**AMOS – Analysis of Moment Structures**Rick Zimmerman, Olga Dekhtyar HIV Prevention Center University of Kentucky**Overview**• Overview of Structural Equation Models (SEM) • Introduction to AMOS • User Interface • AMOS Graphics • Examples of using AMOS • Predictors of Condom Use using latent variables**Structural Equation Modeling (SEM)**• An extension of Regression and general Linear Models • Also can fit more complex models, like confirmatory factor analysis and longitudinal data.**Structural Equation Modeling**Ability to fit • non-standard models, • databases with autocorrelated error structures • time series analysis • Latent Curve Models, • databases with non-normally distributed variables • databases with incomplete data.**Family Tree of SEM**Latent Growth Curve Analysis T-test ANOVA Multi-way ANOVA Repeated Measure Designs Growth Curve Analysis Structural Equation Modeling Multiple Regression Path Analysis Bivariate Correlation Confirmatory Factor Analysis Next Workshop: November 9 See you there! Factor Analysis Exploratory Factor Analysis**Structural Equation Modeling (SEM)**• Exogenous variables=independent • Endogenous variables =dependent • Observed variables =measured • Latent variables=unobserved**.10**: R2 Error .15 Structural Equation Graphs Observed Variable : Loading Latent Variable**Observed variables for**Impulsive decision making Peer norms about condoms Condom attitude Example: Condom Use Model Respondent Sex IDMA1R IDMC1R IDME1R IDMJ1R SEX1 Impulsive Impulsive Decision Making FRBEHB1 ISSUEB1 Legend SXPYRC1 Latent Variables Observed Variables .15 Condom Use Loadings**Dependent**Dependent Dependent Example: Condom Use Model Independent IDMA1R IDMC1R IDME1R IDMJ1R SEX1 Impulsive Independent FRBEHB1 ISSUEB1 Legend SXPYRC1 Latent Variables Observed Variables .15 Dependent Loadings**Example: Condom Use Model**eidm1 eidm2 eidm4 eidm2 IDMA1R IDMC1R IDME1R IDMJ1R SEX1 Impulsive FRBEHB1 efr1 eiss ISSUEB1 Legend SXPYRC1 Latent Variables Observed Variables eSXYRC1 .15 Loadings**Example: Condom Use Model**eidm1 eidm2 eidm4 eidm2 IDMA1R IDMC1R IDME1R IDMJ1R SEX1 Impulsive FRBEHB1 efr1 eiss ISSUEB1 Legend SXPYRC1 Latent Variables Observed Variables eSXYRC1 .15 Loadings**.28 .24 .48 .45**.03 .05 .15 Example: Condom Use Model eidm1 eidm2 eidm4 eidm2 IDMA1R IDMC1R IDME1R IDMJ1R .49 .69 .67 SEX1 -.06 .53 Impulsive -.19 -.10 -.15 .13 FRBEHB1 efr1 eiss ISSUEB1 .11 .38 Legend SXPYRC1 Latent Variables Observed Variables eSXYRC1 .15 Loadings**SEM Assumptions**A Reasonable Sample Size • a good rule of thumb is 15 cases per predictor in a standard ordinary least squares multiple regression analysis. [ “Applied Multivariate Statistics for the Social Sciences”, by James Stevens] • researchers may go as low as five cases per parameter estimate in SEM analyses, but only if the data are perfectly well-behaved [Bentler and Chou (1987)] • Usually 5 cases per parameter is equivalent to 15 measured variables.**SEM Assumptions (cont’d)**• Continuously and Normally Distributed Endogenous Variables • NOTE: At this time AMOS CANNOT handle not continuously distributed outcome variables**SEM Assumptions (cont’d)**Model Identification P is # of measured variables [P*(P+1)]/2 Df=[P*(P+1)]/2-(# of estimated parameters) If DF>0 model is over identified If DF=0 model is just identified If DF<0 model is under identified**Missing data in SEM**Types of missing data • MCAR • Missing Completely at Random • MAR • Missing at Random • MNAR • Missing Not at Random**Handling Missing data in SEM**• Listwise • Pairwise • Mean substitution • Regression methods • Expectation Maximization (EM) approach • Full Information Maximum Likelihood (FIML)** • Multiple imputation(MI)** • The two best methods: FIML and MI**SEM Software**• Several different packages exist • EQS, LISREL, MPLUS, AMOS, SAS, ... • Provide simultaneously overall tests of • model fit • individual parameter estimate tests • May compare simultaneously • Regression coefficients • Means • Variances even across multiple between-subjects groups**AMOS Advantages**• Easy to use for visual SEM ( Structural Equation Modeling). • Easy to modify, view the model • Publication –quality graphics**AMOS Components**• AMOS Graphics • draw SEM graphs • runs SEM models using graphs • AMOS Basic • runs SEM models using syntax**Starting AMOS Graphics**Start Programs Amos 5 Amos Graphics**Reading Data into AMOS**• File Data Files • The following dialog appears:**Reading Data into AMOS**• Click onFile Nameto specify the name of the data file • 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, versions 7.0.2 through 13.0 (both raw data and matrix formats)**Reading Data into AMOS**• Example USED for this workshop: • Condom use and what predictors affect it • DATASET: AMOS_data_valid_condom.sav**To draw an observed variable, click "Diagram" on the top**menu, and click "Draw Observed." 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. You can also use in the tool box to draw observed variables. 2. Unobserved variables can be drawn similarly. Click "Diagram" and "Draw Unobserved." Unobserved variables are shown as circles. You may also use in the toolbox to draw unobserved variables. Drawing in AMOS • In Amos Graphics, a model can be specified by drawing a diagram on the screen**Drawing in AMOS**• To draw a path, Click “Diagram” on the top menu and click “Draw Path”. • Instead of using the top menu, you may use the Tool Box buttons to draw arrows ( and ).