‘STRUCTURAL EQUATION MODELING’: Introduction and Application to HIV Risk Behaviors…and other stuff… James Alan Neff, Ph.D., M.P.H. Professor and Director

‘STRUCTURAL EQUATION MODELING’: Introduction and Application to HIV Risk

Behaviors…and other stuff…

‘STRUCTURAL EQUATION MODELING’: Introduction and Application to HIV Risk

Behaviors…and other stuff…

James Alan Neff, Ph.D., M.P.H.James Alan Neff, Ph.D., M.P.H.Professor and DirectorProfessor and Director

Substance Abuse Research Development Program Substance Abuse Research Development Program University of Texas at Austin School of Social WorkUniversity of Texas at Austin School of Social Work

First Friday PresentationFirst Friday PresentationFebruary 7February 7thth, 2003, 2003

Overview of PresentationOverview of Presentation

Goal: Provide a non-technical overviewGoal: Provide a non-technical overview User orientedUser oriented Presumes working knowledge of Multiple Regression Presumes working knowledge of Multiple Regression

LogicLogic Emphasize linkage of theory to analysisEmphasize linkage of theory to analysis

Overview:Overview: TerminologyTerminology History and DevelopmentHistory and Development LogicLogic Substantive ExamplesSubstantive Examples

I. TerminologyI. Terminology

Related Concepts:Related Concepts: Causal Modeling Causal Modeling Path Analysis Path Analysis Structural Equation ModelingStructural Equation Modeling LISREL (Linear Structural Relations)LISREL (Linear Structural Relations) Covariance Structure Analysis***Covariance Structure Analysis***

***Key clue to what is involved and how it is ***Key clue to what is involved and how it is done!done!

Common Elements:Common Elements:

Involves a Involves a systemsystem of variables. of variables. Variables are Variables are orderedordered in (theoretically derived?) sequence in (theoretically derived?) sequence

(structure) or model, implying assumedly causal (structure) or model, implying assumedly causal relationships (structural relationships or relationships (structural relationships or pathspaths).).

System of System of relationshipsrelationships between variables is specified by between variables is specified by series of equations (structural or prediction), like multiple series of equations (structural or prediction), like multiple regression analysis.regression analysis.

Structural equations define ‘model’ to be tested for ‘fit’ Structural equations define ‘model’ to be tested for ‘fit’ Fit assessed generally against dataFit assessed generally against data Fit may be assessed against alternative modelsFit may be assessed against alternative models

Modeling involves testing of hypothesized Modeling involves testing of hypothesized model.model.Modeling involves testing of hypothesized Modeling involves testing of hypothesized model.model.

Two levels of significance testing:Two levels of significance testing: Testing individual Testing individual structural parametersstructural parameters for significance for significance

Is a specific path regression coefficient significantIs a specific path regression coefficient significant Like Multiple Regression (Ho: Like Multiple Regression (Ho: ββ = 0) = 0)

Look at Look at fit fit of overall ‘of overall ‘modelmodel’ to data (goodness of fit). ’ to data (goodness of fit). Variety of models can be posited; how well do they fit? Variety of models can be posited; how well do they fit? Posited relationships use to estimate expected correlation Posited relationships use to estimate expected correlation

matrix; compare via chi-square test with observed matrix; compare via chi-square test with observed correlation matrix.correlation matrix.

““Fit” is a property of the Fit” is a property of the system system of variables (i.e., the model).of variables (i.e., the model). Model implies tests of presence and/or absence of paths.Model implies tests of presence and/or absence of paths.

Comparison with Multiple RegressionComparison with Multiple Regression Multiple RegressionMultiple Regression Causal ModelingCausal Modeling

X1

X2

X3

X4

X5

Y

Q: How well do predictorspredict (explain variances) inY? What are independenteffects when effects of other variables are controlled?

X1

X3 X4

X2 X5

Y

Q: How well do predictorsrelate with regard to ultimateprediction of Y?

Multiple RegressionMultiple Regression

X1

X2

X3

X4

X5

Y

Y = a + β1X1 + β2X2 + β3X3 + β4X4 + β5X5 + e

•Y viewed as a linear function of combination of Xi

•βi represent partial regression coefficients•Only one Dependent Variable (Y) in system

•Hierarchical Regression: Subsets of predictors added sequentially, but one DV

The Multiple Regression Fallacy:The Multiple Regression Fallacy:

What if only XWhat if only X44 and X and X55 are ‘significant’ in are ‘significant’ in

the regression?the regression? Are XAre X11, X, X22, and X, and X33 causally irrelevant? causally irrelevant?

Non-significant predictors may not have ‘direct Non-significant predictors may not have ‘direct effects’ on DVeffects’ on DV May not be May not be proximalproximal causes causes May be May be distaldistal causes via ‘indirect effects’ causes via ‘indirect effects’

Multiple Regression glosses larger ‘causal system’Multiple Regression glosses larger ‘causal system’ E.G. If causal structure (below) were “true”, what

would MR analysis show? Which predictors would be significant?

Causal modeling can be viewed as series of MR analyses.

