Structural equation modeling can be defined as a class of methodologies that seeks to represent hypotheses about the means, variances, and covariances of observed data in terms of a smaller number of ‘structural’ parameters defined by a hypothesized underlying conceptual or theoretical model.
From: International Encyclopedia of the Social & Behavioral Sciences, 2001
Related terms:
D. Kaplan, in International Encyclopedia of the Social & Behavioral Sciences, 2001
7 Pros and Cons of Structural Equation Modeling
Structural equation modeling is, without question, one of the most popular methodologies in the quantitative social sciences. Its popularity can be attributed to the sophistication of the underlying statistical theory, the potential for addressing important substantive questions, and the availability and simplicity of software dedicated to structural equation modeling. However, despite the popularity of the method, it can be argued that ‘first-generation’ use of the method is embedded in a conventional practice that precludes further statistical as well as substantive advances.
The primary problem with the ‘first-generation’ practice of structural equation modeling lies in attempting to attain a ‘well-fitting’ model. That is, in conventional practice, if a model does not fit from the standpoint of one statistical criterion (e.g., the likelihood ratio chi-squared test), then other conceptually contradictory measures are usually reported (e.g., the NNFI). In addition, it is not uncommon to find numerous modifications made to an ill-fitting model to bring it in line with the data, usually supplemented by post hoc justification for how the modification fit into the original theoretical framework.
Perhaps the problem lies in an obsession with null hypothesis testing—certainly an issue that has received considerable attention. Or perhaps the practice of ‘first-generation’ structural equation modeling is embedded in the view that only a well-fitting model is worthy of being interpreted. Regardless, it has been argued by Kaplan (2000) that this conventional practice precludes learning valuable information about the phenomena under study that could otherwise be attained if the focus was on the predictive ability of a model. Such a focus on the predictive ability of the model combined with a change of view toward strict hypothesis testing might lead to further substantive and statistical developments. Opportunities for substantive development and model improvement emerge when the model does not yield accurate or admissible predictions derived from theory. Opportunities for statistical developments emerge when new methods are developed for engaging in prediction studies and evaluating predictive performance.
That is not to say that there are no bright spots in the field of structural equation modeling. Indeed, developments in multilevel structural equation modeling, growth curve modeling, and latent class applications suggest a promising future with respect to statistical and substantive developments. However, these ‘second-generation’ methodologies will have to be combined with a ‘second-generation’ epistemology so as to realize the true potential of structural equation modeling in the array of quantitative social sciences.
S.I. Novikova, … D. Hall, in International Review of Research in Developmental Disabilities, 2013
6.2.2 Data Analysis
Structural equation modeling (SEM) techniques were used in testing our model of SIB via MPlus (Muthén and Muthén, 2008). SEM consists of a set of multivariate techniques that are confirmatory rather than exploratory in testing whether models fit data (Byrne, 2011). SEM has three major advantages over traditional multivariate techniques: (1) explicit assessment of measurement error; (2) estimation of latent (unobserved) variables via observed variables; and (3) model testing where a structure can be imposed and assessed as to fit of the data. Most multivariate techniques inadvertently ignore measurement error by not modeling it explicitly, whereas SEM models estimate these error variance parameters for both independent and dependent variables (Byrne, 2011). In addition, SEM permits the estimation of latent variables from observed variables; thu, the creation of composites takes into account measurement error. Finally, fully developed models can be tested against the data using SEM as a conceptual or theoretical structure or model and can be evaluated for fit of the sample data. As an advanced statistical technique, SEM requires sample sizes of at least 200 to examine basic models. More complex models would require even larger samples in order to achieve statistical power. Thus, NDAR provides researchers the opportunity to utilize these sophisticated statistical techniques through the availability of larger sample sizes.