**Drawing in AMOS**• To draw Error Term to the observed and unobserved variables. • Use “Unique Variable” button in the Tool Box. 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.)**Drawing in AMOS**• Let us draw:**Naming the variables in AMOS**• double click on the objects in the path diagram. The Object Propertiesdialog box appears. • OR Click on the Texttab and enter the name of the variable in the Variable namefield:**Naming the variables in AMOS**• Example: Name the variables**Constraining a parameter in AMOS**• The scale of the latent variable or variance of the latent variable has to be fixed to 1. Double click on the arrow between EXPYA2 and SXPYRA2. The Object Propertiesdialog appears. Click on the Parameterstab and enter the value “1” in the Regression weightfield:**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 Moveicon 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 Movefunction 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 Graphicswill automatically redraw all connecting arrows.**Improving the appearance of the path diagram**• To change the size and shape of an object, first press the Change the shape of objectsicon on the toolbar. • You will notice that the word “shape”appears under the mouse pointer to let you know the Shapefunction is active. • Click and hold down your left mouse button on the object you wish to re-shape. Change the shape of the object to your liking and release the mouse button. • Change the shape of objectsalso works on two-headed arrows. Follow the same procedure to change the direction or arc of any double-headed arrow.**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 Eraseand Undofunctions. • To erase an object, simply click on the Eraseicon and then click on the object you wish to erase. • To undo your last drawing activity, click on theUndoicon and your last activity disappears. • Each time you click Undo, your previous activity will be removed. • If you change your mind, click on Redoto restore a change.**Performing the analysis in AMOS**• View/Set ® Analysis Propertiesand click on the Output tab. • There is also an Analysis Propertiesicon you can click on the toolbar. Either way, the Outputtab gives you the following options:**Performing the analysis in AMOS**• For our example, check the Minimization history, Standardized estimates, and Squared multiple correlationsboxes. (We are doing this because these are so commonly used in analysis). • To run AMOS, click on the Calculate estimatesicon on the toolbar. • AMOS will want to save this problem to a file. • if you have given it no filename, the Save Asdialog box will appear. Give the problem a file name; let us say, tutorial1:**Results**• When AMOS has completed the calculations, you havetwo options for viewing the output: • text output, • graphics output. • For text output, click the View Text( or F10)icon on the toolbar. • Here is a portion of the text output for this problem:**Estimate**Estimate S.E. C.R. P FRBEHB1 <--- SEX1 -.10 FRBEHB1 <--- SEX1 -.28 .09 -2.98 .00 ISSUEB1 <--- SEX1 .12 ISSUEB1 <--- SEX1 .30 .08 3.79 *** FRBEHB1 <--- IDM -.11 FRBEHB1 <--- IDM -.38 .11 -3.29 *** ISSUEB1 <--- IDM -.19 ISSUEB1 <--- IDM -.57 .10 -5.94 *** SXPYRC1 <--- ISSUEB1 .11 SXPYRC1 <--- ISSUEB1 .16 .05 3.42 *** SXPYRC1 <--- FRBEHB1 .38 SXPYRC1 <--- FRBEHB1 .49 .04 12.21 *** Results for Condom Use Model(see handout) The model is recursive. Sample size = 893 Chi-square=12.88 Degrees of Freedom =3 Maximum Likelihood Estimates Standardized Regression Weights: (Group number 1 - Default model)**Estimate**Estimate S.E. C.R. P Label SEX1 SEX1 <--> <--> IDM IDM -.08 -.02 .01 -2.48 .01 Results for Condom Use Model Covariances: (Group number 1 - Default model) Correlations: (Group number 1 - Default model)**Viewing the graphics output in AMOS**• To view the graphics output, click the View outputicon next to the drawing area. • Chose to view either unstandardizedor (if you selected this option) standardizedestimates by click one or the other in the Parameter Formatspanel next to your drawing area:**0.15 is the squared multiple**correlation between Condom use and ALL OTHER variables Viewing the graphics output in AMOSUnstandardized Standardized**How to read the Output in AMOS**See the handout_1**Modification of the Model**• Search for the better model • Suggestions from: 1) theory 2) modification indices using AMOS**Modifying the Model using AMOS**• View/Set ® Analysis Propertiesand click on the Output tab. • Then check the Modification indices option**M.I.**Par Change Modifying the Model using AMOS Modification Indices (Group number 1 - Default model) Covariances: (Group number 1 - Default model) Parameter increase eiss <--> efr1 9.909 .171 Chi-square decrease**Modifying the Model using AMOS**2.38, .17 -.02 1.45, .25 IDM SEX1 -.28 -.57 -.38 .30 3.74 5.58 FRBEHB1 ISSUEB1 1 1 0, 1.36 0, 1.94 .16 efr1 .49 eiss .17 3.08 SXPYRC1 1 0, 2.80 eSXPYRC1 SEE Handout # 2 for the whole output**Examples using AMOS**• Condom Use Model with missing values • Confirmatory Factor Analysis for Impulsive Decision Making construct • Multiple group analysis • How to deal with non-normal data**Missing data in AMOS**• Full Information Maximum Likelihood estimation • View/Set -> Analysis Propertiesand click on the Estimationtab. • Click on the button Estimate Means and Intercepts.This uses FIML estimation Recalculate the previous example with data “AMOS_data.sav” with some missing values