Structural Structural Equation Equation Modeling Modeling WorkshopWorkshop

PIREPIRE

August 6-7, 2007August 6-7, 2007

Section 1Section 1

Introduction to SEMIntroduction to SEM

Definitions of Structural Definitions of Structural Equation Models/ModelingEquation Models/Modeling

““Structural equation modeling (SEM) does not Structural equation modeling (SEM) does not designate a single statistical technique but designate a single statistical technique but instead refers to a family of related procedures. instead refers to a family of related procedures. Other terms such as covariance structure Other terms such as covariance structure analysis, covariance structural modeling, or analysis, covariance structural modeling, or analysis of covariance structures are essentially analysis of covariance structures are essentially interchangeable. Another term…is causal interchangeable. Another term…is causal modeling, which is used mainly in association modeling, which is used mainly in association with the technique of path analysis. This with the technique of path analysis. This expression may be somewhat dated, however, as expression may be somewhat dated, however, as it seems to appear less often in the literature it seems to appear less often in the literature nowadays.” (Kline, 2005)nowadays.” (Kline, 2005)

History of SEMHistory of SEM

Sewall Wright and Path AnalysisSewall Wright and Path Analysis Duncan and Path AnalysisDuncan and Path Analysis EconometricsEconometrics Joreskog and LISRELJoreskog and LISREL Bentler and EQSBentler and EQS Muthen and MplusMuthen and Mplus

Sewall WrightSewall Wright

GeneticistGeneticist Principle of Path Analysis provides Principle of Path Analysis provides

algorithm for decomposing algorithm for decomposing correlations of 2 variables into correlations of 2 variables into structural relations among a set of structural relations among a set of variablesvariables

Created the path diagramCreated the path diagram Applied path analysis to genetics, Applied path analysis to genetics,

psychology, and economicspsychology, and economics

DuncanDuncan

Applied path analysis methods to the Applied path analysis methods to the area of social stratification area of social stratification (occupational attainment)(occupational attainment)

Key papers by Duncan & Hodge Key papers by Duncan & Hodge (1964) and Blau & Duncan (1967) (1964) and Blau & Duncan (1967)

Developed one of the first texts on Developed one of the first texts on path analysispath analysis

EconometricsEconometrics

Goldberger added the importance of Goldberger added the importance of standard errors and links to statistical standard errors and links to statistical inferenceinference

Showed how ordinary least squares Showed how ordinary least squares estimates of parameters in overidentified estimates of parameters in overidentified systems of equations were more efficient systems of equations were more efficient than averages of multiple estimates of than averages of multiple estimates of parametersparameters

Combined psychometric and Combined psychometric and econometric componentseconometric components

Indirect EffectsIndirect Effects

Duncan (1966, 1975)—applying Duncan (1966, 1975)—applying tracing rulestracing rules

Reduced-form equations (Alwin & Reduced-form equations (Alwin & Hauser, 1975)Hauser, 1975)

Asymptotic distribution of indirect Asymptotic distribution of indirect effects (Sobel, 1982)effects (Sobel, 1982)

JoreskögJoreskög

Maximum Likelihood estimator was Maximum Likelihood estimator was an improvement over 2 and 3 stage an improvement over 2 and 3 stage least squares methodsleast squares methods

Joreskög made structural equation Joreskög made structural equation modeling more accessible (if only modeling more accessible (if only slightly!) with the introduction of slightly!) with the introduction of LISREL, a computer programLISREL, a computer program

Added model fit indicesAdded model fit indices Added multiple-group modelsAdded multiple-group models

BentlerBentler

Refined fit indicesRefined fit indices Added specific effects and brought Added specific effects and brought

SEM into the field of psychology, SEM into the field of psychology, which otherwise was later than which otherwise was later than economics and sociology in its economics and sociology in its introduction to SEMintroduction to SEM

MuthénMuthén

Added latent growth curve analysisAdded latent growth curve analysis Added hierarchical (multi-level) Added hierarchical (multi-level)

modelingmodeling

Other DevelopmentsOther Developments

Models for dichotomous and ordinal Models for dichotomous and ordinal variablesvariables

Various combinations of hierarchical Various combinations of hierarchical (multi-level) modeling, latent growth (multi-level) modeling, latent growth curve analysis, multiple-group curve analysis, multiple-group analysesanalyses

Use of interaction termsUse of interaction terms

Quips and Quotes Quips and Quotes (Wolfle, 2003)(Wolfle, 2003)

““Here I was doing elaborate, cross-lagged, multiple-Here I was doing elaborate, cross-lagged, multiple-partial canonical correlations involving dozens of partial canonical correlations involving dozens of variables, and that eminent sociologist [Paul variables, and that eminent sociologist [Paul Lazarsfeld] was still messing around with chi square Lazarsfeld] was still messing around with chi square tables! What I did not appreciate was that his little tables! What I did not appreciate was that his little analyses were generally more informative than my analyses were generally more informative than my elaborate ones, because he had the ‘right’ variables. elaborate ones, because he had the ‘right’ variables. He knew his subject matter. He was aware of the He knew his subject matter. He was aware of the major alternative explanations that had to be major alternative explanations that had to be guarded against and took that into account when he guarded against and took that into account when he decided upon the four or five variables that were decided upon the four or five variables that were crucial to include. His work represented the state of crucial to include. His work represented the state of the art in model building, while my work represented the art in model building, while my work represented the state of the art in number crunching.” (Cooley, the state of the art in number crunching.” (Cooley, 1978)1978)

Quips and Quotes (cont.)Quips and Quotes (cont.)

““All models are wrong, but some are All models are wrong, but some are useful.” (Box, 1979)useful.” (Box, 1979)

““Analysis of covariance structures…is Analysis of covariance structures…is explicitly aimed at complex testing of explicitly aimed at complex testing of theory, and superbly combines methods theory, and superbly combines methods hitherto considered and used separately. hitherto considered and used separately. It also makes possible the rigorous It also makes possible the rigorous testing of theories that have until now testing of theories that have until now been very difficult to test adequately.” been very difficult to test adequately.” (Kerlinger, 1977)(Kerlinger, 1977)

Quips and Quotes (cont.)Quips and Quotes (cont.)

““The government are very keen on The government are very keen on amassing statistics. They collect them, amassing statistics. They collect them, add them, raise them to the nth power, add them, raise them to the nth power, take the cube root and prepare take the cube root and prepare wonderful diagrams. But you must wonderful diagrams. But you must never forget that every one of these never forget that every one of these figures come in the first instance from figures come in the first instance from the village watchman, who just puts the village watchman, who just puts down what he damn pleases.” (Sir J. down what he damn pleases.” (Sir J. Stamp, 1929)Stamp, 1929)

Family Tree of SEMFamily Tree of SEM

T -te s t

L a te n tG ro w thC u rv e

A n a lys is

A N O V A

M u lti-w a yA N O V A R e p e a te d

M e a s u reD e s ig n s

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A n a lys is

B iv a ria teC o rre la tio n

M u ltip leR e g re s s io n

P a thA n a lys is

S truc tura lE qua tio nM o de ling

F a c to rA n a lys is

C o n firm a to ryF a c to r

A n a lys is

E x p lo ra to ryF a c to r

A n a lys is

Defining SEMDefining SEM

““a melding of factor analysis and a melding of factor analysis and path analysis into one comprehensive path analysis into one comprehensive statistical methodology” (Kaplan, 2000)statistical methodology” (Kaplan, 2000)

Simultaneous equation modelingSimultaneous equation modeling Does implied covariance matrix match Does implied covariance matrix match

up with observed covariance matrixup with observed covariance matrix Degree to which they match represents Degree to which they match represents

goodness of fitgoodness of fit

Types of SEM ModelsTypes of SEM Models

Path Analysis ModelsPath Analysis Models Confirmatory factory analysis Confirmatory factory analysis

modelsmodels Structural regression modelsStructural regression models Latent change modelsLatent change models

How SEM and traditional How SEM and traditional approaches are differentapproaches are different

Multiple equations can be estimated Multiple equations can be estimated simultaneouslysimultaneously

Non-recursive models are possibleNon-recursive models are possible Correlations among disturbances are possibleCorrelations among disturbances are possible Formal specification of a model is requiredFormal specification of a model is required Measurement and structural relations are Measurement and structural relations are

separated, with relations among latent separated, with relations among latent variables rather than measured variablesvariables rather than measured variables

Assessing of model fit is not as Assessing of model fit is not as straightforwardstraightforward

Why Use SEM?Why Use SEM?

