1. Jiyoon An Kiran Pedada Structural Equation Modeling

2. Agenda Part 1 (Presented by Jiyoon An) - SEM and latent variable - Find a model from dataset Part 2 (Presented by Kiran Pedada) - SEM Structural model and measurement models - How to use Lavaan - Addressing missing values - Path Diagrams

3. Part 1 (Presented by Jiyoon An)

4. Structural equation modeling (SEM) Test and estimate the (causal) relationships among observable measures and non-observable theoretical (or latent) variables, and further to describe relationships between the latent variables themselves with directed arrows Source: http://davidakenny.net/

5. Why latent variable? A latent variable, a random variable, differs from a fixed process parameter Measuring a person’s characteristics (e.g. dominance) Everyone has a different level of dominance. Some are less dominant and some are more dominant We cannot measure dominance directly and need a latent variable Source: Borsboom, D., Mellenbergh, G. J., & Van Heerden, J. (2003), The theoretical status of latent variables, Psychological review, 110(2), 203.

6. Measuring ‘dominance’ by using latent variable Latent variable Manifest variables X 1 : “I would like a job where I have power over others” X 2 : “I would make a good military leader” X 3 : “I try to control others” Dominance XiXi Source: Borsboom, D., Mellenbergh, G. J., & Van Heerden, J. (2003), The theoretical status of latent variables, Psychological review, 110(2), 203.

7. When do you have a latent variable? A latent variable is defined as a random variable whose realizations cannot be observed directly Remind an example of “ROA” Assess of true measure against measurement error (e.g. age) Source: Borsboom, D. (2008), Latent variable theory, Measurement 6, 25-53, Howell, R. D. (2014), course materials from MKT 6355 Theory Testing

8. SEM case in point: Student evaluation Infer from data structure to variable structure How to conceptualize latent variables? What are their causal relationships? Source: Borsboom, D. (2008), Latent variable theory, Measurement 6, 25-53, Howell, R. D. (2014), course martials from MKT 6355 Theory Testing

9. How to conceptualize latent variables? Perceived instructor competence (R1, R3, R7, R8, R9, R10) Perceived instructor interaction (R6, R4, R5) Perceived course quality (R11, R12, R13, R14, R15, R16) R2 is removed

10. Factor analysis and SEM EFA - Find a latent variable which affects observed variables - Without prior assumption, all loadings are free to vary CFA - Some loadings are forced to be zero by the researcher - Factors are allowed to correlated - No direct arrows between factors (Measured model) SEM - Test and estimate the (causal) relationships

11. Where is latent variable? R1R2R3R4R5R6R7R8R9R10R11R12R13R14R15R16 F1 (Competence) F2 (Interaction) F3 (Course) Student 1 Student 2 … Student n Comp.Inter.Course Student n R1R1 R3R3 R7R7 R8R8 R9R9 R 10 R6R6 R4R4 R5R5 R 11 R 12 R 13 R 14 R 15 R 16 e1e1 e6e6 e7e7 e8e8 e9e9 e 10 e3e3 e4e4 e5e5 e 11 e 12 e 13 e 14 e 15 e 16

12. What are their causal relationships? Criteria for classifying an explanation as causal - Temporal sequentiality, nonspurious correlation, and common sense logic # of people of drowning and ice cream consumption Source: Hunt, S. D. (2010), Foundations of marketing theory: Toward a general theory of marketing, ME Sharpe

13. Applying criteria for choosing a model Latent variables: Perceived course quality, perceived instructor competence, and perceived instructor interaction Discussion: What are our DV(s) and IV(s)?

14. A model that does not make sense A student forms an opinion about interaction, which influences his/her opinion about competence, which in turn influences his/her opinion about course quality. Remember criteria of causality Course Inter. Comp.

15. A model that makes more sense A student forms his/her opinion on interaction and competence simultaneously, which influences perceived course quality Opinions on interaction and competence are correlated because they come from the same student How the instructor offers and what the instructor offers influence perceived quality of course Course Inter. Comp. Source: Grönroos, C. (1984), A service quality model and its marketing implications, European Journal of marketing, 18(4), 36-44.