In SEM, models are first evaluated for fit. Upon satisfying fit, individual paths may be evaluated. Four statistics were considered to evaluate model fit. Comparative fit index (CFI) and Tucker Lewis index (TLI; also known as the non-normed fit index) values were evaluated, with values of 0.90 and above indicating an acceptable level of model fit (Weston and Gore, 2006). Finally, the standardized root mean residual value was considered in evaluating model fit such that values of 0.08 or less were considered indicative of acceptable model fit (Hu and Bentler, 1999; Schermelleh-Engel, Moosbrugger, & Müller, 2003).
Missing data were handled using full-information maximum likelihood (FIML) as the method of estimation in testing the model. FIML does not provide an imputation of missing data values, but rather estimates coverage of missing data at the covariance matrix level (Allison, 2003). As an extension of maximum likelihood, FIML uses all possible data points during data analyses. FIML is also known as raw data maximum likelihood, where observations are classified into different missing data patterns, “with all patterns subsequently being analyzed into a multiple group design with appropriate constraints across the groups,” (Stoel, van den Wittenboer, & Hox, 2003). Enders and Bandalos (2001) have indicated that FIML is superior to listwise, pairwise, and similar response pattern imputations in handling missing data that may be considered ignorable. Multiple imputation methods were utilized in estimating missing data for the variable of intelligence. Missing data techniques could not be utilized without the availability of larger sample sizes such as through NDAR.
L. Harrison, … K. Friston, in Statistical Parametric Mapping, 2007
Structural equation modelling
Structural equation modelling (SEM), or path analysis, is a multivariate method used to test hypotheses regarding the influences among interacting variables. Its roots go back to the 1920s, when path analysis was developed to quantify unidirectional causal flow in genetic data and developed further by social scientists in the 1960s (Maruyama, 1998). It was criticized for the limitations inherent in the least squares method of estimating model parameters, which motivated a general linear modelling approach from the 1970s onwards. It is now available in commercial software packages, including LISREL, EQS and AMOS. See Maruyama (1998) for an introduction to the basic ideas. Researchers in functional imaging started to use it in the early 1990s (McIntosh and Gonzalez-Lima, 1991, 1992a, b, 1994). It was applied first to animal autoradiographic data and later to human PET data where, among other experiments, it was used to identify task-dependent differential activation of the dorsal and ventral visual pathways (McIntosh et al., 1994). Many investigators have used SEM since then. An example of its use to identify attentional modulation of effective connectivity between prefrontal and premotor cortices can be found in Rowe et al. (2002)).
An SEM is a linear model with a number of modifications, which are illustrated in Figure 38.6: the coupling matrix, β, is ‘pruned’ to include only paths of interest. Critically, self-connections are precluded. The data matrix, Y, contains responses from regions of interest and possibly experimental or bilinear terms. The underlying model is a general linear model:
FIGURE 38.6. An SEM is used to estimate path coefficients for a specific network of connections, after ‘pruning’ the connectivity matrix. The graphic illustrates a particular sparsity structure, which is usually based on prior anatomical knowledge. yt may contain physiological or psychological data or bilinear terms (to estimate the influence of ‘contextual’ input). The innovations ɛ are assumed to be independent, and can be interpreted as driving inputs to each node.
38.14Y=Yβ+ɛ
where the free parameters, β, are constrained, according to the specified pruning or sparsity structure of connections. To simplify the model, the residuals e are assumed to be independent. They are interpreted as driving each region stochastically from one measurement to another and are sometimes called innovations.
Instead of minimizing the sum of squared errors, the free parameters are estimated using the sample covariance structure of the data. The rationale for this is that the covariance reflects the global behaviour of the data, i.e. capturing relationships among variables, in contrast to the former, which reflects the goodness of fit from the point of view of each region. Practically, an objective function is constructed from the sampled and implied covariance, which is optimized with respect to the parameters. The implied covariance, Σ(β), is computed easily by rearranging Eqn. 38.14 and assuming some value for the covariance of the innovations, (ɛTɛ):
38.15Y(I−β)=ɛY=ɛ(1−β)−1∑=〈YTY〉=(1−β)−T〈ɛTɛ〉(1−β)−1
The sample covariance is:
S=1n−1YTY
where n is the number of observations and the maximum likelihood objective function is:
38.16FML=ln|∑|−tr(S∑−1)−ln|S|
This is simply the Kullback-Leibler divergence between the sample and the covariance implied by the free parameters. A gradient descent, such as a Newton-Raphson scheme, is generally used to estimate the parameters, which minimize this divergence. The starting values can be estimated using ordinary least square (OLS) (McIntosh and Gonzalez-Lima, 1994).