Test full theoretical model Test full theoretical model ELM as argued by Stiff & Mongeau (1993)ELM as argued by Stiff & Mongeau (1993)

Simultaneous (full information) Simultaneous (full information) estimationestimation consistent with SEM statistical theoryconsistent with SEM statistical theory Analyze systems of equationsAnalyze systems of equations Assumptions about data distributionAssumptions about data distribution But…error spread throughout modelBut…error spread throughout model

Latent VariablesLatent Variables Divorce measurement errorDivorce measurement error True systematic relationship between variablesTrue systematic relationship between variables

Ways to Increase Ways to Increase Confidence in Causal Confidence in Causal

ExplanationsExplanations Conduct experiment if possibleConduct experiment if possible If not:If not:

Control for additional potential confounding Control for additional potential confounding (independent or mediating) variables(independent or mediating) variables

Control for measurement error (as in SEM)Control for measurement error (as in SEM) Make sure statistical power is adequate to Make sure statistical power is adequate to

detect effects or test modeldetect effects or test model Use theory, carefully conceptualize variables, Use theory, carefully conceptualize variables,

and carefully select variables for inclusionand carefully select variables for inclusion Compare models rather than merely assessing Compare models rather than merely assessing

one modelone model Collect data longitudinally if possibleCollect data longitudinally if possible

Section 2:Section 2:Review of Review of

CorrelationCorrelationand Regressionand Regression

Factors Affecting the size Factors Affecting the size of rof r

Arithmetic operations: generally no Arithmetic operations: generally no effecteffect

Distributions of X and YDistributions of X and Y Reliability of variablesReliability of variables Restriction of rangeRestriction of range

Definitions of semi-partial Definitions of semi-partial and partial correlation and partial correlation

coefficientscoefficients Correlation between Y and XCorrelation between Y and X1 1 where where

effects of Xeffects of X2 2 have been removed from Xhave been removed from X1 1

but not from Y is semi-partial but not from Y is semi-partial correlation (a or b in the Venn Diagram)correlation (a or b in the Venn Diagram)

Squared partial correlation answers the Squared partial correlation answers the question, How much of Y that is not question, How much of Y that is not estimated by the other IVs is estimated estimated by the other IVs is estimated by this variable? a/(a+e) or b/(b+e)by this variable? a/(a+e) or b/(b+e)

Components of Explained Components of Explained Variance in 2-independent Variance in 2-independent

variable Casevariable Case

A

B C

D

Partial = B, Part = B & D for Partial = B, Part = B & D for purple predictor. purple predictor.

Partial = C, Part = C & D for Partial = C, Part = C & D for yellow predictor.yellow predictor.

Interpretation of Part Interpretation of Part CorrelationsCorrelations

1.1. Part correlation (semi partial) Part correlation (semi partial) squared is the unique amount of squared is the unique amount of total variance explained. total variance explained.

2.2. Sum of part correlations squared Sum of part correlations squared does NOT equal Rdoes NOT equal R22 because of because of overlapping variance.overlapping variance.

3.3. The part correlationThe part correlation22 does tell you does tell you how much Rhow much R22 would decrease if would decrease if that predictor was eliminated.that predictor was eliminated.

Ways to account for Ways to account for shared varianceshared variance

A Partial regression coefficient is the A Partial regression coefficient is the correlation between a specific predictor and correlation between a specific predictor and the criterion when statistical control has the criterion when statistical control has occurred for all other variables in the occurred for all other variables in the analysis, meaning all the variance for the analysis, meaning all the variance for the other predictors is completely removed.other predictors is completely removed.

A Part (semi partial) regression coefficient A Part (semi partial) regression coefficient is the correlation between a specific is the correlation between a specific predictor and the criterion when all other predictor and the criterion when all other predictors have been partialed out of that predictors have been partialed out of that predictor, but not out of the criterion.predictor, but not out of the criterion.

Possible RelationshipsPossible Relationshipsamong Variablesamong Variables

SuppressionSuppression

The relationship between the independent The relationship between the independent or causal variables is hiding or or causal variables is hiding or suppressing their real relationships with suppressing their real relationships with Y, which would be larger or possibly of Y, which would be larger or possibly of opposite sign were they not correlated.opposite sign were they not correlated.

The inclusion of the suppressor in the The inclusion of the suppressor in the regression equation removes the regression equation removes the unwanted variance in Xunwanted variance in X1 1 in effect in effect enhanced the relationship between Xenhanced the relationship between X1 1 and and Y.Y.

Effects of Specification Effects of Specification ErrorError

Specification Error when variables Specification Error when variables are omitted from the regression are omitted from the regression equationequation

Effects can be inflated or diminished Effects can be inflated or diminished regression coefficients of the regression coefficients of the variables in the model, and a variables in the model, and a reduced Rreduced R22

MulticollinearityMulticollinearity

Existence of substantial correlation Existence of substantial correlation among a set of independent among a set of independent variables.variables.

Problems of interpretation and Problems of interpretation and unstable partial regression unstable partial regression coefficientscoefficients

Section 3Section 3

Data Screening: Fixing Data Screening: Fixing Distributional Problems, Distributional Problems,

Missing Data, Missing Data, MeasurementMeasurement

MulticollinearityMulticollinearity Existence of substantial correlation Existence of substantial correlation

among a set of independent variables.among a set of independent variables. Problems of interpretation and unstable Problems of interpretation and unstable

partial regression coefficientspartial regression coefficients Tolerance = 1 – RTolerance = 1 – R22 of X with all other X of X with all other X VIF = 1/ToleranceVIF = 1/Tolerance VIF < 8.0 not a bad indicatorVIF < 8.0 not a bad indicator How to fix:How to fix:

Delete one or more variablesDelete one or more variables Combine several variablesCombine several variables

Standardized vs. Standardized vs. Unstandardized Regression Unstandardized Regression

CoefficientsCoefficients Standardized coefficients can be Standardized coefficients can be

compared across variables within a modelcompared across variables within a model Standardized coefficients reflect not only Standardized coefficients reflect not only

the strength of the relationship but also the strength of the relationship but also variances and covariances of variables variances and covariances of variables included in the model as well of variance included in the model as well of variance of variables not included in the model and of variables not included in the model and subsumed under the error termsubsumed under the error term

As a result, standardized coefficients are As a result, standardized coefficients are sample-specific and cannot be used to sample-specific and cannot be used to generalize across settings and populationsgeneralize across settings and populations

Standardized vs. Standardized vs. Unstandardized Regression Unstandardized Regression

Coefficients (cont.)Coefficients (cont.) Unstandardized coefficients, however, Unstandardized coefficients, however,

remain fairly stable despite differences in remain fairly stable despite differences in variances and covariances of variables in variances and covariances of variables in different settings or populationsdifferent settings or populations

A recommendation: Use std. coeff. to A recommendation: Use std. coeff. to compare effects within a given population, compare effects within a given population, but unstd. coeff. to compare effects of given but unstd. coeff. to compare effects of given variables across populations.variables across populations.

In practice, when units are not meaningful, In practice, when units are not meaningful, behavioral scientists outside of sociology behavioral scientists outside of sociology and economics use standardized and economics use standardized coefficients in both cases.coefficients in both cases.

Fixing Distributional Fixing Distributional ProblemsProblems

Analyses assume normality of individual Analyses assume normality of individual variables and multivariate normality, variables and multivariate normality, linearity, and homoscedasticity of linearity, and homoscedasticity of relationshipsrelationships

Normality: similar to normal distributionNormality: similar to normal distribution Multivariate normality: residuals of Multivariate normality: residuals of

prediction are normally and prediction are normally and independently distributedindependently distributed

Homoscedasticity: Variances of residuals Homoscedasticity: Variances of residuals do not vary across values of Xdo not vary across values of X

Transformations:Transformations:Ladder of Re-ExpressionsLadder of Re-Expressions PowerPower Inverses (roots)Inverses (roots) LogarithmsLogarithms ReciprocalsReciprocals

Suggested Suggested TransformationsTransformations

Distributional ProblemDistributional Problem TransformationTransformation

Mod. Pos. SkewMod. Pos. Skew Square rootSquare root

Substantial pos. skewSubstantial pos. skew Log (x+c)*Log (x+c)*

Severe pos. skew, L-shapedSevere pos. skew, L-shaped 1/(x+c)*1/(x+c)*

Mod. Negative skewMod. Negative skew Square root (k-Square root (k-x)x)

Substantial neg. skewSubstantial neg. skew Log (k-x)Log (k-x)

Severe. Neg. skew, J Severe. Neg. skew, J shapedshaped

1/(k-x)1/(k-x)

Dealing with OutliersDealing with Outliers Reasons for univariate outliers:Reasons for univariate outliers:

Data entry errors--correctData entry errors--correct Failure to specify missing values correctly--Failure to specify missing values correctly--

correctcorrect Outlier is not a member of the intended Outlier is not a member of the intended

population--deletepopulation--delete Case is from the intended population but Case is from the intended population but

distribution has more extreme values than a distribution has more extreme values than a normal distribution—modify valuenormal distribution—modify value

3.29 or more SD above or below the mean a 3.29 or more SD above or below the mean a reasonable dividing line, but with large sample reasonable dividing line, but with large sample sizes may need to be less inclusivesizes may need to be less inclusive

Multivariate outliersMultivariate outliers

Cases with unusual patterns of Cases with unusual patterns of scoresscores

Discrepant or mismatched casesDiscrepant or mismatched cases Mahalanobis distance: distance in Mahalanobis distance: distance in

SD units between set of scores for SD units between set of scores for individual case and sample means individual case and sample means for all variablesfor all variables

Linearity and Linearity and HomoscedasticityHomoscedasticity

Either transforming variable(s) or Either transforming variable(s) or including polynomial function of including polynomial function of variables in regression may correct variables in regression may correct linearity problemslinearity problems

Correcting for normality of one or Correcting for normality of one or more variables, or transforming one more variables, or transforming one or more variables, or collapsing or more variables, or collapsing among categories may correct among categories may correct heteroscedasticity. “Not fatal,” but heteroscedasticity. “Not fatal,” but weakens results. weakens results.