Structural Structural EquationsEquations::

XX33 = X = X11 + X + X22

XX44 = X = X22 + X + X33

XX55 = X = X22

Y = XY = X44 + X + X55

X1

X3 X4

X2 X5

Y

PATH ANALYSIS: Theory specifies existence of paths between variables Two kinds of ‘effects’

Direct effectsX3 involves direct effects of X1 and X2

XX33 = X = X11 + X + X22

XX44 = X = X22 + X + X33

XX55 = X = X22

Y = XY = X44 + X + X55

X1

X3 X4

X2 X5

Y

PATH ANALYSIS: Theory specifies existence of paths between variables Two kinds of ‘effects’

Direct effectsX3 involves direct effects of X1 and X2

X4 involves direct effects of X2 and X3

XX33 = X = X11 + X + X22

XX44 = X = X22 + X + X33

XX55 = X = X22

Y = XY = X44 + X + X55

X1

X3 X4

X2 X5

Y

Two kinds of ‘effects’ Direct effects


X5 involves direct effect of X2 only

XX33 = X = X11 + X + X22

XX44 = X = X22 + X + X33

XX55 = X = X22

Y = XY = X44 + X + X55

X1

X3 X4

X2 X5

Y



X5 involves direct effect of X2 only

Y involves direct effects of X4 and X5

XX33 = X = X11 + X + X22

XX44 = X = X22 + X + X33

XX55 = X = X22

Y = XY = X44 + X + X55

X1

X3 X4

X2 X5

Y



X5 involves direct effect of X2 onlyY involves direct effects of X4 and X5

Indirect effects: Effect of a variable through another variable

X1hypothesized to influence X4 via indirect effect through X3

XX33 = X = X11 + X + X22

XX44 = X = X22 + X + X33

XX55 = X = X22

Y = XY = X44 + X + X55

X1

X3 X4

X2 X5

Y



X5 involves direct effect of X2 onlyY involves direct effects of X4 and X5

Indirect effects: Effect of a variable through another variableX1hypothesized to influence X4 via indirect effect through X3

X1 posited to influence Y through indirect effects through X3 and X4

XX33 = X = X11 + X + X22

XX44 = X = X22 + X + X33

XX55 = X = X22

Y = XY = X44 + X + X55

X1

X3 X4

X2 X5

Y

II. History of SEM and Causal ModelingII. History of SEM and Causal Modeling 1920’s - 1930’s: The Early Years1920’s - 1930’s: The Early Years

Sewall Wright Sewall Wright (1934) - Genetic inheritance combined with path (1934) - Genetic inheritance combined with path schemes for differing mating systems to yield calculation of schemes for differing mating systems to yield calculation of genetic correlations among relatives of stated degrees of genetic correlations among relatives of stated degrees of relationship.relationship.

1960’s - 1970’s: The Rise of Computers1960’s - 1970’s: The Rise of Computers

Hubert Blalock Hubert Blalock (1964) - (1964) - Causal inferences in Non-experimental Causal inferences in Non-experimental ResearchResearch Early applications to sociological phenomena; Early applications to sociological phenomena; development of development of directdirect vs. vs. indirectindirect effects. effects. Single-Indicator models (Multiple Regression)Single-Indicator models (Multiple Regression)

1970’s - Present: The “Latent Variable” Era1970’s - Present: The “Latent Variable” Era Karl Karl JoreskogJoreskog (1970; 1973) - Path analysis with latent variables; (1970; 1973) - Path analysis with latent variables; incorporation of incorporation of factor analysisfactor analysis (measurement model and (measurement model and path path analysis analysis (Structural relationships).(Structural relationships). Multiple-Indicator models (beyond Multiple Regression)Multiple-Indicator models (beyond Multiple Regression)

Rise of Computer Technology Crucial to Rise of Sophisticated AnalysisRise of Computer Technology Crucial to Rise of Sophisticated Analysis

Much of early path analysis done by handMuch of early path analysis done by hand Analyses based upon working with correlation Analyses based upon working with correlation

matricesmatrices Correlation matrices still central to approachCorrelation matrices still central to approach Path Analysis Path Analysis NOTNOT a simple matter of seeing if a simple matter of seeing if

variables are correlated!!!variables are correlated!!! Correlation Correlation ≠ Causation≠ Causation Some early research involved drawing path diagrams Some early research involved drawing path diagrams

and putting in correlation coefficients as evidence of and putting in correlation coefficients as evidence of causationcausation

III. The Logic of Path Analysis—Structural Equation Modeling via Multiple RegressionIII. The Logic of Path Analysis—Structural Equation Modeling via Multiple Regression

Path coefficients represent standardized partial regression Path coefficients represent standardized partial regression coefficients (coefficients (ββ))..

Review of basic regression and multiple regression:Review of basic regression and multiple regression: CovarianceCovariance (x , y) = (x , y) = (y - (y - y) (x - y) (x - x)x) reflects deviation of reflects deviation of

x and y around respective meansx and y around respective means Larger values denote more ‘shared variation’ between x and yLarger values denote more ‘shared variation’ between x and y

Linear Regression Coefficient Pearson’s Correlation Coefficient

byx = (y - y) (x - x)

(x - x)2

r = (y - y) (x - x) (y - y)2 (x - x)2

Asymetric measure (use x to predict y; note byx)

Symmetric measure (neither variable regarded as IV or DV)

Equal (intuitively) to Covariance (x,y) ‘standardized’ by Var(x)

Equal (intuitively) to Covariance (x,y) ‘standardized’ by SD(x) * SD(y)