Inferences about changes in the parameters or path coefficients rest on the notion of nested, or stacked, models. A nested model consists of a free-model within which any number of constrained models is ‘nested’. In a free model, all parameters are free to take values that optimize the objective function, whereas a constrained model has one, or a number of parameters omitted, constrained to be zero or equal across models (i.e. attention and non-attention). By comparing the goodness of fit of each model against the others, χ2 statistics can be derived (Bollen, 1989). Hypotheses testing proceeds using this statistic. For example, given a constrained model, which is defined by the omission of a pathway, evidence for or against the pathway can be tested by ‘nesting’ it in the free model. If the difference in goodness of fit is unlikely to have occurred by chance, the connection can be declared significant. An example of a nested model that was tested by Büchel and Friston (1997)) is shown in Figure 38.7.
FIGURE 38.7. Inference about changes in connection strengths proceeds using nested models. Parameters from free and constrained models are compared with a χ2 statistic. This example compares path coefficients during attention (A) and non-attention (NA), testing the null hypothesis that the V1 to V5 connections are the same under both levels of attention.
SEM can accommodate bilinear effects by including them as an extra node. A significant connection from a bilinear term represents a modulatory effect in exactly the same way as in a PPI. Büchel and Friston (1997)) used bilinear terms in an SEM of the visual attention data set, to establish the modulation of connections by prefrontal cortex. In this example, the bilinear term comprised measurements of activity in two brain regions. An interesting extension of SEM has been to look at models of connectivity over multiple brains (i.e. subjects). The nice thing about this is that there are no connections between brains, which provide sparsity constraints on model inversion (see Mechelli et al., 2002).
SEM shares the same limitations as the linear model approach described above, i.e. temporal information is discounted.1 However, it has enjoyed relative success and become established over the past decade due, in part, to its commercial availability as well as its intuitive appeal. However, it usually requires a number of rather ad hoc procedures, such as partitioning the data to create nested models, or pruning the connectivity matrix to render the solution tractable. These problems are confounded with an inability to capture non-linear features and temporal dependencies. By moving to dynamic models, we acknowledge the effect of an input’s history and embed a priori knowledge into models at a more plausible and mechanistic level. These issues will be addressed in the following section.
S. Sinharay, in International Encyclopedia of Education (Third Edition), 2010
Structural-Equation Modeling
Structural-equation modeling is an extension of factor analysis and is a methodology designed primarily to test substantive theory from empirical data. For example, a theory may suggest that certain mental traits do not affect other traits and that certain variables do not load on certain factors, and that structural equation modeling can be used to test the theory. (A mental trait is a habitual pattern of behavior, thought and emotion.) A structural-equation model (SEM) is a system of linear equations among several unobservable variables (constructs) and observed variables. An SEM is composed of two parts: a structural part, linking the constructs to each other (usually, this part expresses the endogenous or dependant constructs as linear functions of the exogenous or independent constructs), and a measurement part, linking the constructs to observed measurements. The second part resembles a confirmatory factor analysis model. The SEMs can be displayed in visual form – these displays are called path diagrams. The full model is then estimated from a data set and inferences drawn.
K.E. Stephan, K.J. Friston, in Encyclopedia of Neuroscience, 2009
Structural Equation Modeling
Structural equation modeling (SEM) is a multivariate, hypothesis-driven technique that is based on a structural model representing a hypothesis about the causal relations among several variables. In the context of fMRI, for example, these variables are the measured blood oxygen level-dependent (BOLD) time series y1, … ,yn of n brain regions and the hypothetical causal relations are based on anatomically plausible connections between the regions. The strength of each connection yi → yj is specified by a so-called path coefficient which, by analogy to a partial regression coefficient, indicates how the variance of yi depends on the variance of yj if all other influences on yj are held constant.