Missing DataMissing Data

How much is too much?How much is too much? Depends on sample sizeDepends on sample size 20%?20%?

Why a problem?Why a problem? Reduce powerReduce power May introduce bias in sample and May introduce bias in sample and

resultsresults

Types of Missing Data Types of Missing Data PatternsPatterns

Missing at random (MAR)—missing Missing at random (MAR)—missing observations on some variable X differ observations on some variable X differ from observed scores on that variable from observed scores on that variable only by chance. Probabilities of only by chance. Probabilities of missingness may depend on observed missingness may depend on observed data but not missing data.data but not missing data.

Missing completely at random (MCAR)—Missing completely at random (MCAR)—in addition to MAR, presence vs. absence in addition to MAR, presence vs. absence of data on X is unrelated to other of data on X is unrelated to other variables. Probabilities of missingness variables. Probabilities of missingness also not dependent on observed ata.also not dependent on observed ata.

Missing not at random (MNAR)Missing not at random (MNAR)

Methods of Reducing Methods of Reducing Missing DataMissing Data

Case DeletionCase Deletion Substituting Means on Valid CasesSubstituting Means on Valid Cases Substituting estimates based on regressionSubstituting estimates based on regression Multiple ImputationMultiple Imputation

Each missing value is replaced by list of simlulated Each missing value is replaced by list of simlulated values. Each of m datasets is analyzed by a complete-values. Each of m datasets is analyzed by a complete-data method. Results combined by averaging results data method. Results combined by averaging results with overall estimates and standard errors.with overall estimates and standard errors.

Maximum Likelihood (EM) method:Maximum Likelihood (EM) method: Fill in the missing data with a best guess under current Fill in the missing data with a best guess under current

estimate of unknown parameters, then reestimate from estimate of unknown parameters, then reestimate from observed and filled-in dataobserved and filled-in data

Checklist for Screening Checklist for Screening DataData

Inspect univariate descriptive statisticsInspect univariate descriptive statistics Evaluate amount/distribution of missing dataEvaluate amount/distribution of missing data Check pairwise plots for nonlinearity and Check pairwise plots for nonlinearity and

heteroscedasticityheteroscedasticity Identify and deal with nonnormal variablesIdentify and deal with nonnormal variables Identify and deal with multivariate outliersIdentify and deal with multivariate outliers Evaluate variables for multicollinearityEvaluate variables for multicollinearity Assess reliability and validity of measuresAssess reliability and validity of measures

Section 4Section 4

Overview of SEM Overview of SEM concepts, path concepts, path

diagrams, diagrams, programsprograms

DefinitionsDefinitions Exogenous variable—Independent variables not presumed to Exogenous variable—Independent variables not presumed to

be caused by variables in the modelbe caused by variables in the model Endogenous variables— variables presumed to be caused by Endogenous variables— variables presumed to be caused by

other variables in the modelother variables in the model Latent variable: unobserved variable implied by the Latent variable: unobserved variable implied by the

covariances among two or more indicators, free of random covariances among two or more indicators, free of random error (due to measurement) and uniqueness associated with error (due to measurement) and uniqueness associated with indicators, measure of theoretical constructindicators, measure of theoretical construct

Measurement model prescribes components of latent Measurement model prescribes components of latent variablesvariables

Structural model prescribes relations among latent variables Structural model prescribes relations among latent variables and/or observed variables not linked to latent variablesand/or observed variables not linked to latent variables

Recursive models assume that all causal effects are Recursive models assume that all causal effects are represented as unidirectional and no disturbance represented as unidirectional and no disturbance correlations among endogenous variables with direct effects correlations among endogenous variables with direct effects between thembetween them

Non-recursive models are those with feedback loopsNon-recursive models are those with feedback loops

Definitions (cont.)Definitions (cont.) Model Specification—Formally stating a model via Model Specification—Formally stating a model via

statements about a set of parametersstatements about a set of parameters Model Identification—Can a single unique value for each Model Identification—Can a single unique value for each

and every free parameter be obtained from the observed and every free parameter be obtained from the observed data: just identified, over-identified, under-identifieddata: just identified, over-identified, under-identified

Evaluation of Fit—Assessment of the extent to which the Evaluation of Fit—Assessment of the extent to which the overall model fits or provides a reasonable estimate of the overall model fits or provides a reasonable estimate of the observed dataobserved data

Fixed (not estimated, typically set = 0), Free (estimated Fixed (not estimated, typically set = 0), Free (estimated from the data), and Constrained Parameters (typically set from the data), and Constrained Parameters (typically set of parameters set to be equal)of parameters set to be equal)

Model Modification—adjusting a specified and estimated Model Modification—adjusting a specified and estimated model by freeing or fixing new parametersmodel by freeing or fixing new parameters

Direct (presumed causal relationship between 2 variables), Direct (presumed causal relationship between 2 variables), indirect (presumed causal relationship via other indirect (presumed causal relationship via other intervening or mediating variables), and total effects (sum intervening or mediating variables), and total effects (sum of direct and indirect effects)of direct and indirect effects)

Path DiagramsPath Diagrams Ovals for latent variablesOvals for latent variables Rectangles for observed variablesRectangles for observed variables Arrows point toward observed Arrows point toward observed

variables to indicate measurement variables to indicate measurement errorerror

Arrows point toward latent variables Arrows point toward latent variables to indicate residuals or disturbancesto indicate residuals or disturbances

Path DiagramsPath Diagrams Straight lines for putative causal Straight lines for putative causal

relationsrelations Curved lines to indicate correlationsCurved lines to indicate correlations

SS

Hours Viewing

TV

PSA Exposure

CSQ

Confirmatory Factor Confirmatory Factor AnalysisAnalysis

The concept and practice of what most of us The concept and practice of what most of us know as factor analysis is now considered know as factor analysis is now considered exploratory factor analysisexploratory factor analysis, that is, with no , that is, with no or few preconceived notions about what the or few preconceived notions about what the factor pattern will look like. There are factor pattern will look like. There are typically no tests of significance for EFAtypically no tests of significance for EFA

Confirmatory factory analysisConfirmatory factory analysis, on the other , on the other hand, is where we have a theoretically or hand, is where we have a theoretically or empirically based conception of the empirically based conception of the structure of measured variables and factors structure of measured variables and factors and enables us to test the adequacy of a and enables us to test the adequacy of a particular “measurement model” to the dataparticular “measurement model” to the data

Structural Regression Structural Regression ModelsModels

Inclusion of measured and latent variablesInclusion of measured and latent variables Assessment both of relationship between Assessment both of relationship between

measured and latent variables measured and latent variables (measurement model) and putative causal (measurement model) and putative causal relationships among latent variables relationships among latent variables (structural model)(structural model)

Controls for measurement error, Controls for measurement error, correlations due to methods, correlations correlations due to methods, correlations among residuals and separates these from among residuals and separates these from structural coefficientsstructural coefficients

Path DiagramsPath Diagrams

Ovals for latent variablesOvals for latent variables Rectangles for observed variablesRectangles for observed variables Straight lines for putative causal Straight lines for putative causal

relationsrelations Curved lines to indicate correlationsCurved lines to indicate correlations Arrows pointing toward observed Arrows pointing toward observed

variables to indicate measurement errorvariables to indicate measurement error Arrows pointing toward latent variables Arrows pointing toward latent variables

to indicate residuals or disturbancesto indicate residuals or disturbances

Steps in SEMSteps in SEM Specify the modelSpecify the model Determine identification of the modelDetermine identification of the model Select measures and collect, prepare and Select measures and collect, prepare and

screen the datascreen the data Use a computer program to estimate the Use a computer program to estimate the

modelmodel Re-specify the model if necessaryRe-specify the model if necessary Describe the analysis accurately and Describe the analysis accurately and

completelycompletely Replicate the results*Replicate the results* Apply the results*Apply the results*