Extension to Multiple RegressionExtension to Multiple Regression

If x and y are standardized, r = b in bivariate caseIf x and y are standardized, r = b in bivariate case In the case of multiple predictors (XIn the case of multiple predictors (X11 and X and X22), b ), b

becomes a partial regression coefficientbecomes a partial regression coefficient Partialing out the correlation with other predictorsPartialing out the correlation with other predictors ββ y1.2y1.2 = = rry1y1 – (r – (ry2y2 * r * r1212))

1 - 1 - rr121222

ββ y2.1y2.1 = = rry2 y2 – (r– (ry1y1 * r * r1212))

1 - 1 - rr121222

The PointThe Point: Partial Regression Coefficients can : Partial Regression Coefficients can be viewed as functions of operations on a be viewed as functions of operations on a correlation matrixcorrelation matrix

The Logic of Model Testing in SEM The Logic of Model Testing in SEM

Begin with an observed correlation matrix for XBegin with an observed correlation matrix for X11, X, X22, and X, and X33

XX11 XX22 XX33

X X11 1.01.0 rr1212 rr1313

X X22 1.01.0 rr2323

X X33 1.01.0

Hypothesize a structural model to testHypothesize a structural model to testX2

X1

X3

This model can be represented by the following equations:ŕ12 = p21

ŕ13 = p31 +p32p21 (direct effect of X1 on X3 plus indirect effect via X2)

ŕ23 = p32 + p31p21 (direct effect of X2 on X3 plus effect of X1 on X3

and X2)

•ŕij represent ‘reconstructed’ or ‘estimated’ correlations based upon the theoretical model

Path coefficients can be estimated using Multiple Path coefficients can be estimated using Multiple Regression methods (Standardized Partial Regression methods (Standardized Partial Coefficients) based upon a given model and can be Coefficients) based upon a given model and can be used to “reconstruct” the correlation matrix.used to “reconstruct” the correlation matrix.

The “estimated” correlations can be compared The “estimated” correlations can be compared with the observed” correlations and with the observed” correlations and chi-squarechi-square will will show whether it fits (show whether it fits (nonnon-significant chi-square -significant chi-square denotes good fit.)denotes good fit.) 2 test involves observed vs. expected 2 test involves observed vs. expected

(“reconstructed”) correlations(“reconstructed”) correlations

2 Goodness of Fit Test2 Goodness of Fit Test

Observed Correlations in DataObserved Correlations in Data XX11 XX22 XX33

X X11 1.01.0 rr12(o)12(o) rr13(o)13(o)

X X22 1.01.0 rr23(o)23(o)

X X33 1.01.0

Reconstructed Correlations based upon Path modelReconstructed Correlations based upon Path model XX11 XX22 XX33

X X11 1.01.0 ŕŕ12(e)12(e) ŕŕ13(e)13(e)

X X22 1.01.0 ŕŕ23(e)23(e) XX33 1.01.0

2 = Σ(rrijij(o) – (o) – ŕŕijij(e))(e))22//ŕŕijij(e))(e))

More similar observed and expected correlations, smaller Chi-square

Note that an alternate model will yield a different set of expected correlations which may fit better or worse.e.g.

X2 r12 = 0 X1 implies: r13 = p31

X3 r23 = p32

Note that an alternate model will yield a different set of expected correlations which may fit better or worse.e.g.

X2 r12 = 0 X1 implies: r13 = p31

X3 r23 = p32

Much causal modeling involves comparison of Much causal modeling involves comparison of alternate model fit. alternate model fit. e.g., What happens if you drop path from Xe.g., What happens if you drop path from X1 1 - X- X22? ?

Run model with and without path; compare chi-Run model with and without path; compare chi-square from each model; does chi-square value square from each model; does chi-square value increase (i.e., worse fit) when path is dropped?increase (i.e., worse fit) when path is dropped?

Model Comparison via Incremental Fit TestModel Comparison via Incremental Fit Test Each model has a Each model has a 2 value based upon a 2 value based upon a

certain degree of freedomcertain degree of freedom If models are nested (ie., identical but MIf models are nested (ie., identical but M22

deletes one parameter found in Mdeletes one parameter found in M11), ),

significance of ‘increment’ or ‘decrement’ significance of ‘increment’ or ‘decrement’ in fit = in fit =

2211 - - 2222 with df = df with df = df11 – df – df22

Simple model comparison exampleSimple model comparison example

MM11: : XX22 rr1212 = p = p2121

XX11 implies:implies: rr1313 = p = p3131

XX33 rr2323 = p = p3232

MM22: : XX22 rr1212 = 0 = 0

XX11 implies:implies: rr1313 = p = p3131

XX33 rr2323 = p = p3232

What does dropping path between X1 and X2 do to model fit?

