1 General Structural Equation (LISREL) Models Week #2 Class #2

2 Today’s class Latent variable structural equations in matrix form (from yesterday) Latent variable structural equations in matrix form (from yesterday) Fit measures Fit measures SEM assumptions SEM assumptions What to write up What to write up LISREL matrices LISREL matrices

3 From yesterday’s lab: Reference indicator: REDUCE Regression Weights: Estimate S.E. C.R. Label REDUCE < Ach NEVHAPP < Ach NEW_GOAL < Ach IMPROVE < Ach ACHIEVE < Ach CONTENT < Ach

4 From yesterday’s lab: Reference indicator: REDUCE Standardized Regression Weights: Estimate REDUCE < Ach NEVHAPP < Ach NEW_GOAL < Ach IMPROVE < Ach ACHIEVE < Ach CONTENT < Ach

5 From yesterday’s lab: Reference indicator: REDUCE

6 Regression Weights: Estimate S.E. C.R. Label REDUCE < Ach NEVHAPP < Ach NEW_GOAL < Ach IMPROVE < Ach ACHIEVE < Ach CONTENT < Ach Standardized Regression Weights: Estimate REDUCE < Ach NEVHAPP < Ach NEW_GOAL < Ach IMPROVE < Ach ACHIEVE < Ach CONTENT < Ach

7 Solution: Use a different reference indicator (Note: REDUCE can be used as a reference indicator in a 2-factor model, though other reference indicators might be better because REDUCE is factorally complex)

8 When to add, when not to add parameters

9 Modification Indices Covariances:M.I. Par Change e1 Ach e1 Cont e6 Ach e5 Cont e5 e e4 e e4 e e3 e e2 e e2 e e2 e e2 e Discrepancy Degrees of freedom8 P

10 Regression Weights:M.I.Par Change REDUCE<--Ach REDUCE<--ACHIEVE REDUCE<--IMPROVE REDUCE<--NEW_GOAL CONTENT<--Ach CONTENT<--ACHIEVE CONTENT<--IMPROVE ACHIEVE<--REDUCE ACHIEVE<--NEVHAPP IMPROVE<--REDUCE IMPROVE<--CONTENT NEW_GOAL<--NEVHAPP NEVHAPP<--REDUCE NEVHAPP<--ACHIEVE NEVHAPP<--NEW_GOAL

11 Choice to add or not to add parameter from Ach1  REDUCE a matter of theoretical judgement. (Note changes in other parameters)

12 Goodness of Fit Measures in Structural Equation Models A Good Reference: Bollen and Long, TESTING STRUCTURAL EQUATION MODELS, Sage, 1993.

13 Goodness of Fit Measures in Structural Equation Models A fit measure expresses the difference between Σ(θ) and S. Using whatever metric it employs, it should register “perfect” whenever Σ(θ) = S exactly. This occurs trivially when df=0 0 to 1 usually thought of as best metric (see Tanaka in Bollen & Long, 1993)

14 Goodness of Fit Measures in Structural Equation Models Early fit measures: Model Χ 2 : Asks the question, is there a statistically significant difference between S and Σ ? If the answer to this question is “no”, we should definitely NOT try to add parameters to the model (capitalizing on change) If the answer to this question is “yes”, we can cautiously add parameters Contemporary thinking is that we need some other measure that is not sample-size dependent

15 Goodness of Fit Measures in Structural Equation Models Model Χ 2 : X 2 = (N-1) * F ml Contemporary thinking is that we need some other measure that is not sample-size dependent An issue in fit measures: “sample size dependency” (not considered a good thing) Chi-square is very much sample size dependent (a direct function of N)

16 Goodness of Fit Measures in Structural Equation Models Model Χ 2 : X 2 = (N-1) * F ml Contemporary thinking is that we need some other measure that is not sample-size dependent An issue in fit measures: “sample size dependency” (not considered a good thing) Chi-square is very much sample size dependent (a direct function of N)

17 Goodness of Fit Measures in Structural Equation Models Problem with Χ 2 itself as a measure (aside from the fact that it is a direct function of N): Logic of trying to “embrace” the null hypothesis. Even if chi-square not used, it IS important as a “cut off” (never add parameters to a model when chi-square is non-signif. Many measures are based on Χ 2

18 Goodness of Fit Measures in Structural Equation Models The “first generation” fit measures: Jöreskog and Sörbom’s Goodness of Fit Index (GFI) [LISREL] Bentler’s Normed Fit Index (NFI) [EQS] These have now been supplemented in most software packages with a wide variety of fit measures

19 Fit Measures GFI = 1 – tr[Σ -1 S – I] 2 tr (Σ -1 S) 2 Takes on value from 0 to 1 Conventional wisdom:.90 cutoff GFI tends to yield higher values than other coefficients GFI is affected by sample size, since in small samples, we would expect larger differences between Σ and S even if the model is correct (sampling variation is larger)