ProgramsPrograms AMOS—assess impact of one parameter on AMOS—assess impact of one parameter on

model; editing/debugging functions; model; editing/debugging functions; bootstrapped estimates; MAR estimatesbootstrapped estimates; MAR estimates

EQS—data editor; wizard to write syntax; various EQS—data editor; wizard to write syntax; various estimates for nonnormal data; model-based estimates for nonnormal data; model-based bootstrapping and handling randomly missing bootstrapping and handling randomly missing data data

LISREL—data entry to analysis. PRELIS screens LISREL—data entry to analysis. PRELIS screens data files; wizard to write syntax; can easily data files; wizard to write syntax; can easily analyze categorical/ordinal variables; analyze categorical/ordinal variables; hierarchical data can also be usedhierarchical data can also be used

MPLUS—latent growth models; wizard for batch MPLUS—latent growth models; wizard for batch analysis; no model diagram input/output; MAR analysis; no model diagram input/output; MAR data; complex sampling designs; hierarchical and data; complex sampling designs; hierarchical and multi-level modelsmulti-level models

Section 5Section 5

Equations for path Equations for path analysis, analysis,

decomposing decomposing correlations, correlations,

mediationmediation

Path EquationsPath Equations

Components of Path Model:Components of Path Model: Exogenous VariablesExogenous Variables Correlations among exogenous Correlations among exogenous

variablesvariables Structural pathsStructural paths Disturbances/residuals/errorDisturbances/residuals/error

Relationship between Relationship between regression coefficients and regression coefficients and

path coefficientspath coefficients When residuals are uncorrelated When residuals are uncorrelated

with variables in the equation in with variables in the equation in which it appears, nor with any of the which it appears, nor with any of the variables preceding it in the model, variables preceding it in the model, the solution for the path coefficients the solution for the path coefficients takes the form of OLS solutions for takes the form of OLS solutions for the standardized regression the standardized regression coefficients.coefficients.

The Tracing RuleThe Tracing Rule If one causes the other, then always start with If one causes the other, then always start with

the one that is the effect. If they are not the one that is the effect. If they are not directly causally related, then the starting directly causally related, then the starting point is arbitrary. But once a start variable is point is arbitrary. But once a start variable is selected, always start there.selected, always start there.

Start against an arrow (go from effect to Start against an arrow (go from effect to cause). Remember, the goal at this point is to cause). Remember, the goal at this point is to go from the start variable to the other go from the start variable to the other variable.variable.

Each particular tracing of paths between the Each particular tracing of paths between the two variables can go through only one two variables can go through only one noncausal (curved, double-headed) path noncausal (curved, double-headed) path (relevant only when there are three or more (relevant only when there are three or more exogenous variables and two or more curved, exogenous variables and two or more curved, double-headed arrows).double-headed arrows).

The Tracing Rule (cont.)The Tracing Rule (cont.)

For each particular tracing of paths, any For each particular tracing of paths, any intermediate variable can be included intermediate variable can be included only once.only once.

The tracing can go back against paths The tracing can go back against paths (from effect to cause) for as far as (from effect to cause) for as far as possible, but, regardless of how far back, possible, but, regardless of how far back, once the tracing goes forward causally once the tracing goes forward causally (i.e., with an arrow from cause to effect), (i.e., with an arrow from cause to effect), it cannot turn back against an arrow.it cannot turn back against an arrow.

Mediation vs. Mediation vs. ModerationModeration

Mediation: Intervening variablesMediation: Intervening variables Moderation: Interaction among Moderation: Interaction among

independent or independent or interventing/mediating variablesinterventing/mediating variables

How to Test for How to Test for MediationMediation

X X Y Y X X M M M M Y Y When M is added to X as predictor of Y, X is no When M is added to X as predictor of Y, X is no

longer significantly predictive of Y (Baron & longer significantly predictive of Y (Baron & Kenny)Kenny)

Assess effect ratio: a X b / c [indirect effect Assess effect ratio: a X b / c [indirect effect divided by direct effect]divided by direct effect]

Direct, Indirect, and Direct, Indirect, and Total EffectsTotal Effects

Total Effect = Direct + Indirect Total Effect = Direct + Indirect EffectsEffects

Total Effect = Direct Effects + Total Effect = Direct Effects + Indirect Effects + Spurious Causes Indirect Effects + Spurious Causes + Unanalyzed due to correlated + Unanalyzed due to correlated causescauses

IdentificationIdentification

A model is identified if:A model is identified if: It is theoretically possible to derive a It is theoretically possible to derive a

unique estimate of each parameterunique estimate of each parameter The number of equations is equal to the The number of equations is equal to the

number of parameters to be estimatednumber of parameters to be estimated It is fully recursiveIt is fully recursive

OveridentificationOveridentification

A model is overidentified if:A model is overidentified if: A model has fewer parameters than A model has fewer parameters than

observationsobservations There are more equations than are There are more equations than are

necessary for the purpose of estimating necessary for the purpose of estimating parametersparameters

UnderidentificationUnderidentification

A model is underidentified or not A model is underidentified or not identified if:identified if: It is not theoretically possible to derive It is not theoretically possible to derive

a unique estimate of each parametera unique estimate of each parameter There is insufficient information for the There is insufficient information for the

purpose of obtaining a determinate purpose of obtaining a determinate solution of parameters.solution of parameters.

There are an infinite number of There are an infinite number of solutions may be obtainedsolutions may be obtained

Necessary but not Necessary but not Sufficient Conditions for Sufficient Conditions for Identification: Counting Identification: Counting

RuleRule Counting rule: Number of estimated Counting rule: Number of estimated parameters cannot be greater than parameters cannot be greater than the number of sample variances and the number of sample variances and covariances. Where the number of covariances. Where the number of observed variables = p, this is given observed variables = p, this is given by by

[p x (p+1)] / 2[p x (p+1)] / 2

Necessary but not Necessary but not Sufficient Conditions for Sufficient Conditions for

Identification: Order Identification: Order ConditionCondition If m = # of endogenous variables in the If m = # of endogenous variables in the

model and k = # of exogenous variables model and k = # of exogenous variables in the model, and kin the model, and kee = # exogenous = # exogenous variables in the model excluded from the variables in the model excluded from the structural equation model being tested structural equation model being tested and mand mii = number of endogenous = number of endogenous variables in the model included in the variables in the model included in the equation being tested (including the one equation being tested (including the one being explained on the left-hand side), being explained on the left-hand side), the following requirement must be the following requirement must be satisfied: ksatisfied: ke e > > mmii-1-1

Necessary but not Necessary but not Sufficient Conditions for Sufficient Conditions for

Identification: Rank Identification: Rank ConditionCondition For nonrecursive models, each For nonrecursive models, each

variable in a feedback loop must variable in a feedback loop must have a unique pattern of direct have a unique pattern of direct effects on it from variables outside effects on it from variables outside the loop.the loop.

For recursive models, an analogous For recursive models, an analogous condition must apply which requires condition must apply which requires a very complex algorithm or matrix a very complex algorithm or matrix algebra.algebra.

Guiding Principles for Guiding Principles for IdentificationIdentification

A fully recursive model (one in which A fully recursive model (one in which all the variables are interconnected) all the variables are interconnected) is just identified.is just identified.

A model must have some scale for A model must have some scale for unmeasured variables unmeasured variables

Where are Identification Where are Identification Problems More Likely?Problems More Likely?

Models with large numbers of Models with large numbers of coefficients relative to the number of coefficients relative to the number of input covariancesinput covariances

Reciprocal effects and causal loopsReciprocal effects and causal loops When variance of conceptual level When variance of conceptual level

variable and all factor loadings linking variable and all factor loadings linking that concept to indicators are freethat concept to indicators are free

Models containing many similar Models containing many similar concepts or many error covariances concepts or many error covariances

How to Avoid How to Avoid UnderidentificationUnderidentification

Use only recursive modelsUse only recursive models Add extra constraints by adding indicatorsAdd extra constraints by adding indicators Fixed whatever structural coefficients are Fixed whatever structural coefficients are

expected to be 0, based on theory, especially expected to be 0, based on theory, especially reciprocal effects, where possiblereciprocal effects, where possible

Fix measurement error variances based on Fix measurement error variances based on known data collection proceduresknown data collection procedures

Given a clear time order, reciprocal effects Given a clear time order, reciprocal effects shouldn’t be estimatedshouldn’t be estimated

If the literature suggests the size of certain If the literature suggests the size of certain effects, one can fix the coefficient of that effect to effects, one can fix the coefficient of that effect to that constantthat constant

How to Test for How to Test for UnderidentificationUnderidentification

If ML solution repeatedly converges to If ML solution repeatedly converges to same set of final estimates given different same set of final estimates given different start values, suggests identificationstart values, suggests identification

If concerned about the identification of a If concerned about the identification of a particular equation/coefficient, run the particular equation/coefficient, run the model once with the coefficient free, once model once with the coefficient free, once at a value thought to be “minimally yet at a value thought to be “minimally yet substantially different” than the estimated substantially different” than the estimated value. If the fit of the model is worse, it value. If the fit of the model is worse, it suggests identification.suggests identification.