2222 - - 221 1 = 76.5 – 51.2 = 25.3 on 1df, p < .001= 76.5 – 51.2 = 25.3 on 1df, p < .001

ConcludeConclude: path from X1 – X2 : path from X1 – X2 ≠ 0; dropping that path ≠ 0; dropping that path significantly decreases model fitsignificantly decreases model fit

More Complex Example of Reconstructing Correlation Matrix (ŕij) in Terms of Indirect and Direct Path Coefficients

More Complex Example of Reconstructing Correlation Matrix (ŕij) in Terms of Indirect and Direct Path Coefficients

ŕ 13 = p31

ŕ 23 = p32

ŕ 14 = p43p31

ŕ 24 = p43p32 + p42

ŕ 52 = p52

ŕ y1 = py4p43p31

ŕ y2 = py4p42 + py4p43p32 + py5p52X1

X3 X4

X2 X5

Y

Rule of Thumb: Reconstructed correlation between two variables is equal to the sum of all possible direct and indirect paths

Obtaining Path Coefficients via Multiple Regression (Observed Variable Models)

Obtaining Path Coefficients via Multiple Regression (Observed Variable Models)

Use SPSS, for example to obtain Standardized Use SPSS, for example to obtain Standardized Betas for each structural equationBetas for each structural equation

XX33 = = ββ3131xx11 + + ββ3232xx22 XX44 = = ββ4141xx11 + + ββ4242xx2 2 + + ββ4343xx3 3 XX55 = = ββ5252xx22 Y = Y = ββy5y5xx55 + + ββy4y4xx4 4 + + ββy3y3xx3 3 ++ ββy2y2xx2 2 ++ ββy1y1xx11

Each regression model includes all predictors Each regression model includes all predictors before it that have arrows leading to it before it that have arrows leading to it Note Note Missing pathsMissing paths where direct effects are not where direct effects are not

includedincluded These may represent These may represent alternative modelsalternative models to test to test

Compute Compute 2 for hypothesized model to assess fit2 for hypothesized model to assess fit

“Theory” Trimming: Adding/Deleting Paths“Theory” Trimming: Adding/Deleting Paths

Step DownStep Down: Run ‘Full’ model, identify non-significant : Run ‘Full’ model, identify non-significant paths; re-run model dropping paths one-by-one; assess paths; re-run model dropping paths one-by-one; assess changes in fit. Empirical approach. Model should be changes in fit. Empirical approach. Model should be theory based.theory based.

Step UpStep Up: Run ‘reduced’ (hypothesized) model ; re-run : Run ‘reduced’ (hypothesized) model ; re-run with missing paths to assess if addition of those paths with missing paths to assess if addition of those paths improves model fit improves model fit

Look at:Look at: Significance of pathsSignificance of paths Overall improvement in model fitOverall improvement in model fit

Should be ‘theory driven’Should be ‘theory driven’

IV. Historical Trends in the LISREL ApproachIV. Historical Trends in the LISREL Approach

1970’s - The Joreskog - Sorbom Approach 1970’s - The Joreskog - Sorbom Approach ( ( http://www.ssicentral.com/lisrel/define.htm)http://www.ssicentral.com/lisrel/define.htm)Parameter matrices Parameter matrices - had to specify which parameters to estimate in each - had to specify which parameters to estimate in each matrix (set those not estimated to zero).matrix (set those not estimated to zero).Greek Notation Greek Notation

The structural equation model: The structural equation model: ηη = = βηβη + + ΓζΓζ + + ξξ

The measurement model for The measurement model for yy: : y = y = ΛΛyy ηη + + εε

The measurement model for The measurement model for xx: : x = x = ΛΛxx ξξ + + δδ

NotNot user-friendly user-friendlySpecify matrices for x’s, y’s, x-y’s, factor loadings,and errorsSpecify matrices for x’s, y’s, x-y’s, factor loadings,and errors

Huh?!?!?!?!?

1980’s - Peter Bentler’s EQS (Structural Equations Approach)

Alternative to cumbersome matrix approach Specification of equations identifying predictors of

each dependent variable in modelY=X1 + X2

1980’s - Peter Bentler’s EQS (Structural Equations Approach)

Alternative to cumbersome matrix approach Specification of equations identifying predictors of

each dependent variable in modelY=X1 + X2

Late 1980’s - 1990’s - “Merging of Late 1980’s - 1990’s - “Merging of Approaches”Approaches”

LISREL and AMOS allow alternative specification LISREL and AMOS allow alternative specification approaches approaches EQS style input in equation formEQS style input in equation form Graphic Interface (draw path diagrams)Graphic Interface (draw path diagrams)

LISREL and SEM: Combine Structural and Measurement ModelsLISREL and SEM: Combine Structural and Measurement Models Structural ModelsStructural Models:: The Causal relationsThe Causal relations ββ’s linking ’s linking

independent and independent and dependentdependent

Measurement ModelsMeasurement Models Factor Structure Factor Structure

involved in latent involved in latent constructsconstructs

SESDEPRESS

ED

INC

OCC

SESββl1

l3

Observed vs. Latent Variable ModelsObserved vs. Latent Variable Models Observed Variable:Observed Variable:

Single indicator or multi-item ‘scale’ scoreSingle indicator or multi-item ‘scale’ score Eg. SES :Eg. SES :

• Measured by ‘EDUCATION’Measured by ‘EDUCATION’• Measured by composite index of EDUCATION, Measured by composite index of EDUCATION,

INCOME, and OCCUPATIONAL STATUSINCOME, and OCCUPATIONAL STATUS Typical Multiple Regression ApproachTypical Multiple Regression Approach Assumes all component variables carry equal weightAssumes all component variables carry equal weight

Latent Variable:Latent Variable: Underlying ‘construct’ with multiple indicatorsUnderlying ‘construct’ with multiple indicators

Eg., SES as a constructEg., SES as a construct Allows modeling factor structureAllows modeling factor structure Components get different weightsComponents get different weights

SES

ED INC OCC

For Those of Us Who at least know Multiple Regression…For Those of Us Who at least know Multiple Regression… ββ’s are standardized partial regression weights’s are standardized partial regression weights Where do Where do ββ’s come from?’s come from?

