Bió Bió 2007 The Use of Structural Equation Modeling in Business Suzanne Altobello Nasco, Ph.D. Assistant Professor of Marketing Southern Illinois University

Bió Bió 2007 Examples of business research questions A CEO wants to examine how her company is perceived, relative to its competitors. She asks respondents to rate the similarity of every possible paired combination of firms to find out which competing firms are similar/dissimilar to her own company. A company is designing a new type of answering machine and wants to know which attributes are most important to consumers in the new product design. They present several product combinations to a focus group and ask respondents to rank order the product combinations. A stockbroker has 50 clients. He wants to organize these clients into groups based on the clients’ responses on several variables that measure risk tolerance, income, age, and years until retirement.

Bió Bió 2007 More business research questions The human resource department wants to predict whether a person should be hired or not, based on all available information from their job application. A firm is examining the effectiveness of its advertising and wants to know whether the type of publication (magazine vs. television show) and the nature of the publication (entertainment vs. news) affect attitudes towards the ad, the brand, and the company. An academic department wants to determine which variables (such as age, grade average and IQ) can differentiate between successful, moderately successful, and not successful students. WHAT DO THESE EXAMPLES HAVE IN COMMON? They all can be answered with MULTIVARIATE STATISTICS

Bió Bió 2007 Multivariate Statistics - Defined All statistical methods that simultaneously analyze multiple (more than 2) measurements on each individual or object under investigation. Multivariate statistics are an extension of univariate and bivariate statistics. Univariate = analyses of single variable distributions Bivariate = analyses of two variables where neither is an Independent Variable or Dependent Variable Multivariate = analyses of multiple I.V.s and D.V.s, all correlated with one another to varying degrees. In other words, their different effects cannot meaningfully be interpreted separately.

Bió Bió 2007 Basic Concepts in Multivariate Statistics The “VARIATE” = The building block of all multivariate statistical analyses A linear combination of variables with empirically determined weights Variate = w 1 X 1 + w 2 X 2 + …. + w n X n The variables (Xs) are specified by the researcher, the weights (ws) are determined by the multivariate technique to meet a specific objective. The result is a single value representing a combination of the entire set of variables that best achieves the goal of the specific multivariate test.

Bió Bió 2007 Important Decision: Variable Measurement The first consideration when choosing the appropriate multivariate method of analysis is how the researcher measured the variables. Two types of data: Non-metric / Qualitative: Categorical, DISCRETE values. If you are in one category, you can not be in the other (can’t be both male and female). Metric / Quantitative: Measured on a scale that changes values smoothly/continuously. Variables can take on any value within the range of the scale and the size of the number reflects the “amount”, “quantity”, “degree” or “magnitude” of the variable.

Bió Bió 2007 Determining the appropriate Multivariate Technique to use Must ask 3 questions of the data Can the variables be divided into independent and dependent variables (based on theory)? How many variables are dependent? How are the independent and dependent variables measured (metric or non-metric)? Answering these 3 questions will lead you to the appropriate multivariate technique to perform However, these questions WILL NOT relate the multivariate technique to your original questions or hypotheses of interest.

Bió Bió 2007 Examples of Interdependent Multivariate Techniques In interdependent techniques, there are no “independent” or “dependent” variables Instead, the researcher is looking for some structure in the data OR wants to reduce the number of variables in the analysis 3 primary interdependent techniques in business Factor analysis (reduce survey questions into fewer factors) Cluster analysis (group respondents or objects) Multidimensional Scaling (identify competitors)

Bió Bió 2007 Examples of Dependent Multivariate Techniques Variables divided into independent and dependent One Metric DV, ≥ 2 metric IVs Regression One Non-Metric DV (2 levels), ≥ 2 metric IVs Logistic Regression One Non-Metric DV (2 or more levels), ≥ 2 metric IVs Discriminant Analysis One metric DV, ≥ 1 categorical IV(s) Analysis of Variance (ANOVA) More than one metric DVs, ≥ 1 categorical IV(s) Multivariate Analysis of Variance (MANOVA)

Bió Bió 2007 Introducing Structural Equation Modeling WHAT is SEM? WHY should a business researcher use this tool? WHEN does a researcher use SEM? HOW does the researcher perform this analysis? HOW is an SEM analysis interpreted?

