fit the data as well or better, an accepted model is only a not-disconfirmed model. Alternative models approach: One may test two or more causal models to determine which has the best fit. There are many goodness-of-fit measures, reflecting different considerations, and usually three or four are reported by the researcher. Although desirable in principle, this AM approach runs into the real-world problem that in most specific research topic areas, the researcher does not find in the literature two well-developed alternative models to test. Model development approach: In practice, much SEM research combines confirmatory and exploratory purposes: a model is tested using SEM procedures, found to be deficient, and an alternative model is then tested based on changes suggested by SEM modification indexes. This is the most common approach found in the literature. The problem with the model development approach is that models confirmed in this manner are post-hoc ones which may not be stable (may not fit new data, having been created based on the uniqueness of an initial dataset). Researchers may attempt to overcome this problem by using a cross-validation strategy under which the model is developed using a calibration data sample and then confirmed using an independent validation sample. Regardless of approach, SEM cannot itself draw causal arrows in models or resolve causal ambiguities. Theoretical insight and judgment by the researcher is still of utmost importance. SEM is a family of statistical techniques which incorporates and integrates path analysis and factor analysis. In fact, use of SEM software for a model in which each variable has only one indicator is a type of path analysis. Use of SEM software for a model in which each variable has multiple indicators but there are no direct effects (arrows) connecting the variables is a type of factor analysis. Usually, however, SEM refers to a hybrid model with both multiple indicators for each variable (called latent variables or factors), and paths specified connecting the latent variables. Synonyms for SEM are covariance structure analysis, covariance structure modeling, and analysis of covariance structures. Although these synonyms rightly indicate that analysis of covariance is the focus of SEM, be aware that SEM can also analyze the mean structure of a model. See also partial least squares regression, which is an alternative method of modeling the relationship among latent variables, also generating path coefficients for a SEM-type model, but without SEM's data distribution assumptions. PLS path modeling is sometimes called "soft modeling" because it makes soft or relaxed assumptions about data...

Contents Key concepts and terms Handling ordinal data Modeling Specification search Model fit measures Multiple group analysis Mixture (latent class) modeling Latent growth curve analysis Mean structure analysis Multilevel models Assumptions SPSS output example Frequently asked questions Bibliography

Key Concepts and Terms

• The structural equation modeling process centers around two steps: validating the measurement model and fitting the structural model. The former is accomplished primarily through confirmatory factor analysis, while the latter is accomplished primarily through path analysis with latent variables. One starts by specifying a model on the basis of theory. Each variable in the model is conceptualized as a latent one, measured by multiple indicators. Several indicators are developed for each model, with a view to winding up with at least two and preferably three per latent variable after confirmatory factor analysis. Based on a large (n>100) representative sample, factor analysis (common factor analysis or principal axis factoring, not principle components analysis) is used to establish that indicators seem to measure the corresponding latent variables, represented by the factors. The researcher proceeds only when the measurement model has been validated. Two or more alternative models (one of which may be the null model) are then compared in terms of "model fit," which measures the extent to which the covariances predicted by the model correspond to the observed covariances in the data. "Modification indexes" and other coefficients may be used by the researcher to alter one or more models to improve fit.

• LISREL, AMOS, and EQS are three popular statistical packages for doing SEM. The first two are distributed by SPSS. LISREL popularized SEM in sociology and the social sciences and is still the package of reference in most articles about structural equation modeling. AMOS (Analysis of MOment Structures) is a more recent package which, because of its user-friendly graphical interface, has become popular as an easier way of specifying structural models. AMOS also has a BASIC programming interface as an alternative. See R. B. Kline (1998). Software programs for structural equation modeling: AMOS, EQS, and LISREL. Journal of Psychoeducational Assessment (16): 343-364.

  The AMOS interface looks like this (large, initially blank area to draw the path diagram on the right is not shown):

  

  In AMOS, the general process of structural modeling is to use the icons above to draw a circle-and-arrow path diagram, associated the diagram with data (a correlation matrix or raw data), then select Analyze, Calculate Estimates from the menu.

• Indicators are observed variables, sometimes called manifest variables or reference variables, such as items in a survey instrument. Four or more is recommended and three is acceptable and common practice. However, two indicators or even a single indicator may be acceptable if the researcher is confident in the measure's validity and reliability. In fact, the prime consideration in selecting indicators is whether they are theoretically sound and reliably measured. Also, allowing one- and two-indicator latents to a model may allow the testing of theoretically important latent-level control relationships which otherwise might not be possible. However, with one indicator, error cannot be modeled but rather one must specify a fixed measurement error variance. Also, models using only two indicators per latent variable are more likely to be underidentified and/or fail to converge and error estimates may be unreliable. By convention, indicators should have pattern coefficients (factor loadings) of .7 or higher on their latent factors.
  

  Example. In the AMOS example above, the latent variable PriorAbility is measured by the indicator variables pretest1 and pretest2. The latent variable PostAbility is measured by the indicator variables posttst1 and posttst2. Indicator and other measured variables are depicted as rectangles by convention. Latent variables are depicted as ovals by convention.The e1 to e4 terms are the error terms associated with each indicator variable. The arrows hypothesize that PostAbility is caused by PreAbility and by the practical performance experience of the exogenous measured variable Perform. The two-headed arrow indicates Perform is thought to be correlated with PriorAbility, which is an exogenous latent variable. As is usual, there is a disturbance or error term, Dist, associated with the endogenous latent variable, PostAbility. The 1's next to certain arrows are the regression weights necessary to set metrics in the model, as discussed below. Also shown is the Data Files window (select File, Data Files from the AMOS menu), showing the associted data file, structur1.sav, which is for this example is a correlation matrix with information on n, standard deviations, and means.

  For this example, correlation matrix input looks like this (though note conventional raw data input is possible also and, indeed, is necessary if certain operations such as Data Recode, discussed below, are requested).

  

• Regression, path, and structural equation models. While SEM packages are used primarily to implement models with latent variables (see below), it is possible to run regression models or path models also. In regression and path models, only observed variables are modeled, and only the dependent variable in regression or the endogenous variables in path models have error terms. Independents in regression and exogenous variables in path models are assumed to be measured without error. Path models are like regression models in having only observed variables w/o latents. Path models are like SEM models in having circle-and-arrow causal diagrams, not just the star design of regression models. Using SEM packages for path models instead of doing path analysis using traditional regression procedures has the benefit that measures of model fit, modification indexes, and other aspects of SEM output discussed below become available.

• Latent variables are the unobserved variables or constructs or factors which are measured by their respective indicators. In the example above, PriorAbility and PostAbility are latent variables. Latent variables include both independent, mediating, and dependent variables. "Exogenous" variables are independents with no prior causal variable (though they may be correlated with other exogenous variables, depicted by a double-headed arrow. In fact it is customary to assume that exogenous variables are correlated (connected by a double-headed covariance arrow) unless there is theoretical reason not to. If two exogenous variables are connected by a covariance arrow, there cannot also be a straight (regression path) arrow, nor can one have a covariance arrow connecting an exogenous variable to an endogenous variable. Exogenous constructs are sometimes denoted by the Greek letter ksi. "Endogenous" variables are mediating variables (variables which are both effects of other exogenous or mediating variables, and are causes of other mediating and dependent variables), and pure dependent variables. Endogenous variables are on the receiving end of single-headed straight arrows indicating a regression path and implying a causal relationship. The path to the endogenous variable may come from an exogenous variable or another endogenous variable. Endogenous constructs are sometimes denoted by the Greek letter eta. Variables in a model may be "upstream" or "downstream" depending on whether they are being considered as causes or effects respectively. The representation of latent variables based on their relation to observed indicator variables is one of the defining characteristics of SEM.
  Warning: Indicator variables cannot be combined arbitrarily to form latent variables. For instance, combining gender, race, or other demographic variables to form a latent variable called "background factors" would be improper because it would not represent any single underlying continuum of meaning. The confirmatory factor analysis step in SEM is a test of the meaningfulness of latent variables and their indicators, but the researcher may wish to apply traditional tests (ex., Cronbach's alpha) or conduct traditional factor analysis (ex., principal axis factoring).

  
  

• Numerical vs. ordered-categorical data in Amos. Prior to Amos version 7, all data were treated as numerical and modeled by default using maximum likelihood estimation (MLE, discussed below). However, starting with Amos version 7.0, an ordinal data type became available in addition to the numerical type, with estimation done by Bayesian methods. Ordered categorical data are now entered as ordinal in type.
  • Entering ordinal data. For example, if a variable was measured as low = <5, medium = 5-10. and high= >10. the actual data enteries for cases might be "<5", "5<<10", and ">10". Note that there is no inequality operator for ">=" or "<=" because variables are assumed to be continuously distributed with 0 probability of being exactly equal to the cutting point. Note that the Tools, Data Recode option in Amos allows one to recode automatically such that a case whose entry is "low" corresponds, for instance, to "<5". For dichotomies, Amos assigns "0" to the category boundary for two-category variables by default but this can be overriden by clicking the Details button in the Data Recode window, a illustrated below. For three-category variables, Amos by default sets the boundaries at 0 and 1, but Details, Data Recode can also override this. Note, however, that it does not matter what boundaries are for a three-category exogenous variable assuming that the intercept and regression weight are unconstrained. If the variable has more than three ordered categories, Amos estimates category boundaries under the assumption that the underlying numeric variable is normally distributed with a mean of 0 and a standard deviation of 1, picking boundaries such that the z-value corresponds to the category frequencies (ex., if the lowest category has 3.51% of the cases, the boundary between low and medium will be -1.811 since 3.51% of the normal curve lies to the left of -1.811). After data are coded and saved, when Amos is invoked, then in the Data Files window next to whatever data file or files the researcher specifies, a check mark is placed to indicate "Allow non-numeric data." This will enable Data Recode on the Tools menu, where automatic recoding can occur. Note that Data Recode is disabled if there has been correlation matrix input rather than raw data input.
    
    • Warnings. The default Amos method of setting boundary points is not appropriate in mixture modeling (a type of latent class analysis supported by Amos), when there are more than for categories for an ordered-categorical vairable. Instead, the Details button under Tools, Data Recode dialog should be used to set boundary values. Alternatively, categories can be combined to reduce categories to three, in which case the exact boundary points no longer matter. Also, if the researcher has reason to know that the underlying variable is not normally distributed, cutting points should be set manually using the Details button under Data Recode. Finally, if multiple items need to have the same boundaries (ex., multiple Likert items ranging from strongly agree to strongly disagree), and a normal underlying distribution cannot be assumed for all items, Amos has no way of automatically setting the same boundaries, so it is important instead to reduce the categories to three.

  • Parameter identification. If a formerly numerical variable is converted into an ordered categorical variable with two categories, the model may become unidentified unless a parameter constraint is imposed to set the scale of the variable. For exogenous dichotomies modeled as ordered-categorical, one must set the mean, the variance, or one of the regression weight for any single-headed arrow emanating from the dichotomy. For endogenous dichotomies, one must constrain the variable's intercept, its error variance, or a regression weight for any single-headed arrow emanating from or arriving at the dichotomy. This constraint sets the scale of the ordered-categorical dichotomy. Setting constraints is discussed further below.

