Fun With Structural Equation Modelling in Psychological Research

1. Fun WithStructural Equation Modellingin Psychological Research Jeremy Miles IBS, Derby University

2. Structural Equation Modelling • Analysis of Moment Structures • Covariance Structure Analysis • Analysis of Linear Structural Relationships (LISREL) • Covariance Structure Models • Path Analysis

3. Normal Statistics • Modelling process • What is the best model to describe a set of data • Mean, sd, median, correlation, factor structure, t-value Data Model

4. SEM • Modelling process • Could this model have led to the data that I have? Model Data

5. Theory driven process • Theory is specified as a model • Alternative theories can be tested • Specified as models Theory B Theory A Data

6. Ooohh, SEM Is Hard • It was. Now its not • Jöreskog and Sörbom developed LISREL • Matrices: lx qd ly qeY Fb G • Variables: X Yh x z • Intercepts: tk

7. The Joy of Path Diagrams Variable Causal Arrow Correlational Arrow

8. Doing “Normal” Statistics Correlation x y

9. Doing “Normal” Statistics T-Test x y

10. Doing “Normal” Statistics One way ANOVA (Dummy coding) x1 x2 y x3

11. Doing “Normal” Statistics Two- way ANOVA (Dummy coding) x1 x2 y x1 * x2

12. Doing “Normal” Statistics Regression x x y x

13. Doing “Normal” Statistics MANOVA y1 x1 y2 x2 y3

14. Doing “Normal” Statistics ANCOVA x y z

15. etc . . .

16. Identification • Often thought of as being a very sticky issue • Is a fairly sticky issue • The extent to which we are able to estimate everything we want to estimate

17. X = 4 Unknown: x

18. x = 4 y = 7 Unknown: x, y

19. x + y= 4 x - y = 1 Unknown: x, y

20. x + y = 4 Unknown: x, y

21. Things We Know = Things We Want to Know x=4 x + y = 4, x - y = 2 Just identified Can never be wrong “Normal” statistics are just identified

22. Things We Know < Things We Want to Know x + y = 7 Not identified Can never be solved

23. Things We Know > Things We Want to Know x + y = 4, x - y = 2, 2x - y = 3 over-identified Can be wrong SEM models areover-identified

24. Identification • We have information • (Correlations, means, variances) • “Normal” statistics • Use all of the information to estimate the parameters of the model • Just identified • All parameters estimated • Model cannot be wrong

25. Over-identification • SEM • Over-identified • The model can be wrong • If a model is a theory • Enables the testing of theories

26. Parameter Identification x - 2 = y x + 2 = y • Should be identified according to our previous rules • it’s not though • There is model identification • there is not parameter identification

27. Sampling Variation and c2 • Equations and numbers • Easy to determine if its correct • Sample data may vary from the model • Even if the model is correct in the population • Use the c2 test to measure difference between the data and the model • Some difference is OK • Too much difference is not OK

28. Simple Over-identification Estimate 1 parameter -just-identified x y Estimate 0 parameters -over-identified x y

29. Example 1 • Rab = 0.3, N = 100 • Estimate = 0.3, SE = 0.105, C.R. = 2.859 • The correlation is significantly different from 0 a b

30. a b • Model • Tests the hypothesis that the correlation in the population is equal to zero • It will never be zero, because of sampling variation • The c2 tells us if the variation is significantly different from zero

31. Example 2 • Test the model • Force the value to be zero • Input parameters = 1 • Parameters estimated = 0 • The model is now over-identified and can therefore be wrong a b

32. The program gives a c2 statistic • The significance of difference between the data and the model • Distributed with df = known parameters - input parameters • c2 = 9.337, df = 1 - 0 = 1, p = 0.002 • So what? A correlation of 0.3 is significant?

33. Hardly a Revelation • No. We have tested a correlation for significance. Something which is much more easily done in other ways • But • We have introduced a very flexible technique • Can be used in a range of other ways

34. 0.15 a b Testing Other Than Zero • Estimated parameters usually tested against zero • Reasonable? • Model testing allows us to test against other values • c2 = 2.3, n.s. • Example 3

35. Example 4: Comparing correlations • 4 variables • mothers' sensitivity • mothers' parental bonding • fathers' sensitivity • fathers' parental bonding • Does the correlation differ between mothers and fathers?

36. 0.1 M PB F S 0.2 0.3 0.5 0.2 M S F PB 0.1

37. Example 4a • analyse with all parameters free • 0 df, model is correct • Example 4b • fix FS-FPB and MS-MPB to be equal. • See if that model can account for the data

38. M PB F S dave dave M S F PB c2 = 1.82, df = 1 p = 0.177 dave = 0.41 (s.e. 0.08)

39. Latent Variables • The true power of SEM comes from latent variable modelling • Variables in psychology are rarely (never?) measured directly • the effects of the variable are measured • Intelligence, self-esteem, depression • Reaction time, diagnostic skill

40. Latent Measured Measuring a Latent Variable • Latent variables are drawn as ellipses • hypothesised causal relationship with measured variables • Measured variable has two causes • latent variable • “other stuff” • random error

41. x = t + e • Reliability is: • the square root of proportion of variance in x that is accounted • the correlation between x and e Error Measured True Score

42. Identification and Latent Variables • 1 measured variable • not (even close to) identified • 4 measured variables • 6 known, 4 estimated • model is identified

43. Need four measured variables to identify the model • Need to identify the variance of the latent variable • fix to 1

44. Why oh why oh why? • Why bother with all these tricky latent variables? • 2 reasons • unidimensional scale construction • attenuation correction

45. Unidimensionality 1.00 0.68 1.00 0.73 0.63 1.00 0.68 0.63 0.69 1.00 • Correlation matrix • c2 = 3.65, df = 2, p = 0.16

46. Attenuation Correction • Why bother? • Gets accurate measure of correlation between true scores • Why bother • theories in psychology are ordinal • attenuation can only cause relationships to lower

47. The Multivariate Case • Much more complex and unpredictable x1 y1 d e c a x2 y2 b

48. Some More Models • Multiple Trait Multiple Method Models (MTMM) • Temporal Stability • Multiple Indicator Multiple Cause (MIMIC)

49. MTMM • Multiple Trait • more than one measure • Multiple Method • using more than one technique • Variance in measured score comes from true score, random error variance, and systematic error variance, associated with the shared methods

50. What? • Example 6 (From Wothke, 1996) • Three traits • Getting along with others (G) • Dedication (D) • Apply learning (L) • Three methods • Peer nomination (PN) • Peer Checklist (PC) • Supervisor ratings (SC)