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SEM Overview
SEM Language
Why use SEM
SEM Assumptions
SEM is an extension of the general linear model (GLM) that enables to test a set of regression equations simultaneously.
SEM is a combination of
Multiple regression analysis
Confirmatory factor analysis (CFA)
Path modelling (path analysis)
Multiple regression
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Relationship between set of independent variables and one dependent variable
Only observed variables are modelled
Only the dependent variable in regression has an error term. Independent variables are assumed to be modelled without error
The regression analysis performed by SPSS is limited in several ways: Multiple dependent or outcome variables are not
permitted
Mediating variables cannot be included in the same single model as predictors
Each predictor is assumed to be measured without error
The error or residual variable is only latent variable permitted in the model
Multicollinearity among the predictors may hinder result interpretation
Analysis of several regression equations simultaneously
Only observed variables without latent variables
Unlike regression models but like structural equation models, independents can be both causes and effects of other variables (e.g., mediation)
Only the endogenous variables in path models have error terms. Exogenous variables in path models are assumed to be measured without error
Path modelling
The purpose of confirmatory factor analysis is to test hypothesis about a factor structure
The theories come first
The model is derived from the theory
The model is tested for consistency with observed data
Confirmatory factor analysis
Independent variables are called exogenous variables
Dependent or mediating variables are called endogenous variables
Observed variables are directly measured by
researchers
Latent variables are not directly measured but are inferred by the relationships or correlations among measured variables in the analysis This statistical estimation is accomplished in much the same way that an
exploratory factor analysis infers the presence of latent factors from shared variance among observed variables.
The interest in SEM is often on theoretical constructs, which are represented by the latent factors
The relationships between the theoretical constructs are represented by regression or path coefficients between the latent factors
Ovals or circles represent latent variables, while rectangles or squares represent measured variables
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Assumptions underlying the statistical analyses are clear and testable, giving the investigator full control and potentially furthering understanding of the analyses
Graphical interface software boosts creativity and facilitates rapid model debugging
SEM programs provide overall tests of model fit and individual parameter estimate tests simultaneously
Regression coefficients, means, and variances may be compared simultaneously, even across multiple between-subjects groups.
Measurement and confirmatory factor analysis models can be used to remove errors, making estimated relationships among latent variables less inflected by measurement error
Sample size 15 cases per predictor (Bentler & Chou, 1987)
Min. 200 cases (Loehlin, 1992) if data normally distributed
Continuously and normally distributed endogenous variables
Complete data or appropriate handling of incomplete data
Theoretical basis for model specification and causality
Amos
SPSS data file
Building CFA model
Interpretation of Amos output
Exercises (CFA and SEM)
To launch the Amos on your computer go to Start All Programs IBM SPSS Statistics IBM SPSS Amos 19 Amos Graphics
Amos is a part of IBM SPSS software thus it can read the SPSS file without converting. To load the data go to File Data Files. The Data Files dialog box then opens. Click on File Name and navigate to the location where the data file is stored. By default, Amos looks for an SPSS file.
The Measure of Criminal Social Identity (Boduszek et al., 2012) 3 factor model (8 items measured on 5 point Likert scale)
312 participants from High Security Prison
To add the observed variables (e.g., scale items) to the diagram, first click on the blue rectangle in the upper left corner of the tool bar
Then in the empty drawing area hold down the left mouse button to draw a rectangle. Left-click seven more times to create a total of eight equally sized boxes.
To add a latent variables, click on the blue oval in the tool box
Left-click two more times to create a total of three equally sized oval boxes. You also have to add another eight oval boxes to represent measurement error specific to each of the observed variable (rectangle boxes)
Move Objects button located on the tool bar and dragging the pieces of the diagram to where you want them. The path diagram will look something like this
Move Objects button
Next, click on the Draw Paths button and click and drag from the factors to the appropriate observed variable.
The next step is to name each of the variables. The easiest way to name the observed variables is to bring up a list of variable names in the loaded data file. Go to View Variables in Dataset.
The Variables in Dataset window then opens. It is now possible to click-and-drag each variable to its corresponding rectangle in the path diagram.