  • Censored data are where there are measurements for a range of data, but either at the high end some measurements are "more than k" (ex., "> 12 months") or at the low end are "less than k" (ex., "<$20,000"). Prior to Amos 7, censored cases had to be treated as missing or coded with an arbitrary number. Now they are coded as ordered-categorical data.

  • Data imputation. Amos can impute a numerical value for a missing ordered-categorical variable or for a censored measurement, just as it can impute missing numerical data. From the Amos menu, select Analyze, Data Imputation. If ordered-categorical data are being modeled, one must elect Bayesian Estimation (if numerical data, Regression and Stochastic Regression imputation are available). A new, imputed dataset is created which can be used as input to other programs that require complete numerical data. Use this option to convert non-numeric (ordered-categorical and censored) data into numeric data.

  • Analysis of ordinal data. If ordinal data are modeled, Bayesian estimation is required. That is, if the Data Files window has "Allow non-numeric data" checked to allow ordered-categorical data, the Analyze, Calculate Estimates menu choice is disabled in Amos. Instead one must select Analyze, Bayesian Estimation.

• The measurement model. The measurement model is that part (possibly all) of a SEM model which deals with the latent variables and their indicators. A pure measurement model is a confirmatory factor analysis (CFA) model in which there is unmeasured covariance between each possible pair of latent variables, there are straight arrows from the latent variables to their respective indicators, there are straight arrows from the error and disturbance terms to their respective variables, but there are no direct effects (straight arrows) connecting the latent variables. Note that "unmeasured covariance" means one almost always draws two-headed covariance arrows connecting all pairs of exogenous variables (both latent and simple, if any), unless there is strong theoretical reason not to do so. The measurement model is evaluated like any other SEM model, using goodness of fit measures. There is no point in proceeding to the structural model until one is satisfied the measurement model is valid. See below for discussion of specifying the measurement model in AMOS.
  • The null model. The measurement model is frequently used as the "null model," differences from which must be significant if a proposed structural model (the one with straight arrows connecting some latent variables) is to be investigated further. In the null model, the covariances in the covariance matrix for the latent variables are all assumed to be zero. Seven measures of fit (NFI, RFI, IFI, TLI=NNFI, CFI, PNFI, and PCFI) require a "null" or "baseline" model against which the researcher's default models may be compared. SPSS offers a choice of four null models, selection among which will affect the calculation of these fit coefficients:
    • Null 1: The correlations among the observed variables are constrained to be 0, implying the latent variables are also uncorrelated. The means and variances of the measured variables are unconstrained. This is the default baseline "Independence" model in most analyses. If in AMOS you do not ask for a specification search (see below), Null 1 will be used as the baseline.

    • Null 2: The correlations among the observed variables are constrained to be equal (not 0 as in Null 1 models). The means and variances of the observed variables are unconstrained (the same as Null 1 models).

    • Null 3: The correlations among the observed variables are constrained to be 0. The means are also constrained to be 0. Only the variances are unconstrained. The Null 3 option applies only to models in which means and intercepts are explicit model parameters.

    • Null 4: The correlations among the observed variables are constrained to be equal. The means are also constrained to be 0. The variances of the observed variables are unconstrained. The Null 4 option applies only to models in which means and intercepts are explicit model parameters.

    • Where to find alternative null models. Alternative null models, if applicable, are found in AMOS under Analyze, Specification Search; then under the Options button, check "Show null models"; then set any other options wanted and click the right-arrow button to run the search. Note there is little reason to fit a Null 3 or 4 model in the usual situation where means and intercepts are not constrained by the researcher but rather are estimated as part of how maximum likelihood estimation handles missing data.
    

• The structural model may be contrasted with the measurement model. It is the set of exogenous and endogenous variables in the model, together with the direct effects (straight arrows) connecting them, any correlations among the exogenous variable or indicators, and the disturbance terms for these variables (reflecting the effects of unmeasured variables not in the model). Sometimes the arrows from exogenous latent constructs to endogenous ones are denoted by the Greek character gamma, and the arrows connecting one endogenous variable to another are denoted by the Greek letter beta. SPSS will print goodness of fit measures for three versions of the structural model.
  • The saturated model. This is the trivial but fully explanatory model in which there are as many parameter estimates as degrees of freedom. This occurs when the model has all possible path arrows. Most goodness of fit measures will be 1.0 for a saturated model, but since saturated models are the most un-parsimonious models possible, parsimony-based goodness of fit measures will be 0. Some measures, like RMSEA, cannot be computed for the saturated model at all.

  • The independence model. The independence model is one which assumes all relationships among measured variables are 0. This implies the correlations among the latent variables are also 0 (that is, it implies the null model, in which all paths in the structural model are zero). Where the saturated model will have a parsimony ratio of 0, the independence model has a parsimony ratio of 1. Most fit indexes will be 0, whether of the parsimony-adjusted variety or not, but some will have non-zero values (ex., RMSEA, GFI) depending on the data.

  • The default model. This is the researcher's structural model, always more parsimonious than the saturated model and almost always fitting better than the independence model with which it is compared using goodness of fit measures. That is, the default model will have a goodness of fit between the perfect explanation of the trivial saturated model and terrible explanatory power of the independence model, which assumes no relationships.

  • MIMIC models are multiple indicator, multiple independent cause models. This means the latent has the usual multiple indicators, but in addition it is also caused by additional observed variables. Diagrammatically, there are the usual arrows from the latent to its indicators, and the indicators have error terms. In addition, there are rectangles representing observed causal variables, with arrows to the latent and (depending on theory) covariance arrows connecting them since they are exogenous variables. Model fit is still interpreted the same way, but the observed causal variables must be assumed to be measured without error.

• Confirmatory factor analysis (CFA) may be used to confirm that the indicators sort themselves into factors corresponding to how the researcher has linked the indicators to the latent variables. Confirmatory factor analysis plays an important role in structural equation modeling. CFA models in SEM are used to assess the role of measurement error in the model, to validate a multifactorial model, to determine group effects on the factors, and other purposes discussed in the factor analysis section on CFA.
  • Two-step modeling. Kline (1998) urges SEM researchers always to test the pure measurement model underlying a full structural equation model first, and if the fit of the measurement model is found acceptable, then to proceed to the second step of testing the structural model by comparing its fit with that of different structural models (ex., with models generated by trimming or building, or with mathematically equivalent models ). It should be noted this is not yet universal practice.
    • Four-step modeling. Mulaik & Millsap (2000) have suggested a more stringent four-step approach to modeling:
      1. Common factor analysis to establish the number of latents
      2. Confirmatory factor analysis to confirm the measurement model. As a further refinement, factor loadings can be constrained to 0 for any measured variable's crossloadings on other latent variables, so every measured variable loads only on its latent. Schumacker & Jones (2004: 107) note this could be a tough constraint, leading to model rejection.
      3. Test the structural model.
      4. Test nested models to get the most parsimonious one. Alternatively, test other research studies' findings or theory by constraining parameters as they suggest should be the case. Consider raising the alpha significant level from .05 to .01 to test for a more significant model.

• Reliability
  • Cronbach's alpha is a commonly used measure testing the extent to which multiple indicators for a latent variable belong together. It varies from 0 to 1.0. A common rule of thumb is that the indicators should have a Cronbach's alpha of .7 to judge the set reliable. It is possible that a set of items will be below .7 on Cronbach's alpha, yet various fit indices (see below) in confirmatory factor analysis will be above the cutoff (usually .9) levels. Alpha may be low because of lack of homogeneity of variances among items, for instance, and it is also lower when there are fewer items in the scale/factor. See the further discussion of measures of internal consistency in the section on standard measures and scales.

  • Raykov's reliability rho, also called reliability rho or composite reliability, tests if it may be assumed that a single common factor underlies a set of variables. Raykov (1998) has demonstrated that Cronbach's alpha may over- or under-estimate scale reliability. Underestimation is common. For this reason, rho is now preferred and may lead to higher estimates of true reliability. Raykov's reliability rho is not to be confused with Spearman's median rho, an ordinal alternative to Cronbach's alpha, discussed in the section on reliability.. The acceptable cutoff for rho would be the same as the researcher sets for Cronbach's alpha since both attempt to measure true reliability. . Raykov's reliability rho is ouput by EQS. See Raykov (1997), which lists EQS and LISREL code for computing composite reliability.Graham (2006) discusses Amos computation of reliability rho.

  • Construct reliability and variance extracted, based on structure loadings, can also be used to assess the extent to which a latent variable is measured well by its indicators. This is discussed below.

• Model Specification is the process by which the researcher asserts which effects are null, which are fixed to a constant (usually 1.0), and which vary. Variable effects correspond to arrows in the model, while null effects correspond to an absence of an arrow. Fixed effects usually reflect either effects whose parameter has been established in the literature (rare) or more commonly, effects set to 1.0 to establish the metric (discussed below) for a latent variable. The process of specifying a model is discussed further below.
  • Model parsimony. A model in which no effect is constrained to 0 is one which will always fit the data, even when the model makes no sense. The closer one is to this most-complex model, the better will be one's fit. That is, adding paths will tend to increase fit. This is why a number of fit measures (discussed below) penalize for lack of parsimony. Note lack of parsimony may be a particular problem for models with few variables. Ways to decrease model complexity are erasing direct effects (straight arrows) from one latent variable to another; erasing direct effects from multiple latent variables to the same indicator variable; and erasing unanalyzed correlations (curved double-headed arrows) between measurement error terms and between the disturbance terms of the endogenous variables. In each case, arrows should be erased from the model only if there is no theoretical reason to suspect that the effect or correlation exists.

  • Interaction terms and power polynomials may be added to a structural model as they can in multiple regression. However, to avoid findings of good fit due solely to the influence of the means, it is advisable to center a main effect first when adding such terms. Centering is subtracting the mean from each value. This has the effect of reducing substantially the collinearity between the main effect variable and its interaction and/or polynomial term(s). Testing for the need for interaction terms is discussed below.

• Metric: In SEM, each unobserved latent variable must be assigned explicitly a metric, which is a measurement range. This is normally done by constraining one of the paths from the latent variable to one of its indicator (reference) variables, as by assigning the value of 1.0 to this path. Given this constraint, the remaining paths can then be estimated. The indicator selected to be constrained to 1.0 is the reference item. Typically one selects as the reference item the one which in factor analysis loads most heavily on the dimension represented by the latent variable, thereby allowing it to anchor the meaning of that dimension. Note that if multiple samples are being analyzed, the researcher should use the same indicator variable in each sample to assign the metric.

  In the illustration below, the AMOS Object Properties window has been opened to the Parameters tab to show 1 entered as the metric for the regression line from the Dist disturbance term to the PostAbility latent variable, whose path is also labeled 1 on the diagram. Object Properties may be opened on any object by right-clicking the object in the diagram, then selecting Object Properties from the context menu. Though the metric of 1 is set automatically, this is where the researcher may constrain any parameter to any value, or, alternatively, erase a setting to free the parameter to be freely estimated.

  

  Alternatively, one may set the factor variances to 1, thereby effectively obtaining a standardized solution. This alternative is inconsistent with multiple group analysis. Note also that if the researcher does not explicitly set metrics to 1.0 but instead relies on an automatic standardization feature built into some SEM software, one may encounter underidentification error messages -- hence explicitly setting the metric of a reference variable to 1.0 is recommended. See step 2 in the computer output example. Warning: LISREL Version 8 defaulted to setting factor variances to 1 if the user did not set the loading of a reference variable to 1.