What to do if a Model is What to do if a Model is UnderidentifiedUnderidentified

Simplify the modelSimplify the model Add indicatorsAdd indicators Eliminate reciprocal effectsEliminate reciprocal effects Eliminate correlations among Eliminate correlations among

residualsresiduals

Introduction toIntroduction to

AMOS, AMOS, Part 1Part 1

AMOS AdvantagesAMOS Advantages

Easy to use for Easy to use for visual SEMvisual SEM ( Structural Equation Modeling). ( Structural Equation Modeling).

Easy to Easy to modify, viewmodify, view the model the model Publication –quality Publication –quality graphicsgraphics

AMOS ComponentsAMOS Components

AMOS GraphicsAMOS Graphics draw SEM draw SEM graphsgraphs runs SEM models using runs SEM models using graphsgraphs

AMOS BasicAMOS Basic runs SEM models using runs SEM models using syntaxsyntax

Starting AMOS GraphicsStarting AMOS Graphics

StartStart Programs Amos 5 Amos Graphics

Reading Data into AMOSReading Data into AMOS

FileFile Data FilesData Files The following dialog appears:The following dialog appears:

Reading Data into Reading Data into AMOSAMOS

Click onClick on File NameFile Name to specify the to specify the name of the data filename 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 AMOSReading Data into AMOS

Example USED for this workshop: Example USED for this workshop: Condom use and what predictors affect Condom use and what predictors affect

itit

DATASET: DATASET: AMOS_data_valid_condom.savAMOS_data_valid_condom.sav

Drawing in AMOSDrawing in AMOS In Amos Graphics, a model can be In Amos Graphics, a model can be

specified by drawing a diagram on specified by drawing a diagram on the screenthe 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 AMOS

To draw a path, Click “To draw a path, Click “DiagramDiagram” on the top ” on the top

menu and click “menu and click “Draw PathDraw Path”.”. Instead of using the top menu, you may use the Instead of using the top menu, you may use the

Tool Box buttons to draw arrows ( and ). Tool Box buttons to draw arrows ( and ).

Drawing in AMOSDrawing in AMOS To draw Error Term to the observed and To draw Error Term to the observed and

unobserved variables. unobserved variables. Use “Use “Unique VariableUnique Variable” button in the Tool Box. ” button in the Tool Box.

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

1

1 1

Drawing in AMOSDrawing in AMOS

Let us draw:Let us draw:

Naming the variables in Naming the variables in AMOSAMOS

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

Example: Name the variables Example: Name the variables

ISSUEB1

SXPYRC1

eSXPYRC1

1

SEX1

eiss

FRBEHB1

efr1

1 1

IDM

Constraining a parameter in Constraining a parameter in AMOSAMOS

The scale of the latent variable or variance of The scale of the latent variable or variance of the latent variable the latent variable has to be fixed to 1has 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 You can change the appearancechange the appearance of your path of your path diagram by diagram by moving objects aroundmoving objects around

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

Then click and hold down your Then click and hold down your left mouse left mouse buttonbutton on the object you wish to move. With the on the object you wish to move. With the mouse button still depressed, mouse button still depressed, move the objectmove the object to to where you want it, and let go of your mouse where you want it, and let go of your mouse button. button. Amos GraphicsAmos Graphics will automatically will automatically redraw all connecting arrows.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, To change the size and shape of an object, first press the first press the Change the shape of objectsChange the shape of objects icon on the toolbar.icon on the toolbar.

You will notice that the word “You will notice that the word “shape”shape” appears under the mouse pointer to let you appears under the mouse pointer to let you know the know the ShapeShape function is active. function is active.

Click and hold down your Click and hold down your left mouse buttonleft mouse button on the object you wish to re-shape. Change on the object you wish to re-shape. Change the shape of the object to your liking and the shape of the object to your liking and release the mouse button.release the mouse button.

Change the shape of objectsChange the shape of objects also works on also works on two-headed arrows. Follow the same two-headed arrows. Follow the same procedure to change the direction or arc of procedure to change the direction or arc of any double-headed arrow.any double-headed arrow.

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

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

To To erase an objecterase an object, simply click on the , simply click on the

EraseErase icon and then click on the object icon and then click on the object you wish to erase.you wish to erase.

To To undo your last drawing activityundo your last drawing activity, , click on theclick on the UndoUndo icon and your last icon and your last activity disappears.activity disappears.

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

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

Performing the Performing the analysis in AMOS analysis in AMOS

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

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

Performing the analysis in Performing the analysis in AMOSAMOS

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

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

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

Results Results When When AMOSAMOS has completed the has completed the

calculations, you havecalculations, you have twotwo options options for viewing the output:for viewing the output: text outputtext output, , graphics outputgraphics output..

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

Here is a portion of the text output Here is a portion of the text output for this problem: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 Results for Condom Use ModelModel

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 Viewing the graphics output in AMOSAMOS

• 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

Section 7Section 7

Putting it All Putting it All TogetherTogether

Section 6Section 6

Model Testing and Fit Model Testing and Fit Indices, Statistical Indices, Statistical

PowerPower

Model SpecificationModel Specification

Use theory to determine variables Use theory to determine variables and relationships to testand relationships to test

Fix, free, and constrain parameters Fix, free, and constrain parameters as appropriateas appropriate

Estimation MethodsEstimation Methods Maximum Likelihood—estimates maximize the likelihood Maximum Likelihood—estimates maximize the likelihood

that the data (observed covariances) were drawn from this that the data (observed covariances) were drawn from this population. Most forms are simultaneous. The fitting population. Most forms are simultaneous. The fitting function is related to discrepancies between observed function is related to discrepancies between observed covariances and those predicted by the model. Typically covariances and those predicted by the model. Typically iterative, deriving an initial solution then improves is iterative, deriving an initial solution then improves is through various calculations.through various calculations.

Generalized and Unweighted Least Squares-- based on Generalized and Unweighted Least Squares-- based on least squares criterion (rather than discrepancy function) least squares criterion (rather than discrepancy function) but estimate all parameters simultaneously.but estimate all parameters simultaneously.

2-Stage and 3-Stage Least Squares—can be used to 2-Stage and 3-Stage Least Squares—can be used to estimate non-recursive models, but estimate only one estimate non-recursive models, but estimate only one equation at a time. Applies multiple regression in two equation at a time. Applies multiple regression in two stages, replacing problematic variables (those correlated stages, replacing problematic variables (those correlated to disturbances) with a newly created predictor to disturbances) with a newly created predictor (instrumental variable that has direct effect on problematic (instrumental variable that has direct effect on problematic variable but not on the endogenous variable).variable but not on the endogenous variable).

Does the model “fit”?Does the model “fit”? Model fit = sample data are Model fit = sample data are

consistent with the implied modelconsistent with the implied model The smaller the discrepancy The smaller the discrepancy

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

Model fit is Achilles’ heel of SEMModel fit is Achilles’ heel of SEM Many fit indexesMany fit indexes None are fallible (though some are None are fallible (though some are

better than others) better than others)

Measures of Model FitMeasures of Model Fit 22 = N-1 * minimization criterion. Just-identified model has = 0, = N-1 * minimization criterion. Just-identified model has = 0,

no df. As chi-square increases, fit becomes worse. Badness of fit no df. As chi-square increases, fit becomes worse. Badness of fit index. Tests difference in fit between given overidentified model index. Tests difference in fit between given overidentified model and just-identified version of it.and just-identified version of it.