Matrix algebra solution rather than a simple formula for Matrix algebra solution rather than a simple formula for each variableeach variable ββ = ( = (XXTTXX))-1-1 ( (XXTTYY))

What are these component parts?What are these component parts? ((XXTTXX))-1 Inverse of the sum of squares of X variables-1 Inverse of the sum of squares of X variables

((XXTTYY) ) Basically, the sum of squares of y variablesBasically, the sum of squares of y variables

Basically, B’s are derived from the Basically, B’s are derived from the variance/covariance matricesvariance/covariance matrices

In MR, we use OLS estimates; In SEM, use Generalized In MR, we use OLS estimates; In SEM, use Generalized Least Squares or Maximum Likelihood estimatesLeast Squares or Maximum Likelihood estimates

LISREL and SEM Special ApplicationsLISREL and SEM Special Applications Confirmatory Factor Analysis (CFA):Confirmatory Factor Analysis (CFA):

‘‘Testing’ hypotheses regarding fit of factor Testing’ hypotheses regarding fit of factor structuresstructures

Test alternative factor modelsTest alternative factor models Multiple Group Models: Moderating EffectsMultiple Group Models: Moderating Effects

Testing to see if parameters differ between Testing to see if parameters differ between groupsgroups

Do magnitude of structural paths differ between Do magnitude of structural paths differ between groupsgroups

Fit IndicesFit Indices Overall Overall 2 a poor measure of fit2 a poor measure of fit

With large n, With large n, 2 will generally be significant2 will generally be significant Alternative fit indices developed to avoid Alternative fit indices developed to avoid

problems with problems with 2 2 Involve improvement in fit compared to some Involve improvement in fit compared to some

reference modelreference model Normed Fit Index (NFI; Bentler and Bonnet)Normed Fit Index (NFI; Bentler and Bonnet)

• Measures improvement in fit over independence model Measures improvement in fit over independence model (no relationships between model variables)(no relationships between model variables)

• Ranges from 0 (no improvement) to 1 (100% Ranges from 0 (no improvement) to 1 (100% improvement)improvement)

Others: Non-Normed fit index (NNFI), Goodness of Others: Non-Normed fit index (NNFI), Goodness of Fit Index (GFI)Fit Index (GFI)

Model Modification Indices Model Modification Indices

LISREL/AMOS provide model fit indicesLISREL/AMOS provide model fit indices Also provide ‘modification indices’Also provide ‘modification indices’

Suggest paths or error co-variances that may Suggest paths or error co-variances that may improve model fitimprove model fit Show amount of change in Show amount of change in 2 that would result2 that would result

Atheoretical and sample-specific, but may Atheoretical and sample-specific, but may make sensemake sense

IV. Substantive SEM ExamplesIV. Substantive SEM Examples

Confirmatory Factor Analysis:Exploring the Dimensionality of Spirituality (Neff, in Progress)

Confirmatory Factor Analysis:Exploring the Dimensionality of Spirituality (Neff, in Progress)

Conceptual Models of Spiritual DimensionsConceptual Models of Spiritual Dimensions ‘‘Religiosity’ vs. ‘Spirituality’ a fundamental Religiosity’ vs. ‘Spirituality’ a fundamental

distinctiondistinction ‘‘ReligiosityReligiosity’: Organized religious involvement ’: Organized religious involvement

or participationor participation ‘‘SpiritualitySpirituality’: Relationship between individual ’: Relationship between individual

and transcendent force (higher power)and transcendent force (higher power) Provides meaning and core values to organize lifeProvides meaning and core values to organize life May involve individual’s relationship to higher May involve individual’s relationship to higher

power, self, others, and life in general power, self, others, and life in general

Fetzer/NIA Multidimensional Measure of Spirituality Set of measures developed by expert panel to tap key Set of measures developed by expert panel to tap key

dimensionsdimensions Organizational ReligiosityOrganizational Religiosity Private Religious PracticesPrivate Religious Practices Self-Rated ReligiositySelf-Rated Religiosity Daily Spiritual ExperienceDaily Spiritual Experience Spiritual Values/BeliefsSpiritual Values/Beliefs ForgivenessForgiveness Positive and Negative Religious CopingPositive and Negative Religious Coping Religious SupportReligious Support

Brief and Extended versions of measures were developedBrief and Extended versions of measures were developed 24 items were included in National GSS Survey in 199824 items were included in National GSS Survey in 1998

Fundamental Questions Regarding Fetzer/NIA Measures

Measures were developed in ad hoc, Measures were developed in ad hoc, atheoretical fashionatheoretical fashion Are ‘dimensions’ unique or overlappingAre ‘dimensions’ unique or overlapping

Parsimony: 9 factors needed?Parsimony: 9 factors needed? Will simpler models fit the dataWill simpler models fit the data

Are dimensions similar (invariant) across Are dimensions similar (invariant) across ethnic groups?ethnic groups?