• Measurement error terms. A measurement error term refers to the measurement error factor associated with a given indicator. Such error terms are commonly denoted by the Greek letter delta for indicators of exogenous latent constructs and epsilon for indicators of endogenous latents. Whereas regression models implicitly assume zero measurement error (that is, to the extent such error exists, regression coefficients are attenuated), error terms are explicitly modeled in SEM and as a result path coefficients modeled in SEM are unbiased by error terms, whereas regression coefficients are not. Though unbiased statistically, SEM path coefficients will be less reliable when measurement error is high.
  • Warning for single-indicator latents: If there is a latent variable in a SEM model which has only a single indicator variable (ex., gender as measured by the survey item "Sex of respondent") it is represented like any other latent, except the error term for the single indicator variable is constrained to have a mean of 0 and a variance of 0, or an estimate based on its reliability. This is because when using a single indicator, the researcher must assume the item is measured without error. AMOS and other packages will give an error message if such an error term is included.

  • Error variance when reliability is known. If the reliability coefficient for a measure has been determined, then error variance = (1 - reliability)*standard deviation squared. In Amos, error variance terms are represented as circles (or ellipses) with arrows to their respective measured variables. One can right-click on the error variance term and enter the computed error variance in the dialog box.

  • Correlated error terms refers to situations in which knowing the residual of one indicator helps in knowing the residual associated with another indicator. For instance, in survey research many people tend to give the response which is socially acceptable. Knowing that a respondent gave the socially acceptable response to one item increases the probability that a socially acceptable response will be given to another item. Such an example exhibits correlated error terms. Uncorrelated error terms are an assumption of regression, whereas the correlation of error terms may and should be explicitly modeled in SEM. That is, in regression the researcher models variables, whereas in SEM the researcher must model error as well as the variables.

• Structural error terms. Note that measurement error terms discussed above are not to be confused with structural error terms, also called residual error terms or disturbance terms, which reflect the unexplained variance in the latent endogenous variable(s) due to all unmeasured causes. Structural error terms are sometimes denoted by the Greek letter zeta.

• Building and Modifying Models
  • Model-building is the strategy of starting with the null model or a simple model and adding paths one at a time. Model-building is followed by model-trimming, discussed below.. As paths are added to the model, chi-square tends to decrease, indicating a better fit and also increasing the chi-square difference. That is, a significant chi-square difference indicates the fit of the more complex model is significantly better than for the simpler one. Adding paths should be done only if consistent with theory and face validity. Modification indexes (MIs), discussed below, indicate when adding a path may improve the model.
    • Model-building versus model trimming. The usual procedure is to overfit the model, then change only one parameter at a time. That is, the researcher first adds paths one at a time based on the modification indexes, then drops paths one at a time based on the chi-square difference test or Wald tests of the significance of the structural coefficients, discussed below. Modifying one step at a time is important because the MIs are estimates and will change each step, as may the structural coefficients and their significance. One many use MIs to add one arrow at a time to the model, taking theory into account. When this process has gone as far as judicious, then the researcher may erase one arrow at a time based on non-significant structural paths, again taking theory into account in the trimming process. More than one cycle of building and trimming may be needed before the researcher settles on the final model.

    • Alpha significance levels in model-building and model-trimming. Some authors, such as Ullman (2001), recommend that the alpha significance cutoff when adding or deleting model parameters (arrows) be set at a more stringent .01 level rather than the customary .05, on the rationale that after having added parameters on the basis of theory, the alpha significance for their alteration should involve a low Type I error rate.

    • Non-hierarchical model comparisons. Model-building and model-trimming involve comparing a model which is a subset of another. Chi-square difference cannot be used directly for non-hierarchical models. This is because model fit by chi-square is partly a function of model complexity, with more complex models fitting better. For non-hierarchical model comparisons, the researcher should use a fit index which penalizes for complexity (rewards parsimony), such as AIC.

    • Modification indices (MI) are related to the Lagrange Multiplier (LM) test or index because MI is a univariate form of LM. The Lagrange multiplier statistic is the mulitvariate counterpart of the MI statistic. MI is often used to alter models to achieve better fit, but this must be done carefully and with theoretical justification. That is, blind use of MI runs the risk of capitalization of chance and model adjustments which make no substantive sense (see Silvia and MacCallum, 1988). Arrows should be added only when the relationship makes theoretical sense, so as to avoid overfitting the model to data noise and sampling fluctuations. Moreover, when n is large, even very small discrepances may trigger an MI flag, so the researcher should also take into account the effect size of the arrow to be added as indicated by the parameter change (discussed below), not just the fact that it is significant by MI. In MI, improvement in fit is measured by a reduction in chi-square (recall a finding of chi-square significance corresponds to rejecting the model as one which fits the data). In AMOS, the modification indexes have to do with adding arrows: high MI's flag missing arrows which might be added to a model. Note: MI output in AMOS requires a dataset with no missing values.
      

      Example. In the illustration above, the highest modification indexes have to do with correlated error terms between pre- and post-tests, particularly between pretest 1 and posttest1 (error terms 1 and 3). Modification indexes are also presented to suggest adding paths (regression lines), such as from posttest1 to pretest1. However, that would violate chronological logic for these data and the path therefore should not be added. In general, one should have sound theoretical reason for adding paths suggested by MIs.

      • MI threshold. You can set how large the reduction in model chi-square should be to have a parameter (path) listed in the MI output. The minimum value would be 3.84, since chi-square must drop that amount simply by virtue of having one less parameter (path) in the model. This is why the default threshold is set to 4. The researcher can set a higher value if wanted. Setting the threshold is done in AMOS under View, Analysis Properties; in the Output tab, enter a value in "Threshold for modification indices" in the lower right.

      • Par change, also called "expected parameter change" (EPC) in some programs, is an effect size measure. AMOS output will list the parameter (which arrow to add or to subtract), the chi-square value (the estimated chi-square value for this path, labeled "M.I."), the probability of this chi-square (significant ones are candidates for change), and the "parameter change," which is the estimated change in the new path coefficient when the model is altered (labeled "Par Change"). 'Par change" is the estimated coefficient change when adding arrows, since no arrow corresponds to a 0 regression coefficient, and the parameter change is the regression coefficient for the added arrow. The actual new parameter value may differ somewhat from the old coefficient + "Par Change". The MI and the parameter change should be looked at jointly. The researcher may decide not to add an arrow indicated by MI if the parameter change is trivial. Likewise, the researcher may wish to add an arrow where the parameter change is large in absolute size even if the corresponding MI is not the largest one.

      • Covariances. In the case of modification indexes for covariances, the MI has to do with the decrease in chi-square if the two error term variables are allowed to correlate. For instance, in AMOS, if the MI for a covariance is 24 and the "Par Change" is .8, this means that if the model is respecified to allow the two error terms to covary their covariance would be expected to change by .8, leading to a reduction of model chi-square by 24 (lower is better fit). If there is correlated error, as shown by high MI's on error covariances, causes may include redundant content of the two items, methods bias (for example, common social desirability of both items), or omission of an exogenous factor (the two indicators share a common cause not in the model). Even if MI and Par Change indicate that model fit will increase if a covariance arrow is added between indicator error terms, the standard recommendation is not to do so unless there are strong theoretical reasons in the model for expecting such covariance (ex., the researcher has used a measure at two time periods, where correlation of error would be predicted). That is, error covariance arrows should not be added simply to improve model fit.

      • Structural (regression) weights. In the case of MI for estimated regression weights, the MI has to do with the change in chi-square if the path between the two variables is restored (adding an arrow).

      • Rules of thumb for MIs. One arbitrary rule of thumb is to consider adding paths associated with parameters whose modification index exceeds 100. However, another common strategy is simply to add the parameter with the largest MI (even if considerably less than 100), then see the effect as measured by the chi-square fit index. Naturally, adding paths or allowing correlated error terms should only be done when it makes substantive theoretical as well as statistical sense to do so. The more model modifications done on the basis of sample data as reflected in MI, the more chance the changed model will not replicate for future samples, so modifications should be done on the basis of theory, not just the magnitude of the MI. LISREL and AMOS both compute modification indexes.

      • Lagrange multiplier statistic, sometimes called "multivariate MI," is a variant in EQS software output, providing a modification index to determine if an entire set of structure coefficients should be constrained to 0 (no direct paths) in the researcher's model or not. Different conclusions might arise from this multivariate approach as compared with a series of individual MI decisions.

  • Model-trimming is deleting one path at a time until a significant chi-square difference indicates trimming has gone too far. A non-significant chi-square difference means the researcher should choose the more parsimonious model (the one in which the arrow has been dropped). The goal is to find the most parsimonious model which is well-fitting by a selection of goodness of fit tests, many of them based on the given model's model-implied covariance matrix not be significantly different from the observed covariance matrix. This is tantamount to saying the goal is to find the most parsimonious model which is not significantly different from the saturated model, which fully but trivially explains the data. After dropping a path, a significant chi-square difference indicates the fit of the simpler model is significantly worse than for the more complex model and the complex model may be retained. However, as paths are trimmed, chi-square tends to increase, indicating a worse model fit and also increasing chi-square difference. In some cases, other measures of model fit for the more parsimonious model may justify its retention in spite of a significant chi-square difference test. Naturally, dropping paths should be done only if consistent with theory and face validity.
    • Critical ratios. One focus of model trimming is to delete arrows which are not significant. The researcher looks at the critical ratios (CR's) for structural (regression) weights. Those below 1.96 are non-significant at the .05 level. However, in SPSS output, the P-level significance of each structural coefficient is calculated for the researcher, making it unnecessary to consult CRs. Is the most parsimonious model the one with the fewest terms and fewest arrows? The most parsimonious model is indeed the one with the fewest arrows, which means the fewest coefficients. However, much more weight should be given to parsimony with regard to structural arrows connecting the latent variables than to measurement arrows from the latent variables to their respective indicators. Also, if there are fewer variables in the model and yet the dependent is equally well explained, that is parsimony also; it will almost always mean fewer arrows due to fewer variables. (In a regression context, parsimony refers to having the fewest terms (and hence fewest b coefficients) in the model, for a given level of explanation of the dependent variable.)

    • Chi-square difference test, also called the likelihood ratio test, LR. It is computed as the difference of model chi-square for the larger model (usually the initial default model) and a nested model (usually the result of model trimming), for one degree of freedom. LR measures the significance of the difference between two SEM models for the same data, in which one model is a nested subset of the other. Specifically, chi-square difference is the standard test statistic for comparing a modified model with the original one. If chi-square difference shows no significant difference between the unconstrained original model and the nested, constrained modified model, then the modification is accepted on parsimony grounds.
      • Warning! Chi-square difference, like chi-square, is sensitive to sample size. In large samples, differences of trivial size may be found to be significant, whereas in small samples even sizable differences may test as non-significant.

      • Definition: nested model. A nested model is one with parameter restrictions compared to a full model. One model is nested compared to another if you can go from one model to the other by adding constraints or by freeing constraints. Constraints may include setting paths to zero, making a given variable independent of others in the model. However, the two models will still have the same variables.

