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    Structural Equation Modeling (Hybrid Models)

    What is it? General method for modeling = () i.e. for modeling covariance

    structure

    Intuitively can be thought of as the combination of confirmatory factor

    analysis (CFA) with path analysis.

    CFA and path analysis are each special cases of structural equation

    modeling

    What is it good for?

    To account for measurement error in modeling relationships between

    variables measured with error (i.e. latent variables)

    As it uses the graphical notation of path analysis, it provides a methodfor describing the assumed causal relationships between observed vari-

    ables, between observed and latent, and between latent and latent.

    To take advantage of multicollinearity in a set of predictors rather than

    seeing it as a hinderance.

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    SEM - becoming ubiquitousSTRUCTURAL EQUATION MODELING, 10(1), 3546

    Copyright 2003, Lawrence Erlbaum Associates, Inc.

    The Growth of Structural EquationModeling: 19942001

    Scott L. HershbergerDepartment of Psychology

    California State University, Long Beach

    This study examines the growth and development of structural equation modeling

    (SEM) from the years 1994 to 2001. The synchronous development and growth of

    the Structural Equation Modeling journal was also examined. Abstracts located on

    PsycINFO were used as the primary source of data. The major results of this investi-

    gation were clear: (a) The number of journal articles concerned with SEM increased;(b) the number of journals publishing these articles increased; (c) SEM acquired he-

    gemony among multivariate techniques; and (d) Structural Equation Modeling be-

    came the primary source of publication for technical developments in SEM.

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    SEM - becoming ubiquitous

    FIGURE 1 Distribution of number of articles and journals by year.

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    SEM - becoming ubiquitous

    Over 1100 Selected Publications that Cite Amos for Structural Equation Mod-eling March, 2004, http://www.amosdevelopment.com/

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    Effect of ignoring measurement errorLet x1 = f1 + 1 and x2 = f2 + 2

    where V ar(1) = 1, V ar(2) = 2, V ar(f1) = 1, V ar(f2) = 2, and Corr(f1, f2) =

    . That is, the true correlation between the variables of interest f1 and f2 is .

    Now, as we did not observe f1 and f2 directly we will instead have to deal with the

    observed x1 and x2.

    What is the Corr(x1, x2)? Is it close to the Corr(f1, f2) = ?

    Corr(x1, x2) =Cov(x1, x2)

    (V arx1) (V arx2)

    = Cov(f1, f2)

    1 + 1

    2 + 2

    =Corr(f1, f2)

    1

    2

    1 + 1

    2 + 2

    = 1

    1 + 1

    2

    2 + 2

    = reliability of x1 reliability of x2

    Correlation between x1 and x2 will be smaller than the true correlation between thevariables we are interested in f1 and f2

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    SEM takes the measurement error into account

    Rather than taking scales with less than perfect reliability and using them as ifthey are perfect measurements of the latent variable, SEM models incorporates

    the measurement error and thus adjusts the correlations and path coefficients

    appropriately. Assuming the model specification is correct (as usual).

    Two nice papers discussing this:

    Charles EP (2005) The Correction for Attenuation Due to Measurement

    Error: Clarifying Concepts and Creating Confidence Sets, Psychological

    Methods 10(2) 206-226.

    DeShon, R. P. (1998). A cautionary note on measurement error corrections

    in structural equation models. Psychological M ethods, 3(4), 412-423.

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    Examples of correction for attenuation

    observed correlation of .3

    .2 .4 .6 .8 1.0

    .2 - - .87 .75 .67

    .4 - .75 .61 .53 .47

    .6 .87 .61 .50 .43 .39

    .8 .75 .53 .43 .38 .33

    1.0 .67 .47 .39 .33 .30

    observed correlation of .5

    .2 .4 .6 .8 1.0

    .2 - - - - -

    .4 - - - .88 .79

    .6 - - .83 .72 .65

    .8 - .88 .72 .63 .56

    1.0 - .79 .65 .56 .50

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    A useful example from CFA

    f1

    x3 e1.93

    x5 e2.62

    x10 e3

    .85

    f2

    x1 e4

    .92

    x7 e5.58

    x8 e6

    .58

    x9 e7

    .57

    .54

    Chi-square = 9.8 d.f. = 13, p-value = .704, Corr(f1, f2) = .54

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    A useful example from CFA

    A natural/practical thing to do with these 7 variables is to create two scales. One

    created from X3, X5, X10, that is Scale1 = X3 + X5 + X10, and one created

    from X1, X7, X8, and X9, that is Scale2 = X1 + X7 + X8 + X9

    Then we can calculate the observed correlation between Scale1 and Scale2 is .45.Obviously this is smaller than the correlation found between the factors usingCFA (i.e. SEM).

