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PBF Zagreb, Croatia, 25.01. 2012

Structural Equation Modeling- data analyzing -

Tatjana Atanasova – Pachemska,

„Goce Delcev” University - Shtip, Macedonia

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Essentials

Purpose of this lecture is to provide a very brief presentation of the things one needs to know about SEM before learning how apply SEM.

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I. Essential Points about SEM

Outline

II. Structural Equation Models: Form and Function

III. Research Examples

What is SEM?

•Structural equation modeling (SEM) is a series of statistical methods that allow complex relationships between one or more independent variables and one or more dependent variables.

•Though there are many ways to describe SEM, it is most commonly thought of as a hybrid between some form of analysis of variance (ANOVA)/regression and some form of factor analysis. In general, it can be remarked that SEM allows one to perform some type of multilevel regression/ANOVA on factors. We should therefore be quite familiar with univariante and multivariate regression/ANOVA as well as the basics of factor analysis to implement SEM for our data.

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• SEM goes beyond factor analysis to test expected relationships between a set of variables and the factors upon which they are expected to load. As such, it is considered to be a confirmatory tool

• SEM also goes beyond multiple regression to demonstrate how those independent variables contribute to explanation

of the dependent variable. It models the direction of

relationships within a multiple regression equation.

• The goal of SEM is to identify a model that makes theoretical sense, is a good fit to the data . The model developed should be theory-driven, or based on past research.

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I. SEM Essentials ( SEM language):

1. SEM is a form of graphical modeling, and therefore, a system in which relationships can be represented in either graphical or equational form.

x1 y1

111graphicalform

y1 = γ11x1 + ζ1equationalform

2. An equation is said to be structural if there exists sufficient evidence from all available sources to support the interpretation that x1 has a causal effect on y1.

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y2y1

x1 y3

ζ1 ζ2

ζ3

Complex

Hypothesis

e.g.y1 = γ11x1 + ζ1

y2 = β 21y1 + γ 21x1 + ζ 2

y3 = β 32y2 + γ31x1 + ζ 3

Corresponding

Equations

3. Structural equation modeling can be defined as the use of two or more structural equations to represent complex hypotheses.

(gamma) used to represent effect of exogenous on endogenous.

(beta) used to represent effect of endogenous on endogenous

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Some preliminary terminology will also be useful. The following definitions regarding the types of variables that occur in SEM allow for a more clear explanation of the procedure: •Variables that are not influenced by another other variables in a model are called exogenous (independent) variables.

•Variables that are influenced by other variables in a model are called endogenous variables.

•A variable that is directly observed and measured is called an indicator (manifest) variable. There is a special name for a structural equation model which examines only manifest variables, called path analysis.

•A variable that is not directly measured is a latent variable. The “factors” in a factor analysis are latent variables.

Drawing our hypothesized model: procedures and notation

 

The most important part of SEM analysis is the causal model we are required to draw before attempting an analysis. The following basic, general rules are used when drawing a model:

•Rule 1. Latent variables/factors are represented with circles and measured/manifest variables are represented with squares.

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Rule 2. Lines with an arrow in one direction show a hypothesized direct relationship between the two variables. It should originate at the causal variable and point to the variable that is caused. Absence of a line indicates there is no causal relationship between the variables.

Rule 3. Lines with an arrow in both directions should be curved and this demonstrates a bi-directional relationship (i.e., a covariance).

Rule 3a. Covariance arrows should only be allowed for exogenous variables.

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Rule 4. For every endogenous variable, a residual term should be added in the model. Generally, a residual term is a circle with the letter E written in it, which stands for error.

Rule 4a. For latent variables that are also endogenous, a residual term is not called error in the lingo of SEM. It is called a disturbance, and therefore the “error term” here would be a circle with a D written in it, standing for disturbance. 