      • Nested comparisons. Modified models are usually nested models with parameter constraints compared with the full unconstrained model. For instance, the subset model might have certain paths constrained to 0 whereas the unconstrained model might have non-zero equivalent paths. In fact, all paths and from a given variable might be constrained to 0, making it independent from the rest of the model. Another type of comparison is to compare the full structural model with the measurement model alone (the model without arrows connecting the latent variables), to assess whether the structural model adds significant information.

      • Hierarchical analysis. Comparison with nested models is called "hierarchical analysis," to which the chi-square difference statistic is confined. Other measures of fit, such as AIC, may be used for non-hierarchical comparisons. Chi-square difference is simply the chi-square fit statistic for one model minus the corresponding value for the second model. The degrees of freedom (df) for this difference is simply the df for the first minus the df for the second. If chi-square difference is not significant, then the two models have comparable fit to the data and for parsimony reasons, the subset model is preferred.

      • Testing for common method variance. Common method variance occurs when correlations or part of them are due not to actual relationships between variables but because they were measured by the same method (ex., self-ratings may give inflated scores on all character variables as compared to ratings by peers or supervisors). To assess common method variance, one must use a multi-method multi-trait (MTMM) approach in which each latent variable is measured by indicators reflecting two or more methods. The researcher creates two models. In the first model, covariance arrows are drawn connecting the error terms of all indicators within any given method, but not across methods. This model is run to get the chi-square. Then the researcher creates a second model by removing the error covariance terms. The second model is run, getting a different chi-square. A chi-square difference test is computed. If the two models are found to be not significantly different (p(chi-squaredifference)>.05), one may assume there is no common method variance and the researcher selects the second model (without covariance arrows connecting the indicator error terms) on parsimony grounds. Also, when running the first model (with error covariance arrows) you can look at the correlation among sets of error terms. The method with the highest correlations is the method contributing the most to common method variance.

    • Wald test. The Wald test is a chi-square-based alternative to chi-square difference tests when determining which arrows to trim in a model. Parameters for which the Wald test has a non-significant probability are arrows which are candidates for dropping. As such the Wald test is analogous to backward stepwise regression.

      
      

• Output. In AMOS, output options are selected under Analyze, Analysis Properties, Output tab, illustrated below.
  
  • Structural or Path Coefficients are the effect sizes calculated by the model estimation program. Often these values are displayed above their respective arrows on the arrow diagram specifying a model. In AMOS, these are labeled "regression weights," which is what they are, except that in the structural equation there will be no intercept term.
    • Types of estimation of coefficients in SEM. Structural coefficients in SEM may be computed any of several ways. Ordinarily, one will get similar estimates by any of the methods. In AMOS, types of estimation are selected under View, Analysis Properties, in the Analysis Properties dialog, which is shown below with default settings.
      
      • MLE. Maximum likelihood estimation (MLE or ML) is by far the most common method. Unless the researcher has good reason, this default should be taken even if other methods are offered by the modeling software. MLE makes estimates based on maximizing the probability (likelihood) that the observed covariances are drawn from a population assumed to be the same as that reflected in the coefficient estimates. That is, MLE picks estimates which have the greatest chance of reproducing the observed data. See Pampel (2000: 40-48) for an extended discussion of MLE.
        • Assumptions. Unlike OLS regression estimates, MLE does not assume uncorrelated error terms and thus may be used for non-recursive as well as recursive models, though some researchers prefer 2SLS estimation for recursive models. Key assumptions of MLE are large samples (required for asymptotic unbiasedness); indicator variables with multivariate normal distribution; valid specification of the model; and continuous interval-level indicator variables ML is not robust when data are ordinal or non-normal (very skewed or kurtotic), though ordinal variables are widely used in practice if skew and kurtosis is within +/- 1.5 [note some use 1.0, others 2.0]. If ordinal data are used with MLE, they should have at least five categories and not be strongly skewed or kurtotic. Ordinal measures modeled as numeric data with MLE (as opposed to modeled as ordered-categorical data with Bayesian estimation) of underlying continuous variables likely incur attenuation and hence may call for an adjusted statistic such as Satorra-Bentler adjustment to model chi-square; error variance estimates will be most affected, with error underestimated.

        • Starting values. Note MLE is an iterative procedure in which either the researcher or the computer must assign initial starting values for the estimates. Poor starting values (ex., opposite in sign to the proper estimates, or outside the data range) may cause MLE to fail to converge on a solution. Sometimes the researcher is wise to override manually computer-generated starting values.

        • MLE estimates of variances, covariances, and paths to disturbance terms. Whereas MLE differs from OLS in estimating structural (path) coefficients relating variables, it uses the same method (i.e., the observed values) as estimates for the variances and covariances of the exogenous variables. Each path from a latent endogenous variable to its disturbance term is set to 1.0, thereby allowing SEM to estimate the variance of the disturbance term.

      • Bayesian Estimation. Starting with Amos 6, Bayesian estimation became available. Bayesian estimation uses prior information about model parameters (ex., marginal distributions) to calculate posterior distributions. Bayesian estimation is often preferred for small samples, when MLE estimation is yielding negative variances, or for ordinal data (in fact, Bayesian estimation is required in Amos if ordered-categorical data are modeled).

      • Other estimation methods do exist and may be appropriate in some atypical situations.
        1. WLS: weighted least squares. Asymptotically distribution-free (ADF) for large samples. If there is definite violation of multivariate normality, WLS may be the choice.

        2. GLS (generalized least squares) is also a popular method when MLE is not appropriate. GLS works well for large samples (n>2500) even for non-normal data. GLS: generalized least squares. Variables can be rescaled. GLS displays asymptotic unbiasedness (unbiasedness can be assumed for large samples) and assumes multivariate normality and zero kurtosis.

        3. OLS: ordinary least squares (traditional regression) . Used sometimes to get initial parameter estimates as starting points for other methods.

        4. ULS: unweighted least squares. No assumption of normality and no significance tests available, this is the only method that is scale-dependent (different estimates for different transforms of variables). Relatively rare.

    • Standardized structural (path) coefficients. When researchers speak of structural or path coefficients in SEM, they often mean standardized ones. Standardized structural coefficient estimates are based on standardized data, including correlation matrixes. Standardized estimates are used, for instance, when comparing direct effects on a given endogenous variable in a single-group study. That is, as in OLS regression, the standardized weights are used to compare the relative importance of the independent variables. The interpretation is similar to regression: if a standardized structural coefficient is 2.0, then the latent dependent will increase by 2.0 standard units for each unit increase in the latent independent. In AMOS, the standardized structural coefficients are labeled "standardized regression weights," which is what they are. In comparing models across samples, however, unstandardized coefficients are used.
      

    • The Critical Ratio and significance of path coefficients. When the Critical Ratio (CR) is > 1.96 for a regression weight, that path is significant at the .05 level (that is, its estimated path parameter is significant). In AMOS, in the Analysis Properties dialog box check "standardized estimates" and the critical ratio will also be printed. The significance of the standardized and unstandardized estimates will be identical. In LISREL, the critical ratio is the z-value for each gamma in the Gamma Matrix of standardized and unstandardized path coefficient estimates for the paths linking the endogenous variables. The "z-values" for paths linking the exogenous to the endogenous variables are in the Beta Matrix. If the z-value is greater than or equal to 1.96, then the gamma coefficient is significant at the .05 level.

    • The Critical Ratio and the significance of factor covariances. The significance of estimated covariances among the latent variables are assessed in the same manner: if they have a c.r. > 1.96, they are significant.

    • Unstandardized structural (path) coefficients. Unstandardized estimates are based on raw data or covariance matrixes. When comparing across groups, indicators may have different variances, as may latent variables, measurement error terms, and disturbance terms. When groups have different variances, unstandardized comparisons are preferred. For unstandardized estimates, equal coefficients mean equal absolute effects on y, whereas for standardized estimates, equal coefficients mean equal effects on y relative to differences in means and variances. When comparing the same effect across different groups with different variances, researchers usually want to compare absolute effects and thus rely on unstandardized estimates.

  • Model Fit Measures are found in AMOS by running the model, then selecting View, Text Output, as illustrated below for the goodness of fit section of output:
    
    • Goodness of fit tests determine if the model being tested should be accepted or rejected. These overall fit tests do not establish that particular paths within the model are significant. If the model is accepted, the researcher will then go on to interpret the path coefficients in the model ("significant" path coefficients in poor fit models are not meaningful). LISREL prints 15 and AMOS prints 25 different goodness-of-fit measures, the choice of which is a matter of dispute among methodologists. Jaccard and Wan (1996 87) recommend use of at least three fit tests, one from each of the first three categories below, so as to reflect diverse criteria. Kline (1998: 130) recommended at least four tests, such as chi-square; GFI, NFI, or CFI; NNFI; and SRMR. Another list of which-to-publish lists chi-square, AGFI, TLI, and RMSEA. Yet another listing recommended IFI and SRMR or RMSEA. This researcher recommends reporting chi-square (CMIN), RMSEA, and one of the baseline fit measures (NFI, RFI, IFI, TLI, CFI); and if there is model comparison, also report one of the parsimony measures (PNFI, PCFI) and one of the information theory measures (AIC, BIC, CAIC, BCC, ECVI, MECVI). There is wide disagreement on just which fit indexes to report. For instance, many consider GFI and AGFI no longer to be preferred. There is agreement that one should avoid the shotgun approach of reporting all of them, which seems to imply the researcher is on a fishing expedition.
      Note not all fit indices can be computed by AMOS and thus will not appear on output when there are missing data. See the section on handling missing data. If missing data are imputed, there there are other problems using AMOS.
      • Warnings about interpreting fit indexes: A "good fit" is not the same as strength of relationship: one could have perfect fit when all variables in the model were totally uncorrelated, as long as the researcher does not instruct the SEM software to constrain the variances. In fact, the lower the correlations stipulated in the model, the easier it is to find "good fit." The stronger the correlations, the more power SEM has to detect an incorrect model. When correlations are low, the researcher may lack the power to reject the model at hand. Also, all measures overestimate goodness of fit for small samples (<200), though RMSEA and CFI are less sensitive to sample size than others (Fan, Thompson, and Wang, 1999).
        In cases where the variables have low correlation, the structural (path) coefficients will be low also. Researchers should report not only goodness-of-fit measures but also should report the structural coefficients so that the strength of paths in the model can be assessed. Readers should not be left with the impression that a model is strong simply because the "fit" is high. When correlations are low, path coefficients may be so low as not to be significant....even when fit indexes show "good fit."

        Likewise, one can have good fit in a misspecified model. One indicator of this occuring is if there are high modification indexes in spite of good fit. High MI's indicate multicollinearity in the model and/or correlated error.

        A good fit doesn't mean each particular part of the model fits well. Many equivalent and alternative models may yield as good a fit -- that is, fit indexes rule out bad models but do not prove good models.Also, a good fit doesn't mean the exogenous variables are causing the endogenous variables (for instance, one may get a good fit precisely because one's model accurately reflects that most of the exogenous variables have little to do with the endogenous variables). Also keep in mind that one may get a bad fit not because the structural model is in error, but because of a poor measurement model.