16. Part 2 (Presented by Kiran Pedada)

17. SEM Structural Model SEM model for the case: Z = B z U + e z Here: Z is the endogenous latent variable, U is a (2x1) matrix of exogenous latent variables B z is a (1x2) matrix of coefficients of exogenous variables, e z is the error associated with the endogenous variable. Source: “Factor Analysis, Path Analysis, and Structural Equations Modeling”, Book extract, Jones and Bartlett publishers. http://www.jblearning.com/samples/0763755486/55485_CH14_Walker.pdf Note: The equation is taken from the above mentioned source. However, the symbols are changed for ease and convenience. Perceived Course Quality Perceived Interaction Perceived Competence Model

18. Exogenous Measurement Model Exogenous measurement model: X = B x U + e x Here: X is a (9 x 1) matrix of exogenous indicators, B x is a (9 x 2) matrix of coefficients from the exogenous variables to exogenous indicators, U is a (2 x 1) matrix of exogenous latent variables, e x is a (9 x 1) matrix for error associated with the exogenous indicators. Source: “Factor Analysis, Path Analysis, and Structural Equations Modeling”, Book extract, Jones and Bartlett publishers. http://www.jblearning.com/samples/0763755486/55485_CH14_Walker.pdf Note: The equation is taken from the above mentioned source. However, the symbols are changed for ease and convenience.

19. Exogenous Measurement Model X = B x U + e x

20. Endogenous Measurement Model Endogenous measurement model: Y = B y Z + e y Here: Y is a (6x1) matrix of endogenous indicators, B y is a (6x1) matrix of coefficients from the endogenous variable to endogenous indicators, Z is a (1x1) matrix of endogenous latent variable, e y is a (6x1) matrix for error associated with the endogenous indicators. Source: “Factor Analysis, Path Analysis, and Structural Equations Modeling”, Book extract, Jones and Bartlett publishers. http://www.jblearning.com/samples/0763755486/55485_CH14_Walker.pdf Note: The equation is taken from the above mentioned source. However, the symbols are changed for ease and convenience.

21. Y = B y Z + e y Endogenous Measurement Model

22. SEM and Analysis of Covariance SEM is based on the analysis of covariances Analysis of covariances allows for estimation of both standardized and unstandardized parameters Source: www.structuralequations.com/resources/SEM+Essentials.pps

23. Example of Analysis of Covariance Structure Source: www.structuralequations.com/resources/SEM+Essentials.pps Compare S denotes the observed covariances (typically the unstandardized covariances) ∑ denotes the model-implied covariances

24. R Packages for SEM – Non-commerical SEM Developer: John Fox (since 2001) For a long time, the only option in R Will not do multiple groups OpenMX Developer: Steven Boker (available at http://openmx.psyc.Virginia.edu/)http://openmx.psyc.Virginia.edu/ Very powerful All parts of OpenMX are open-source, except for the NPSOL optimizer, which is closed- source Somewhat idiosyncratic syntax Lavaan Developer: Yves Rosseel (http://lavaan.ugent.be/)http://lavaan.ugent.be/ First public release – May 2010. On 1 st Oct’14 version 0.5-17 has been released on CRANCRAN Uses a more compact notation that sem Will work on multiple groups Source: Rosseel, Yves. "lavaan: An R package for structural equation modeling."Journal of Statistical Software 48.2 (2012): 1-36. Source 2: https://personality-project.org/revelle/syllabi/454/wk6.lavaan.pdf

25. Why lavaan? A free, open-source for latent variable modeling Easy and intuitive to use Results are typically very close, to the results of Mplus Powerful, easy-to-use text-based syntax describing the model Fairly complete Source: Rosseel, Yves. "lavaan: An R package for structural equation modeling."Journal of Statistical Software 48.2 (2012): 1-36.

26. Data #Data Data = read.csv(file.choose(), header=T) attach(Data) #Responses 1 to 16 evals=as.matrix(cbind(RESP_1,RESP_2,RESP_3,RESP_4,RESP_5,RESP_ 6,RESP_7,RESP_8,RESP_9,RESP_10,RESP_11,RESP_12,RESP_13,RES P_14,RESP_15,RESP_16))

27. Formulae and Operators Formula type OperatorMnemonic Latent variable=~is manifested by Regression~is regressed on Covariance~~is correlated with Defined parameter: =is defined as Equality constraint==is equal to Inequality constraint<is smaller than Inequality constraint>is larger than Source: Rosseel, Yves. "lavaan: An R package for structural equation modeling."Journal of Statistical Software 48.2 (2012): 1-36.