20 Fit Measures GFI is an “absolute” fit measure There are “incremental” fit measures that compare the model against some baseline. - one such baseline is the “Independence Model - Independence Model: models only the variances of manifest variables (no covariances) [=assumpt. all MVs independent] “Independence Model chi- square” (usually very large) - Σ will have 0’s in the off-diagonals

21 Fit Measures NFI = (Χ 2 b -Χ 2 m )/ Χ 2 b Normed Fit Index (Bentler) (subscript b = baseline m=model) Both NFI and GFI will increase as the number of model parameters increases and are affected by N (though not as a simple *N or *N-1 function). GFI = widely used in earlier literature since it was the only measure (along with AGFI) available in LISREL NFI (along with NNFI) only measure available in early versions of EQs

22 Fit Measures Thinking about fit indices: Desirable properties: 1.Normed (esp. to 0  1) Some measures only approx: TLI Arbitrary metric: AIC (Tanaka: AIC could be normed) 2.Not affected by sample size (GFI, NFI are) 3.“Penalty function” for extra parameters (no inherent advantage to complex models) – “Parsimony” indices deal with this 4.Consistent across estimation techniques (ML, GLS, other methods)

23 Fit Measures Bollens delta-2 (Χ 2 b – Χ 2 m )/ Χ 2 b – df m RMR – root mean residual (only works with standardized residuals) SRMR - standardized RMR Parsimony GFI 2df/p * (p+1) * GFI AGFI = 1 – [1(q+1) / 2df ] [1 – GFI] RNI (Relative Noncentrality Index) = [(Χ 2 b – df b ) – (X 2 m - df m )] / (Χ 2 b – df b ) CFI = 1 – max[(X 2 m - df m ),0] / max[(X 2 m - df m ), (X 2 b - df b ),0] RMSEA = sqrt (MAX[(X 2 m - df m ),(n-1),0) / df m

24 Fit Measures Some debate on conventional.90 criterion for most of these measures Hu & Bentler, SEM 6(1), 1999 suggest: Use at least 2 measures Use criterion of >.95 for 0-1 measure, <.06 for RMSEA or SRMR

25 SEM Assumptions F ml estimator: 1. No Kurtosis 2. Covariance matrix analysed * 3. Large sample 4. H0: S = Σ(θ) holds exactly

26 SEM Assumptions F ml estimator: 1. Consistent 2. Asymptotically efficient 3. Scale invariant 4. Distribution approximately normal as N increases

27 SEM Assumptions F ml estimator: Small Samples 1980s simulations: - Not accurate N< highly recommended - “large sample” usually in small samples, chi-square tends ot be too large

28 Writing up results from Structural Equation Models What to Report, What to Omit

29 Writing up results from Structural Equation Models Reference: Hoyle and Panter chapter in Hoyle. Reference: Hoyle and Panter chapter in Hoyle. Important to note that there is a wide variety of reporting styles (no one “standard”). Important to note that there is a wide variety of reporting styles (no one “standard”).

30 Writing up results from Structural Equation Models A Diagram A Diagram Construct Equation Model Construct Equation Model Measurement Equation model Measurement Equation model Some simplification may be required. Adding parameter estimates may clutter (but for simple models helps with reporting). Alternatives exist (present matrices).

31 Reporting Structural Equation Models “Written explanation justifying each path and each absence of a path” (Hoyle and Panter) “Written explanation justifying each path and each absence of a path” (Hoyle and Panter) (just how much journal space is available here? ) It might make more sense to try to identify potential controversies (with respect to inclusion, exclusion).

32 Controversial paths?

33 What to report and what not to report….. Present the details of the statistical model Present the details of the statistical model Clear indication of all free parameters Clear indication of all free parameters Clear indication of all fixed parameters Clear indication of all fixed parameters  It should be possible for the reader to reproduce the model 4. Describe the data 1. Correlations and standard errors (or covariances) for all variables ?? Round to 3-4 digits and not just 2 if you do this

34 What to report and what not to report… 4. Describing the data (continued) Distributions of the data Distributions of the data Any variable highly skewed? Any variable highly skewed? Any variable only nominally continuous (i.e., 5-6 discrete values or less)? Any variable only nominally continuous (i.e., 5-6 discrete values or less)? Report Mardia’s Kurtosis coefficient (multivariate statistic) Report Mardia’s Kurtosis coefficient (multivariate statistic) Dummy exogenous variables, if any Dummy exogenous variables, if any 5. Estimation Method If the estimation method is not ML, report ML results.