Methodology

Secondary analysis of 1998 GSS dataSecondary analysis of 1998 GSS data National probability sample of 2,832National probability sample of 2,832 Fetzer data available on Fetzer data available on ~1,400 adults~1,400 adults

75.8% non-Hispanic white (n = 1,012)75.8% non-Hispanic white (n = 1,012) 14.1% African-American (n = 188)14.1% African-American (n = 188) 6.3% Hispanic/Latino (n = 84)6.3% Hispanic/Latino (n = 84) 3.8% ‘Other’ (n = 51)3.8% ‘Other’ (n = 51)

Analysis Strategy Confirmatory Factor Analyses (CFA) using Confirmatory Factor Analyses (CFA) using

LISREL 8.50LISREL 8.50 Focus upon non-Hispanic Whites and African-Focus upon non-Hispanic Whites and African-

AmericansAmericans Compare the ‘fit’ of alternative models using fit Compare the ‘fit’ of alternative models using fit

statisticsstatistics Bentler and Bonnet normed fit index (NFI) and non-Bentler and Bonnet normed fit index (NFI) and non-

normed fit index (NNFI) reflects improvement of fit of normed fit index (NNFI) reflects improvement of fit of model compared to null reference modelmodel compared to null reference model

Values of .90 or better denote ‘acceptable’ modelsValues of .90 or better denote ‘acceptable’ models Identify best fitting parsimonious modelsIdentify best fitting parsimonious models Test for invariance of factor loadings and factor Test for invariance of factor loadings and factor

correlations between non-Hispanic Whites and correlations between non-Hispanic Whites and African Americans African Americans

A Priori Alternative ModelsA Priori Alternative Models

MM11: Single ‘Spirituality’ Factor Reference Model: Single ‘Spirituality’ Factor Reference Model All 24 Fetzer items load on a single latent variable All 24 Fetzer items load on a single latent variable

reflecting ‘spirituality’reflecting ‘spirituality’ MM22: 3 Factor Chatters Model (Religious : 3 Factor Chatters Model (Religious

involvement vs. Spirituality)involvement vs. Spirituality) Split out Organizational, Non-organizational Split out Organizational, Non-organizational

religiosity, and remaining items as ‘spirituality’ factorreligiosity, and remaining items as ‘spirituality’ factor MM33: Multi-factor Fetzer Model: Multi-factor Fetzer Model

Perceived Religiosity, Public/Private Religiosity, Perceived Religiosity, Public/Private Religiosity, Positive coping, Negative coping, Religious values, Positive coping, Negative coping, Religious values, Forgiveness, Daily Spiritual InvolvementForgiveness, Daily Spiritual Involvement

NNFIIa: Constrained .85Ib: Loadings Free .85

M2: Chatters Model

M1: Single Factor Spirituality Model

Spirituality

Private Spirituality

: NNFIIIa: Constrained .86IIb: Loadings Free .85IIc: Loadings/Corr .85

M3: Modified Fetzer Model[Six Factor dropping Negative Coping]

: NNFIIIIa: Constrained .89IIIb: Loadings Free .90IIIc: Loadings/Corr .89

Summary of Spirituality CFA Models

* *

Public

Multiple Group Comparisons: Testing the Invariance of Stress-Distress Relationships for Whites and African Americans (Neff, 1985)

Multiple Group Comparisons: Testing the Invariance of Stress-Distress Relationships for Whites and African Americans (Neff, 1985)

Research QuestionResearch Question

Given history of oppression, slavery, and Given history of oppression, slavery, and discrimination, would African-Americans discrimination, would African-Americans manifest higher vulnerability to life stress manifest higher vulnerability to life stress than Whites?than Whites? Differential vulnerability operationalized in Differential vulnerability operationalized in

terms of differences in magnitude of terms of differences in magnitude of relationship between life stress and relationship between life stress and psychological distress constructs.psychological distress constructs.

SEM Question: Does race SEM Question: Does race moderatemoderate the effect the effect of life stress upon distress?of life stress upon distress?

MethodologyMethodology

Data on 1000 adult Florida residentsData on 1000 adult Florida residents 829 usable respondents:829 usable respondents:

658 Whites; 171 African Americans658 Whites; 171 African Americans Measures:Measures:

SESSES: Latent variable measured by Education and : Latent variable measured by Education and IncomeIncome

Stressful EventsStressful Events: Latent variable measured by 1) : Latent variable measured by 1) number of undesirable events and 2) number of events number of undesirable events and 2) number of events rated as producing changerated as producing change

DistressDistress: Latent variable measured by 3 subscales : Latent variable measured by 3 subscales tapping 1) nervous upset, 2) depressive affect, and 3) tapping 1) nervous upset, 2) depressive affect, and 3) psychopathologypsychopathology

NFI = .95

M5: Stress path allowed to varyM6: Stress path forced to be equal for B &W

2 difference (M5 – M6) = .75 on 1 df; ns.

Strategy:

Fit model for Whites and Blacks. Test invariance of stress-distress path (β21).