Bió Bió 2007 WHAT is Structural Equation Modeling? Structural Equation Modeling (SEM) is “a comprehensive statistical approach to testing hypotheses about relations among observed and latent variables” (Hoyle, 1995) SEM is an extension of several multivariate techniques Multiple regression, Factor analysis, Canonical Correlation, MANOVA, Mediational analysis Also called: simultaneous equation modeling analysis of covariance structures confirmatory factor analysis causal modeling causal analysis path analysis

Bió Bió 2007 WHY should business researchers use SEM? SEM can be used to test existing theories or help to develop new theories SEM can examine several dependent relationships simultaneously. Other bivariate & multivariate techniques can only examine one dependent variable at a time. SEM can test relationships between one or more IVs (either continuous or discrete) and one or more DVs (either continuous or discrete). Both IVs and DVs can be either previously- detected factors (via factor analysis) or can be measured variables (e.g., items on a survey).

Bió Bió 2007 WHEN should a researcher use SEM? When the researcher wants to estimate multiple and interrelated dependence relationships And has “a priori” theory When the researcher wants to represent unobserved (unmeasured or latent) concepts in these relationships When the researcher wants to account for any measurement error in the estimation process And has “multiple measures” for each latent construct

Bió Bió 2007 WHEN the researcher has any (or all!) of the following questions? Does the original data “fit” the proposed model? Does the data validate a particular theory? Or, does the data suggest a different theory? How good is my measurement model? Do my measured items represent the underlying latent construct reliably? How good is my structural model? Are all the proposed paths significant and in the predicted direction? Can some paths be removed without hurting the model? Is mediation present in my model? Do different groups (ex: small firms vs. large firms) need different models? How much “better” is the model for one group over another?

Bió Bió 2007 An Example Attitude Perceived Behavioral Control Subjective Norm Intention ATT 1 ATT 2 ATT 3 PBC 1 PBC 2 PBC 3 SN 1 SN 2 INT 1 INT 2 INT 3 Behavior Actual Behavior

Bió Bió 2007 HOW does a researcher perform SEM? Draw your proposed model by hand Pick a statistical package (LISREL, EQS, AMOS) Use the raw data or input a correlation / covariance matrix of all of your MEASURED (manifest) variables Within the program, draw your model precisely OR write lines of programming code that represent relationships in your model Run the model via the computer program Analyze the results

Bió Bió 2007 HOW does a researcher perform SEM? Model specification Drawing the Path Diagram to represent the measurement and structural models Identification counting parameters and degrees of Freedom Estimation of Model Evaluation of overall model fit Interpretation of the parameter estimates Model modifications Either those suggested by the computer program or suggested by competing theories Communicating SEM results

Bió Bió 2007 HOW to draw the proposed model? MODEL = A statistical statement about relationships among variables Undirected relationships: correlational Directed relationships: causal TWO parts of every SEM model:  “Structural Model” The underlying pattern of dependent relationships (among unobservable constructs)  “Measurement Model” The specific rules of correspondence between manifest and latent variables

Bió Bió 2007 An Example Attitude Perceived Behavioral Control Subjective Norm Intention ATT 1 ATT 2 ATT 3 PBC 1 PBC 2 PBC 3 SN 1 SN 2 INT 1 INT 2 INT 3 Behavior Actual Behavior

Bió Bió 2007 An Example Attitude Perceived Behavioral Control Subjective Norm Intention ATT 1 ATT 2 ATT 3 PBC 1 PBC 2 PBC 3 SN 1 SN 2 INT 1 INT 2 INT 3 Behavior Actual Behavior

Bió Bió 2007 HOW to draw the “Path Diagram”? Relationships between variables are indicated by lines Straight lines with one arrow: direct (causal) relationship between two variables Curved line with 2 arrows: correlational relationship between variables Measured variables: manifest variables or indicators that are represented by squares or rectangles ● Latent variables: constructs, factors, or unobserved variables that are represented by circles or ovals Factor 1 v1 Typically described as an item on a questionnaire; Denoted with all lowercase letters u1 Unique unobserved variable; typically used to represent measurement disturbance/error unique to the manifest variable it is affecting