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SEM Process

A suggested approach to SEM analysis proceeds through the following process:

• review the relevant theory and research literature to support model specification

• specify a model (e.g., diagram, equations)

• determine model identification (e.g., if unique values can be found for parameter estimation; the number of degrees offreedom df, for model testing is positive)

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• select measures for the variables represented in the model

• collect data

• conduct preliminary descriptive statistical analysis (e.g., scaling, missing data, collinearity issues, outlier detection)

• estimate parameters in the model

• assess model fit

• respecify the model if meaningful

• interpret and present results.

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Examples

Figure 1. Regression Model (math achievement at age 10, reading comprehension achievement at age 12, and mother’s educational level predicting math achievement at age 12).

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Figure 2. Revised model (math achievement at age 10, reading comprehension at age 12 predict math achievement at age 12;indirect effect of mother’s educational level and math achievement at age 10).

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Figure 3. Structural Equation Model - Relationship between academic and job constructs

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A Grossly Oversimplified History of SEM

Wright(1918)

Pearson(1890s) Fisher

(1922)

Joreskog(1973)

Lee(2007)

Neyman & E. Pearson(1934)

Spearman(1904)

Bayes & LaPlace(1773/1774)

MCMC(1948-)

test

ing

alt.

mod

els

likeli

hood

r, chi-square

factor analysis

path analysisSEM

Contemporary

Conven-tionalStatistics

BayesianAnalysisRaftery

(1993)

Note that SEM is a framework and incorporates new statistical techniques as they become available (if appropriate to its purpose)

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The LISREL Synthesis

Karl Jöreskog1934 - present

Key Synthesis paper- 1973

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The Methodological Side of SEM

0

20

40

60

80

100

softwarehyp testingstat modelingfactor analysisregression

modified from Starfield and Bleloch (1991)

How do data relate to learning?

Understanding of Processes

univariate descriptive statistics

exploration, methodology and

theory development

realistic predictive models

abstract models

multivariate descriptive statistics

more detailed theoretical models

univariate data modeling

multivariate data modeling

Data

SEM

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SEM is one of the few applications of statistical inference where the results of estimation are frequently “you have the wrong model!”. This feedback comes from the unique feature that in SEM we compare patterns in the data to those implied by the model. This is an extremely important form of learning about systems.

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AMOS Graphics

AMOS (Analysis of MOments Structures) is a statistical package specializing in structural equation modeling.

AMOS builds measurement, structural or full structural models. It tests, modifies and retests models. AMOS also tests alternate models, equivalence across groups or samples, as well as hypotheses about means and intercepts. It handles missing datausing Maximum Likelihood (ML) estimation and provides bootstrapping procedures.

Results obtained in AMOS are comparable to those obtained through other SEM packages.

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Five Steps to SEM

• Model specification; • Model identifiability;• Measure selection, data collection, cleaning and

preparation; • Model analysis and evaluation; • Model respecification

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Model specification involves mathematically or diagrammatically expressing hypothesized relationships among a set of variables.

The challenge at this step is to include all endogenous and exogenous variables, (including moderators and mediators), that are expected to contribute to central endogenous variables. Exclusion of important variables may result in the misestimation of endogenous variables. The extent of misestimation increases with the strength of the correlation between missing and endogenous variables. Whilst it is impossible to include all variables that contribute to the prediction of endogenous variables, it is possible to identify the main ones through careful examination of relevant theory and past research

A second challenge is to determine the direction of relationships between pairs of variables in the SEM model. Actual direction is debatable, especially where manifest variables are measured at the same point in time

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Step 2: Model Identifiability

Specified models need to be checked for identifiability. A model is theoretically identifiable if there is a unique solution possible for it and each of its parameters. If a model is not identifiable, then it has no unique solution and SEM software will fail to converge. Such models need to be respecified to be identifiable.

The maximum number of parameters that can be specified in the model is equivalent to the number of unique variances and covariances that can be found in its underlying covariance matrix.

If, for example, there are four variables (say: A, B, C, and D), a covariance matrix has four unique variances (one for eachvariable) along with six unique covariances (AB, AC, AD, BC, BD and CD), giving a total of ten unique parameters. (See figure).