        All other things equal, a model with fewer indicators per factor will have a higher apparent fit than a model with more indicators per factor. Fit coefficients which reward parsimony, discussed below, are one way to adjust for this tendency.

      • Fit indexes are relative to progress in the field: Although there are rules of thumb for acceptance of model fit (ex., that CFI should be at least .90), Bollen (1989) observes that these cut-offs are arbitrary. A more salient criterion may be simply to compare the fit of one's model to the fit of other, prior models of the same phenomenon. For example, a CFI of .85 may represent progress in a field where the best prior model had a fit of .70.

      • Equivalent models exist for almost all models. Though systematic examination of equivalent models is still rare in practice, such examination is increasingly recommended. Kline, for instance, strongly encourages all SEM-based articles to include demonstration of superior fit of preferred models over selected, plausible equivalent models. Likewise, Spirtes notes, "It is important to present all of the simplest alternatives compatible with the background knowledge and data rather than to arbitrarily choose one" (Spirtes, Richardson, Meek, Scheines, and Glymour, 1998: 203).
        • Replacing rules (see Lee and Hershberger, 1990; Hershberger, 1994; Kline, 1998: 138-42) exist to help the researcher respecify his or her model to construct mathematically equivalent models (ones which yield the same model-predicted correlations and covariances). Also, Spirtes and his associates have created a software program which implements an algorithm for searching for covariance-equivalent models, TETRAD, downloadable with documentation from the TETRAD Project.

    • The maximum likelihood function, LL is not a goodness-of-fit test itself but is used as a component of many. It is a function which reflects the difference between the observed covariance matrix and the one predicted by the model.
      • Baseline log likelihood is the likelihood when there are no independents, only the constant, in the equation.

      • Model log likelihood is the log likelihood when the independents are included in the model also. The bigger the difference of baseline LL minus model LL, the more the researcher is sure that the independent variables do contribute to the model by more than a random amount. However, it is necessary to multiply this difference by -2 to give a chi-square value with degrees of freedom equal to the number of independent variables (including power and interaction terms). This value, -2LL, is called "model chi-square," discussed below.

    • Goodness-of-fit tests based on predicted vs. observed covariances (absolute fit indexes):
      This set of goodness-of-fit measures are based on fitting the model to sample moments, which means to compare the observed covariance matrix to the one estimated on the assumption that the model being tested is true. These measures thus use the conventional discrepancy function.
      • Model chi-square. Model chi-square, also called discrepancy , discrepancy function, likelihood ratio chi-square, chi-square fit index, chi-square goodness of fit, or simply chi-square is the most common fit test, printed by all computer programs. Model chi-square approximates for large samples what in small samples and loglinear analysis is called G², the generalized likelihood ratio. AMOS outputs model chi-square as CMIN. The chi-square value should not be significant if there is a good model fit, as in the illustration below. A significant chi-square indicates lack of satisfactory model fit. That is, chi-square is a "badness of fit" measure in that a finding of significance means the given model's covariance structure is significantly different from the observed covariance matrix. If model chi-square < .05, the researcher's model is rejected by this criterion. However, because the model chi-square is so conservative (prone to Type II error), researchers may well discount a negative model chi-square finding if other model fit measures support the model.
        

        There are three ways, listed below, in which the chi-square test may be misleading. Because of these reasons, many researchers who use SEM believe that with a reasonable sample size (ex., > 200) and good approximate fit as indicated by other fit tests (ex., NNFI, CFI, RMSEA, and others discussed below), the significance of the chi-square test may be discounted and that a significant chi-square is not a reason by itself to modify the model.

        1. The more complex the model, the more likely a good fit. In a just-identified model, with as many parameters as possible and still achieve a solution, there will be a perfect fit. Put another way, chi-square tests the difference between the researcher's model and a just-identified version of it, so the closer the researcher's model is to being just-identified, the more likely good fit will be found.
        2. The larger the sample size, the more likely the rejection of the model and the more likely a Type II error (rejecting something true). In very large samples, even tiny differences between the observed model and the perfect-fit model may be found significant.
        3. The chi-square fit index is also very sensitive to violations of the assumption of multivariate normality. When this assumption is known to be violated, the researcher may prefer Satorra-Bentler scaled chi-square, which adjusts model chi-square for non-normality.

      • Relative chi-square, also called normal or normed chi-square, is the chi-square fit index divided by degrees of freedom, in an attempt to make it less dependent on sample size. Carmines and McIver (1981: 80) state that relative chi-square should be in the 2:1 or 3:1 range for an acceptable model. Ullman (2001) says 2 or less reflects good fit. Kline (1998) says 3 or less is acceptable. Some researchers allow values as high as 5 to consider a model adequate fit (ex., by Schumacker & Lomax, 2004: 82), while others insist relative chi-square be 2 or less. Less than 1.0 is poor model fit. AMOS lists relative chi-square as CMIN/DF, as in the illustration above.

      • Satorra-Bentler scaled chi-square: Sometimes called Bentler-Satorra chi-square, this is an adjustment to chi-square which penalizes chi-square for the amount of kurtosis in the data. That is, it is an adjusted chi-square statistic which attempts to correct for the bias introduced when data are markedly non-normal in distribution. As of 2008, this statistic was only available in the EQS model-fitting program, not AMOS.

      • FMIN is the minimum fit function. It can be used as an alternative to CMIN to compute CFI, NFI, NNFI, IFI, and other fit measures. It was used in earlier versions of LISREL but is little used today.
        

      • Goodness-of-fit index, GFI, also called gamma-hat or Jöreskog-Sörbom GFI. GFI = 1 - (chi-square for the default model/chi-square for the null model). GFI varies from 0 to 1 but theoretically can yield meaningless negative values. A large sample size pushes GFI up. Though analogies are made to R-square, GFI cannot be interpreted as percent of error explained by the model. Rather it is the percent of observed covariances explained by the covariances implied by the model. That is, R² in multiple regression deals with error variance whereas GFI deals with error in reproducing the variance-covariance matrix. By convention, GFI should by equal to or greater than .90 to accept the model. As GFI often runs high compared to other fit models, many (ex., Schumacker & Lomax, 2004: 82) now suggest using .95 as the cutoff. LISREL and AMOS both computed GFI. However, because of problems associated with the measure, GFI is no longer a preferred measure of goodness of fit. Recently is has been dropped by AMOS.

        Also, when degrees of freedom are large relative to sample size, GFI is biased downward except when the number of parameters (p) is very large. Under these circumstances, Steiger recommends an adjusted GFI (GFI-hat). GFI-hat = p / (p + 2 * F-hat), where F-hat is the population estimate of the minimum value of the discrepancy function, F, computed as F-hat = (chisquare - df) / (n - 1), where df is degrees of freedom and n is sample size. GFI-hat adjusts GFI upwards. Also, GFI tends to be larger as sample size increases; correspondingly, AGFI may underestimate fit for small sample sizes, according to Bollen (1990).

      • Adjusted goodness-of-fit index, AGFI. AGFI is a variant of GFI which adjusts GFI for degrees of freedom: the quantity (1 - GFI) is multiplied by the ratio of your model's df divided by df for the baseline model, then AGFI is 1 minus this result. AGFI can yield meaningless negative values. AGFI > 1.0 is associated with just-identified models and models with almost perfect fit. AGFI < 0 is associated with models with extremely poor fit, or based on small sample size. AGFI should also be at least .90. Many (ex., Schumacker & Lomax, 2004: 82) now suggest using .95 as the cutoff. Like GFI, AGFI is also biased downward when degrees of freedom are large relative to sample size, except when the number of parameters is very large. Like GFI, AGFI tends to be larger as sample size increases; correspondingly, AGFI may underestimate fit for small sample sizes, according to Bollen (1990). AGFI is related to GFI: AGFI = 1 - [ (1 - GFI) * ( p * (p + 1) / 2*df ) ], where p is the number of parameters and df is degrees of freedom. Lisrel and Amos both computed AGFI. AGFI's use has been declining and it is no longer considered a preferred measure of goodness of fit. AMOS has now dropped it.

      • Root mean square residuals, or RMS residuals, or RMSR, or RMR. RMR is the mean absolute value of the covariance residuals. Its lower bound is zero but there is no upper bound, which depends on the scale of the measured variables. The closer RMR is to 0, the better the model fit. One sees in the literature such rules of thumb as that RMR should be < .10, or .08, or .06, or .05, or even .04 for a well-fitting model. These rules of thumb are not unreasonable, but since RMR has no upper bound, an unstandardized RMR above such thresholds does not necessarily indicate a poorly fitting model. As RMR is difficult to interpret, SRMR is recommended instead. Unstandardized RMR is the coefficient which results from taking the square root of the mean of the squared residuals, which are the amounts by which the sample variances and covariances differ from the corresponding estimated variances and covariances, estimated on the assumption that your model is correct. Fitted residuals result from subtracting the sample covariance matrix from the fitted or estimated covariance matrix. LISREL computes RMSR. AMOS does also, but calls it RMR.

      • Standardized root mean square residual, Standardized RMR (SRMR): SRMR is the average difference between the predicted and observed variances and covariances in the model, based on standardized residuals. Standardized residuals are fitted residuals (see above) divided by the standard error of the residual (this assumes a large enough sample to assume stability of the standard error). The smaller the SRMR, the better the model fit. SRMR = 0 indicates perfect fit. A value less than .05 is widely considered good fit and below .08 adequate fit. In the literature one will find rules of thumb setting the cutoff at < .10, .09, .08, and even .05, depending on the authority cited. . Note that SRMR tends to be lower simply due to larger sample size or more parameters in the model. To get SRMR in AMOS, select Analyze, Calculate Estimates as usual. Then Select Plugins, Standardized RMR: this brings up a blank Standardized RMR dialog. Leave this blank dialog open and then re-select Analyze, Calculate Estimates, and the Standardized RMR dialog will display SRMR in the previously blank dialog box.

      • Hoelter's critical N issued to judge if sample size is adequate. By convention, sample size is adequate if Hoelter's N > 200. A Hoelter's N under 75 is considered unacceptably low to accept a model by chi-square. Two N's are output, one at the .05 and one at the .01 levels of significance. This throws light on the chi-square fit index's sample size problem. AMOS and LISREL compute Hoelter's N. For the .05 level, Hoelter's N is computed as (((2.58+(2df - 1)**2)**.5)/((2chisq)/(n-1)))+1, where chisq is model chi-square, df is degrees of freedom, and n is the number of subjects.

    • Information theory goodness of fit measures are also absolute fit indexes, though of a different type. Measures in this set are appropriate when comparing models which have been estimated using maximum likelihood estimation. As a group, this set of measures is less common in the literature, but that is changing. All are computed by AMOS. They do not have cutoffs like .90 or .95. Rather they are used in comparing models, with the lower value representing the better fit.
      • AIC is the Akaike Information Criterion. AIC is a goodness-of-fit measure which adjusts model chi-square to penalize for model complexity (that is, for lack of parsimony and overparameterization). Thus AIC reflects the discrepancy between model-implied and observed covariance matrices. AIC is used to compare models and is not interpreted for a single model. It may be used to compared models with different numbers of latent variables, not just nested models with the same latents but fewer arrows. The absolute value of AIC has no intuitive value, except by comparison with another AIC, in which case the lower AIC reflects the better-fitting model. Unlike model chi-square, AIC may be used to compare non-hierarchical as well as hierarchical (nested) models based on the same dataset, whereas model chi-square difference is used only for the latter. It is possible to obtain AIC values < 0. AIC close to zero reflects good fit and between two aic measures, the lower one reflects the model with the better fit. AIC can also be used for hierarchical (nested) models, as when one is comparing nested modifications of a model. in this case, one stops modifying when AIC starts rising. AIC is computed as (chisq/n) + (2k/(n-1)), where chisq is model chi-square, n is the number of subjects, and k is (.5v(v+1))-df, where v is the number of variables and df is degrees of freedom. See Burnham and Anderson (1998) for further information on AIC and related information theory measures.
        
