STRUCTURAL EQUATION MODELING IN AMOS

Balaji.P

Basic Concept Partial Correlation:Correlation between Y and X1 where effects ofX2 have been removed from X1 but not from Yis partial correlation

Interpretation of Partial correlation

Part correlation squared is the unique amount of total variance explained.

Sum of part correlations squared does NOT equal R square because of overlapping variance.

Multi collinearity: Existence of substantial correlation among a set of independent variables.

Latent variableLatent variables: representation of the variance shared among the variables

common variance without error or specific variance

Structural Equation Modeling Structural equation modeling (SEM), as a concept, is

a combination of statistical techniques such as exploratory factor analysis and multiple regression.

The purpose of SEM is to examine a set of relationships between one or more Independent Variables (IV) and one or more Dependent Variables (DV).

Goals of SEM To understand the patterns of

correlation/covariance among a set of variables.

To explain as much of their variance as possible with the model specified .

How SEM is different from traditional approach? Multiple equations can be estimated simultaneously Non-recursive models are possible Correlations among disturbances are possible Formal specification of a model is required Measurement and structural relations are

separated, with relations among latent variables rather than measured variables

Assessing of model fit is not as straightforward

Types of SEM models Path analysis Confirmatory factor analysis. Structural regression model Latent change model

Approach to SEM analysis Review the relevant theory and research literature to support

model specification Specify a model (e.g., diagram, equations) Determine model identification Collect data Conduct preliminary descriptive statistical analysis Estimate parameters in the model (Model Estimation) Assess model fit Model Respecification Interpret and present results.

Components of SEMLatent variables, factors, constructs

Observed variables, measures, indicators, manifest variables

Direction of influence, relationship from one variableto another

Association not explained within the model

Important Definition A measured variable (MV) is a variables that is

directly measured. A latent variable could be defined as whatever

its multiple indicators have in common with each other. It isn't measured directly.

Relationships between variables are of three types such as Association (Correlation, covariance), direct effect and indirect effect.

Path Analysis Extension of multiple regression allowing us to consider

more than one dependent variable at a time and more importantly, allowing variables to be both Dependent and Independent variables.

B is dependent as well as independent variable (mediating variable).

Path AnalysisOnce the data is available, conduction of path analysis is straightforward:

Draw a path diagram according to the theory. Conduct one or more regression analyses. Compare the regression estimates (B) to the theoretical

assumptions or (Beta) other studies. If needed, modify the model by removing or adding

connecting paths between the variables and redo stages 2 and 3.

Examples of Path Analysis

illness = p5p3 fitness + p5p4 stress + p5p1 exercise + p5p2 hardy + e5;

Examples of Path Analysis

fitness = p3p1 exercise + e3,stress = p4p2 hardy + e4,illness = p5p3 fitness + p5p4 stress + e5;

Software’s (SEM) LISREL AMOS EQS MPLUS SAS

AMOS (Analysis of Moment Structures) Starting AMOS Graphics

Reading Data in to AMOS

Filename Data.sav

AMOS can read Currently AMOS reads the following data

file formats:  Access dBase 3 – 5 Microsft Excel 3, 4, 5, and 97 FoxPro 2.0, 2.5 and 2.6 Lotus wk1, wk3, and wk4 SPSS *.sav files,

Data File

List variable in data set

Drawing in AMOS (Draw observed variable)

Move the cursor to the place where you want to place an observed variable and click your mouse. Drag the box in order to adjust the size of the box.

Click Draw unobserved

Direct effectClick path icon of direct effect and click respective independent variable drag up to dependent variable

Touch up variable

It gives neat look to our model. Click the touch up variable and click respective observed or unobserved which you want to look neat.

Add unique variable

Click and then click a box or a circle to which you want to add errors or a unique variables. (When you use "Unique Variable" button, the path coefficient will be automatically constrained to 1.)

Naming the variables in AMOS

Click list all the variable. Drag the variable from list and put directly in to observed variable.

Naming the variables in AMOS double click on the objects in the path

diagram. The Object Properties dialog box appears. And

Click on the Text tab and enter the name of the variable in the Variable name field.

Regression Weight

Normally for error value, regression weight takes the value of 1.

Performing the analysis in AMOS For our example, check the Minimization

history, Standardized estimates, and Squared multiple correlations boxes.

To run AMOS, click on the Calculate estimates icon on the toolbar.

AMOS will want to save this problem to a file.

Results When AMOS has completed the

calculations, you have two options for viewing the output:

text output, graphics output.

For text output, click the View Text icon on the toolbar.

Text output

Viewing the graphics output in AMOS

To view the graphics output, click the View output icon next to the drawing area.

Chose to view either unstandardized or standardized estimates by click one or the other in the Parameter Formats panel next to your drawing area

Standardized vs. Unstandardized Standardized coefficients can be compared across variables

within a model. Standardized coefficients reflect not only the strength of the

relationship but also variances and covariance's of variables included in the model as well of variance of variables not included in the model and subsumed under the error term.