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A B C DA Var(A)B Cov(AB) Var(B)C Cov(AC) Cov(BC) Var(C )D Cov(AD) Cov(BD) Cov(CD) Var(D)

A Covariance Matrix With Four Variables, A, B, C and D.

Note: For four variables, there are four unique variances and six unique covariances, giving a maximum of tenparameters estimable with SEM.

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Step 3: Measure Selection, Data Collection, Cleaning and Preparation

Step 3 has four substeps: measure selection, data collection, data cleaning and data preparation

Step 3a - Measure SelectionManifest variables are estimates of the underlying latent constructs they purport to measure. It is therefore recommended that each latent construct be measured by at least two manifest variables.

Measures selected need to demonstrate good psychometric properties. They need to be both “reliable” and “valid” measure.

• Coefficients of 0.8 or above suggest good reliability, whilst those in the range of 0.7 to 0.8 suggest adequacy. Coefficients below 0.5 should be avoided or improved before use in research.

• Validity is assessed by examining its content, criterion-related, convergent or discriminant validities

• Content validity exists when experts agree that the measure is tapping into the relevant domain.

• Criterion-related validity assesses whether a measure taps into a particular domain, as assessed against some set criteria

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• Step 3b - Data Collection

• A sufficiently large sample needs to be drawn in order to analyse the model specified at Step 1. The sample drawn should be ten times the number of model parameters to be estimated, with a minimum of 200 cases. If planning to divide the sample in two for model development and testing purposes, then each half sample needs to be sufficiently large. Moreover, expected response rates should be factored into consideration when drawing the sample.

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• Step 3c - Data “Cleaning”

• The acronym GIGO (Garbage In, Garbage Out) highlights the importance of checking the veracity and integrity of data entry. In statistical terms, doing so ensures that data is “clean” before proceeding further.

• Checking each datapoint of a large dataset may be tedious. However, it is possible to check (and correct) the first five or ten cases and extrapolating their accuracy rate to the remaining cases in the dataset. If accuracy is less than, say, 95%, the data could be reentered using a double entry method.

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II. Structural Equation Models: Form and Function

A. Anatomy of Observed Variable Models

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x1

y1

y2

1

2

Some Terminology

exogenous

variableendogenous

variables

21

11 21

path

coefficients

direct effect of x1 on y2

indirect effect of x1 on y2

is 11 times 21

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recursive (the term recursive refers to the mathematical property thateach item in a series is directly determined by the preceding item).

nonrecursive

y1x2

x1 y2

ζ1

ζ2

C

y1x2

x1 y2

ζ1

ζ2

D

x1 y1

ζ1

y2

A

ζ2

x1 y1 y2

B

ζ1ζ2

model B, which has paths between all variables is “saturated” (vs A, which is “unsaturated”)

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First Rule of Path Coefficients: the path coefficients for

unanalyzed relationships (curved arrows) between exogenous variables are simply the correlations (standardized form) or covariances (unstandardized form).

x1

x2

y1.40

x1 x2 y1

-----------------------------

x1 1.0

x2 0.40 1.0

y1 0.50 0.60 1.0

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x1 y1 y2

11 = .50 21 = .60

Second Rule of Path Coefficients: when variables areconnected by a single causal path, the pathcoefficient is simply the standardized or unstandardized regression coefficient (note that a standardized regression coefficient = a simple correlation.)

x1 y1 y2

-------------------------------------------------x1 1.0y1 0.50 1.0y2 0.30 0.60 1.0

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Third Rule of Path Coefficients: strength of a compound path is the product of the coefficients along the path.

x1 y1 y2

.50 .60

Thus, in this example the effect of x1 on y2 = 0.5 x 0.6 = 0.30

Since the strength of the indirect path from x1 to y2 equals the

correlation between x1 and y2, we say x1 and y2 are

conditionally independent.

