AMOS-nov1_1

AMOS – Analysis of AMOS – Analysis of Moment StructuresMoment Structures

HIV Prevention CenterHIV Prevention CenterUniversity of KentuckyUniversity of Kentucky

Rick Zimmerman, Olga DekhtyarRick Zimmerman, Olga Dekhtyar

OverviewOverview

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 ModelsStructural Equation Models

Structural Equation Modeling Structural Equation Modeling (SEM) (SEM)

An extension of Regression and general Linear Models

Also can fit more complex models, like confirmatory factor analysis and longitudinal data.

Structural Equation Structural Equation ModelingModeling

Ability to fitAbility to fit non-standard modelsnon-standard models, databases with autocorrelated error autocorrelated error structuresstructures time series analysistime series analysis Latent Curve ModelsLatent Curve Models,

databases with non-normally distributed non-normally distributed variablesvariables databases with incomplete dataincomplete data.

Family Tree of SEMFamily Tree of SEMT-test

ANOVAMulti-wayANOVA Repeated

MeasureDesigns Growth

CurveAnalysis

BivariateCorrelation

MultipleRegression

PathAnalysis

StructuralEquationModeling

FactorAnalysis

ExploratoryFactor

Analysis

ConfirmatoryFactor

Analysis

LatentLatentGrowthGrowthCurveCurve

AnalysisAnalysis

Next Next Workshop:Workshop:

November 9November 9See you there!See you there!

Structural Equation Modeling Structural Equation Modeling (SEM)(SEM)

ExogenousExogenous

variables=independentindependent

Endogenous variables

=dependentdependent

Observed variables =measuredmeasured

Latent variables=unobservedunobserved

Structural Equation Structural Equation GraphsGraphs

Observed Observed VariableVariable

LatentLatentVariableVariable

.15

ErrorError

: Loading Loading

.10 : RR22

Example: Condom Use Example: Condom Use ModelModel

SEX1

FRBEHB1ISSUEB1

SXPYRC1

Impulsive

IDMC1R

Observed Variables

Latent Variables

.15 Loadings

LegendLegend

IDMA1R IDME1R IDMJ1R

Respondent Sex

Condom Use

Condom attitude

Peer norms about

condoms

Impulsive Decision Making

Observed variables for Impulsive decision making

Example: Condom Use Example: Condom Use ModelModel

SEX1

FRBEHB1ISSUEB1

SXPYRC1

Impulsive

IDMC1R

Observed Variables

Latent Variables

.15 Loadings

LegendLegend

IDMA1R IDME1R IDMJ1R

Independent

Dependent

DependentDependent

Independent

Dependent

Example: Condom Use Example: Condom Use ModelModel

SEX1

FRBEHB1ISSUEB1

SXPYRC1

Impulsive

eSXYRC1

efr1eiss

IDMC1R

eidm4eidm2eidm2eidm1

Observed Variables

Latent Variables

.15 Loadings

LegendLegend

IDMA1R IDME1R IDMJ1R

Example: Condom Use Example: Condom Use ModelModel

SEX1

FRBEHB1ISSUEB1

SXPYRC1

Impulsive

eSXYRC1

efr1eiss

IDMC1R

eidm4eidm2eidm2eidm1

Observed Variables

Latent Variables

.15 Loadings

LegendLegend

IDMA1R IDME1R IDMJ1R

Example: Condom Use Example: Condom Use ModelModel

SEX1

FRBEHB1ISSUEB1

SXPYRC1

Impulsive

eSXYRC1

efr1eiss

IDMC1R

eidm4eidm2eidm2eidm1

Observed Variables

Latent Variables

.15 Loadings

LegendLegend

IDMA1R IDME1R IDMJ1R

.11.38

-.06

-.10-.19

-.15 .13

.53.49 .69 .67

.15

.05

.28 .24 .48 .45

.03

SEM AssumptionsSEM AssumptionsA Reasonable Sample SizeA Reasonable Sample Size

a good rule of thumb is 15 cases per predictor15 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 five cases per per parameterparameter estimate estimate in SEM analyses, but only if the data are perfectly well-behaved

[Bentler and Chou (1987)]

Usually 5 cases per parameter5 cases per parameter is equivalent to 15 15 measured variablesmeasured variables.