RMSEA—parsimony adjusted index to correct for model RMSEA—parsimony adjusted index to correct for model complexity. Approximates non-central chi-square distribution, complexity. Approximates non-central chi-square distribution, which does not require a true null hypothesis, i.e., not a perfect which does not require a true null hypothesis, i.e., not a perfect model. Noncentrality parameter assesses the degree of falseness model. Noncentrality parameter assesses the degree of falseness of the null hypothesis. Badness of fit index, with 0 best and of the null hypothesis. Badness of fit index, with 0 best and higher values worse. Amount of error of approximation per higher values worse. Amount of error of approximation per model df. RMSEA model df. RMSEA << .05 close fit, .05-.08 reasonable, .05 close fit, .05-.08 reasonable, >> .10 poor .10 poor fitfit

CFI—Assess fit of model compared to baseline model, typically CFI—Assess fit of model compared to baseline model, typically independence or null model, which assumes zero population independence or null model, which assumes zero population covariances among the observed variablescovariances among the observed variables

AIC—used to select among nonhierarhical modelsAIC—used to select among nonhierarhical models

Model FitModel Fit

22 Goodness of Fit test Goodness of Fit test Historically usedHistorically used Desire a nonsignificant Desire a nonsignificant pp-value, i.e., -value, i.e., pp>.05>.05 Adversely affected by sample sizeAdversely affected by sample size

(N-1)*minimization function(N-1)*minimization function Badness of fit indexBadness of fit index Tests difference in fit between Tests difference in fit between

overidentified model and its just-identified overidentified model and its just-identified version.version.

Mixed opinions on its value in reporting.Mixed opinions on its value in reporting.

Model FitModel Fit

CFICFI Fit determined by comparing Fit determined by comparing

implied model to a baseline model implied model to a baseline model which assumes zero population which assumes zero population covariances among the observed covariances among the observed variablesvariables

Initially, Bentler CFI > .90Initially, Bentler CFI > .90 Hu & Bentler (1998, 1999) CFI Hu & Bentler (1998, 1999) CFI

> .95.> .95.

Model FitModel FitRMSEARMSEA Root Mean Squared Error of ApproximationRoot Mean Squared Error of Approximation Adjusts fit index to correct for model complexityAdjusts fit index to correct for model complexity Based on noncentrality parameter which assesses Based on noncentrality parameter which assesses

the degree of falseness of the null hypothesis. the degree of falseness of the null hypothesis. Badness of fit index; 0 best & higher values Badness of fit index; 0 best & higher values

worse. worse. Amount of error of approximation per model df.Amount of error of approximation per model df. RMSEA RMSEA << .05 close fit .05 close fit .05-.08 reasonable and .05-.08 reasonable and >> .10 poor fit .10 poor fit ALWAYS REPORT CONFIDENCE INTERVAL!ALWAYS REPORT CONFIDENCE INTERVAL!

Model FitModel Fit

Many other fit indexesMany other fit indexes IdeallyIdeally

Nonsignificant Nonsignificant 22 Goodness of Fit test Goodness of Fit test CFI > .95CFI > .95 RMSEA > .08RMSEA > .08

IF model fits, then look at pathsIF model fits, then look at paths

Model Fit & Model Fit & RespecificationRespecification

What if the model does NOT fit?What if the model does NOT fit? Model trimming and buildingModel trimming and building

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

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

Consider equivalent modelsConsider equivalent models

Model RespecificationModel Respecification

Model trimming and buildingModel trimming and building Empirical vs. theoretical Empirical vs. theoretical

respecificationrespecification Consider equivalent modelsConsider equivalent models

Comparison of Models Comparison of Models

Hierarchical Models: Hierarchical Models: Difference of Difference of 22 test test

Non-hierarchical Models:Non-hierarchical Models: Compare model fit indicesCompare model fit indices

Sample Size GuidelinesSample Size Guidelines Small (under 100), Medium (100-200), Large Small (under 100), Medium (100-200), Large

(200+) [try for medium, large better](200+) [try for medium, large better] Models with 1-2 df may require samples of Models with 1-2 df may require samples of

thousands for model-level power of .8.thousands for model-level power of .8. When df=10 may only need n of 300-400 for When df=10 may only need n of 300-400 for

model level power of .8.model level power of .8. When df > 20 may only need n of 200 for power When df > 20 may only need n of 200 for power

of .8of .8 20:1 is ideal ratio for # cases/# free parameters, 20:1 is ideal ratio for # cases/# free parameters,

10:1 is ok, less than 5:1 is almost certainly 10:1 is ok, less than 5:1 is almost certainly problematicproblematic

For regression, N For regression, N >> 50 + 8m for overall R 50 + 8m for overall R22, with , with m = # IVs and N m = # IVs and N > > 104 + m for individual 104 + m for individual predictorspredictors

Statistical PowerStatistical Power Use power analysis tables from Cohen to Use power analysis tables from Cohen to

assess power of specific detecting path assess power of specific detecting path coefficient.coefficient.

Saris & Satorra: use Saris & Satorra: use χχ22 difference test difference test using predicted covariance matrix using predicted covariance matrix compared to one with that path = 0compared to one with that path = 0

McCallum et al. (1996) based on RMSEA McCallum et al. (1996) based on RMSEA and chi-square distribution for close fit, and chi-square distribution for close fit, not close fit and exact fitnot close fit and exact fit

Small number of computer programs that Small number of computer programs that calculate power for SEM at this pointcalculate power for SEM at this point

Power Analysis for testing Power Analysis for testing DATA-MODEL fitDATA-MODEL fit

HH00: : εε00≥ 0.05≥ 0.05

The Null hypothesis: The data-model fit is The Null hypothesis: The data-model fit is unacceptableunacceptable

HH11: : εε11< 0.05 < 0.05

The Alternative hypothesis: The data-model fit is The Alternative hypothesis: The data-model fit is acceptable acceptable

If If RMSEA from the model fit is less than 0.05, then RMSEA from the model fit is less than 0.05, then the null hypothesis containing unacceptable the null hypothesis containing unacceptable population data-model fit is rejected population data-model fit is rejected

Post Hoc Power Analysis for Post Hoc Power Analysis for testing Data-Model fit testing Data-Model fit

If If εε11 is close to 0 is close to 0 Power increases Power increases

If N (sample size) increases If N (sample size) increases Power Power increasesincreases

If df ( degree of freedom) increasesIf df ( degree of freedom) increases Power Power increasesincreases

Post Hoc Power Analysis for Post Hoc Power Analysis for testing Data-Model fittesting Data-Model fit

Examples Using Appendix B calculate Examples Using Appendix B calculate power power

for for εε11 =0.02, df=55, N=400 =0.02, df=55, N=400 Power ? Power ?

for for εε11 =0.04, df=30, N=400 =0.04, df=30, N=400 Power ? Power ?

Section 7:Section 7:

Confirmatory Factor Confirmatory Factor AnalysisAnalysis

Factor AnalysisFactor Analysis

Single Measure in Path AnalysisSingle Measure in Path Analysis Measurement error is higherMeasurement error is higher

Multiple Measures in Factor Analysis Multiple Measures in Factor Analysis correspond to some type of correspond to some type of HYPOTHETICAL CONSTRUCTHYPOTHETICAL CONSTRUCT

Reduce the overall effect of measurement Reduce the overall effect of measurement errorerror

Latent ConstructLatent Construct

Theory guides through the scale Theory guides through the scale development process (DeVellis,1991; development process (DeVellis,1991; Jackson, 1970)Jackson, 1970)

Unidimensional vs Multidimensional Unidimensional vs Multidimensional constuctconstuct

Reliability and Validity of constructReliability and Validity of construct

Reliability - consistency, precision, Reliability - consistency, precision, repeatability repeatability

Reliability concerns with RANDOM ERRORReliability concerns with RANDOM ERROR

Types of reliability:Types of reliability: test-retesttest-retest alternate formalternate form interrater interrater split-half and internal consistency split-half and internal consistency

Validity of constructValidity of construct

4 types of validity4 types of validity content content criterion-related criterion-related convergent and discriminantconvergent and discriminant constructconstruct

Factor analysisFactor analysis

Indicators: Indicators: continuouscontinuous Measurement error are independent Measurement error are independent

of each other and of the factorsof each other and of the factors All associations between the factors All associations between the factors

are unanalyzedare unanalyzed

Two Classes of Factor Two Classes of Factor AnalysisAnalysis

Exploratory Factor AnalysisExploratory Factor Analysis Exploring possible factorsExploring possible factors Factor analysis you’re probably used toFactor analysis you’re probably used to

Confirmatory Factor AnalysisConfirmatory Factor Analysis Testing possible models of factor structureTesting possible models of factor structure Using previous findingsUsing previous findings

Identification of CFAIdentification of CFA

Can estimate v*(v+1)/2 of Can estimate v*(v+1)/2 of parametersparameters

Necessary Necessary # of free parameters <= # of # of free parameters <= # of

observationsobservations Every latent variable should be Every latent variable should be scaledscaled

Additional: fix the unstandardized residual path of the error to 1. (assign a scale of the unique variance of its indicator)

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)

OR

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

Both methods of scaling result in the same overall

fit of the model

Identification of CFAIdentification of CFA

Sufficient :Sufficient : At least three (3) indicators per factor At least three (3) indicators per factor

to make the model identifiedto make the model identified Two-indicator rule – prone to estimation Two-indicator rule – prone to estimation

problems (esp. with small sample size)problems (esp. with small sample size)

Interpretation of the Interpretation of the estimatesestimates

Unstandardized solutionUnstandardized solution Factor loadings =unstandardized regression Factor loadings =unstandardized regression

coefficientcoefficient Unanalyzed association between factors or errors= Unanalyzed association between factors or errors=

covariances covariances

•Standardized solutionStandardized solution Unanalyzed association between factors or errors= Unanalyzed association between factors or errors=

correlationscorrelations Factor loadings =standardized regression coefficient Factor loadings =standardized regression coefficient

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

the explained ( common) indicator variance, the explained ( common) indicator variance, RR22(squared multiple correlation)(squared multiple correlation)

Problems in estimation Problems in estimation of CFAof CFA

Heywood cases – negative variance estimated or Heywood cases – negative variance estimated or correlations > 1.correlations > 1.