    Note that the Cronbachs alpha for Scale 1 is 0.827 and for Scale 2 is 0.751.

    Might consider fixing up the correlation between the scales by their estimated

    reliabilities. That is, rewriting derivation from two pages ago we have that (also

    page 197 of Kline)

    =Corr(x1, x2)

    reliability ofx1reliability of x2 So can calculate .45

    .827.751

    = .571

    Notice that it overadjusted, this estimate is actually larger than the true cor-relation of .54. This may be expected since Cronbachs alpha underestimates

    reliability when factor loadings are not equal.

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    Using a single indicator of a latent factor to adjust forunreliability

    scale1 scale2

    .45

    4.89

    scale1

    13.84

    scale2

    3.68

    The variance of scale 1 is 4.89, the variance of scale2 is 13.84.

    Notice that the simple correlation between the scales is .45 which is smaller

    than .54.

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    Using a single indicator of a latent factor to adjust forunreliability

    To adjust for the unreliability, fix the variance of the error terms to be equal to

    Variance of scale time (1-reliability). Here the Cronbachs alpha for scale 1 is

    .827 and for scale 2 is .751.

    f1f2

    scale1

    4.89*(1-.827)

    e1

    1

    1

    scale2

    13.84*(1-.751)

    e2

    1

    1

    f1 f2

    scale1

    .84597

    e1

    1

    1

    scale2

    3.44616

    e2

    1

    1

    f1f2

    scale1

    e1

    .42

    scale2

    e2

    .50

    .57

    Notice the correlation now has been adjusted for the unreliability

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    Example of Structural equation modeling

    From Neumark-Sztainer D, Wall MM, Story M, Perry C (2003) Correlates of

    unhealthy weight-control behaviors among adolescents: Implications for pre-

    vention programs, Health Psychology, 22(1), 88-98.

    Figure 1. Proposed model: Correlates of unhealthy weight-control behaviors among adolescents.

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    Example of SEM - Measuring the latent variables

    Table 2

    Results From Confirmatory Factor Analysis Including Standardized Factor Loadings and

    Correlation Between the Factors

    Model and factors

    Factor

    loading Correlation between factors

    Personal measurement model 1 2 31. Weightbody concerns .73 .05

    Weight concerns .68Weight importance .54

    Body dissatisfaction .752. Psychological well-being .73 .22

    Self-esteem .91

    Depressive mood

    .563. Healthnutrition attitudes .05 .22

    Concern about health .77Perceived benefits of healthy eating .34

    Socioenvironmental measurement model 4 5 64. Familypeer weight norms .26 .06

    Parental concernsbehaviors .71Peer dieting .395. Weight teasing .26 .21

    Frequency of teasing .83Source of teasing .72

    6. Family connectedness .06 .21 Family communication .85Atmosphere at family meals .55

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    Could create scales and do Path Analysis - Ignoring mea-surement error

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    Instead, use full SEM - Incorporate CFA into the PathAnalysis - thus accounting for measurement error

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    Final results of SEM

    Figure 4. Final model testing among adolescent girls: Correlates of unhealthy weight-control behaviors.

    BMI body mass index. * p .01. Figure 5. Final model testing among adolescent boys: Correlates of unhealthy weight-control behaviors.BMI body mass index. * p .01.

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    Common to use 2-step approach to SEM

    1. Develop measurement model (CFA) relating observed variables to latentvariables. Examine goodness of fit of this model on its own. Examine

    correlations between all variables (usually latent variables) of interest by

    looking at correlations between factors from CFA.

    2. Develop full structural equation model. That is, change the spuriously

    correlated relationships in the CFA to impose theoretical causal direct

    effects between variables and drop relationships not assumed by theory.

    Examine goodness of fit of this model as a whole.Common reference advocating this approach is

    Anderson