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What does it mean when two separated variables

are not conditionally independent?

x1 y1 y2

-------------------------------------------------

x1 1.0

y1 0.55 1.0

y2 0.50 0.60 1.0

x1 y1 y2

r = .55 r = .60

0.55 x 0.60 = 0.33, which is not equal to 0.50

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The inequality implies that the true model is

x1

y1

y2

Fourth Rule of Path Coefficients: when variables are

connected by more than one causal pathway, the path

coefficients are "partial" regression coefficients.

additional process

Which pairs of variables are connected by two causal paths?

answer: x1 and y2 (obvious one), but also y1 and y2, which are connected by the joint influence of x1 on both of them.

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And for another case:

x1

x2

y1

A case of shared causal influence: the unanalyzed relation

between x1 and x2 represents the effects of an unspecified

joint causal process. Therefore, x1 and y1 connected by two

causal paths x2 and y1 likewise.

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x1

y1

y2

.40

.31

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How to Interpret Partial Path Coefficients: The Concept of Statistical Control

The effect of y1 on y2 is controlled for the joint effects of x1.

Grace, J.B. and K.A. Bollen 2005. Interpreting the results from multiple regression and structural equation models. Bull. Ecological Soc. Amer. 86:283-295.

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x1

y1

y2

Fifth Rule of Path Coefficients: paths from error variables are correlations or covariances.

R2 = 0.16

.92

R2 = 0.44

.73

2

1

.31

.40 .48

21iyR

equation for path

from error variable

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alternative is to

show values for zetas,

which = 1-R2

.84

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x1

y1

y2

R2 = 0.16

R2 = 0.25

2

1

.50

.40x1 y1 y2

-------------------------------

x1 1.0

y1 0.40 1.0

y2 0.50 0.60 1.0

Now, imagine y1 and y2

are joint responses

Sixth Rule of Path Coefficients: unanalyzed residual correlations between endogenous variables are partial correlations or covariances.

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x1

y1

y2

R2 = 0.16

R2 = 0.25

2

1

.50

.40

.40

This implies that some other factor is influencing y1 and y2

the partial correlation between y1 and y2 is typically

represented as a correlated error term

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Seventh Rule of Path Coefficients: total effect one variable has on another equals the sum of its direct and indirect effects.

y1x2

x1 y2

ζ1

ζ2.80

.15

.64

-.11.27

x1 x2 y1

-------------------------------

y1 0.64 -0.11 ---

y2 0.32 -0.03 0.27

Total Effects:

Eighth Rule of Path Coefficients:

sum of all pathways between two variables (causal and noncausal) equals the correlation/covariance.

note: correlation between

x1 and y1 = 0.55, which

equals 0.64 - 0.80*0.11

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Suppression Effect - when presence of another variable causes path coefficient to strongly differ from bivariate correlation.

x1 x2 y1 y2 -----------------------------------------------x1 1.0x2 0.80 1.0y1 0.55 0.40 1.0y2 0.30 0.23 0.35 1.0

y1x2

x1 y2

ζ1

ζ2.80

.15

.64

-.11.27

path coefficient for x2 to y1 very different from correlation,

(results from overwhelming influence from x1.)

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II. Structural Equation Models: Form and Function

B. Anatomy of Latent Variable Models

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Latent Variables

Latent variables are those whose presence we suspect or theorize, but for which we have no direct measures.

Intelligence IQ score

*note that we must specify some parameter, either error,

loading, or variance of latent variable.

ζ

latent variable observed indicator errorvariable

1.0

fixed loading*

1.0

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Latent Variables (cont.)

Purposes Served by Latent Variables:

(2) Allow us to estimate and correct for measurement error.

(3) Represent certain kinds of hypotheses.

(1) Specification of difference between observed data and processes of interest.