Ratio of the sample size to the free parameters – Ratio of the sample size to the free parameters – 10:1 ( better 20:1)10:1 ( better 20:1)

Nonnormality – affects ML estimationNonnormality – affects ML estimation

Suggestions by March and Hau(1999)when Suggestions by March and Hau(1999)when sample size is small: sample size is small:

indicators with high standardized loadings( >0.6) indicators with high standardized loadings( >0.6) constrain the factor loadingsconstrain the factor loadings

Testing CFA modelsTesting CFA models

Test for a single factor with the theory or notTest for a single factor with the theory or not If reject HIf reject H0 0 of good fit - try two-factor of good fit - try two-factor

model…model… Since one-factor model is restricted version of Since one-factor model is restricted version of

the two -factor model , then compare one-the two -factor model , then compare one-factor model to two-factor model using Chi-factor model to two-factor model using Chi-square test . If the Chi-square is significant – square test . If the Chi-square is significant – then the 2-factor model is better than 1-factor then the 2-factor model is better than 1-factor model.model.

Check RCheck R22 of the unexplained variance of the of the unexplained variance of the indicators.indicators.

Respecification of CFARespecification of CFA

IFIF lower factor loadings lower factor loadings

of the indicator of the indicator (standardized<=0.2) (standardized<=0.2)

High loading on High loading on more than one factormore than one factor

High correlation of High correlation of the residuals the residuals

High factor High factor correlationcorrelation

THENTHEN Specify that indicator on a Specify that indicator on a

different factordifferent factor

Allow to load on one more Allow to load on one more than one factorthan one factor

(multidimensional vs unidimensional)(multidimensional vs unidimensional)

Allow error Allow error measurements measurements to to covarycovary

Too many factors specifiedToo many factors specified

Other testsOther tests

Indicators:Indicators: congeneric – measure the same constructcongeneric – measure the same construct

if model fits , then if model fits , then

-tau-equivalent – constrain all unstandardized -tau-equivalent – constrain all unstandardized loadings to 1loadings to 1

if model fit, thenif model fit, then

- parallelism – equality of error variances- parallelism – equality of error variances

All these can be tested by All these can be tested by χχ2 2 difference testdifference test

Nonnormal distributionsNonnormal distributions Normalize with transformationsNormalize with transformations Use Use corrected normal theory method, corrected normal theory method, e.g. use e.g. use

robust standard errors and corrected test robust standard errors and corrected test statistics, ( Satorra-Bentler statistics)statistics, ( Satorra-Bentler statistics)

Use Asymptotic distribution free or arbitrary Use Asymptotic distribution free or arbitrary distribution function (ADF) - no distribution distribution function (ADF) - no distribution assumption - Need large sampleassumption - Need large sample

Use elliptical distribution theory – need only Use elliptical distribution theory – need only symmetric distributionsymmetric distribution

Mean-adjusted weighted least squares (MLSW) Mean-adjusted weighted least squares (MLSW) and variance-adjusted weighted least square and variance-adjusted weighted least square (VLSW) - MPLUS with categorical indicators(VLSW) - MPLUS with categorical indicators

Use normal theory with nonparametric Use normal theory with nonparametric bootstrappingbootstrapping

Remedies to Remedies to nonnormalitynonnormality

Use a parcel which is a linear Use a parcel which is a linear composite of the discrete scores, as composite of the discrete scores, as continuous indicators continuous indicators

Use parceling ,when underlying Use parceling ,when underlying factor is unidimentional.factor is unidimentional.

Section 8:Section 8:

Putting it All Together:Putting it All Together:Structural Regression Structural Regression

ModelsModels

Testing Models with Testing Models with Structural and Structural and

Measurement ComponentsMeasurement Components Identification IssuesIdentification Issues

For the structural portion of SR model to be For the structural portion of SR model to be identified, its measurement portion must be identified, its measurement portion must be identified.identified.

Use the two-step rule: Respecify the SR model Use the two-step rule: Respecify the SR model as CFA with all possible unanalyzed as CFA with all possible unanalyzed associations among factors. Assess associations among factors. Assess identificaiton.identificaiton.

View structural portion of the SR model and View structural portion of the SR model and determine if it is recursive. If so, it is determine if it is recursive. If so, it is identified. If not, use order and rank identified. If not, use order and rank conditions.conditions.

The 2-Step ApproachThe 2-Step Approach

Anderson & Gerbing’s approachAnderson & Gerbing’s approach Saturated model, theoretical model of Saturated model, theoretical model of

interestinterest Next most likely constrained and Next most likely constrained and

unconstrained structural modelsunconstrained structural models Kline and others’ 2-step approach:Kline and others’ 2-step approach:

Respecify SR as CFA. Then test various Respecify SR as CFA. Then test various SR models.SR models.

The 4-Step ApproachThe 4-Step Approach

Factor ModelFactor Model Confirmatory Factor ModelConfirmatory Factor Model Anticipated Structural Equation Anticipated Structural Equation

ModelModel More Constrained Structural More Constrained Structural

Equation ModelEquation Model

Constraint InteractionConstraint Interaction When chi-square and parameter estimates differ When chi-square and parameter estimates differ

depending on whether loading or variance is depending on whether loading or variance is constrained.constrained.

Test: If loadings have been constrained, change Test: If loadings have been constrained, change to a new constant. If variance constrained, fix to to a new constant. If variance constrained, fix to a constant other than 1.0. If chi-square value for a constant other than 1.0. If chi-square value for modified model is not identical, constraint modified model is not identical, constraint interaction is present. Scale based on interaction is present. Scale based on substantive grounds.substantive grounds.

Single Indicators in Single Indicators in Partially Latent SR ModelsPartially Latent SR Models

Estimate proportion of variance of Estimate proportion of variance of variable due to error (unique variable due to error (unique variance). Multiply by variance of variance). Multiply by variance of measured variable.measured variable.

Section 9Section 9Multiple-Group Models,Multiple-Group Models,

a Word about Latent Growth a Word about Latent Growth Models,Models,

Pitfalls, Critique and Pitfalls, Critique and

Future Directions for SEM Future Directions for SEM

Multiple-Group ModelsMultiple-Group Models

Main question addressed: do values of Main question addressed: do values of model parameters vary across groups?model parameters vary across groups?

Another equivalent way of expressing this Another equivalent way of expressing this question: does group membership question: does group membership moderate the relations specified in the moderate the relations specified in the model?model?

Is there an interaction between group Is there an interaction between group membership and exogenous variables in membership and exogenous variables in effect on endogenous variables?effect on endogenous variables?

Cross-group equality Cross-group equality constraintsconstraints

One model is fit for each group, with One model is fit for each group, with equal unstandardized coefficients for equal unstandardized coefficients for a set of parameters in the modela set of parameters in the model

This model can be compared to an This model can be compared to an unconstrained model in which all unconstrained model in which all parameters are unconstrained to be parameters are unconstrained to be equal between groupsequal between groups

Latent Growth ModelsLatent Growth Models

Latent Growth Models Latent Growth Models in SEM are often in SEM are often structural regression structural regression models with mean models with mean structuresstructures

Mean StructuresMean Structures

Means are estimated by regression Means are estimated by regression of variables on a constant of variables on a constant

Parameters of a mean structure Parameters of a mean structure include means of exogenous include means of exogenous variables and intercepts of variables and intercepts of endogenous variables.endogenous variables.

Predicted means of endogenous Predicted means of endogenous variables can be compared to variables can be compared to observed means.observed means.

Principles of Mean Principles of Mean Structures in SEMStructures in SEM

When a variable is regressed on a predictor When a variable is regressed on a predictor and a constant, the unstandardized and a constant, the unstandardized coefficient for the constant is the intercept.coefficient for the constant is the intercept.

When a predictor is regressed on a When a predictor is regressed on a constant, the undstandardized coefficient is constant, the undstandardized coefficient is the mean of the predictor.the mean of the predictor.