SEM Assumptions (cont’d)SEM Assumptions (cont’d)

Continuously and Normally Distributed Endogenous Variables

NOTE: NOTE: At this time AMOS CANNOT handle notnot continuously distributed outcome variables

SEM Assumptions (cont’d)SEM Assumptions (cont’d)Model IdentificationModel Identification

P P is # of measured variables [P*(P+1)]/2[P*(P+1)]/2

Df=[P*(P+1)]/2-(# of estimated parameters)Df=[P*(P+1)]/2-(# of estimated parameters)If DF>0DF>0 model is over identifiedover identifiedIf DF=0DF=0 model is just identifiedjust identifiedIf DF<0DF<0 model is under identifiedunder identified

Missing data in SEMMissing data in SEM

Types of missing dataTypes of missing data

MCARMCAR Missing Completely at Random

MARMAR Missing at Random

MNARMNAR Missing Not at Random

Handling Missing data in Handling Missing data in SEM SEM

ListwisePairwiseMean substitution

Regression methods

Expectation Maximization (EM) approach

Full Information Maximum Likelihood (FIML)**

Multiple imputation(MI)**

The two best methods: FIML and MI

SEM SoftwareSEM Software Several different packages exist EQS, LISREL, MPLUS, AMOS AMOS, SAS, ...

Provide simultaneouslysimultaneously overall tests of model fit individual parameter estimate tests

May compare simultaneouslysimultaneously Regression coefficients Means Variances

even across multiple between-subjects groupseven across multiple between-subjects groups

Introduction toIntroduction to

AMOSAMOS

AMOS AdvantagesAMOS Advantages

Easy to use for visual SEMvisual SEM ( Structural Equation Modeling). Easy to modify, viewmodify, view the modelPublication –quality graphicsgraphics

AMOS ComponentsAMOS Components

AMOS GraphicsAMOS Graphics draw SEM graphsgraphs runs SEM models using graphs

AMOS Basic runs SEM models using syntax

Starting AMOS GraphicsStarting AMOS Graphics

Start Programs Amos 5 Amos Graphics

Reading Data into AMOS

FileFile Data FilesData Files The following dialog appears:

Reading Data into AMOSClick on File NameFile Name to 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 SPSSSPSS *.sav files*.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 Condom use and what predictors

affect itaffect it

DATASET: AMOS_data_valid_condom.sav

Drawing in AMOSIn Amos Graphics, a model can be specified by drawing a diagram on the screen 1. To draw an observed variable, click

"Diagram" on the top menu, and click "Draw ObservedDraw 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 UnobservedDraw Unobserved." Unobserved variables are shown as circles. You may also use      in the toolbox to draw unobserved variables.

Drawing in AMOSDrawing in AMOSTo draw a path, Click “DiagramDiagram” on the top

menu and click “Draw PathDraw Path”. Instead of using the top menu, you may use the Tool Box buttons to draw arrows ( and ).

Drawing in AMOSTo draw Error Term to the observed and unobserved variables. Use “Unique VariableUnique 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.)

1

1 1

Drawing in AMOS

Let us draw:

Naming the variables in Naming the variables in AMOSAMOSdouble click on the objects in the path diagram. The Object PropertiesObject Properties dialog box appears.

• OR

Click on the TextText tab and enter the name of the variable in the Variable Variable namename field:

Naming the variables in AMOS

Example: Name the variables

ISSUEB1

SXPYRC1

eSXPYRC1

1

SEX1

eiss

FRBEHB1

efr1

1 1

IDM

Constraining a parameter in AMOSConstraining 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 PropertiesObject Properties dialog appears.

Click on the ParametersParameters tab and enter the value “11” in the Regression weightRegression weight field:

Improving the appearance Improving the appearance of the path diagramof the path diagram

You can change the appearance of your path diagram by moving objects aroundTo move an object, click on the MoveMove 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 MoveMove 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 GraphicsAmos Graphics will automatically redraw all connecting arrows.