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Range of Examples

single-indicator

Elevationestimate

from map

multi-method

SoilOrganic

soil C

loss onignition

TerritorySize

singing range, t1

singing range, t2

singing range, t3

repeated measures

CaribouCounts

observer 1

observer 2

repeatability

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The Concept of Measurement Error

the argument for universal use of latent variables

1. Observed variable models, path or other, assume all independent variables are measured without error.

2. Reliability - the degree to which a measurement is repeatable (i.e., a measure of precision).

error in measuring x is ascribed to error in predicting/explaining y

x y0.60

R2 = 0.300

5

10

15

20

25

0 0.5 1 1.5 2

x

y

illustration

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Example

Imagine that some of the observed variance in x is due to error of measurement.

calibration data set based on repeated measurement trials

plot x-trial1 x-trial2 x-trial31 1.272 1.206 1.2812 1.604 1.577 1.6713 2.177 2.192 2.1044 1.983 2.080 1.999. ........ ........ .......n 2.460 2.266 2.418

average correlation between trials = 0.90therefore, average R-square = 0.81

reliability = square root of R2

measurement error variance =(1 - R2) times VARx

imagine in this case VARx = 3.14, so error variance = 0.19 x 3.14 = 0.60

LV1x LV2 y.90 .65 1.0

.60

R2 = .42

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II. Structural Equation Models: Form and Function

C. Estimation and Evaluation

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1. The Multiequational Framework

(a) the observed variable model

We can model the interdependences among a set of predictors and responses using an extension of the general linear model that accommodates the dependences of response variables on other response variables.

y = p x 1 vector of responses

α = p x 1 vector of intercepts

Β = p x p coefficient matrix of ys on ys

Γ = p x q coefficient matrix of ys on xs

x = q x 1 vector of exogenous predictors

ζ = p x 1 vector of errors for the elements of y

Φ = cov (x) = q x q matrix of

covariances among xs

Ψ = cov (ζ) = q x q matrix of covariances among errors

y = α + Βy + Γx + ζ

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The LISREL Equations

Jöreskög 1973

(b) the latent variable model

η = α + Β η + Γξ + ζ

x = Λxξ + δy = Λyη + ε

where: η is a vector of latent responses,ξ is a vector of latent predictors,Β and Γ are matrices of coefficients,ζ is a vector of errors for η, andα is a vector of intercepts for η

(c) the measurement model

where: Λx is a vector of loadings that link observed x variables to latent predictors,Λy is a vector of loadings that link observed y variables to latent responses, andδ and ε are vectors are errors

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Estimation Methods

(a) decomposition of correlations (original path analysis)

(b) least-squares procedures (historic or in special cases)

(c) maximum likelihood (standard method)

(d) Markov chain Monte Carlo (MCMC) methods (including Bayesian applications)

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Bayesian References:

Bayesian Networks: Neopolitan, R.E. (2004). Learning Bayesian Networks. Upper

Saddle River, NJ, Prentice Hall Publs.

Bayesian SEM: Lee, SY (2007) Structural Equation Modeling: A Bayesian

Approach. Wiley & Sons.

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SEM is Based on the Analysis of Covariances!

Why? Analysis of correlations represents loss of information.

0

20

40

60

80

100

0 10 20 30

x

y

0

20

40

60

80

100

0 10 20 30

x

y

A B

r = 0.86 r = 0.50

Illustration with regressions having same slope and intercept

Analysis of covariances allows for estimation of both standardized and unstandardized parameters.

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Σ = {σ11

σ12 σ22

σ13 σ23 σ33

}Model-Implied CorrelationsObserved Correlations*

{1.0.24 1.0.01 .70 1.0}S =

* typically the unstandardized correlations, or covariances

2. Estimation (cont.) – analysis of covariance structure

The most commonly used method of estimation over the past 3 decades has been through the analysis of covariance structure (think – analysis of patterns of correlations among variables).

compare

Σ = {σ11

σ12 σ22

σ13 σ23 σ33

}Implied Covariance Matrix

compareAbsolute Model Fit

x1

y1

y2

Hypothesized Model Observed Covariance Matrix

{1.3.24 .41.01 9.7 12.3}S =

+

Estimation and Evaluation

ParameterEstimates

estimation

LS, ML, and BA

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Model Identification - Summary

2. Several factors can prevent identification, including:a. too many paths specified in modelb. certain kinds of model specifications can make parameters unidentifiedc. multicollinearityd. combination of a complex model and a small sample

1. For the model parameters to be estimated with unique values, they must be identified. As in linear algebra, we have a requirement that we need as many known pieces of information as we do unknown parameters.