28. Specifying the Model model <- ' # Defining the Latent Variables Competence =~ RESP_1 + RESP_3 + RESP_7 + RESP_8 + RESP_9 + RESP_10 Course =~ RESP_11 + RESP_12 + RESP_13 + RESP_14 + RESP_15 + RESP_16 Interaction =~ RESP_6 + RESP_4 + RESP_5 #Regression Course ~ Interaction + Competence #covariance of latent variables Interaction ~~ Competence '

29. Install Packages Install.packages(“lavaan”) Install.packages(“semplot”)

30. Running the Model require("lavaan") #Fitting the data fit <- sem(model, data = evals, missing = "FIML")

31. Dealing with Missing Values in Lavaan “listwise” - cases with missing data removed listwise (before analysis) “fiml” - the package offers estimation using all available data.This is also called “case-wise” maximum likelihood estimation. Source: http://cran.r-project.org/web/packages/lavaan/lavaan.pdf

32. Examining the Results #Examining the results summary(fit, fit.measure=TRUE, standardized = TRUE)

33. Examining the Results Used Total Number of observations 7828 7830 Number of missing patterns 92 Estimator ML Minimum Function Test Statistic 6068.046 Degrees of freedom 87 P-value (Chi-square) 0.000 Parameter estimates: Information Observed Standard Errors Standard

34. Examining the Results Estimate Std.err Z-value P(>|z|) Std.lv Std.all Latent variables: Competence =~ RESP_1 1.000 0.778 0.902 RESP_3 1.038 0.009 121.814 0.000 0.807 0.889 RESP_7 1.072 0.009 114.296 0.000 0.834 0.867 RESP_8 0.957 0.008 114.973 0.000 0.745 0.871 RESP_9 1.026 0.009 110.423 0.000 0.798 0.855 RESP_10 0.695 0.007 94.256 0.000 0.541 0.792 Course =~ RESP_11 1.000 0.853 0.869 RESP_12 0.971 0.009 110.946 0.000 0.829 0.891 RESP_13 0.947 0.009 107.388 0.000 0.808 0.879 RESP_14 0.766 0.008 90.252 0.000 0.654 0.805 RESP_15 0.829 0.009 90.857 0.000 0.707 0.808 RESP_16 0.890 0.010 88.775 0.000 0.760 0.795 Interaction =~ RESP_6 1.000 0.612 0.822 RESP_4 1.151 0.012 97.686 0.000 0.704 0.910 RESP_5 1.196 0.012 100.429 0.000 0.731 0.922 Regressions: Course ~ Interaction 0.075 0.019 4.059 0.000 0.054 0.054 Competence 0.929 0.016 56.843 0.000 0.847 0.847 Covariances: Competence ~~ Interaction 0.394 0.008 48.130 0.000 0.828 0.828

35. Plotting the SEM Path Diagram #SEM path diagram Require(“semplot”) # Plot input path diagram semPaths(fit,title=FALSE, curvePivot = TRUE, exoVar = FALSE, exoCov = FALSE) # Plot output path diagram with standardized parameters semPaths(fit, "std", edge.label.cex = 1.0, curvePivot = TRUE)

36. Input Path Diagram

37. Output Path Diagram

38. Relating to the Results Estimate Std.err Z-value P(>|z|) Std.lv Std.all

39. Relating to the Results Estimate Std.err Z-value P(>|z|) Std.lv Std.all

40. Relating to the Results Estimate Std.err Z-value P(>|z|) Std.lv Std.all

41. References Borsboom, D., Mellenbergh, G. J., & Van Heerden, J. (2003), The theoretical status of latent variables, Psychological review, 110(2), 203. Borsboom, D. (2008), Latent variable theory, Measurement 6, 25-53. Grönroos, C. (1984), A service quality model and its marketing implications, European Journal of marketing, 18(4), 36-44. Howell, R. D. (2014), course materials from MKT 6355 Theory Testing. Hunt, S. D. (2010), Foundations of marketing theory: Toward a general theory of marketing, ME Sharpe. Rosseel, Yves. "lavaan: An R package for structural equation modeling."Journal of Statistical Software 48.2 (2012): 1-36

42. Thank You