Structural Equation Modeling And Related Techniques

Structural Equation Modeling

And Related Techniques

Michael Biderman

Department of Psychology

University of Tennessee at Chattanooga

Why do this?

Altruistic Reasons
To familiarize faculty and students with new techniques.
Selfish Reasons
“You don’t really understand something until you’ve taught it.”
To force myself to address issues that I’ve glossed over in my own reading.
To get to know faculty and students.
To assuage loneliness.
Why learn structural equation modeling techniques?
The following reasons will be much more meaningful after you’ve had the workshop.
Provides a single analytic technique for testing a) multiple hypotheses, involving b) multiple variables including c) latent variables and possibly involving d) multiple stages.
Latent variable: A variable believed to exist but one which is not directly observable.
Latent variables are typically indicated by several observed variables, each of which gives partial information about the latent variable.
,
Written resources
In order of relevance to this workshop
Arbuckle, James L., & Wothke, Werner. (1999). Amos 4.0 User’s Guide. Chicago, IL: SmallWaters Corporation. ISBN 1-56827-264-2. AMOS is at V5, so there may be a V5 User’s Guide.
Byrne, Barbara M. (2001). Structural equation modeling with AMOS: basic concepts, applications, and programming. Mahwah, NJ: Erlbaum. ISBN 0-8058-3322-6.
Loehlin, John C. (1998). Latent variable models: an introduction to factor, path, and structural analysis. 3^rd Ed. Mahwah, NJ: Erlbaum. ISBN 0-8058-2831-1 (Paperback), 0-8058-2830-3 (Cloth).
Maruyama, Geoffrey M. (1998). Basics of structural equation modeling. Thousand Oaks, CA: Sage. ISBN 0-8039-7408-6 (Cloth), 0-8039-7409-4 (Pbk.)
Duncan, Terry E., Duncan, Susan C., Strycker, Lisa A., Fuzhong, Li, & Alpert, Anthony. (1999). An introduction to latent variable growth curve modeling: concepts, issue, and applications. Mahway, NJ: Erlbaum. ISBN 0-8058-3060-X
Kline, Rex B. (198). Principles and practice of structural equation modeling. New York: Guilford Press. ISGN 1-57230-336-0 (Cloth), 1-57230-337-9 (Pbk). Lots of words, few diagrams.
Byrne, Barbara M. (1994). Structural equation modeling with EQS and EQS/Windows. 1000 Oaks, CA: Sage. ISBN 0-8039-5092-6.
Byrne, Barbara M. (1998). Structural equation modeling with LISREL, PRELIS, and SIMPLIS: basic concepts, applications, and programming. Mahway, NJ: Erlbaum. ISGN 0-8058-2924-5.
Other resources
Statistical Analysis with Latent Variables by Bengt O. Muthén
http://www.ats.ucla.edu/stat/seminars/ed231e/default.htm
This is the web site of a course being offered in Spring 2004. The course lectures are available as streaming video. It goes quite a bit beyond what will be covered in this workshop.
Ed Rigdon's Structural Equation Modeling Page
http://www.gsu.edu/~mkteer/
This page contains a compendium of links to other sites dealing with structural equation modeling. (I found it by googling structural equation modeling.)
Working Group Structural Equation Modeling
http://www.uni-muenster.de/SoWi/struktur/
From the web page: “This is the Web-Page of the Working Group Structural Equation Modeling. This group has now its 10 years anniversary and is going to create information pages for its members and other people who are interested in developments and applications of structural equation models.” Note: The group offers a workshop on Amos. Cost is $190. Two catches: It’s in Germany and it’s in German.
Some computer programs for SEM
In order of appearance on the scene
LISREL K.G. Jőreskog
The gold standard.

Reputation for being difficult to use.

Typically used via a programming language.
EQS. Peter Bentler.

Easier to use than LISREL.

Tyypically used via a programming language.
AMOS James Arbuckle

Very easy to use (as a structural equation modeling program can be)

Typically programmed using path diagrams.

A programming language is available.

Ties with SPSS, Inc.
SAS, PROC CALIS
I don’t know much about this proc.
MPLUS. Bengt O. Muthén

Latent variable analyses with continuous and discrete variables.

Touted as integrating a variety of analyses that go beyond traditional SEM.

May be the program of the future.

Relationship of SEM to other types of analyses

A hierarchy of analytic techniques.
Each analytic technique can perform all the essential analyses of the prior techniques.
t-tests are a special case of the Analysis of Variance. ANOVA does t-tests and more.
All analysis of variance tests can be performed using multiple regression analyses. MR does ANOVA and more.
All multiple regression analyses can be performed using structural equation modeling techniques and programs. SEM does multiple regression and more.
Who know what the next omnibus technique will be.