Improving the appearance of the Improving the appearance of the path diagrampath diagram

To change the size and shape of an object, first press the Change the shape of objectsChange the shape of objects icon on the toolbar. You will notice that the word “shape” appears under the mouse pointer to let you know the ShapeShape function is active. Click and hold down your left mouse buttonleft 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 objectsChange the shape of objects also 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 mistakemake a mistake, there are always three icons on the toolbar to quickly bail you out: the EraseErase and UndoUndo functions.

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

To undo your last drawing activityundo your last drawing activity, click on the UndoUndo icon and your last activity disappears.Each time you click Undo,Undo, your previous activity will be removed.

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

Performing the Performing the analysis in AMOS analysis in AMOS

View/Set View/Set Analysis Analysis PropertiesProperties and click on the Output Output tab.

There is also an Analysis PropertiesAnalysis Properties icon you can click on the toolbar. Either way, the OutputOutput tab gives you the following options:

Performing the analysis in AMOS

For our example, check the Minimization Minimization history, Standardized estimateshistory, Standardized estimates, and Squared Squared multiple correlationsmultiple correlations boxes. (We are doing this because these are so commonly used in analysis).

To run AMOSAMOS, click on the Calculate estimatesCalculate estimates icon on the toolbar. AMOSAMOS will want to save this problem to a file. if you have given it no filename, the Save AsSave As

dialog box will appear. Give the problem a file name; let us say, tutorial1tutorial1:

Results Results When AMOSAMOS has completed the calculations, you have two options for viewing the output:

text outputtext output, graphics outputgraphics output.

For text output, click the View TextView Text ( or F10F10) icon on the toolbar.

Here is a portion of the text output for this problem:

The model is recursive. Sample size = The model is recursive. Sample size = 893893Chi-square=12.88 Degrees of Freedom =3Chi-square=12.88 Degrees of Freedom =3

Maximum Likelihood EstimatesMaximum Likelihood Estimates         EstimateEstimate S.E.S.E. C.R.C.R. PP

FRBEHB1FRBEHB1 <---<--- SEX1SEX1 -.28-.28 .09.09 -2.98-2.98 .00.00   ISSUEB1ISSUEB1 <---<--- SEX1SEX1 .30.30 .08.08 3.793.79 ******   FRBEHB1FRBEHB1 <---<--- IDMIDM -.38-.38 .11.11 -3.29-3.29 ******   ISSUEB1ISSUEB1 <---<--- IDMIDM -.57-.57 .10.10 -5.94-5.94 ******   SXPYRC1SXPYRC1 <---<--- ISSUEB1ISSUEB1 .16.16 .05.05 3.423.42 ******   SXPYRC1SXPYRC1 <---<--- FRBEHB1FRBEHB1 .49.49 .04.04 12.2112.21 ******   

Standardized Regression Weights: (Group number 1 - Default model)         EstimateEstimate

FRBEHB1FRBEHB1 <---<--- SEX1SEX1 -.10-.10ISSUEB1ISSUEB1 <---<--- SEX1SEX1 .12.12FRBEHB1FRBEHB1 <---<--- IDMIDM -.11-.11ISSUEB1ISSUEB1 <---<--- IDMIDM -.19-.19SXPYRC1SXPYRC1 <---<--- ISSUEB1ISSUEB1 .11.11SXPYRC1SXPYRC1 <---<--- FRBEHB1FRBEHB1 .38.38

Results for Condom Use Model(see handout)

Results for Condom Use Model

Covariances: (Group number 1 - Default model)Covariances: (Group number 1 - Default model)

         EstimateEstimate S.E.S.E. C.R.C.R. PP LabelLabel

SEX1SEX1 <--><--> IDMIDM -.02-.02 .01.01 -2.48-2.48 .01.01   

Correlations: (Group number 1 - Default Correlations: (Group number 1 - Default modelmodel))

         EstimateEstimate

SEX1SEX1 <--><--> IDMIDM -.08-.08

Viewing the graphics output in AMOSViewing the graphics output in AMOS

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

• Chose to view either unstandardizedunstandardized or (if you selected this option) standardizedstandardized estimates by click one or the other in the Parameter FormatsParameter Formats panel next to your drawing area:

.06

ISSUEB1

.15

SXPYRC1

eSXPYRC1

.11

SEX1

eiss

.02

FRBEHB1

efr1 .38

-.10.12

IDM

-.11

-.19

-.08

Viewing the graphics output in AMOSViewing the graphics output in AMOS

UnstandardizedUnstandardized StandardizedStandardized

ISSUEB1

SXPYRC1

2.80

eSXPYRC1

1

.16

.25

SEX1

1.36

eiss

FRBEHB1

1.94

efr1

1 1

.49

-.28.30

.17

IDM

-.38

-.57

-.02

0.150.15 is the squared multiple correlation between Condom use and ALL OTHER ALL OTHER variablesvariables

How to read the How to read the Output in AMOSOutput in AMOS

See the handout_1See the handout_1

Modification of the Model

Search for the better model

Suggestions from: 1) theory2) modification

indices using AMOS

Modifying the Model using AMOSView/Set View/Set Analysis Analysis PropertiesProperties and click on the Output Output tab. Then check the Modification indicesModification indices option

Modifying the Model using AMOS

Modification Indices (Group number 1 - Default model)Modification Indices (Group number 1 - Default model)

Covariances: (Group number 1 - Default model)Covariances: (Group number 1 - Default model)

eisseiss <--><--> efr1efr1 9.9099.909 .171.171

         M.I.M.I. Par ChangePar Change

Chi-square decrease

Parameter increase

Modifying the Model using AMOS

3.74

ISSUEB1

3.08 SXPYRC1

0, 2.80

eSXPYRC1

1

.16

1.45, .25

SEX1

0, 1.36eiss

5.58FRBEHB1

0, 1.94efr1

1 1.49

-.28.30

2.38, .17

IDM

-.38-.57

-.02

.17

SEE Handout # 2 for the whole output SEE Handout # 2 for the whole output

Examples using AMOSExamples 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 PropertiesView/Set -> Analysis Properties and click on the EstimationEstimation tab.

• Click on the button Estimate Means Estimate Means and Intercepts.and Intercepts. This uses FIML estimation

Recalculate the previous example with data “AMOS_data.savAMOS_data.sav” with some missing values

Missing data in AMOSMissing data in AMOS

The standardized graphical output.

.05

ISSUEB1

.14

SXPYRC1

eSXPYRC1

.08

SEX1

eiss

.02

FRBEHB1

efr1 .37

-.09.12

IDM

-.10

-.18

-.10

Missing data in AMOS

Example: see the handout #3

Confirmatory Factor Analysis withConfirmatory Factor Analysis with Impulsive Decision Making scale Impulsive Decision Making scale

Need to fix either the variance of the IDM1 Need to fix either the variance of the IDM1 factor or one of the loadings to factor or one of the loadings to 11..

0,0,idm1idm1

IDMA1RIDMA1R

0,0,e1e1

11

11

IDMC1RIDMC1R

0,0,e2e2

11

IDME1RIDME1R

0,0,e3e3

11

IDMJ1RIDMJ1R

0,0,e4e4

11

Confirmatory Factor Analysis Confirmatory Factor Analysis withwith Impulsive Decision Making Impulsive Decision Making scalescale

idm1

.30

IDMA1R

e1

.55

.26

IDMC1R

e2

.51

.47

IDME1R

e3

.69

.47

IDMJ1R

e4

.69

Chi-square = 11.621 Degrees of freedom = 2, Chi-square = 11.621 Degrees of freedom = 2, p=0.003p=0.003CFI=0.994, RMSEA=0.042CFI=0.994, RMSEA=0.042

Multiple Correlation

Factor Loading

s

Confirmatory Factor Analysis Confirmatory Factor Analysis withwith Impulsive Decision Making Impulsive Decision Making scalescale

What if want to compare two NESTED models for Impulsive Decision Making Model?

1) error variances equal for all 4 measured variables

2) error variances are different

Confirmatory Factor Analysis with Impulsive Decision Making scale: the error variances are the same

Need to give names to the error variances, by double clicking on the error variance. The Object properties will appearObject properties will appear, click on the ParameterParameter and type the name for the error variance( e1, e2...) in the Variance boxVariance box.