3. Good news is that most software (AMOS,…) checks for identification (in something called the information matrix) and lets you know which parameters are not identified.

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The most commonly used fitting function in maximum likelihood estimation of structural equation models is based on the log likelihood ratio, which compares the likelihood for a given model to the likelihood of a model with perfect fit.

Fitting Functions

qptrFML SΣSΣ logˆˆlog 1

Note that when sample matrix and implied matrix are equal, terms 1 and 3 = 0 and terms 2 and 4 = 0. Thus, perfect model fit yields a value of FML of 0.

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Maximum likelihood estimators, such as FML, possess several important properties: (1) asymptotically unbiased, (2) scale invariant, and (3) best estimators.

Assumptions: (1) and S matrices are positive definite (i.e., that they do not have a singular determinant such as might arise from a negative variance estimate, an implied correlation greater than 1.0, or from one row of a matrix being a linear function of another), and

(2) data follow a multinormal distribution.

Fitting Functions (cont.)

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One of the most commonly used approaches to performing such tests (the model Χ2 test) utilizes the fact that the maximum likelihood fitting function FML follows a X2 (chi-square) distribution.

The Χ2 Test

X2 = n-1(FML)

Here, n refers to the sample size, thus X2 is a direct function of sample size.

Assessment of Fit between Sample Covariance and Model- Implied Covariance Matrix

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Illustration of the use of Χ2

X2 = 3.64 with 1 df and 100 samplesP = 0.056

X2 = 7.27 with 1 df and 200 samplesP = 0.007

x

y1

y2

1.00.4 1.00.35 0.5 1.0

rxy2 expected to be 0.2 (0.40 x 0.50)

X2 = 1.82 with 1 df and 50 samplesP = 0.18

correlation matrix

issue: should there be a path from x to y2?

0.40 0.50

Essentially, our ability to detect significant differences from our base model, depends as usual on sample size.

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Alternatives when data are extremely nonnormal

Robust Methods: Satorra, A., & Bentler, P. M. (1988). Scaling corrections for

chi-square statistics in covariance structure analysis. 1988 Proceedings of the Business and Economics Statistics Section of the American Statistical Association, 308-313.

Bootstrap Methods:Bollen, K. A., & Stine, R. A. (1993). Bootstrapping goodness-

of-fit measures in structural equation models. In K. A. Bollen and J. S. Long (Eds.) Testing structural equation models. Newbury Park, CA: Sage Publications.

Alternative Distribution Specification: - Bayesian and other:

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Residuals: Most fit indices represent average of residuals between observed and predicted covariances. Therefore, individual residuals should be inspected.

Correlation Matrix to be Analyzed y1 y2 x -------- -------- --------y1 1.00y2 0.50 1.00 x 0.40 0.35 1.00

Fitted Correlation Matrix y1 y2 x -------- -------- --------y1 1.00y2 0.50 1.00 x 0.40 0.20 1.00

residual = 0.15

Diagnosing Causes of Lack of Fit (misspecification)

Modification Indices: Predicted effects of model modification on model chi-square.

1. It is a “model-oriented” method, not a null-hypothesis-oriented method.

Rewue of SEM

2. Highly flexible modeling toolbox.

3. Can be applied in either confirmatory (testing) or exploratory (model building) mode.

4. Variety of estimation approaches can be used, including likelihood and Bayesian.

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Where You can Learn More about SEM

Grace (2006) Structural Equation Modeling and Natural Systems. Cambridge Univ. Press.

Shipley (2000) Cause and Correlation in Biology. Cambridge Univ. Press.

Kline (2005) Principles and Practice of Structural Equation Modeling. (2nd Edition) Guilford Press.

Bollen (1989) Structural Equations with Latent Variables. John Wiley and Sons.

Lee (2007) Structural Equation Modeling: A Bayesian Approach. John Wiley and Sons.

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Thank you for your attention