To name the three latent variables double click inside the oval box to obtain Object Properties box. Then click on the Text tab and enter the name of your latent variable in the Variable name box. Amos applies the change immediately to the path diagram. It is also possible, if desired, to add a label describing the variable.
To name the unique factors representing measurement error follow the same process. Name these e1 through e8 yielding the following path diagram:
Without introducing some additional constraints the scales of the latent variables are meaningless and the model is not identified. A common procedure for setting the latent variable scale is to constrain a factor loading to equal one. Do this for the arrow pointing from Centrality to CI1C by double clicking directly on the arrow. In the Object Properties click on the Paramaters and tab and enter 1 in the field labelled Regression weight.
Follow the same steps for the arrows connecting In-Group Affect and CI4IA, In-Group Ties and CI6IT and each of the error terms with their respective indicators. You can also correlate your latent variables by selecting correlation button and connecting oval boxes.
Note that there are several ways to draw a path diagram in Amos.
A more efficient means of creating the same diagram may have
been to draw the oval representing the common latent variable,
clicking the Draw a latent variable or add an indicator to a
latent variable button, placing the curser inside the oval,
and clicking three times to create first latent variable (Centrality).
Amos adds three rectangles representing observed indicators
along with ovals representing measurement error. The scales of
the unique factors are automatically set by constraining the
regression weights to equal one. The variables could then be
named as described above (or Plugins option are used)
Before estimating the model it is possible to choose the information that will be provided in the output by going to View → Analysis Properties.
Click on the Output tab and choose the following options: Minimization history, Standardized estimates, Squared multiple correlations, and Modification indices
To start the estimation choose Analyze Calculate Estimates.
After the estimation is completed it is possible to view the parameter estimates
in the path diagram by clicking the View the output path diagram button.
By default the unstandardized estimates will display. To bring up the standardized estimates, click on the Standardized estimates option in the column between the tools and the drawing space.
Standardized
estimates
The screen should look like this:
We can observe that the all
items related to their respective latent factors indicate excellent loading values ranging from .83 to.97.
Additionally, the three latent factors are moderately correlated.
However, in order to get more information about the model than what appears in the path diagram go to View Text output.
This opens an output window giving information about the raw data, the model, estimation, model fit, and any additional information requested with the Analysis Properties box utilized earlier.
For now consider the Notes for the Model, which are displayed in the output window as follows:
This output provides information about Chi-square statistic (and degrees of freedom) and probability level. If the probability level is greater than .05, then we can be sure that the data fits the model (Note: Chi-square statistic is sensitive to large sample)
Computation of degrees of freedom (Default model)
Number of distinct sample moments: 44
Number of distinct parameters to be estimated: 27
Degrees of freedom (44 - 27):17
Result (Default model)
Minimum was achieved
Chi-square = 21.588
Degrees of freedom = 17
Probability level = .201
In order to check for more fit indices go to Model Fit output. The Amos produces very extensive model fit summary however, there are only few things that have to be reported:
Fit indices for baseline comparisons (Most important are CFI and TLI – values above .95 indicate a good model fit)
RMSEA (value below .05 indicate a good model fit)
AIC (only when you are comparing two or more models – smaller value suggest better model)
If the model is “working” we are now interested in details regarding estimates (go to Estimates output).
Again the estimates output provides lots of information however you are interested only in few of them.
Standardized regression weights:
Used when comparing direct effects on a given endogenous variable in a single-group study
Unstandardized regression weights are based on raw data
or covariance matrixes.
When comparing across groups (across samples) and groups have different variances, unstandardized comparisons are preferred.
When the critical ratio (CR) is > 1.96 for a regression weight,
that path is significant at the .05 level or better
In the P column, three asterisks (***) indicate significance smaller than .001.
We know that Criminal Social Identity has 3 factors
To test Factor structure of Rosenberg Self Esteem Scale
1 factor model (all items)
2 factor model (positive SE and negative SE)
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To combine both Confirmatory Factor Analysis and investigate the relationship between latent factors:
between self esteem (2 latent factors) and criminal social identity (3 latent factors)
Graphical representation of proposed structural model with latent variables
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