Maximum Likelihood Estimates of β21: Whites (.26) Blacks (.38)

Theory Articulation: Health Belief Model and HIV Risk Behaviors

The Health Belief Model (HBM): (Rosenstock, Becker)

Theory Articulation: Health Belief Model and HIV Risk Behaviors

The Health Belief Model (HBM): (Rosenstock, Becker) Four components in original model:Four components in original model:

Perceived susceptibility Perceived susceptibility (risk of contracting)(risk of contracting) Perceived seriousness Perceived seriousness (severity)(severity) Benefits of preventive activityBenefits of preventive activity BarriersBarriers

Susceptibility and seriousness barriers provide Susceptibility and seriousness barriers provide motivationmotivation; Benefits/barriers provide ; Benefits/barriers provide directiondirection

The Original HBM (Rosenstock and Becker, 1974)The Original HBM (Rosenstock and Becker, 1974)

Slide 31 of 59

What does the HBM imply Theoretically?What does the HBM imply Theoretically? HBM ‘theory’ not clear!HBM ‘theory’ not clear! Should susceptibility or seriousness relate directly Should susceptibility or seriousness relate directly

to behavior or should they be mediated?to behavior or should they be mediated? Mediation via Benefits and Barriers?Mediation via Benefits and Barriers?

How do Benefits and Barriers relate to Partners?How do Benefits and Barriers relate to Partners? Direct effects or does Barriers mediate effect of Direct effects or does Barriers mediate effect of

Benefits? Benefits? Are effects of education and income direct or Are effects of education and income direct or

indirect?indirect? Does theory drive the modeling procedure or does Does theory drive the modeling procedure or does

the procedure drive theory?the procedure drive theory?

The HBM and Analytic StrategiesWalter et al (1993) “Factors associated with AIDS-related

behavioral intentions among high school students...”Health Education Quarterly 20: 409-420.

The HBM and Analytic StrategiesWalter et al (1993) “Factors associated with AIDS-related

behavioral intentions among high school students...”Health Education Quarterly 20: 409-420.

Simple application of Multiple Regression

What “model” is implied by simple MR analysis?What “model” is implied by simple MR analysis?

SusceptibilitySusceptibility

SeriousnessSeriousness

BenefitsBenefits

BarriersBarriers

AIDSBehavioralIntentions

•Ignores possible relationships between HBM variables! What does HBM specify?

•If MR shows that Benefits and Barriers are only significant predictors, are Susceptibility and Seriousness irrelevant? Or Mediated?

Chen and Land (1990) “SES and the Health Belief Model: LISREL analysis of uni-dimensional vs. multi-dimensional formulations”. Journal of Social Behavior and Personality 5:262-284.

Chen and Land (1990) “SES and the Health Belief Model: LISREL analysis of uni-dimensional vs. multi-dimensional formulations”. Journal of Social Behavior and Personality 5:262-284.

Examined preventive dental visits in a national Examined preventive dental visits in a national samplesample

Elaborate General Model: Begin to allow for Elaborate General Model: Begin to allow for direct and indirect effects of HBM model direct and indirect effects of HBM model variablesvariables

VI. My Study: HBM and HIV Risk Behaviors (Bulletin of HIV/AIDS in Social Work, Forthcoming)

VI. My Study: HBM and HIV Risk Behaviors (Bulletin of HIV/AIDS in Social Work, Forthcoming)

Study of alcohol use and HIV risk behaviors Study of alcohol use and HIV risk behaviors in community sample of 1,392 adults (439 in community sample of 1,392 adults (439 Anglos, 318 Blacks, and 601 Mexican Anglos, 318 Blacks, and 601 Mexican Americans)Americans)

Key HypothesesKey Hypotheses:: How well do HBM variables predict HIV How well do HBM variables predict HIV

risk behaviors (number of partners in past year)risk behaviors (number of partners in past year) Does model fit as well among minorities as Does model fit as well among minorities as

among Anglos? among Anglos?

INCOME

SUSCEPTIBILITY

BENEFITS

EDUCATION

SERIOUSNESS

BARRIERS

PARTNERS

M1

M3

M4

M7

M9

M8

M5

M6

M2

Figure 1. Graphical Presentation of HBM Model Relationships

Note: M1-M9 in Figure denote alternative models being tested. See text for explanation.

Core Model: Mediated effects of HBM variables (Susceptibility and Seriousness)

INCOME

SUSCEPTIBILITY

BENEFITS

EDUCATION

SERIOUSNESS

BARRIERS

PARTNERS

M1

M3

M4

M7

M9

M8

M5

M6

M2

What alternative Models Represent?


M1 – M6: Direct paths between demographics and model variables

INCOME

SUSCEPTIBILITY

BENEFITS

EDUCATION

SERIOUSNESS

BARRIERS

PARTNERS

M1

M3

M4

M7

M9

M8

M5

M6

M2

What alternative models represent?


M7: Direct effect of Susceptibility upon Partners

INCOME

SUSCEPTIBILITY

BENEFITS

EDUCATION

SERIOUSNESS

BARRIERS

PARTNERS

M1

M3

M4

M7

M9

M8

M5

M6

M2

What Alternative Models Represent?


M8: Direct Effect of Seriousness on Partners

INCOME

SUSCEPTIBILITY

BENEFITS

EDUCATION

SERIOUSNESS

BARRIERS

PARTNERS

M1

M3

M4

M7

M9

M8

M5

M6

M2

What Alternative Models Represent?