Confirmatory Factor Analysis Confirmatory Factor Analysis withwith Impulsive Decision Making Impulsive Decision Making scalescale

0,

idm1idm1

IDMA1R

0, e1

e1

1

1

IDMC1R

0, e2

e2

1

IDME1R

0, e3

e3

1

IDMJ1R

0, e4

e4

1

Confirmatory Factor Analysis with Impulsive Decision Making scale: error variances are the same

Click MODEL FITMODEL FIT , then Manage ModelsManage ModelsIn the Manage ModelsManage Models window, click on NewNew. In the Parameter ConstraintsParameter Constraints segment of the window type “e1=e2=e3=e4”

Now there are two nested models

0, .19

idm1

2.18

IDMA1R

0, .45

e1

1.00

1

2.44

IDMC1R

0, .58

e2

1.03

1

2.24

IDME1R

0, .47

e3

1.48

1

2.28

IDMJ1R

0, .43

e4

1.40

1

0, .19

idm1

2.18

IDMA1R

0, .48

e1

1.00

1

2.44

IDMC1R

0, .48

e2

1.15

1

2.24

IDME1R

0, .48

e3

1.50

1

2.28

IDMJ1R

0, .48

e4

1.36

1

Confirmatory Factor Analysis with Impulsive Decision Making scale

error variances are the sameerror variances are different

Chi-square = Chi-square = 11.621, df=3, 11.621, df=3, p=0.003p=0.003

Chi-square = 56.826, Chi-square = 56.826, df=5, p=0.000df=5, p=0.000

Confirmatory Factor Analysis with Confirmatory Factor Analysis with Impulsive Decision Making scale:Impulsive Decision Making scale:error variances are the sameerror variances are the sameCompare Nested Models using Chi-square difference test:

Model1 ( errors are different)Chi-square = 11.621, df=3, p=0.003

Model2( errors the same)

Chi-square = 56.826,

df=5, p=0.000

Chi-squaredifference=56.826-11.621=45.205

df=5-3=2

Chi-squarecritical value=5.99 Significant Model 2 with Equal error variances fits

WORSE than Model 1

Nested Model Comparisons

Assuming model Error are freeError are free to be correct:

Confirmatory Factor Analysis with Confirmatory Factor Analysis with Impulsive Decision Making scale:Impulsive Decision Making scale:error variances are the sameerror variances are the same

Model DF CMIN PNFI

Delta-1IFI

Delta-2RFI

rho-1TLI

rho2

Errors are the same 3 45.205 .000 .026 .026 .032 .032

Multiple group analysisMultiple group analysisWHY: test the equality/invariance of the factor

loadings for two separate groups HOW : 1) test the model to both groups separately to check

the entire model2) the same model by multiple group analysismultiple group analysis

Example: Do Males and Females can be fitted to the same Condom USE modelCondom USE model?

Need to have 2 separate data files for each group. data_boysdata_boys and data_girlsdata_girls.

Multiple group analysisMultiple group analysis• Select Manage Groups...Manage Groups... from the Model FitModel Fit menu.

• Name the first group “GirlsGirls”.

• Next, click on the NewNew button to

add a second group to the analysis.

• Name this group “Boys”“Boys”. • AMOS 4.0 will allow you to consider up to 16 groups per analysis.

• Each newly created group is represented by its own path diagram

Multiple group analysisMultiple group analysis

• Select File->Data File->Data Files...Files... to launch the Data Data Files dialog boxFiles dialog box.

• For each group, specify the relevant data file name.

• For this example, choose the data_girlsdata_girls SPSS database for the girls' group; • choose the data_boysdata_boys SPSS database for the boys' group.

Multiple group analysisMultiple group analysis

Click Model FitModel Fit and Multiple GroupsMultiple Groups. This gives a name to every parameter in the model

in each group.

The following models fit to both groups (see handout) : Unconstrained – all parameters are different in each group

Measurement weights – regression loadings are the same in both groups

Measurement intercepts – the same intercepts for both groups

Structural weights – the same regression loadings between the latent var.

Structural intercepts – the same intercepts for the latent variables

Structural covariates – the same variances/covariance for the latent var.

