_Applying Structural Equation Modeling to POvert Solutions Models

8/3/2019 _Applying Structural Equation Modeling to POvert Solutions Models

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The purpose of this paper is to introduce structuralequation modeling (SEM), utilizing poverty solutions

theoretical model examples.

The following topics will be introduced:

1. Purpose and description of SEM;

2. Basics of SEM;

3. Steps in SEM process;

4. Observed and latent variable models;

5. Multi-sample SEM; 6. Multilevel SEM; and

7. Assumptions, and power, sample size, and effect sizeconsiderations.


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1. Purpose and Description of SEM

The purpose of SEM is to provide aquantitative test of one or more theoreticalmodels (e.g., the influence of poverty onparental involvement).

In SEM the researcher hypothesizes certain

relationships among a set of theoreticallyimportant variables and then empiricallytests those relationships.


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Here is a very simple theoretical model based onsome poverty research studies:

Poverty Parental Involvement Student Achievement

Bottom line question in SEM:

Do the sample data support the theoretical model?


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2. Basics of SEM

A. Latent variables (or constructs or factors):these are theoretically-based variables that arenot directly observed or measured.

examples: Poverty; Parental Involvement;Student Achievement.


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B. Observed variables (or indicators or

measures): these are measurable variables withactual scores from instruments that are used todefine the latent variables in a particular way.

examples: Expenses for Food, Clothing, andShelter(measures of Poverty); Parent

Attendance at Socialand School Activities(measures of Parental Involvement);Math andLiteracy Achievement subtests (measures ofStudent Achievement).


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C. Independent variables: variables thatare not influenced by any other variable inthe model (Poverty in our example).

D. Dependent variables: variables that areinfluenced by some other variable in themodel (Parental Involvement and Student

Achievement in our example).


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3. Steps in SEM Process

A. Model Specification

This is where the researcher develops and hypothesizes atheoretical model (or models) from available theory and

research (from literally any type of research that examinesrelations among variables).

This is the hard part of SEM as it comes from the literaturereview (i.e., not generated by your data).

This is done prior to any data analysis (and often done prior todata collection so that the strongest measures can be selected).


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In model specification you decide which variables to include in themodel and how these variables are related to one another.

A model is properly specified when the true population model thatgenerated the sample data is consistent with the theoretical modelactually tested.

Otherwise the model is misspecified. Every model is incomplete, ormisspecified, to some degree.

Misspecified models can be due to errors of omission and/or inclusionof any variable and/or parameter.

Misspecified models can result in biased parameter estimates (this biasis known as misspecification error); also the misspecified model may notfit the data according to global model fit indices.


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B. Model Identification

Do we have sufficient information in the sample data touniquely estimate all of the parameters in the model?

In other words, can we obtain a unique value for everyparameter estimate? For a VERY simple example:X+ Y= 11.

Every potential parameter in a model is either (a) free (thatyou wish to estimate), (b) fixed (not free, but fixed to aspecific value such as 0 or 1), or (c) constrained (fixed to beequal to some other free parameter).

Since the results of a nonidentified model cannot be trusted,you must be sure the model is identified before proceedingwith the analysis.


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Some commonly used methods to estimate modelparameters include:

OLS (ordinary least squares);

GLS (generalized least squares);

ML (maximum likelihood).

C. Model Estimation


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D. Model Testing

a) Use global fit indices to assess fit of the entire model:

unfortunately there is no single most powerful fit index (likethe Fin ANOVA).

some commonly used SEM global fit indices include (a) chi-square, (b) GFI (goodness-of-fit index), (c) RMSEA (rootmean-square error of approximation), (d) SRMR

(standardized root mean-square residual), and (e) CFI(comparative fit index).

it is recommended to report multiple global fit indices.


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b) To assess the fit of individual parameters, examine

(1) t values (ratio of estimate to standard error);

(2) significance oft (whether estimates are significantlydifferent from zero);

(3) sign of the estimate (in hypothesized direction);

(4) whether the estimate makes sense (e.g., no negativevariances; no correlations beyond 1.0).


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E. Model Modification

If the global fit of your initial model is not acceptable,which is typical, then detective work is necessary to lookfor ways to modify the initial theoretical model to achieve abetter fit.

Searching for a more properly specified model is known asa specification search.

In any specification search, substantive knowledgemust be the #1 priority.

In other words, do not add additional parameters tosimply achieve better model fit if they do not makesubstantive sense.