Standardized parameter estimates are transformations of unstandardized estimates that remove scaling and can be used for informal comparisons of parameters throughout the model.

Standardized vs. Unstandardized Unstandardized parameter estimates retain

scaling information of variables and can only be interpreted with reference to the scales of the variables.

A correlation matrix standardizes values and loses the metric of the scales.

Therefore for correlation matrix, both standardizes and unstandardized are same.

Graphics output

Improving the appearance of the path diagram

You can change the appearance of your path diagram by moving objects around

To move an object, click on the Move icon on the toolbar. You will notice that the picture of a little moving truck appears below your mouse pointer when you move into the drawing area. This lets you know the Move function is active.

Then click and hold down your left mouse button on the object you wish to move. With the mouse button still depressed, move the object to where you want it, and let go of your mouse button. Amos Graphics will automatically redraw all connecting arrows.

Improving the appearance of the path diagram

If you make a mistake, there are always three icons on the toolbar to quickly bail you out: the Erase and Undo functions.

To erase an object, simply click on the Erase icon and then click on the object you wish to erase.

To undo your last drawing activity, click on the Undo icon and your last activity disappears.

Each time you click Undo, your previous activity will be removed.

If you change your mind, click on Redo to restore a change.

SEM could impacted by the requirement of sufficient sample size. A desirable goal is to

have a 20:1 ratio for the number of subjects to the number of model parameters . However, a 10:1 may be a realistic target. If the ratio is less than 5:1, the estimates may be unstable.

measurement instruments multivariate normality parameter identification outliers missing data interpretation of model fit indices

Model Identification A model is identified if:

It is theoretically possible to derive a unique estimate of each parameter

The number of equations is equal to the number of parameters to be estimated

It is fully recursive (No feedback loop)

Model identification A model is over identified if:

A model has fewer parameters than observations.

There are more equations than are necessary for the purpose of estimating parameters

Model identification A model is under identified or not

identified if: It is not theoretically possible to derive a unique

estimate of each parameter There is insufficient information for the purpose of

obtaining a determinate solution of parameters. There are an infinite number of solutions may be

obtained

Model identification Determine the # of parameters you have. Formula: (v(v+1) / 2), where v= # of observed variables Use of this formula, allows to see if trying to guess more

than the number of parameters the existing data allows. Do not want to be JUST identified (cause lack of fit

indices) or UNDER identified, therefore looking to be OVER-identified.

Being OVER identified essentially means that there are more available parameters than trying to estimate.

Example

Example

Example

Example

Model Estimation Maximum Likelihood Generalized and UnGeneralized least

square 2 stage and 3 stage least square

Model fit Model fit = sample data are consistent with

the implied model The smaller the discrepancy between the

implied model and the sample data, the better the fit.

Many fit indexes None are fallible (though some are better than others)

Fit indexes

Fit indexes

Model RespecificationWhat if the model does NOT fit? Model trimming and building

LaGrange Multiplier test (add parameters) Wald test (drop parameters)

Empirical vs. theoretical respecification What justification do you have to respecify?

Consider equivalent models

Confirmatory factor analysis How it differs from the more commonly

encountered forms of factor analysis. What is factor analysis (FA)?

have many variables and want to examine if they can be explained by a smaller number of factors.

No a priori hypothesis (impossible to even indicate a hunch to the program) as to which variables will cluster together on which factor.

Confirmatory factor analysis The major difference is that an a priori

hypothesis is essential: which variables grouped together as

manifestations of an underlying construct and fits the model.

Like with path analysis, it can be helpful to draw hypothesized relations in a diagram.

CFA is not model building With CFA, you stipulate where you think the variables

should load. Then, the program simply tells you whether your model fits the data.

If no fit, then there are few clues to guide you how to shuffle the variables around to make the model better fit the data.

Note: Even if the model does fit, it does not guarantee that a new arrangement of variables would be an even better fit.

Therefore, one must really use theory, knowledge, or previous research to guide your model, rather than rely on statistical criteria.

Scaling Scaling factor: constrain one of the

factor loadings to 1 ( that variables called – reference variable, the factor has a scale related to the explained variance of the reference variable).

fix factor variance to a constant ( ex. 1), so all factor loadings are free parameters

CFA

Duplicate objectSelect object which you want to duplicate. Click wherever you wanted same variable.

Unobserved

Click unobserved icon and draw like observed icon

Correlation or covariance

Unstandardized and Standardized estimates

Unstandardized solution Factor loadings =unstandardized regression coefficient Unanalyzed association between factors or errors= covariances

Standardized solution Unanalyzed association between factors or errors= correlations Factor loadings =standardized regression coefficient ( structure

coefficient) The square of the factor loadings = the proportion of the

explained ( common) indicator variance, R2(squared multiple correlation)

Standardized regression model Inclusion of observed and latent

variables Assessment both of relationship between

observed and latent variables.

Latent growth Analysis Can change in responses be tracked over

time? Latent Growth Curve Analysis

Thank you