The statistical model of standard SEM can be summarized by the equation
[9]y=Ay+u
where y is an n × s matrix of n area-specific time series with s scans each, A is an n × n matrix of path coefficients (with zeros for absent connections), and u is an n × s matrix of zero mean Gaussian error terms, which are driving the modeled system (‘innovations’; see eqn [10]). Parameter estimation is achieved by minimization of the difference between the observed and the modeled covariance matrix Σ. For any given set of parameters, Σ can be computed by transforming eqn [9]:
[10]y=(I−A)−1uΣ=yyT=(I−A)−1uuT(I−A)−1T
where I is the identity matrix. The first line of eqn [10] can be understood as a generative model of how system function results from the system’s connectional structure: the measured time series y results by applying a function of the interregional connectivity matrix – that is, (I−A)−1 – to the Gaussian innovations u.
In the special case of fMRI, the path coefficients (i.e., the parameters in A) describe the effective connectivity of the system across the entire experimental session. What one would often prefer to know, however, is how the coupling between certain regions changes as a function of experimentally controlled context (e.g., differences in coupling between two different tasks). Notably, SEM does not account for temporal order: if all regional time series were permuted in the same fashion, the estimated parameters would not change. In case of blocked designs, this makes it possible to proceed as if one were dealing with PET data – that is, to partition the time series into condition-specific subseries and fit separate SEMs to them. These SEMs can then be compared statistically to test for condition-specific differences in effective connectivity. An alternative approach is to augment the model with bilinear terms (cf. eqn [9]) which represent the modulation of a given connection by an experimental condition. In this case, only a single SEM is fitted to the entire time series.
A problem with SEM is that one is obliged to use models of relatively low complexity since models with reciprocal connections and loops are often nonidentifiable. As a simple example, consider a fully connected model: in this case, the number of free parameters exceeds the number of observed covariances, which means there are no degrees of freedom. Although some heuristics for dealing with complex models have been proposed, this problem, together with the neglect of temporal order, is a critical limitation in the application of SEM to neural systems.
Jenny M. Porritt, … Sarah R. Baker, in Dentine Hypersensitivity, 2015
Analysis
Structural equation modeling (SEM) using AMOS 18.0 was used to test the proposed model (Figure 14.1). The path analysis technique used measures to the extent that the model fit a data set and allowed testing of interrelationships between a range of variables simultaneously. A bootstrapping technique was conducted using the data because this procedure has been advocated as the best approach when sample sizes are small to medium (<200).31 In addition, the bias-corrected 95% confidence interval (CI) bootstrap percentiles were used because these have been shown to be more accurate when dealing with smaller sample sizes and mediation effects.31,32 A preselection criterion was used for the path analysis and only baseline predictors of follow-up OHRQoL (DHEQ) that had P<0.20 were entered into the model (based on Spearman and Pearson correlations). Maximum likelihood was used and adequacy of overall model fit was assessed using five fit indices including the following: chi-square test statistic, which should not be significantly different from the observed data; chi-square divided by degrees of freedom (CMIN/df), which should be lower than 2.0; root mean-squared error of approximation (RMSEA), which should be less than 0.08; incremental fit index (IFI), which should be more than 0.95; and standardized root mean square residual (SRMR), which should be less than 0.08.33–35 The error variances between illness beliefs were allowed to correlate freely. Missing data were replaced by the item’s median score to generate total scores. However, if more than 50% of the values for any given questionnaire were missing, then total scores were not calculated. Within the SEM analysis, the regression imputation technique handled this missing data.
Figure 14.1. Direct pathways hypothesized between clinical variables, illness beliefs, pain-related coping and follow-up quality of life impacts experienced by adults with dentine hypersensitivity tested within model 1.
Note: Variables in pale grey not entered into final model because these were nonsignificant predictors of the primary outcome variable (follow-up OHRQoL).