        • AIC₀. Following Burnham and Anderson (1998: 128), the AMOS Specification Search tool by default rescales AIC so when comparing models, the lowest AIC coefficient is 0. For the remaining models, the Burnham-Anderson interpretation is: AIC₀ <= 2, no credible evidence the model should be ruled out; 2 - 4, weak evidence the model should be ruled out; 4 - 7, definite evidence; 7 - 10 strong evidence; > 10, very strong evidence the model should be ruled out.

        • Schumacker & Jones (2004: 105) point out that EQS uses a different AIC formula from Amos or LISREL, and therefore may give different coefficients.

        • AICC is a version of AIC corrected for small sample sizes.

      • BIC_p. BIC can be rescaled so Akaike weights/Bayes factors sum to 1.0. In AMOS Specification Search, this is done in a checkbox under Options, Current Results tab. BIC_p values represent estimated posterior probabilities if the models have equal prior probabilities. Thus if BIC_p = .60 for a model, it is the correct model with a probability of 60%. The sum of BIC_p values for all models will sum to 100%, meaning 100% probability the correct model is one of them, a trivial result but one which points out the underlying assumption that proper specification of the model is one of the default models in the set. Put another way, "correct model" in this context means "most correct of the alternatives."

      • BIC_L. BIC can be rescaled so Akaike weights/Bayes factors have a maximum of 1.0. In AMOS Specification Search, this is done in a checkbox under Options, Current Results tab. BIC_L values of .05 or greater in magnitude may be considered the most probable models in "Occam's window," a model-filtering criterion advanced by Madigan and Raftery (1994).

      • CAIC is Consistent AIC, which penalizes for sample size as well as model complexity (lack of parsimony). The penalty is greater than AIC or BCC but less than BIC. As with AIC, the lower the CAIC measure, the better the fit.

      • BCC is the Browne-Cudeck criterion, also called the Cudeck & Browne single sample cross-validation index. It should be close to .9 to consider fit good. It is computed as (chisq/n) + ((2k)/(n-v-2)), where chisq is model chi-square, n is number of subjects, v is number of variables, and k is (.5v(v+1))-df, where df is degrees of freedom. BCC penalizes for model complexity (lack of parsimony) more than AIC.

      • ECVI , the expected cross-validation index, in its usual variant is useful for comparing non-nested models, as in multisample analysis of a development sample model with the same model using a validation dataset, in cross-validation. .Like AIC, it reflects the discrepancy between model-implied and observed covariance matrices. Lower ECVI is better fit. When comparing nested models, chi-square difference is normally used. ECVI if used for nested models differs from chi-square difference in that ECVI penalizes for number of free parameters. This difference between ECVI and chi-square difference could affect conclusions if the chi-square difference is a substantial relative to degrees of freedom.
        

      • MECVI, the modified expected cross-validation index, is a variant on BCC, differing in scale factor. Compared to ECVI, a greater penalty is imposed for model complexity. Lower is better between models.

      • CVI, the cross-validation index, less used, serves the same cross-validation purposes as ECVI and MECVI. A value of 0 indicates the model-implied covariance matrix from the calibration sample is identical to the sample covariance matrix from the validation sample. There is no commonly accepted rule of thumb on how close to zero is "close enough." However, when comparing alternative models, the one with the lowest CVI has the greatest validity. "Double cross-validation" with CVI is computing CVI twice, reversing the roles (calibration vs. validation) of the two samples.

      • BIC is the Bayesian Information Criterion, also known as Akaike's Bayesian Information Criterion (ABIC) and the Schwarz Bayesian Criterion (SBC). Like CAIC, BIC penalizes for sample size as well as model complexity. Specifically, BIC penalizes for additional model parameters more severely than does AIC. In general , BIC has a conservative bias tending toward Type II error (thinking there is poor model fit when the relationship is real). Put another way, compared to AIC, BCC, or CAIC, BIC more strongly favors parsimonious models with fewer parameters. BIC is recommended when sample size is large or the number of parameters in the model is small.

        BIC is an approximation to the log of a Bayes factor for the model of interest compared to the saturated model. BIC became popular in sociology after it was popularized by Raftery in the 1980s. See Raftery (1995) on BIC's derivation. Recently, however, the limitations of BIC have been highlighted. See Winship, ed. (1999), on controversies surrounding BIC. BIC uses sample size n to estimate the amount of information associated with a given dataset. A model based on a large n but which has little variance in its variables and/or highly collinear independents may yield misleading model fit using BIC.

        • BIC₀. Following Burnham and Anderson (1998: 128), the AMOS Specification Search tool by default rescales BIC so when comparing models, the lowest BIC coefficient is 0. For the remaining models, the Raftery (1995) interpretation is: BIC₀ <= 2, weak evidence the model should be ruled out; 2 - 4, positive evidence the movel should be ruled out; 6 - 10, strong evidence; > 10, very strong evidence the model should be ruled out.

    • Non-centrality based goodness of fit measures alter tests to test the proposition that chi-square is greater than 0 rather than the usual null hypotheses that chi-square is zero. By flipping the test this way, one uses the non-central chi-square distribution rather than the chi-square distribution and uses the degrees of freedom associated with the perfect saturated model rather than the independence model. Non-centrality goodness of fit measures are influenced by sample size, with large sample size yielding higher fit indexes for the same model.
      • Noncentrality parameter, NCP, also called the McDonald noncentrality parameter index and DK, is chi-square penalizing for model complexity. It is computed as ((chisqn-dfn)-(chisq-df))/(chisqn-dfn), where chisqn and chisq are model chi-squares for the null model and the given model, and dfn and df are the corresponding degrees of freedom. To force it to scale to 1, the conversion is exp(-DK/2). NCP is used with a table of the noncentral chi-square distribution to assess power and as a basis for computing RMSEA, CFI, RNI, and CI model fit coefficients. Raykov (2000, 2005) and Curran et al. (2002) have argued that these fit measures based on noncentrality are biased.
        

      • Relative non-centrality index, RNI, penalizes for sample size as well as model complexity. It should be greater than .9 for good fit. The computation is ((chisqn/n -dfn/n)-DK)/(chisqn/n-dfn/n), where chisqn and chisq are model chi-squares for the null model and the given model, dfn and df are the corresponding degrees of freedom, n is the number of subjects, and DK is the McDonald noncentrality index. There is also a McDonald relative non-centrality index, computed as 1 - ((chisq-df)/(chisqn-dfn)). Note Raykov (2000, 2005) and Curran et al. (2002) have argued that RNI, because based on noncentrality, is biased as a model fit measure.

      • Centrality index, CI: CI is a function of model chi-square, degrees of freedom in the model, and sample size. By convention, CI should be .90 or higher to accept the model.

    • Goodness-of-fit tests comparing the given model with a null or an alternative model. This set of goodness of fit measures compare the researcher's model to the fit of another model, usually the . independence model. Since the fit of the independence model is the worst case (chi-square is maximum), comparing your model to it will generally make the researcher's model look good but may not serve research purposes. AMOS computes all of measures in this set.
      • The comparative fit index, CFI: Also known as the Bentler Comparative Fit Index. CFI compares the existing model fit with a null model which assumes the latent variables in the model are uncorrelated (the "independence model"). That is, it compares the covariance matrix predicted by the model to the observed covariance matrix, and compares the null model (covariance matrix of 0's) with the observed covariance matrix, to gauge the percent of lack of fit which is accounted for by going from the null model to the researcher's SEM model. Note that to the extent that the observed covariance matrix has entries approaching 0's, there will be no non-zero correlation to explain and CFI loses its relevance. CFI is similar in meaning to NFI (see below) but penalizes for sample size. CFI and RMSEA are among the measures least affected by sample size (Fan, Thompson, and Wang, 1999). CFI varies from 0 to 1 (if outside this range it is reset to 0 or 1). CFI close to 1 indicates a very good fit. CFI is also used in testing modifier variables (those which create a heteroscedastic relation between an independent and a dependent, such that the relationship varies by class of the modifier). By convention, CFI should be equal to or greater than .90 to accept the model, indicating that 90% of the covariation in the data can be reproduced by the given model. It is computed as (1-max(chisq-df,0))/(max(chisq-df),(chisqn-dfn),0)), where chisq and chisqn are model chi-square for the given and null models, and df and dfn are the corresponding degrees of freedom. Note Raykov (2000, 2005) and Curran et al. (2002) have argued that CFI, because based on noncentrality, is biased as a model fit measure.
        

      • The incremental fit index, IFI, also known as Bollen's IFI, BL89. or Delta2 (Δ₂). IFI = (chi-square for the null model - chi-square for the default model)/(chi-square for the null model - degrees of freedom for the default model). By convention, IFI should be equal to or greater than .90 to accept the model. IFI can be greater than 1.0 under certain circumstances. IFI is relatively independent of sample size and is favored by some researchers for that reason.

      • The normed fit index, NFI, also known as the Bentler-Bonett normed fit index, or simply Delta1. NFI was developed as an alternative to CFI, but one which did not require making chi-square assumptions. "Normed" means it varies from 0 to 1, with 1 = perfect fit. NFI = (chi-square for the null model - chi-square for the default model)/chi-square for the null model. NFI reflects the proportion by which the researcher's model improves fit compared to the null model (random variables, for which chi-square is at its maximum. For instance, NFI = .50 means the researcher's model improves fit by 50% compared to the null model. Put another way, the researcher's model is 50% of the way from the null (independence baseline) model to the saturated model. By convention, NFI values above .95 are good (ex., by Schumacker & Lomax, 2004: 82), between .90 and .95 acceptable, and below .90 indicates a need to respecify the model. Some authors have used the more liberal cutoff of .80. NFI may underestimate fit for small samples, according to Ullman (2001). Also, NFI does not reflect parsimony: the more parameters in the model, the larger the NFI coefficient, which is why NNFI below is now preferred.

      • The Bentler-Bonett index, BBI (not to be confused with the Bentler-Bonett normed fit index, NFI, discussed below): is the model chi-square for the given model minus model chi-square for the null model, this difference divided by model chi-square for the null model. BBI should be greater than .9 to consider fit good.