?????

Structural Equation Modeling

Multiple Regression /General Linear Model

ANOVA

t-tests

Basic SEM Concepts
Observed Variable:
A variable whose values are directly observable without error.
The scores in a column of the SPSS data editor. So these are what we work with every day in SPSS.
UGPA, GREV, GREQ, GREA, IQ scores (but not intelligence), Depression scores (but not depression).
Most current SEM analyses involve continuous variables.
All SEM analyses begin and end with observed variables.
Latent Variable:
A variable whose values cannot be directly observed without error.
Values of latent variables are indicated by observed variables.
The indicators are assumed to imperfectly represent the values of the latent variable.
Latent Variable Indicators
Job Satisfaction Score on Job Satisfaction Survey,

Score on Job Descriptive Index

Score on MN Satisfaction Questionnaire.
Intelligence Score on Wonderlic,

Score on Stanford Binet

Score on Wechsler

Depression Score on Beck Depression Inventory

Hamilton Depression Rating Scale

Center for Epidemiological Studies-D Scale

Exogenous Variable:
A variable whose variation is assumed to exist but is not accounted for in an analysis.
Equivalent to independent variables in traditional analyses.
May be observed or latent.
Endogenous Variable
A variable whose variation is completely explained by the other variables in an analysis.
Equivalent to dependent variables in traditional analyses.
May be observed or latent.
Path Diagrams
It is very common to represent the analyses performed using SEM using path diagrams.
Although all the analyses can be represented as sets of equations, many, perhaps most analysts prefer to represent models with path diagrams.
Some programs, for example, AMOS, were written to be programmed using path diagrams. (How cool is that?)
We will make extensive use of path diagrams to represent relationships, models, and analyses in this workshop.

Basic Path Analytic / SEM Notation

Observed variables are symbolized by squares or rectangles.

103

84

121

76

. . .

97

81

Observed

Variable

Latent Variables are symbolized by Circles or ellipses.


Note that the values of the latent variable are represented as dimmed – because they’re not directly observable.


Correlations between variables are represented by double-headed arrows

"Correlation"

Arrow

Latent

Variable A

Latent

Variable B

"Causal" or "Predictive" relationships between variables are represented by single-headed arrows





Exogenous Observed Variables: Variables whose variation originates outside the setting and is unexplained.

Exogenous observed variable connect to other variables in the model through either a “causal” arrow or a correlation arrow.

Exogenous Latent Variables: Latent Variables whose variation originates outside the setting.

As is the case with exogenous observed variables, exogenous latent variables connect to other variables in the model through either a “causal” arrow or a correlation

Endogenous Observed Variables - Endogenous Latent Variable

Endogenous variables connect to other variables in the model by being on the “receiving” end of “causal” arrows. Specifically, endogenous variables are typically represented as being “caused” by 1) other variables in the theory and 2) random error. Thus, 100% of the variation in every endogenous variable is accounted for by either other variables in the model or random error. This means that random error is an exogenous latent variable in SEM diagrams.

Values associated with symbols

Our SEM program, AMOS, prints means and variances above and to the right. Typically the mean and variance of latent variables are fixed at 0 and 1 respectively, although there are exceptions to this in advanced applications.

New shoes for old feet: Old analyses done using new techniques
SEM programs can be used to conduct any analysis that can be performed using multiple regression techniques. And this means that they can conduct any analysis that can be done using MR, such as ANOVA, t-tests.
“Can be used” should not be confused with “Should be used.”
SEM program output is not designed for regression analyses, so much of the useful stuff printed by regression programs may be difficult or impossible to obtain.
SEM programs use different statistical tests than most regression programs, so p-values will often not agree with those of SPSS, for example.
SEM programs have little diagnostic output.
So don’t throw away your regression procedure.
The same goes for ANOVA and for t-tests.

AMOS representation of correlation of two variables

Data are records of 190 psych graduate students admitted between 1991 and 2002.
SPSS Representation




AMOS Unstandardized Estimates View

Covariance




Mean

Variance

AMOS Standardized Estimates View


Means and variances of observed variables are not represented in Standardized Estimates view because they’re assumed to be 0 and 1 respectively.







AMOS representation of correlations of many variables.
Data are records of 190 psych graduate students admitted between 1991 and 2002.
S

PSS Representation

A
Argh!!
MOS Unstandardized Estimates View



AMOS Standardized Estimates View


Much prettier.

AMOS representation of simple regression analysis
Data are records of 190 psych graduate students admitted between 1991 and 2002.

P511G is proportion of required points in 1^st semester graduate statistics course.