B. Mišić, A.R. McIntosh, in Brain Mapping, 2015
Causal Model
As with SEM, an initial model of causal influence is defined by specifying a set of regions, which may be chosen based on hypotheses or analyses. Each region in the model is composed of neuronal subpopulations intrinsically coupled to each other. Neuronal populations at each region are extrinsically coupled to each other, forming a network. The activity of each neuronal population is governed by a set of coupled stochastic or ordinary differential equations that relate the rate of change of activity ∂x∂t to current activity (x):
[3]∂x∂t=fx,u,θc
Synaptic coupling between populations is mathematically represented by introducing terms for the state of one population into the equation for the state of another population, thereby allowing the former to influence the latter. The speed with which one population influences another is described by a set of coupling parameters (θc). The coupling parameters θc are unknown, and the purpose of DCM is to infer them, analogous to path coefficients in SEM but focusing on the observed time series. Exogenous influences (u), representing experimental manipulations, manifest as external inputs that induce changes in an individual population or in the coupling between populations.
Anjali Raja Beharelle, Steven L. Small, in Neurobiology of Language, 2016
64.3.3.1 SEM Method
The purpose of structural equation modeling (SEM) is to define a theoretical causal model consisting of a set of predicted covariances between variables and then test whether it is plausible when compared to the observed data (Jöreskog, 1970; Wright, 1934). In neuroimaging, these causal models consist of the brain activity signal of interest in a subset of ROIs and the pattern of directional influences among them (McIntosh & Gonzalez-Lima, 1991, 1994). The influences are constrained anatomically so that a direct connection between two regions is only possible if there is a known white matter pathway between them.
The first step in defining an SEM is to specify the brain regions, which are treated as variables, and the causal influences between them in terms of linear regression equations. There is always one equation for each dependent variable (activity in the ROI), and some variables can be included in more than one equation. This system of equations can be expressed in matrix notation as Y=βY+ψ, where Y contains the variances of the regional activity for the ROIs, β is a matrix of connection strengths that defines the anatomical network model, and ψ contains residual effects, which can be thought of as either the external influences from other brain regions that cannot be stipulated in the model or the influence of the brain region on itself. Because the model is underspecified, having more unknown than known parameters, it is not possible to construct the model in a completely data-driven manner, and thus some constraints are needed on the model parameters. The most common approach is to arbitrarily restrict some elements of the residual matrix ψ to a constant, usually 35–80% of the variance for a given brain region, and to set the covariances between residuals to zero (McIntosh & Gonzalez-Lima, 1994). It is also common in neuroimaging to keep the path coefficients in both directions equal for regions that have mutually coupled paths.
The main idea of SEM is that the system of equations takes on a specific causal order, which can be used to generate an implied covariance matrix (McArdle & McDonald, 1984). Unlike in multiple regression models, where the regression coefficients are derived from the minimization of the sum of squared differences from the observed and predicted dependent variables, SEM minimizes the difference between the observed covariance structure and the one implied by the structural or path model. This is done by modifying the path coefficients and residual variances iteratively until there is no further improvement in fit. In most cases, a method such as maximum likelihood estimation or weighted least-squares is used to establish a fit criterion that needs to be maximized. The identified best-fitting path coefficient has a meaning similar to a semipartial correlation in that it reflects the influence of one region onto a second region with the influences from all other regions to the second region held constant. SEM can be conceptualized as a method that uses patterns of functional connectivity (covariances) to derive information about effective connectivity (path coefficients) (McIntosh & Mišić, 2013).