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4. Observed and Latent Variable Models

A. Regression Models an observed variable model with asingle equation or dependent variable; thus only simpletheoretical models are possible; measurement error is not takeninto account.

ShelterExpenses

SchoolActivities


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B. Path Models an observed variable model with multipleequations or dependent variables; thus more complex theoreticalmodels are possible; measurement error is not taken into account.

ShelterExpenses

SchoolActivities

LiteracyAchieve


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C. Confirmatory Factor Analysis Models a latent variable model withmultiple observed variables measuring each factor; latent variables are not

related to one another other than perhaps by a correlation (or covariance).

Poverty

ParentalInvolve

Achieve

e3

e2

e1

e4

e5

e6

e7

Food

Clothing

Shelter

Social

School

Math

Literacy


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D. Structural Equation Analysis Models a latent variable model whichcombines a confirmatory factor analysis model and a path model; latent

variables can influence one another.

Poverty

ParentalInvolve

Achieve

e3

e2

e1

e4

e5

e6

e7

Food

Clothing

Shelter

Social

School

Math

Literacy


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5. Multi-sample SEM

To this point we have only been concerned withmodels involving a single sample.

We can also examine any type of SEM model usingmultiple samples, sub-samples, populations, ortreatment groups (e.g., is the same poverty modelappropriate for different national samples, or fordifferent levels of poverty?).


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First there is multiple sample SEM (MS-SEM) where

we consider the invariance (equality) of parametersacross samples.

Applications of MS-SEM might involve (a) validating a

particular model across different samples, (b) takingrandom subsamples from the original sample, or (c)comparing multiple groups (e.g., treatment groups,levels of poverty, levels of education).


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Second there is structured means SEM (SM-SEM) where

you are interested in group mean differences. Here you arealso looking at mean structures, whereas all previousmodels were only looking at covariance structures.

For example, suppose we wish to compare samples ofchildren who have attended preschool versus those whohave not. Here we can consider mean differences on latentvariables (e.g., Student Achievement), or in a measurementequation (e.g., a particular measure of Student

Achievement, such as Math Achievement), or in astructural equation (e.g., the prediction of StudentAchievement).


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6. Multilevel SEM

To this point we have only considered models wheredata have been gathered and analyzed at a single level(e.g., family level).

There is also multilevel SEM (ML-SEM) where data arehierarchical in nature. For example, families arenested within neighborhoods, within cities, withinstates, etc.

For our example model, we might have data at both thefamily and the neighborhood levels, which is a two-level model.


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In ML-SEM we are interested in the relations among

the variables, perhaps both within and between levels.

ML-SEM allows us to (a) properly take into accountthe nested nature of the design in terms of parameterestimation, (b) test the same or a different theoreticalmodel at each level, (c) use repeated measures data onthe same subjects with latent growth curve models, (d)use of covariates as in ANCOVA, (e) test nonlinearmodels, or (f) test models with categorical outcomes.


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7. Assumptions, and power, sample

size, and effect size considerations There are two important assumptions in SEM: normality

and linearity.

Thus the population data should be normally distributedand the observed variables should be linearly related.

We have to assess normality and linearity, and subsequentlydeal with any nonnormality and nonlinearity; theseviolations can affect SEM results.


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With all inferential statistics, it is important to have sufficientpower (i.e., our ability to correctly reject the null hypothesis).

Power is largely a function of effect size, selected alpha level,and sample size.

Since sample size is the most important aspect from a designperspective, most researchers focus on determining what a

large enough sample needs to be in order to have sufficientpower in a study (i.e., a priori power).

A priori power is typically determined using either statisticalsoftware (e.g., G*Power), statistical tables, or rules of thumb

(e.g., need at least 100 to 150 for a basic SEM model).

Effect size deals with the magnitude of the relations amongthe variables in your model. In SEM, effect size is typicallyassessed via the noncentrality parameter or by delta.


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Shumow & Lomax (2002). In Parenting: Science and Practice

Poverty Solutions SEM Examples


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Palomar et al. (2005). In Social Indicators Research


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Additional Information

For additional information on SEM, one source is:

Schumacker, R.E. & Lomax, R.G. (2010). A beginnersguide to structural equation modeling (3rd ed.). NewYork: Routledge.

Email address: [email protected]
mailto:[email protected]:[email protected]

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_Applying Structural Equation Modeling to POvert Solutions Models