35 What to report and what not to report… 6. Treatment of Missing Data How big is the problem? How big is the problem? Treatment method used? Treatment method used? Pretend there are no missing data Pretend there are no missing data Listwise deletion Listwise deletion Pairwise deletion Pairwise deletion FIML estimation (AMOS, LISREL >=8.5) FIML estimation (AMOS, LISREL >=8.5) Nearest neighbor imputation (LISREL >=8.1) Nearest neighbor imputation (LISREL >=8.1) EM algorithm (covariance matrix imputation ) (LISREL >=8.5) EM algorithm (covariance matrix imputation ) (LISREL >=8.5)

36 What to report and what not to report… 7. Fit criterion Hoyle and Panter suggest “.90; justify if lower”. Hoyle and Panter suggest “.90; justify if lower”. Choice of indices also an issue. Choice of indices also an issue. There appears to be “little consensus on the best index” (H & P recommend using multiple indices in presentations) Standards: Bollen’s delta 2 (IFI) Comparative Fit Index RMSEA

37 Fit indices Older measures: Older measures: GFI (Joreskog & Sorbom) GFI (Joreskog & Sorbom) Bentler’s Normed Fit index Bentler’s Normed Fit index Model Chi-Square Model Chi-Square

38 What to report & what not to report…. 8. Alternative Models used for Nested Comparisons (if appropriate)

39 9. Plausible explanation for correlated errors [“these things were just too darned big to ignore”] Generally assumed when working with panel model with equivalent indicators across time: Generally assumed when working with panel model with equivalent indicators across time:

40 What to report 10. Interpretation of regression-based model Present standardized and unstandardized coefficients (usually) Present standardized and unstandardized coefficients (usually) Standard errors? (* significance test indicators?) Standard errors? (* significance test indicators?) R-square for equations R-square for equations Measurement model too? Measurement model too? (expect higher R-squares) (expect higher R-squares)

41 What to report. Problems and issues Problems and issues Negative error variances or other reasons for non-singular parameter covariance matrices Negative error variances or other reasons for non-singular parameter covariance matrices How dealt with? Does the final model entail any “improper estimates”? How dealt with? Does the final model entail any “improper estimates”? Convergence difficulties, if any Convergence difficulties, if any LISREL: can look at F ml across values of given parameter, holding other parameters constant LISREL: can look at F ml across values of given parameter, holding other parameters constant Collinearity among exogenous variables Collinearity among exogenous variables Factorially complex items Factorially complex items

42 What to report & what not to report…. General Model Limitations, Future Research issues: General Model Limitations, Future Research issues: Where the number of available indicators compromised the model Where the number of available indicators compromised the model 2-indicator variables? (any constraints required?) 2-indicator variables? (any constraints required?) Single-indicator variables? (what assumptions made about error variances?) Single-indicator variables? (what assumptions made about error variances?) Indicators not broadly representative of the construct being measured? Indicators not broadly representative of the construct being measured? Where the distribution of data presented problems Where the distribution of data presented problems Larger sample sizes can help Larger sample sizes can help

43 What to report & what not to report…. General Model Limitations, Future Research issues: General Model Limitations, Future Research issues: Missing data (extent of, etc.) Missing data (extent of, etc.) Cause-effect issues, if any (what constraints went into non-recursive model? How reasonable are these?) Cause-effect issues, if any (what constraints went into non-recursive model? How reasonable are these?)

44 Matrix form: LISREL M EASUREMENT MODEL MATRICES Manifest variables: X’s Measurement errors: DELTA ( δ ) Coefficients in measurement equations: LAMBDA ( λ ) Sample equation: X 1 = λ 1 ξ 1 + δ 1 MATRICES: LAMBDA-x THETA-DELTAPHI

45 Matrix form: LISREL M EASUREMENT MODEL MATRICES A slightly more complex example:

46 Matrix form: LISREL M EASUREMENT MODEL MATRICES Labeling shown here applies ONLY if this matrix is specified as “diagonal” Otherwise, the elements would be: Theta-delta 1, 2, 5, 9, 15. OR, using double-subscript notation: Theta-delta 1,1 Theta-delta 2,2 Theta-delta 3,3 Etc.

47 Matrix form: LISREL M EASUREMENT MODEL MATRICES While this numbering is common in some journal articles, the LISREL program itself does not use it. Two subscript notations possible: Single subscriptDouble subscript

48 Matrix form: LISREL M EASUREMENT MODEL MATRICES Models with correlated measurement errors:

49 Matrix form: LISREL M EASUREMENT MODEL MATRICES Measurement models for endogenous latent variables (ETA) are similar: Manifest variables are Ys Measurement error terms: EPSILON ( ε ) Coefficients in measurement equations: LAMBDA (λ) same as KSI/X side to differentiate, will sometimes refer to LAMBDAs as Lambda-Y (vs. Lambda-X) Equations Y 1 = λ 1 η 1 + ε 1

50 Matrix form: LISREL M EASUREMENT MODEL MATRICES Measurement models for endogenous latent variables (ETA) are similar:

51 LISREL MATRIX FORM An Example:

52 LISREL MATRIX FORM An Example:

53 LISREL MATRIX FORM An Example:

54 LISREL MATRIX FORM An Example: + theta-epsilon, 8 x 8 matrix with parameters in diagonal and 0s in off diagonals (a “diagonal” matrix)

55 Class Exercise #1 Provide labels for each of the variables

56 #2

57 #1 delta epsilon ksi eta zeta

58 #2

59 Lisrel Matrices for examples. No Beta Matrix in this model

60 Lisrel Matrices for examples.

61 Lisrel Matrices for examples (example #2)

62 Lisrel Matrices for examples (example #2)