Model inference is done in SEM by comparing the goodness-of-fit between the model implied covariance matrix and the empirical covariance matrix using a χ2 test. It is also possible to compare model fits using a χ2 difference test, and this can be done to examine whether one or more causal influences change as the result of a task or group effect. To this end, models are combined in a single multigroup or stacked run. The null hypothesis is that the effective connections do not differ between groups or task conditions and the null model is constructed so that path coefficients are set to be equal across groups or task conditions. The alternative hypothesis is that the effective connections are significantly different between groups or task conditions. Implied covariance matrices are generated for each group- or task-specific model. An alternative χ2 that is significantly lower (better fitting) than the null χ2 implies a significant group or task effect on the effective connections that were specified differently in the models. It is possible that the omnibus test can indicate a poor overall fit, but the difference test shows a significant change from one task to another. SEM has been shown to be robust in these cases and is able to detect changes in effective connectivity, even if the absolute fit of the model is insufficient (Protzner & McIntosh, 2006). Finally, it is possible to use an alternative approach to model selection, where nodes of the network are selected a priori, but the paths are connected in a data-driven manner (see Bullmore et al., 2000).
The advantage of SEM is that one can identify directionality in the influence of activity from one region to that of another. In addition, SEM allows the researcher to test the validity of a theoretical model regarding network interactions among regions supporting the task under investigation.
W. Wu, T.D. Little, in Encyclopedia of Adolescence, 2011
Longitudinal Data Analysis
The advances in MLM and SEM provide convenient and flexible ways to analyze longitudinal data. Longitudinal data involve repeated observations or measures over time (e.g., repeated measures on academic achievement over grades). Longitudinal data allow researchers to measure change which is a fundamental concern of practically all scientific disciplines. Longitudinal data can be analyzed in three ways: (a) as traditional repeated measures of individual differences, or panel models; (b) as growth curve models of the individual differences in the intraindividual (within person) trends of change, for example, the individual change in academic achievement; or (c) as within person models of each person, or p-technique analyses. Space limit does not allow a comprehensive discussion of those methods. In the following sections, we highlight some of the basic models and basic concepts related to longitudinal modeling, focusing on the second approach.
The second approach can be implemented using a two-level MLM because the repeated measures (level 1) are nested within individuals (level 2). Intraindividual change is modeled at first level. The interindividual differences are captured at the second level. Interestingly, such models can also be specified in the SEM framework as a latent growth curve model with equivalent results under most of the conditions. A variety of change trajectories can be modeled, ranging from linear, curvilinear (e.g., quadratic) to nonlinear (e.g., s-shaped curve). More waves of repeated measures are usually required to estimate more complex change trajectories. If distinct periods of growth exist over the course of a study (e.g., before and after an intervention), the growth trajectories should be divided into pieces representing each period. Different growth curves are then fit to the pieces simultaneously. This type of model is very useful and often referred to as spline or piecewise growth curve model.
Covariates or predictors of intraindividual and interindividual changes can also be included. Covariates that explain intraindividual change must vary across time and individuals, (i.e., time-varying covariates). For example, stress level can be a time-varying covariate to the change in depression over time. Covariates that explain individual difference in intraindividual change are constant across time but different across individuals (i.e., time-constant covariate). For example, adolescents’ perception of their connection with their parents and teachers predicts their growth in math achievement from grades 8 to 12.
In the SEM framework, one can parallel the change process of two or more outcome variables and examine how the change processes covary with one another (e.g., change series in adolescent and peer alcohol use were found positively related to each other). Longitudinal panel models of parallel constructs are useful in examining longitudinal mediation effects to answer questions such as whether change in M mediates the effect of change in X on the change in Y or whether the mediation effect of M varies across time (e.g., a prevention program increased the rate of change for perceived importance of a team leader, which in turn increased the change rate of nutrition behaviors of high school football players across years).