The mean of an endogenous variable is a The mean of an endogenous variable is a function of three parameters: the intercept, function of three parameters: the intercept, the unstandardized path coefficient, and the the unstandardized path coefficient, and the mean of the exogenous variable.mean of the exogenous variable.

Requirements for LGMRequirements for LGMwithin SEMwithin SEM

continuous dependent variable measured continuous dependent variable measured on at least three different occasionson at least three different occasions

scores that have the same units across scores that have the same units across time, can be said to measure the same time, can be said to measure the same construct at each assessment, and are not construct at each assessment, and are not standardizedstandardized

data that are time structured, meaning data that are time structured, meaning that cases are all tested at the same that cases are all tested at the same intervals (not need be equal intervals)intervals (not need be equal intervals)

Pitfalls--SpecificationPitfalls--Specification

Specifying the model after data Specifying the model after data collectioncollection

Insufficient number of indicators. Insufficient number of indicators. Kenny: “2 might be fine, 3 is better, 4 is Kenny: “2 might be fine, 3 is better, 4 is best, more is gravy”best, more is gravy”

Carefully consider directionalityCarefully consider directionality Forgetting about parsimonyForgetting about parsimony Adding disturbance or measurement Adding disturbance or measurement

errors without substantive justificationerrors without substantive justification

Pitfalls--DataPitfalls--Data

Forgetting to look at missing data Forgetting to look at missing data patternspatterns

Forgetting to look at distributions, Forgetting to look at distributions, outliers, or non-linearity of outliers, or non-linearity of relationshipsrelationships

Lack of independence among Lack of independence among observations due to clustering of observations due to clustering of individualsindividuals

Pitfalls—Analysis/Pitfalls—Analysis/RespecificationRespecification

Using statistical results only and not Using statistical results only and not theory to respecify a modeltheory to respecify a model

Failure to consider constraint interactions Failure to consider constraint interactions and Heywood cases (illogical values for and Heywood cases (illogical values for parameters)parameters)

Use of correlation matrix rather than Use of correlation matrix rather than covariance matrixcovariance matrix

Failure to test measurement model firstFailure to test measurement model first Failure to consider sample size vs. model Failure to consider sample size vs. model

complexitycomplexity

Pitfalls--InterpretationPitfalls--Interpretation

Suggesting that “good fit” proves the Suggesting that “good fit” proves the modelmodel

Not understanding the difference between Not understanding the difference between good fit and high Rgood fit and high R22

Using standardized estimates in comparing Using standardized estimates in comparing multiple-group resultsmultiple-group results

Failure to consider equivalent or Failure to consider equivalent or (nonequivalent) alternative models(nonequivalent) alternative models

Naming fallacyNaming fallacy Suggesting results prove causalitySuggesting results prove causality

CritiqueCritique The multiple/alternative models problemThe multiple/alternative models problem The belief that the “stronger” method The belief that the “stronger” method

and path diagram proves causalityand path diagram proves causality Use of SEM for model modification Use of SEM for model modification

rather than for model testing. Instead:rather than for model testing. Instead: Models should be modified before SEM is Models should be modified before SEM is

conducted orconducted or Sample sizes should be large enough to Sample sizes should be large enough to

modify the model with half of the sample modify the model with half of the sample and then cross-validate the new model with and then cross-validate the new model with the other halfthe other half

Future DirectionsFuture Directions

Assessment of interactionsAssessment of interactions Multiple-level modelsMultiple-level models Curvilinear effectsCurvilinear effects Dichotomous and ordinal variablesDichotomous and ordinal variables

Final ThoughtsFinal Thoughts

SEM can be useful, especially to:SEM can be useful, especially to: separate measurement error from structural separate measurement error from structural

relationshipsrelationships assess models with multiple outcomesassess models with multiple outcomes assess moderating effects via multiple-sample assess moderating effects via multiple-sample

analysesanalyses consider bidirectional relationshipsconsider bidirectional relationships

But be careful. Sample size concerns, lots of But be careful. Sample size concerns, lots of model modification, concluding too much, model modification, concluding too much, and not considering alternative models are and not considering alternative models are especially important pitfalls.especially important pitfalls.

AMOS, Part 2AMOS, Part 2

Modification of the Modification of the ModelModel

Search for the better modelSearch for the better model

Suggestions from: 1) theorySuggestions from: 1) theory

2) modification 2) modification indices indices using using AMOSAMOS

Modifying the Model using Modifying the Model using AMOSAMOS

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

Then check the Then check the Modification indicesModification indices optionoption

Modifying the Model using Modifying the Model using AMOSAMOS

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 Modifying the Model using AMOSAMOS

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 ModelCondom Use Model with missing with missing valuesvalues

Confirmatory Factor AnalysisConfirmatory Factor Analysis for for Impulsive Decision Making constructImpulsive Decision Making construct

Multiple groupMultiple group analysis analysis

How to deal with How to deal with non-normal datanon-normal data

Missing data in AMOSMissing data in AMOS

Full Information Maximum Full Information Maximum Likelihood estimation 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 The standardized graphical output.output.

.05

ISSUEB1

.14

SXPYRC1

eSXPYRC1

.08

SEX1

eiss

.02

FRBEHB1

efr1 .37

-.09.12

IDM

-.10

-.18

-.10

Missing data in AMOSMissing data in AMOS

Example: see the handout #3Example: 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 factor Need to fix either the variance of the IDM1 factor or one of the loadings to 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 Confirmatory Factor Analysis withAnalysis with

Impulsive Decision Impulsive Decision Making scaleMaking scale

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 Confirmatory Factor Analysis withAnalysis with

Impulsive Decision Making Impulsive Decision Making scale scale What if want to What if want to compare two compare two

NESTED modelsNESTED models for for Impulsive Impulsive Decision Making ModelDecision Making Model??

1)1) error variances equal for all 4 measured error variances equal for all 4 measured variablesvariables

2)2) error variances are different error variances are different

Confirmatory Factor Analysis Confirmatory Factor Analysis withwith

Impulsive Decision Making Impulsive Decision Making scale: scale:

the error variances are the the error variances are the samesame

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

Confirmatory Factor Confirmatory Factor Analysis withAnalysis with

Impulsive Decision Making Impulsive Decision Making scale scale

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 Confirmatory Factor Analysis withwith

Impulsive Decision Making Impulsive Decision Making scale: scale:

error variances are the sameerror variances are the same Click Click MODEL FITMODEL FIT , , then then Manage ModelsManage Models In the In the Manage ModelsManage Models window, click on window, click on NewNew. . In the In the Parameter ConstraintsParameter Constraints segment of the segment of the

windowwindow type “ type “e1=e2=e3=e4”e1=e2=e3=e4”

Now there are Now there are two nested modelstwo 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 Confirmatory Factor Analysis withAnalysis with

Impulsive Decision Impulsive Decision Making scaleMaking scaleerror 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 Confirmatory Factor Analysis with with

Impulsive Decision Making Impulsive Decision Making scale:scale:

error variances are the sameerror variances are the same Compare Nested Models using Chi-Compare Nested Models using Chi-

square difference testsquare 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 Confirmatory Factor Analysis with with

Impulsive Decision Making Impulsive Decision Making scale: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 analysis

WHYWHY: test the equality/invariance of the factor : test the equality/invariance of the factor loadings for two separate groups loadings for two separate groups

HOW HOW : : 1) test the model to 1) test the model to both groupsboth groups separately separately to check to check

the entire modelthe entire model2) the same model by 2) the same model by multiple group analysismultiple group analysis

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

Need to have 2 separate data files for each Need to have 2 separate data files for each group.group. data_boysdata_boys and 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 Click Model FitModel Fit and and Multiple GroupsMultiple Groups..

This gives a name to every parameter in the model This gives a name to every parameter in the model in each group. 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 Example: Multiple group analysis for Condom use Model analysis 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 Example: Multiple group analysis for Condom use Modelanalysis for Condom use Model

see handoutsee handout

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

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

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

Chi-squareChi-squarediff diff =68.901-65.119=2.282=68.901-65.119=2.282

df=29-26=3 df=29-26=3 NOT SIGNIFICANT NOT SIGNIFICANT

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

Handling non-normal data:Handling non-normal data:

Verify that your variables are not Verify that your variables are not distributed joint multivariate normaldistributed joint multivariate normal

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

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

Handling non-normal data: Handling non-normal data: checking for normalitychecking 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: Handling non-normal data: checking for normalitychecking 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

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

11 3.8413.841

22 5.9915.991

33 7.8157.815

44 9.4889.488

55 11.07011.070

66 12.59212.592

77 14.06714.067

88 15.50715.507

99 16.91916.919

1010 18.30718.307

1111 19.67519.675

Upper critical values of chi-square distribution