[pic]University of Illinois at Chicago
College of Business Administration
Department of Information & Decision Sciences
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IDS 578 Research Methodology II
Instructor Stanley L. Sclove
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The Sears, Roebuck Case
These notes Copyright © 2005 Stanley Louis Sclove
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HyperTable of Contents

0. Introduction; Key Terms 1. What is Structural Equation Modeling? 2. A Simple Example of SEM 3. The Role of Theory in Structural Equation Modeling 4. Developing A Modeling Strategy 5. Estimating a Path Model with Structural Equation Modeling

References
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0. Introduction; Key Terms

Preview

In this course we consider Structural Equation Models. This note anticipates much material which will be discussed in more detail in the course or in the text (Schumacker and Lomax 2004). The note is based on a Harvard case, together with an example from Hair et al. and some introductory material.

Key Terms

The diagrams used in SEM are called path diagrams. The response variables are endogenous; the explanatory variables, exogenous. The modeling proceeds in terms of concepts, represented by constructs, measured by variables called indicators of the constructs. The model relating the indicators to the constructs is called the measurement model. The model relating the constructs to one another is called the structural model.
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1. What is Structural Equation Modeling?

Structural equation models (SEMs) include path analysis and factor analysis. SEMs include the econometrician's Simultaneous Equations Models.
However, SEMs may include unobserved ("latent") variables (constructs) representing unobserved concepts.
Multiple regression analysis often raises more questions than it answers. It examines data via correlations without establishing causation. Path analysis seeks causal pathways; in fact, it can be called causal pathway modeling.

1.1. Accommodating Multiple Interrelated Dependence Relationships

SEM models a set of m dependent variables (Y's) in terms of a set of n explanatory ("independent") variables (X's). There is a separate equation for each Y, and other Y's may be on the righthand side.

1.2. Incorporating Variables That We Do Not Measure Directly

We represent concepts or "constructs" in terms of observable indicators which measure aspects of them.

2. A Simple Example of SEM

Example (Hair et al.). HATCO's personnel department identified three employee attitudes they felt most important: job satisfaction, organizational commitment, and probability of employee turnover. They then modeled each of these as follows, in terms of
Exogenous variables:-- X1: Coworker Attitude X2: Work Environment X3: Pay Level
Endogenous variables:-- Intermediate-level dependent variables:-- Y1, Job Satisfaction = f (Coworker Attitude, Work Environment) Y2, Organizational commitment = f (Job satisfaction, Pay Level) Final level dependent variable:-- Y3, Probability of employee turnover = f (Job Satisfaction, Organizational commitment) Exercise. Make a path diagram for this situation.

Sears Case

This is an example of the use of SEM at Sears, Roebuck & Company. (Rucci, Kirn and Quinn 1998; pages 89-93 are perhaps especially relevant to this course.) This effort was overseen by Anthony Rucci. Tony was then Sears' executive VP for administration and "Chief Learning Officer". Later he was for a short time Dean of the CBA at UIC and then in Oct., 1999, went to Cardinal Health, Inc. as Exec VP, HR. The Sears' model is not unlike the HATCO example, except that, very importantly, it takes things a step further and relates employee and customer satisfaction to financial performance. Financial Performance in quarter t is modelled with Customer Satisfaction in quarter t-1 and Employee Satisfaction in quarter t-2. This yields a predictive model for Financial Performance. Data on the other variables were obtained by such devices as a note on randomly selected cash-register receipts which told the customers that if they would dial the given 800 number and answer a few questions using telephone push buttons, they would receive five dollars.

|SEARS' PATH MODEL | | |
|Employee Attitudes |Customers |Financial Performance |
|about the job |customer retention |revenue growth |
|about the company |recommending Sears to others | |

Executives' pay bonuses were then related not only to Financial Performance but also to Employee Attitudes and Customer Satisfaction generated by those who report to them.
Epilogue. Since this case was published in the late 1990s, Sears has undergone a number of changes, both in top-level personnel and otherwise. In November, 2004, Sears was in the process of being acquired by K-Mart in a $11 billion deal.

3. The Role of Theory in Structural Equation Modeling

So, to do a path analysis, the researcher begins with an assumed model, indicating the direction and sources of causality among the variables.
Path Analysis plays a role in causal analysis. It is not, however, by itself a method of discovering causal laws but rather a procedure for giving a quantitative interpretation to an assumed causal system. In the example, everything is relative to the relationships assumed and the corresponding diagram. E.g., in the example above, another modeler might have put in an arrow directly from Pay Level to Job Satisfaction. In the Sears example, Sears execs specified the model; the consulting firm of econometric statisticians (Claes Fornel International/CFI, headquartered in Ann Arbor), "ran the numbers."
Consider the simplest case, one Y and two X's. In path analysis, the contribution from X1 and X2 is split into four components or "paths" which are
-- direct paths from X1 to Y and X2 to Y; and
-- indirect paths from X1 through X2 to Y and X2 through X1 to Y.
The four paths which are thus defined are as follows.
X1 --> Y X2 --> Y X1 --> X2 --> Y X2 --> X1 --> Y
Each of these may or may not be in the diagram used by the researcher to model a given situation. (What would be all the possible paths with three X's and one Y?)

4. Developing A Modeling Strategy

Model conceptually, generating the relevant constructs. Then find suitable indicators for the constructs. Consider forming a construct's indicators into a single factor. Test to see whether the indicators obey a single-factor model; if not, use two or three factors to represent the construct. Steps are listed below.