M9: Indirect Effect of Benefits on Partners via Barriers

Approach:Approach:

Opted to utilize single indicator approachOpted to utilize single indicator approach Constructed composite scales of HBM variablesConstructed composite scales of HBM variables

Began by fitting ‘full’ model including all possible Began by fitting ‘full’ model including all possible direct effects (Mdirect effects (M11 – M – M99))

Successively drop out paths (MSuccessively drop out paths (M11 – M – M99) and test ) and test incremental change in model fitincremental change in model fit

2 tests2 tests Changes in NFI (Normed Fit Index)Changes in NFI (Normed Fit Index)

Separate analyses for Males and FemalesSeparate analyses for Males and Females Finally, compare parameter estimates for Whites, Finally, compare parameter estimates for Whites,

African Americans, and Mexican Americans African Americans, and Mexican Americans LISREL also allows formal test of equivalence LISREL also allows formal test of equivalence

parameters between groupsparameters between groups

Table 3. Goodness of fit statistics for alternative models for MALES Males Model: 2 df P NFI 2 diff P M0 Full Unconstrained 258.88 16 .00 .16 MM0 Added Error Covariance 30.69 15 .00 .90 228.19 <.001 M 1 Income-Risk=0 31.78 18 .01 .90 1.09 ns M2 Education-Risk=0 30.81 15 .01 .90 .12 ns M3 Income-Benefits=0 32.91 15 .01 .89 2.22 ns M4 Income-Barriers=0 32.90 18 .01 .89 2.21 ns M5 Education-Benefits=0 32.38 18 .01 .90 1.69 ns M6 Education-Barriers=0 33.82 15 .00 .89 3.13 ns M7 Susceptibility-Risk=0 60.57 15 .00 .80 29.88 <.001 M8 Seriousness-Risk=0 33.12 15 .00 .89 2.43 ns M9 Benefit-Barrier>0 29.96 9 .00 .90 .73 ns

Simplified Bottom Line?Simplified Bottom Line?

For Males, Mediated Model generally For Males, Mediated Model generally works…works… No direct effects of demographic variables No direct effects of demographic variables

upon number of partners are neededupon number of partners are needed No direct effect of Seriousness upon number of No direct effect of Seriousness upon number of

partners is neededpartners is needed However, dropping direct effect of However, dropping direct effect of

Susceptibility significantly impairs model fit.Susceptibility significantly impairs model fit. Overall model fit good (NFI ~ .90)Overall model fit good (NFI ~ .90)

SumSum SEM has evolved into an art form over timeSEM has evolved into an art form over time

Rooted in Multiple Regression logicRooted in Multiple Regression logic Forces conceptual clarity—beyond laundry list of Forces conceptual clarity—beyond laundry list of

‘predictors’‘predictors’ Potential to promote theory articulationPotential to promote theory articulation

Computer revolution has made incomprehensible Computer revolution has made incomprehensible ‘Greek’ methodology accessible‘Greek’ methodology accessible Equation and graphic model specificationEquation and graphic model specification

New advances: Latent Variable Hierarchical Linear New advances: Latent Variable Hierarchical Linear Modeling in LISREL 8.5Modeling in LISREL 8.5

Potential to take Social Work Research to a higher Potential to take Social Work Research to a higher level of theoretical and analytic sophistication…level of theoretical and analytic sophistication…

Resources Resources LISREL (Scientific Software, Inc): Online Information

regarding SEM and free download http://www.ssicentral.com/lisrel/define.htmhttp://www.ssicentral.com/lisrel/define.htm

AMOS homepage (Information and free student version available) AMOS homepage (Information and free student version available) http://www.smallwaters.com/amos/http://www.smallwaters.com/amos/

UT Austin online tutorials:UT Austin online tutorials: http://www.utexas.edu/cc/stat/tutorials/amos/http://www.utexas.edu/cc/stat/tutorials/amos/ http://www.utexas.edu/cc/stat/tutorials/lisrel/index.htmlhttp://www.utexas.edu/cc/stat/tutorials/lisrel/index.html

Online Reference Listing: Online Reference Listing: http://www.upa.pdx.edu/IOA/newsom/semrefs.htmhttp://www.upa.pdx.edu/IOA/newsom/semrefs.htm

U Michigan Summer Session Syllabus:U Michigan Summer Session Syllabus: http://www.icpsr.umich.edu/TRAINING/Biblio97/baer.html http://www.icpsr.umich.edu/TRAINING/Biblio97/baer.html

David Kenny’s SEM Course Website:David Kenny’s SEM Course Website: http://users.erols.com/dakenny/causalm.htmhttp://users.erols.com/dakenny/causalm.htm

Ken Bollen’s SEM Course Syllabus:Ken Bollen’s SEM Course Syllabus: http://www.unc.edu/courses/soci317/syllabus.htmlhttp://www.unc.edu/courses/soci317/syllabus.html Page on Fit measures: http://users.erols.com/dakenny/fit.htmPage on Fit measures: http://users.erols.com/dakenny/fit.htm

Online Annotated Bibliography on SEMOnline Annotated Bibliography on SEM http://moon.ouhsc.edu/dthompso/sem/sembib.htmhttp://moon.ouhsc.edu/dthompso/sem/sembib.htm

Documents

‘STRUCTURAL EQUATION MODELING’: Introduction and Application to HIV Risk Behaviors…and other stuff… James Alan Neff, Ph.D., M.P.H. Professor and Director