Brian S. Everitt, in Comprehensive Clinical Psychology, 1998
3.13.4.1.2 Stability of alienation
As a further example of structural equation modeling a study reported by Wheaton, Muthen, Alwin, and Summers (1977) is used. The study was concerned with the stability over time of attitudes such as alienation and their relationships to background variables such as education and occupation. Data were collected on attitude scales from 932 people in two rural regions in Illinois at three points in time (1966, 1967, and 1971). Only that part of the data collected in 1967 and 1971 will be of concern here and Table 24 shows the covariances between six observed variables. The anomia and powerlessness subscales are taken to be indicators of a latent variable, alienation, and the two background variables, education (years of schooling completed) and Duncan’s socioeconomic index (SEI) are assumed to relate to a respondent’s socioeconomic status. The path diagram for the model postulated to explain the covariances between the observed variables is shown in Figure 7. The model involves a combination of a confirmatory factor analysis model with a regression model for the latent variables. One of the important questions here involves the size of the regression coefficient of alienation in 1971 on alienation in 1967, since this reflects the stability of the attitude over time. Note that the error terms of anomia and powerlessness are allowed to be correlated over time to account for possible memory or other retest effects. Some of the results of fitting the proposed model are shown in Table 25.
Table 24. Covariance of manifest variables in the stability of alienation example.
| 1 | 2 | 3 | 4 | 5 | 6 | |
| 1 | 11.834 | |||||
| 2 | 6.947 | 9.364 | ||||
| 3 | 6.819 | 5.09 | 12.532 | |||
| 4 | 4.783 | 5.028 | 7.495 | 9.986 | ||
| 5 | −3.839 | −3.889 | −3.841 | −3.625 | 9.610 | |
| 6 | −2.190 | −1.883 | −2.175 | −1.878 | 3.552 | 4.503 |
Note. 1 = Anomia 67, 2 = Powerlessness 67, 3 = Anomia 71, 4 = Powerlessness 71, 5 = Education, 6 = Duncan’s Socioeconomic Index.
Figure 7. Causal model for stability of alienation.
Table 25. Regression coefficients for stability of the alienation model in Figure 5.
| Alienation 67 on socioeconomic status: −1.500, SE = 0.124 |
| Alienation 71 on alienation 67: 0.607, SE = 0.051 |
| Alienation 71 on socioeconomic status: −0.592, SE = 0.131 |
Table 26. Identification examples.
| (i)
Consider three variables, x1, x2, and x3, with correlation matrix R given by (ii) Suppose we are interested in fitting a single-factor model, that is x1=λ1f+e1x2=λ2f+e2x3=λ3f+e3 (iii) There are seven parameters to be estimated, namely λ1,λ2,λ3,var(f),var(e1),var(e2),var(e3) (iv) There are, however, only six statistics for use in parameter estimation: var(x1), var(x2), var(x3), corr(x1, x2), corr(x1, x3), corr(x2, x3) (v) Consequently, the model is underidentified (vi) If var(f) is set equal to one then the model is just identified—there are exactly the same number of parameters to estimate as there are informative sample statistics (vii) Equating observed and expected variances and correlations will give the required estimates: λˆ1λˆ2=0.83λˆ1λˆ3=0.78λˆ2λˆ3=0.67vaˆr(e1)=1.0−λˆ12vaˆr(e2)=1.0−λˆ22vaˆr(e3)=1.0−λˆ32 (where “hats” indicate estimates) (ix) Solving these equations leads to the estimates λˆ1=0.99,λˆ2=0.84,λˆ3=0.79,var(e1)=0.02,var(e2)=0.30,var(e3)=0.38 (x) Now consider an analogous measurement model with four observed variables and again set var(f) = 1 (this model is now the one used on the string data and the examination scores data) (xi) Equating observed and expected variances and correlations in this case will lead to more than a single unique estimate for some of the parameters. The model is over identified and represents a genuinely more parsimonious description of the structure of the data. Here a better strategy for estimation is clearly needed (see Table 9) |
The chi-squared goodness-of-fit statistic takes a value of 4.73 with four degrees of freedom and suggests that the proposed model fits the observed covariances extremely well. The estimated regression coefficient of alienation on socioeconomic status in both 1967 and 1971 is negative, as might have been expected since higher socioeconomic status is likely to result in lower alienation and vice versa. The estimated regression coefficient for alienation in 1971 on alienation in 1967 is positive and highly significant. Clearly the attitude remains relatively stable over the time period.
An example of the application of structural equation modeling in clinical psychology is given in Taylor and Rachman (1994).