5. Assessing the Identification of the Structural Model

The number of distinct parameters in a covariance matrix in terms of the number v of variables is v(v+1)/2. When there are p X's and q Y's, v = p+q . (Hair's book uses p and q for the numbers of observed endogenous and exogenous variables, resp., and m and n for the numbers of underlying variables.) The number of d.f. is the number of correlation coefficients, minus the number t of est'd coeffs in the proposed model; this gives the expression (1/2)[(p+q)(p+q+1)] - t in Hair's book. For a correlation matrix the number of distinct parameters (correlation coefficients) is v(v-1)/2.

6. Estimating a Path Model with SEM

Competing Models

Suppose we consider three competing models, COMPMOD1 , COMPMOD2 and COMPMOD3. It is possible that COMPMOD2 has the best AIC. COMPMOD1 has the best (lowest) chi-square value. But it used more parameters to achieve this. A figure-of-merit for competing models that combines chi-square and the number of parameters is AIC (Akaike's Information Criterion),
-2 LL + 2 parms, where • -2LL is -2 times the log maximum likelihood (sometimes called the deviance) • parms is the number of parameters fitted in that model

That is, a penalty of 2 units of -2LL is paid for each additional parameter estimated. The model with the lowest AIC wins. Here -2LL can be replaced by the chi-square for testing the model, which is -2 ln(Wilks' lambda), where Wilks' lambda is Lreduced/Lfull . Another such criterion is BIC (Bayesian Information Criterion), due to Schwarz (1978). In BIC the penalty term is
(ln N) parms, that is, there is a penalty of ln N units of -2LL for each additional parameter estimated. Since for N > 8, ln N exceeds 2, the penalty in BIC is greater than the penalty in AIC, and SIC will tend to choose models with fewer parameters than does AIC. See Sclove (1987) for further discussion and some other applications.
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Exercises

1. For the system X1 --> Y, X2 --> Y, X1 X2

write down the path equations and their solution in terms of the three correlation coefficients.

2. For the system X --> Y, X --> T, T --> Y

write down the path equations and their solution in terms of the three correlation coefficients.

3. For the system F --> X1, F --> X2, write down the path equations and their solution in terms of the three correlation coefficients.

4. For the system F --> X1, F --> X2, F --> X3, write down the path equations and their solution in terms of the three correlation coefficients.

5. EXAMPLE OF A TWO-COMPONENT MODEL
In this exercise it is assumed that the three variables X, Y and Z obey a particular two-component model where the components are correlated. Then it is to be shown that the two correlated components can be replaced by two uncorrelated components. Suppose the three variables X, Y and Z can be expressed as linear combinations of the two variables V and W according to the following equations. • X = V + W • Y = V + 2W • Z = V + 3W
Suppose further that Var(V) = 9, Var(W) = 4, and Cov(V,W) = 3.

(a) Compute the coefficient of regression of W on V. [Hint: The regression coefficient beta is Cov(W,V)/Var(V).] (b) Find linear combinations F and G of V and W that are uncorrelated and have unit variance. [Hint: (W - beta*V) and V are uncorrelated.] (c) Express V and W in terms of F and G. (d) Express X, Y and Z as linear combinations of F and G.

References

General

Hair, J. F., Jr., R. E. Anderson, R. L. Tatham, and W. C. Black (1998). Multivariate Data Analysis. 5th ed. Prentice Hall. (A new edition comes out every six or seven years; a 6th ed. is coming out soon.)

Schumacker, Randall E., and Richard G. Lomax (2004). A Beginner's Guide to Structural Equation Modeling, 2nd ed. Lawrence Erlbaum Associates, Mahwah, NJ.
Model-Selection Criteria
Schwarz, Gideon (1978). Estimating the dimension of a model. Annals of Statistics 6, 461-464.
Sclove, Stanley L. (1987). Application of model-selection criteria to some problems in multivariate analysis. Psychometrika 52, 333-343.

Sears, Roebuck Case

FORTUNE Magazine. "Bringing Sears into the New World." Smart Managing Section. 13-Oct-1997. Stratford Sherman interviews Anthony Rucci. On line but will presumably disappear in time as space is needed for newer material.
Harvard Business School. "Sears, Roebuck and Company (A): Turnaround." Case #9-898-007. "Sears, Roebuck and Company (B): Transformation." Case #9-898-077. Rev. 3-February 3, 1998.
Rucci, Anthony J., Kirn, Steven P, and Quinn, Richard T. "The Employee-Customer-Profit Chain at Sears." Harvard Business Review, January-February 1998. Reprint 98109.

Software

AMOS Reference Manual. Available through SPSS.

Wolfle, Lee M., and Ethington, Corinna A. GEMINI: Program for Analysis of Structural Equations with Standard Errors of Indirect Effects. USER'S MANUAL. Manuscript, Dept. of Educational Psychology, College of Education, UIC. Professors Wolfle and Ethington were at UIC but moved to another university. GEMINI was available in the CMS operating system at UIC until it was closed on 31-December-1999.
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APPENDIX: Software

Here we mention some implementations of SEM.
LISREL (LInear Structural RELations) was perhaps the first implementation of SEM. It is available with SPSS. LISREL is the implementation of SEM that is discussed in Hair's book.
At UIC, LISREL 7 can be run on the mainframe through the LISREL command in SPSS. PC software has been developed to implement path analysis; among such software, in addition to LISREL, is EQS and a package from Systat.
GEMINI is a UIC-developed mainframe program for SEM which is user friendly. (The authors of GEMINI have left UIC. The IBM CMS Operating System was removed from the UIC mainframe on 31-December-1999.)
AMOS is a newer program for SEM. It is available through SPSS. It is perhaps more user-friendly than is LISREL.
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Created 1998: Oct 15 latest update 2005: Jan 15
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