Structural residuals – the same disturbances

Measurement residuals – the same errors-THE MOST RESTRICTIVE MODEL

Example: Multiple group Example: Multiple group analysis for Condom use analysis for Condom use Model Model

UNCONSTRAINED MODELUNCONSTRAINED MODEL

0, 1.13

3.06

ISSUEB1

2.16

SXPYRC1

0, 2.56

eSXPYRC1

1

.26eiss

4.12

FRBEHB1

0, 1.81

efr1

1 1

.62

0, .16

Impulsive

2.21

IDMA1R

0, .47

eidm1

1.00

12.41

IDMC1R

0, .62

eidm2

1.14

12.36

IDME1R

0, .44

eidm3

1.56

12.40

IDMJ1R

0, .39

eidm4

1.45

1

-.28 -.38

2.72

ISSUEB1

3.63

SXPYRC1

0, 2.95

eSXPYRC1

1

.11

0, 1.50

eiss

4.35

FRBEHB1

0, 2.12

efr1

1 1

.40

0, .18

Impulsive

2.33

IDMA1R

0, .48

eidm1

1.00

12.60

IDMC1R

0, .65

eidm2

1.04

12.39

IDME1R

0, .47

eidm3

1.58

12.43

IDMJ1R

0, .47

eidm4

1.41

1

-.62 -.64

BoysBoys GirlsGirls

Example: Multiple group analysis Example: Multiple group analysis for Condom use Model for Condom use Model

3.06

ISSUEB1

2.16

SXPYRC1

0, 2.56

eSXPYRC1

1

.26

0, 1.12

eiss

4.12

FRBEHB1

0, 1.81efr1

1 1

.62

0, .16Impulsive

2.21

IDMA1R

0, .47

eidm1

1.00

12.41

IDMC1R

0, .63

eidm2

1.08

12.36

IDME1R

0, .43

eidm3

1.57

12.40

IDMJ1R

0, .40

eidm4

1.42

1

-.45-.50

2.72

ISSUEB1

3.62

SXPYRC1

0, 2.95

eSXPYRC1

1

.11

0, 1.51

eiss

4.35

FRBEHB1

0, 2.14efr1

1 1

.40

0, .18Impulsive

2.33

IDMA1R

0, .48

eidm1

1.00

12.60

IDMC1R

0, .64

eidm2

1.08

12.39

IDME1R

0, .48

eidm3

1.57

12.43

IDMJ1R

0, .46

eidm4

1.42

1

-.45-.50

Boys Measurement weightsMeasurement weights Girls

Example: Multiple group analysis Example: Multiple group analysis for Condom use Modelfor Condom use Model

see handoutsee handout

Since Measurement WeightsMeasurement Weights model is nested within Unconstrained Unconstrained . .

Chi-square difference test computed to test the null hypothesis that the regression weights for boys and regression weights for boys and girlsgirls are the same. However, the variances and variances and covariancecovariance are different different across groups.

Example: Multiple group analysis Example: Multiple group analysis for Condom use Modelfor Condom use Model

Chi-squarediff =68.901-65.119=2.282

df=29-26=3 NOT SIGNIFICANT NOT SIGNIFICANT

FIT of the FIT of the Measurement WeightsMeasurement Weights model model is not significantly worse than is not significantly worse than UnconstrainedUnconstrained

Handling non-normal data:

Verify that your variables are not distributed joint multivariate normal

Assess overall model fit using the Bollen-Stine corrected p-value

Use the bootstrap to generate parameter estimates, standard errors of parameter estimates, and significance tests for individual parameters

Handling non-normal data: checking for normality

To verify that the data is not normal. Check the Univariate Univariate SKEWNESSSKEWNESS and KURTOSISKURTOSIS for each variable .

• View/Set -> Analysis View/Set -> Analysis PropertiesProperties

and click on the OutputOutput tab.

•Click on the button Tests for Tests for normality and outliersnormality and outliers

Handling non-normal data: checking for normality

Assessment of normality

Variable min max skew c.r. kurtosis c.r.