      • TLI, also called the (Bentler-Bonett) non-normed fit index, NNFI (in EQS), the Tucker-Lewis index, TLI (this is the label in AMOS), , the Tucker-Lewis rho index, or RHO2. TLI is similar to NFI, but penalizes for model complexity. Marsh et al. (1988, 1996) found TLI to be relatively independent of sample size. TLI is computed as (chisqn/dfn - chisq/df)/(chisqn/dfn - 1), where chisq and chisqn are model chi-square for the given and null models, and df and dfn are the associated degrees of freedom. NNFI is not guaranteed to vary from 0 to 1, but if outside the 0 - 1 range may be arbitrary reset to 0 or 1. It is one of the fit indexes less affected by sample size. A negative NNFI indicates that the chisquare/df ratio for the null model is less than the ratio for the given model, which might occur if one's given model has very few degrees of freedom and correlations are low.

        NNFI close to 1 indicates a good fit. Rarely, some authors have used the a cutoff as low as .80 since TLI tends to run lower than GFI. However, more recently, Hu and Bentler (1999) have suggested NNFI >= .95 as the cutoff for a good model fit and this is widely accepted (ex., by Schumacker & Lomax, 2004: 82) as the cutoff. . NNFI values below .90 indicate a need to respecify the model.

      • The Bollen86 Fit Index is identical to NNFI except the "-1" term is omitted in the foregoing equation. It should be greater than .9 for a good fit.

      • The relative fit index, RFI, also known as RHO1, is not guaranteed to vary from 0 to 1. RFI = 1 - [(chi-square for the default model/degrees of freedom for the default model)/(chi-square for the null model/degrees of freedom for the default model)]. RFI close to 1 indicates a good fit.

      • Warning: LISREL vs. AMOS and EQS. Schmukle & Hardt (2005) have demonstrated that the different method employed by LISREL compared to most other SEM statistical packages, such as AMOS and EQS, results in different values for incremental fit indexes which compare the fitted model with the null model. Consequently, using LISREL estimates of such measures as NFI, TLI, and CFI will result in Type 1 error (false positives), because LISREL estimates run higher and should not be used with traditional cutoffs for these goodness of fit measures. (Specifically, since LISREL version 8.52, LISREL has used a reweighted least squares rather than maximum-likelihood least squares fitting function -while both generate the same parameter estimates, the chi-square values differ, affecting incremental goodness of fit measures.)

    • Goodness-of-fit tests penalizing for lack of parsimony. Parsimony measures penalize for lack of parsimony on the rationale that more complex models will, all other things equal, generate better fit than less complex ones. Parsimony measures do not use the same cutoffs as their counterparts (ex., PCFI does not use the same cutoff as CFI) but rather will be noticeably lower in most cases. Used when comparing models, the higher parsimony measure represents the better fit. There is controversy surrounding use of parsimony measures, with some researchers opposed to penalizing models with more parameters. This school suggests using other goodness of fit measures to assess acceptable models, then using parsimony measures to select among the set of acceptable models.
      • The parsimony ratio (PRATIO) is the ratio of the degrees of freedom in your model to degrees of freedom in the independence (null) model. PRATIO is not a goodness-of-fit test itself, but is used in goodness-of-fit measures like PNFI and PCFI which reward parsimonious models (models with relatively few parameters to estimate in relation to the number of variables and relationships in the model). See also the parsimony index, below.
        

      • The parsimony normed fit index, PNFI,also known as PNFI1, is equal to the PRATIO times NFI (see above). The closer your model is to the (all-explaining but trivial) saturated model, the more NFI is penalized. There is no commonly agreed-upon cutoff value for an acceptable model. Parsimony-adjusted coefficients are lower than their non-adjusted counterparts, and the .95 cutoffs do not apply. There is no accepted cut-off level for a good model. When comparing nested models, the model with the lower PNFI is better

      • PNFI2 is similar to PNFI1, but is based on adjusting IFI rather than NFI for parsimony.

      • The parsimony comparative fit index, PCFI, is equal to PRATIO times CFI (see above).The closer your model is to the saturated model, the more CFI is penalized. There is no commonly agreed-upon cutoff value for an acceptable model. When comparing nested models, the model with the lower PCFI is better

      • The parsimony goodness of fit index, PGFI. PGFI is a variant of GFI which penalizes GFI by multiplying it times the ratio formed by the degrees of freedom in your model divided by degrees of freedom in the independence model.

      • The parsimony index is the parsimony ratio times BBI, the Bentler/Bonett index, discussed above.It should be greater than .9 to assume good fit.

      • Root mean square error of approximation, RMSEA, is also called RMS or RMSE or discrepancy per degree of freedom. By convention (ex.,Schumacker & Lomax, 2004: 82) there is good model fit if RMSEA less than or equal to .05. There is adequate fit if RMSEA is less than or equal to .08. More recently, Hu and Bentler (1999) have suggested RMSEA <= .06 as the cutoff for a good model fit. RMSEA is a popular measure of fit, partly because it does not require comparison with a null model and thus does not require the author posit as plausible a model in which there is complete independence of the latent variables as does, for instance, CFI. also, RMSEA has a known distribution, related to the non-central chi-square distribution, and thus does not require bootstrapping to establish confidence intervals. confidence intervals for RMSEA are reported by some statistical packages. It is one of the fit indexes less affected by sample size, though for smallest sample sizes it overestimates goodness of fit (Fan, Thompson, and Wang, 1999). RMSEA is computed as ((chisq/((n-1)df))-(df/((n-1)df)))*.5, where chisq is model chi-square, df is the degrees of freedom, and n is number of subjects. Note Raykov (2000, 2005) and Curran et al. (2002) have argued that RMSEA, because based on noncentrality, is biased as a model fit measure.
        

        It may be said that RMSEA corrects for model complexity, as shown by the fact that df is in its denominator. However, degrees of freedom is an imperfect measure of model complexity. Since RMSEA computes average lack of fit per degree of freedom, one could have near-zero lack of fit in both a complex and in a simple model and RMSEA would compute to be near zero in both, yet most methodologists would judge the simpler model to be better on parsimony grounds. Therefore model comparisons using RMSEA should be interpreted in the light of the parsimony ratio, which reflects model complexity according to its formula, PR = df(model)/df(maximum possible df). Also, RMSEA is normally reported with its confidence intervals. In a well-fitting model, the lower 90% confidence limit includes or is very close to 0, while the upper limit is less than .08.

        • PCLOSE tests the null hypothesis that RMSEA is no greater than .05. If PCLOSE is less than .05, we reject the null hypothesis and conclude that the computed RMSEA is greater than .05, indicating lack of a close fit. LISREL labels this the "P-Value for Test of Close Fit."

  • Pattern coefficients (also called factor loadings. factor pattern coeffiicents, or validity coefficients): The latent variables in SEM are similar to factors in factor analysis, and the indicator variables likewise have loadings on their respective latent variables. These coefficients are the ones associated with the arrows from latent variables to their respective indicator variables. By convention, the indicators should have loadings of .7 or higher on the latent variable (ex., Schumacker & Lomax, 2004: 212). The loadings can be used, as in factor analysis, to impute labels to the latent variables, though the logic of SEM is to start with theory, including labeled constructs, and then test for model fit in confirmatory factor analysis. Loadings are also used to assess the reliability of the latent variables, as described below.

    • Factor structure is the term used to collectively refer to the entire set of pattern coefficients (factor loadings) in a model.

    • Communalites. The squared factor loading is the communality estimate for a variable. The communality measures the percent of variance in a given variable explained by its latent variable (factor) and may be interpreted as the reliability of the indicator.

    • Construct reliability, by convention, should be at least .70 for the factor loadings. Let sl_i be the standardized loadings for the indicators for a particular latent variable. Let e_i be the corresponding error terms, where error is 1 minus the reliability of the indicator, which is the square of the indicator's standardized loading.
      reliability = [(SUM(sl_i))²]/[(SUM(sl_i))² + SUM(e_i))].

    • Variance extracted, by convention, should be at least .50. Its formula is a variation on construct reliability:
      variance extracted = [(SUM(sl_i²)]/[(SUM(sl_i²) + SUM(e_i))].

  • R-squared, the squared multiple correlation. There is one R-squared or squared multiple correlation (SMC) for each endogenous variable in the model. It is the percent variance explained in that variable. In Amos, enter $smc in the command area to obtain squared multiple correlations. In the AMOS Analysis Properties dialog box check squared multiple correlation if in the graphical mode, or if in the BASIC mode, enter $smc in the command area.

  • Quantile or Q-Plots order the standardized residuals by size and their percentage points in the sample distribution are calculated. Then the residuals are plotted against the normal deviates corresponding to these percentage points, called normal quantiles. Stem-and-leaf plots of standardized residuals are also available in LISREL.
    1. Squared multiple correlations for the Y variable: This is the portion of LISREL output which gives the percent of the variance in the dependent indicators attributed to the latent dependent variable(s) rather than to measurement error.

    2. Squared multiple correlations for the X variables: This is the portion of LISREL output which gives the percent of the variance in the independent indicators attributed to the latent independent variables rather than to measurement error.

    3. Squared multiple correlations for structural equations: This is the portion of LISREL output which gives the percent of the variance in the latent dependent variable(s) accounted for by the latent independent variables.

  • Completely standardized solution: correlation matrix of eta and KSI: In LISREL output this is the matrix of correlations of the latent dependent and latent independent variables. Eta is a coefficient of nonlinear correlation.

    
    

• Specification search is provided by AMOS as an automated alternative to manual model-building and model-trimming discussed above. In general, the researcher opens the Specification Search toolbar, chooses which arrows to make optional or required, sets other options, presses the Perform Specification Search button (a VCR-type right-arrow 'play' icon), then views output of alternative default and null models, with various fit indices. The researcher selects the best-fitting model which is also consistent with theory. Note specification search is not available for multigroup models, discussed below. See Arbuckle (1996) for a complete description.
  • Specification Search toolbar is opened in AMOS under Analyze, Specification Search.
    

    Example. In the example above, the path from Perform to PriorAbility has been made optional (here shown in green, but yellow in AMOS). Therefore the specification search generates two default models, one with and one without the optional arrow. In the output, the original full model with the optional arrow is the one with the larger number of parameters (12). Various fit measures are shown, explained below. By most (but not all) measures, such as AIC, the original model is best-fitting.

    • Make Arrows Optional tool is the first tool on the Specification Search toolbar. It is represented by a dotted-line icon. Click the Optional tool, then click on an arrow in the graphic model. It will turn yellow, indicating it is now optional (it can be made to turn dashed by selecting View, Interface Properties; Accessibility tab; check Alternative to Color checkbox). For instance, if the researcher were unsure which way to draw the structural arrows connecting two endogenous variables, the researcher could draw two arrows, one each way, and make both optional.
      • Show/Hide All Optional Arrows tool. These two tools turn the optional arrows on and off in the graphic diagram. Both icons are path trees, with the "Show" tool having blue optional lines and the "Hide" tool not.

      • Make Arrow Required tool turns an optional arrow back to a required arrow. This tool's icon is a straight black line. When used, the arrow turns black in the graphic diagram.

    • Options button. The Options button leads to the Options dialog box, which contains three tabs: Current Results, Next Search, and Appearance.

      • Current Results tab. Here one can select which fit and other coefficients to display in the output; whether to show saturated and null models; what criteria to use for AIC, BIC, and BCC (Raw, 0, P, and L models are possible, with 0 the default; see the discussion of BIC, below); whether to ignore inadmissability and instability; and a Reset button to go back to defaults.