Bió Bió 2007 More Modeling Terms to Know Exogenous constructs: not “caused’ or predicted by any other variables Like an independent variable in regression No arrows pointing to them Endogenous constructs: constructs predicted by one or more constructs; Like a dependent variable in regression Arrows in the path diagram lead “to” endogenous constructs

Bió Bió 2007 HOW to design an SEM study? Sample Size: SEM is a “large-sample technique” Consider “number of subjects per estimated parameter” (10 subjects per parameter) Usually want at least 200 subjects How many indicators (variables) should be used to represent each construct? Minimum=1, but 3 is the preferred minimum (allows for empirical estimation of reliability) with an upper limit of 5-7 Can use Correlation matrix OR Covariance matrix (among all measured variables) as input

Bió Bió 2007 HOW to evaluate SEM output? Chi Square:  2 (want value to be non-significant) For models with about 75 to 200 cases, this is a reasonable measure of fit. But for models with more cases, the chi square is almost always significant. Normed Chi Square:  2 /df (want between 1 and 2-3) Root Mean Square Error of Approximation (RMSEA) (want.05 or less) Takes an average of the residuals between the observed and estimated matrices Many “_FI” measures (want greater than.90) GFI, AGFI, CFI, Normed Fit Index (NFI), NNFI We want convergence on multiple fit indices to claim our model is “good”

Bió Bió 2007 An Example of a nested model Attitude Perceived Behavioral Control Subjective Norm Intention ATT 1 ATT 2 ATT 3 PBC 1 PBC 2 PBC 3 SN 1 SN 2 INT 1 INT 2 INT 3 Behavior Actual Behavior

Bió Bió 2007 How do you know your model is “right”? 1. CONFIRMATORY STRATEGY Researcher specifies a single model and SEM is used to assess its statistical significance All or nothing approach; confirmation bias 2. COMPETING MODELS STRATEGY Nested model: same number of constructs and indicators but number of estimated relationships (parameters) changes. Not all competing models are nested!!

Bió Bió 2007 HOW to make model modifications? Comparing alternate models Compare the  2 of “null” model with your current model (we WANT the difference to be significant, meaning your model is significantly better than null) Can also look at the  2 difference in “nested” models For non-nested models, compare AIC values (from EQS) Examining individual paths for model changes Use Lagrange Multiplier Test (LM) to see if model will improve with the addition of more parameters & use Wald Test (W) to determine if the model will improve if you remove a parameter Model modifications must be made judiciously, with respect to your original theory and the goal of SEM (theory-testing, exploration, confirmation)

Bió Bió 2007 HOW to use SEM to build theory? SEM is the only multivariate technique that is (almost) completely theory-driven If your Fit Indices are all good, your parameter estimates match your predictions, your structural model fits as predicted AND your measurement model is good, then you can say you have strong support for your model…..HOWEVER, There is no single “correct” model; no model is unique in the level of fit achieved For any model with an acceptable “fit”, there are a number of alternative models with the same level of model fit!

Bió Bió 2007 Several Important SEM Articles Kenny, David A. and Deborah A. Kashy (1992). Analysis of the Multitrait-Multimethod Matrix by Confirmatory Factor Analysis. Psychological Bulletin, 112(1), Bagozzi, Richard P. and Youjae Yi (1989). On the Use of Structural Equation Models in Experimental Designs. Journal of Marketing Research, 26 (August), Bagozzi, Richard P. (1978). Salesforce Performance and Satisfaction as a Function of Individual Difference, Interpersonal, and Situational Factors. Journal of Marketing Research, 15 (November), Fornell, Claes and David F. Larcker (1981). Evaluating Structural Equation Models with Unobservable Variables and Measurment Error. Journal of Marketing Research, 18 (February), MacCullum, Robert C. and James T. Austin (2000). Applications of Structural Equation Modeling in Psychological Research. Annual Review of Psychology, 51,

Bió Bió 2007 Several good SEM websites Ed Rigdon's (Department of Marketing, Georgia State University) SEM Frequently Asked Questions Dave Kenny's (Department of Psychology, University of Connecticut) SEM tutorial site Good introduction (manuals, tutorials) of LISREL program, maintained by University of Texas at Austin. Excellent LISREL site with tutorials, maintained by SSI Scientific Software International. Homepage for EQS software