IDM 1.182 3.727 .381.381 4.649 .496.496 3.025

SEX1 1.000 2.000 .182.182 2.222 -1.967-1.967 -11.997

FRBEHB1 1.000 6.000 -.430-.430 -5.245 -.778-.778 -4.748

ISSUEB1 1.000 4.000 -.431-.431 -5.259 -1.387-1.387 -8.462

SXPYRC1 2.000 7.000 -.937-.937 -11.436 -.715-.715 -4.360

MultivariateMultivariate -3.443-3.443 -6.149

Critical ratio of +/- 2Critical ratio of +/- 2 for skewness and kurtosis

statistical significance of NON-NORMALLITY

Multivariate kurtosis >10Multivariate kurtosis >10 Severe Non-normality

Handling non-continuous data:Bootstrapping

Use Bootstrapping

Bootstrapping generates an estimate of the sampling distribution from the available data and computes the p-values and construct confidence intervals.

Bootstrapping in AMOS generates random covariance matrices from the sample covariance matrix assuming multivariate normality

Handling non-continuous data:Bootstrapping

Bootstrapping is useful for estimating standard errors for statistics with complex distributions, for which there is no practical approximate However, Some limitations include:

The “population” in nonparametric bootstrapping is merely the researcher’s sample

If the researcher’s sample is small, unrepresentative, or the observations are not independent, resembling from it can

magnify the effects of these features (see Rodgers, 1999)

Bootstrap analyses are probably biased in small samples (just as

they are in other methods)—that is, bootstrapping is not a “cure”

Handling non-normal data:Bollen-Stine bootstrapping p-value

•View/Set -> Analysis PropertiesView/Set -> Analysis Properties

and click on the BootstrapBootstrap tab.

Check Perform bootstrapPerform bootstrap and Bollen-Stine bootstrapBollen-Stine bootstrap

BOLLEN_STINE BOOTSTRAP BOLLEN_STINE BOOTSTRAP performed only for dataset performed only for dataset without any missing valueswithout any missing values

(see handout #6: (see handout #6: amos_data_valid_condom.sav)amos_data_valid_condom.sav)

Handling non-normal data:Bollen-Stine bootstrapping p-value

The model fits better than expected in 496 samples out of 500 samples

(500-496)/500=0.010

So, p-value=0.01 < 0.05 - Model does not fit to the data very well

Handling non-normal data:Bollen-Stine bootstrapping p-value

Overall Model Fit:Chi-square=12.88; Chi-square=12.88; Degrees of freedom = 3Degrees of freedom = 3

The expected(mean) value of Chi-square is 2.929.

The mean value of Chi-Chi-squaresquare (2.929) serves as the critical chi-square value against which the obtained chi-square of 12.88 is compared

In our example, results from the Bollen-Stine are the same as resultsIn our example, results from the Bollen-Stine are the same as results

for the overall model. for the overall model.

Handling non-normal data: Bootstrapping Standard Errors

Bootstrapping can be used to evaluate the estimatesestimates, by computing the Standard ErrorsStandard Errors of the estimates

UnSELECT Bollen-Stine Bootstrap

and Select Percentile Percentile Confidence Intervals Confidence Intervals and Bias-corrected Bias-corrected confidence intervalsconfidence intervals

Handling non-normal data: Bootstrapping Standard Errors

Relationship between

Condom use and Peer Condom use and Peer Norms about Norms about CondomCondom is 0.487, with S.E.=0.04,

Almost the same estimate produced by Bootstrap, 0.488 with S.e=0.042

Estimates using ML

Bootstrap estimates

Handling non-normal data: Bootstrapping Standard Errors

90% Percentile Method 90% Bias Corrected Percentile method

Hope to see similar results for the estimatesHope to see similar results for the estimates

NOTE: BOOTSTRAP option works ONLY with COMPLETE data

Handling non-normal data: Bootstrapping Standard Errors

NOTE: BOOTSTRAP option works ONLY with COMPLETE data

if missing is less than 5% , it is defensible to use LISTWISE deletion

Sample size should be reasonably large with 200 for SEMs that contain latent variables ( by Nevitt and Hancock, 1998)

Thank You!Thank You!

See you in a week!See you in a week!

Degree of freedomDegree of freedom Chi-square critical valueChi-square critical value

1 3.841

2 5.991

3 7.815

4 9.488

5 11.070

6 12.592

7 14.067

8 15.507

9 16.919

10 18.307

11 19.675

Upper critical values of chi-square distribution