      • Next Search tab. Here one can specify to "Retain only the best ____ models" to specify maximum number of models explored (this can speed up the search but may prevent normalization of Akaike weights and Bayes factors so they do not sum to 1 across models); one can specify Forward, Backward, Stepwise, or All Subsets searching; and one can specify which benchmark models are to be used (ex., Saturated, Null 1,Null 2).

      • Appearance tab. This tab lets you set Font; Text Color; Background Color; and Line Color.

    • Perform Specification Search button . After this button is clicked and after a period of computational time, the Specification Search toolbox window under default settings will display the results in 12 columns:
      1. Model: Null 1....Null n, as applicable; 1.....n for n default models; Sat for the Saturated model.
      2. Name: Null 1....Null n, as application; "Default model" for 1...n default models; "[Saturated]"
      3. Params: Number of parameters in the model; lower is more parsimonious.
      4. df: Degrees of freedom in the model.
      5. C: Model chi-square, a.k.a. likelihood ratio chi-square, CMIN; higher is better.
      6. C - df: C with a weak penalty for lack of parsimony; higher is better.
      7. AIC₀: The Akaike Information Criterion; AIC is rescaled so the smallest AIC value is 0 (assuming the default under Options, Current Results tab is set to "Zero-based"), and lower is better. As a rule of thumb, a well-fitting model has AIC 0 < 2.0. Models with 2 < AIC 0 < 4 may be considered weakly fitting. Note: AIC is not default output but must be selected under Options, Current Results.
      8. BCC₀: The Browne-Cudeck Criterion; .BCC is rescaled so the smallest BCC value is 0 (assuming the default under Options, Current Results tab is set to "Zero-based"), and lower is better. As a rule of thumb, a well-fitting model has BCC 0 < 2.0. models with 2 < BCC 0 < 4 may be considered weakly fitting.
      9. BIC₀: Bayesian Information Criterion; BIC is rescaled so the smallest BIC value is 0 (assuming the default under Options, Current Results tab is set to "Zero-based"), and lower is better.
      10. C/df: Relative chi-square (stronger penalty for lack of parsimony); higher is better.
      11. p: The probability for the given model of getting as large a value of model chi-square as would occur in the present sample assuming a correctly specified perfectly-fitting model. The p value tests model fit, and larger p values reflect better models.
      12. Notes: As applicable, often none. If a model is marked "Unstable" this means it involves recursivity with regression coefficient values which are such that coefficient estimation fails to converge on a set of stable coefficients. That is, unstable models are characterized by infinite regress in the iterative estimation process, and the regress does not stabilize on a reliable set of coefficient values. Rather, for unstable models, the parameter values represent an unknown degree of unreliability.

    • Other tools include Show Summary (column page icon); Increase/Decrease Decimal Places icons (up or down arrow icons with ".00"); Short List (icon with small page with up/down arrows under), which shows the best model for any given number of parameters; Show Graphs (scatterplot icon), which shows a type of scree plot with all the models by Number of Parameters on the X axis and any of six Fit Measures on the Y axis (click on a point to reveal which model it is); Show Path Diagram (blue rectangle icon), shows selected model in main graphic workspace; Show Parameter Estimates on Path Diagram (gamma icon); Copy Rows to Clipboard (double sheets copy icon); Print (printer icon); Print Preview; About this Program; and Help.

• Testing models for convergent and divergent validity. Validity is discussed in a separate section.
  • Convergent validity is demonstrated by showing that indicators for latent variables correlate with each other to an acceptable degree. Acceptable goodness of fit measures for a model indicate convergent validity when it is also the case that the pattern coefficients, discussed above, are at least .70 for all indicators using a common rule of thumb. .

  • Divergent validity is demonstrated by showing that the indicators are better associated with their respective latent variables than they are with other latent variables. Discriiminant validity is indicated when modification index coefficients do not indicate the need for cross-loadings (arrows to indicators from latent variables other than their own latent). An additional confirmation of divergent validity, suggested by Dunn, Seaker, & Waller (1994), is to compare the fit of the best fitting model with that of a model with one general trait factor associated with all the indicator variables (accomplished by constraining the correlations among latents to be 1.0), demonstrating that the general trait model has inferior fit to a degree found significant by the chi-square difference test.. In a more rigorous variant, Anderson & Gerbing (1988) recommend each pair of latents be tested separately, first with the correlation unconstrained and then with the correlation constrained to 1.0, demonstrating for each pair that the constrained model is significantly inferior in fit.

  
  

• Multiple Group Analysis or multi-sample SEM analysis is used for cross-validation (compare model calibration/development sample with a model validation sample); experimental research (compare treatment group with control group); and longitudinal analysis (compare an earlier sample with a sample at a later time), as well as simply to compare two groups in a cross-sectional sample (ex., males v. females).
  • Data setup for basic multigroup analysis. Multigroup analysis in AMOS requires that the researcher first enter multiple group names under Analyze, Manage Groups. The group names also appear on the AMOS display area, as illustrated below:
    

    The next step is to associate the group names with actual data files under Data, Data Files. as illustrated below, using the Data Files button as usual to add the files, here males.sav and females.sav,

    

    Before testing for measurement invariance across groups, the researcher first checks to see if the model as drawn has acceptable fit for each of the multiple groups (in this example, for males and females). Often the researcher tests one-sample models separately first. For instance one might test the model separately for a male sample and for a female sample. Separate testing provides an overview of how consistent the model results are, but it does not constitute testing for significant differences in the model's parameters between groups. If consistency is found, then the researcher will proceed to multigroup testing. First a baseline chi-square value is derived by computing model fit for the pooled sample of all groups. To accomplish this, one simply selects Analyze, Calculate Estimates. View, Text Output, will reveal (1) the usual overall goodness of fit measures, which should show the model has acceptable fit; and (2) separate regression parameter estimates for each of the groups, and these parameters should be significant for all groups. These findings establish that the given path model is plausible for the multiple groups and set the stage for testing measurement invariance across groups.

    First, the researcher can click the "View output path" icon as illustrated below, then alternately select the groups to display the path parameter estimates to verify that they do not differ strongly by group. If the path parameters seem similar, the researcher has reason to suspect that measurement invariance is upheld (or structural invariance in the case of path parameters pertaining to the structural model).

    

  • Testing for measurement invariance across groups (multigroup modeling). Often the researcher wishes to determine if the same SEM model is applicable across groups (ex., for men as well as women; for Catholics, Protestants, and Jews; for time 1 versus time 2; etc.). The general procedure is to test for measurement invariance between the unconstrained model for all groups combined (as just discussed above), then for a model where certain parameters are constrained to be equal between the groups. If the chi-square difference statistic does not reveal a significant difference between the original and the constrained-equal models, then the researcher concludes that the model has measurement invariance across groups (that is, the model applies across groups). See Vandenberg (2002) and Vandenberg & Lance (2000).
    • Measurement invariance may be defined with varying degrees of stringency, depending on which parameters are constrained to be equal. One may test for invariance on number of factors; for invariant factor loadings; and for invariant structural relations (arrows) among the latent variables. While possible also to test for equality of error variances and covariances across groups, "the testing of equality constraints bearing on error variances and covariances is now considered to be excessively stringent..." (Byrne, 2001: 202n). Assuming a multiple group structure has been set up in AMOS under File, Data Files, then selecting Analyze, Multiple-Group Analysis brings up the dialog below, where up to eight models of progressive degrees of stringency of invariance testing may be specified, as illustrated below. When comparing models, it is usual to test in the order shown in the Multiple-Group Analysis window below. Thus measurement model invariance is to be upheld before structural model invariance, and both before residual invariance.
      

      The parameters noted in the illustration above are:

      1. Measurement weights. Regression weights for the paths from the latent variables to their respective indicator variables (the paths in the measurement model). These correspond to factor loadings in factor analysis.
      2. Measurement intercepts. The intercepts for the paths from the latent variables to their respective indicator variables.
      3. Structural weights. Regression weights for the paths from one latent variable to another (the paths in the structural model).
      4. Structural intercepts. The intercepts for the paths between latent variables.
      5. Structural means. The means of exogenous latents in the structural model.
      6. Structural covariances. The variances and covariances in the structural model. These correspond to factor variances and covariances in a factor analysis model.
      7. Structural residuals. The variances and covariances of the residual (error) variables in the structural model
      8. Measurement residuals. The variances and covariances of residual (error) variables in the measurement model

      The selection of parameters to constrain corresponds to the researcher's purpose. For instance, in testing measurement invariance for a factor analysis model, the weights to constrain to be equal across groups would be the regression weights from the latent variables (the factors) to the indicator variables.

    • It is common to define measurement invariance as being when the factor loadings of indicator variables on their respective latent factors do not differ significantly across groups (that is, to constrain measurement weights). If lack of measurement invariance is found, this means that the meaning of the latent construct is shifting across groups or over time. Interpretational confounding can occur when there is substantial measurement variance because the factor loadings are used to induce the meaning of the latent variables (factors). That is, if the loadings differ substantially across groups or across time, then the induced meanings of the factors will differ substantially even though the researcher may retain the same factor label. As explained in the factor analysis section on tests of factor invariance, the researcher may constrain factor loadings to be equal across groups or across time.

      After selecting the constraint model(s) using Analyze, Multiple-Group Analysis from the AMOS menu, the researcher selects Analyze, Calculate Estimates. In the main AMOS display as illustrated below, each of the models is evaluated. The "OK" next to each model indicates it could be fitted (was not unidentified).

      

    • Manual constraint-setting. The researcher may add constraints manually by specifying that various model parameters must be equal across groups. This is done, as illustrated below, by assigning a name to the parameter (ex., a regression weight) and then checking the box, "All groups". Then the model is fitted, yielding a chi-square value for the constrained model. A chi-square difference test is then applied to see if the difference is significant. If it is not significant, the researcher concludes that the constrained-equal model is the same as the unconstrained multigroup model, leading to the conclusion that the model does apply across groups and does display measurement invariance or other types of invariance depending on what is constrained.
      

      Then the researcher selects View, Text Output, to see the output, including the (excerpted) goodness of fit measures illustrated below. Each measure is shown for the unconstrained model and each constrained model selected under Analyze, Multiple-Group Analysis. In this example, only three such models were requested. The P value for CMIN is the model chi-square test, which for these models is non-significant, meaning that none of the models are significantly different from the saturated (perfect explanation) model and all are acceptable. By GFI, fit was acceptable for all three, meaning that constraining the groups to be equal still yielded acceptable model fit.

      

      Cheung & Rensvold (2002) examined 20 goodness of fit measures for use when testing for measurement invariance across multiple groups, recommending the use of CFI, NCP, and GFI because these measures were independent of model complexity and sample size and were uncorrelated with model chi-square.

      • Critical ratios of differences test. If the researcher asks for "critical ratios for differences" in the Output tab of View, Analysis Properties in Amos, a table is generated for the unconstrained model in which both the rows and columns are the same list of parameters as illustrated below. Critical ratios are parameter estimates divided by their corresponding standard errors. This means that CR > 1.96 indicates a parameter estimate (ex., a regression coefficient) which is significantly different from 0 at the .05 significance level, if variables are normally distributed. On this assumption, parameters with CR above more than 1.96 indicate a significant difference between groups. The illustration below contains default parameter names, but the table will be much easier to read if