Formula is 200*UGPA + (GREV+GREQ)/2

S





PSS Representation

Mean and variance of residual variance are arbitrarily set = 0 and 1 respectively. (Math voodoo.)

Mean, Variance

Intercept

B (Reg Coeff)

AMOS Unstandardized Estimates View


A
r-squared.
ll variance of an endogenous variable (P511G) must be accounted for. That which is not related to the observed exogenous variables (UGPA) is assumed to be accounted for by a latent variable representing residual variation (Other).
AMOS Standardized Estimates View


The standardized relationship of P511G with other factors.

(= √.1-r-squared)

Beta, Standardized regression coefficient. (Also r in a simple regression.)

AMOS Representation of Multiple Regression

Data are records of 190 psych graduate students admitted between 1991 and 2002.

P

Standardized Regression weights (Betas)

Analyses more easily done than in SPSS

PSY 513 is taken immediately following PSY 511.

There is a positive relationship (B=.69)of P511G to P513G controlling for Formula. And there is a small, positive relationship (B=.0001) of Formula to P513G controlling for P511G.

The output below shows that all coefficients are significantly different from 0. (AMOS prints *** when a p-value is less than .001.) The fact that the Formula-to- P513G coefficient is significantly different from 0 indicated that P511G only partially mediates the relationship between P513G and Formula.
Excerpt from Text Output

			Estimate	S.E.	C.R.	P
P511G	<---	FallOther	.0650	.0033	19.4422	***
P511G	<---	FORMULA	.0004	.0001	7.9551	***
P513G	<---	FORMULA	.0001	.0001	2.5154	.0119
P513G	<---	P511G	.6943	.0662	10.4812	***
P513G	<---	SpringOther	.0592	.0030	19.4422	***

AMOS standardized view


R²

The Standardized Formula-to-P511G coefficient of .50 means that a difference of one standard deviation in formula score is associated with about a .50 SD increase in P511G score.

The .62 means that a 1 SD difference in P511G is associated with a .62 SD increase in P513G, controlling for Formula scores.

The .15 means that a 1 SD increase in Formula score is associated with a .15 SD increase in P513G even when controlling for P511G.
What’s it all mean?
P511G: Immediate past performance is the best predictor of future performance. Plus the P511G score probably reflects aspects of the graduate school experience not represented in the Formula score – adjustment to UTC, roommates, financial situations, etc. – that carries over to the spring P513 course.

Formula: Even among persons equal in performance in the fall course, P513G performance is related to UGPA and GRE scores as embodied in the Formula score. This might reflect difference in content between 511 (review of basic statistics, research methods) and 513 (emphasis on multiple regression), or differential weighting of intellectual and motivational characteristics between fall and spring.

Regression with a Latent Dependent Variable
Quionna Caldwell Thesis
Quionna studied the relationship of perceptions of organizations’ diversity efforts to liking for the organization.
Sample: 200 African American female managers
Independent Variables
IV1. Perception of organization’s inclusion of diverse elements.
Management here encourages the formation of employee network support groups.

There is a mentoring program in use here that identifies and prepares all minority and female employees for promotion.

The “old boys’ network” is alive and well here. ®

The company spends enough money and time on diversity awareness and related training.

Managers here have a track record of hiring and promoting employees objectively, regardless of their race, sex, religion, or age.

Managers here give feedback and evaluate employees fairly, regardless of the employee’s ethnicity, gender, age, or social background.

Manager’s here make layoff decisions fairly, regardless of factors such as employees’ race, sex, age, or social background.

Managers interpret human resource policies (such as sick leave) fairly for all employees.

Managers here give assignments based on the skills and abilities of employees.

Caldwell Thesis Dependent Variables

All in all, how satisfied are you with the persons on your work group?

All in all, how satisfied are you with your supervisor?

All in all, how satisfied are you with this organization, compared to most others?

How satisfied do you feel with the progress you have made in this organization up to now?

Considering your skills and the effort you put into the work, how satisfied are you with your pay?

All in all, how satisfied are you with your job?
DV2: Affective Commitment Scale
I would be very happy to spend the rest of my career with this organization.

I really feel as if this organization’s problems are my own.

I do not feel like “part of the family” at my organization. ®

I do not feel “emotionally attached” to this organization. ®

This organization has great deal of personal meaning for me.

I do not feel a strong sense of “belonging” to my organization. ®

DV3: Turnover intention Scale
I think a lot about leaving the organization.

I am actively searching for a substitute for the organization.

As soon as it is possible I will leave the organization.
Scale scores for the three dependent variables were computed in a fashion analogous to those for the independent variables.

Caldwell Thesis: Analysis of Observed Dependent Variables Separately

Caldwell Thesis: Analysis of Liking for Organization as a latent Dependent Variable