o Kline, R. B. (2013b). Exploratory and confirmatory factor analysis. In Y. Petscher & C. Schatsschneider. (Eds.), Applied quantitative analysis in the social.
Structural equation modeling
o Rex B Kline Concordia
QICSS Set D
D1
CFA models
Resources o
o o
Bollen, K. A., & Hoyle, R. H. (2012). Latent variable models in structural equation modeling. In R. H. Hoyle (Ed.), Handbook of structural equation modeling (pp. 56–67). New York: Guilford. Fabrigar, L. R., & Wegener, D. T. (2012). Exploratory factor analysis. New York: Oxford University Press. Kline, R. B. (2013b). Exploratory and confirmatory factor analysis. In Y. Petscher & C. Schatsschneider (Eds.), Applied quantitative analysis in the social sciences (pp. 171–207). New York: Routledge.
D2
EFA o Phases: 1. 2. 3. 4.
Specification Extraction Retention Rotation D3
Extraction methods 1. Principle components analysis (PCA) 2. Principle axis factoring (PAF) 3. Alpha factoring 4. ML factoring D4
PCA X1
X2
A
X3
X4
X5
X6
B
D5
PAF E1
X1
X2
A
E2
X3
E3
E4
X4
X5
E5
E6
X6
B
D6
Indicator variance Unique
Common
Specific
Error
Systematic
1 − rXX
D7
EFA o Retention: No need to specify But best by theory D8
EFA o Retention: Parallel analysis Scree plots D9
4
Eigenvalue
3
2
1
0 1
2
3
4
5
6
7
8
Factor
D10
EFA o Rotation: 1. Orthogonal 2. Oblique D11
EFA o Orthogonal: 1. Varimax 2. Quartimax 3. Equamax D12
EFA o Oblique: 1. Promax 2. Oblimin D13
EFA o Rotation: Infinite Not identified D14
a) EFA (unrestricted; rotation)
E1
X1
E2
X2
E3
X3
E4
X4
E5
X5
A
E6
X6
B
b) CFA (restricted; no rotation)
E1 1
X1
E2 1
X2
1
E3
E4
1
X3
1
X4
E5 1
X5
E6 1
X6
1
A
B
D15
CFA after EFA o Does not “confirm” EFA: Restricted vs. unrestricted Items are “noisy” Follow EFA with EFA D16
CFA after EFA o
o
Osborne, J. W., & Fitzpatrick, D. C. (2012). Replication analysis in exploratory factor analysis: What it is and why it makes your analysis better. Practical Assessment, Research & Evaluation, 17. Retrieved from http://pareonline.net/pdf/ v17n15.pdf van Prooijen, J.-W., & van der Kloot, W. A. (2001). Confirmatory analysis of exploratively obtained factor structures. Educational and Psychological Measurement, 61, 777–792.
D17
D18
EG
1
1
Gender
EE
EA
1
Age
Ethnic
1
Background
D19
E1
X1
1
1
X2
1
E3
E2
1
+
X3
−
A
D20
CFA specification o Standard model: Continuous indicators (X) A→X←E D21
Reflective measurement
X =T+E 2 X
2 T
σ =σ +σ
rXX
2 E
2 T 2 X
σ = σ
D22
Reflective measurement
1− rXX but rXX estimates a single source
D23
CFA specification o Standard model: Independent E A
B D24
CFA specification o Unidimensional: Simple indicator (A → X only) No Ei
Ej D25
CFA specification o Unidimensional: Precise test Convergent validity Discriminant validity D26
CFA specification o Multidimensional: Complex indicator Ei
Ej D27
CFA specification o Ei
Ej: Indicators share something Repeated measures D28
CFA specification o Multidimensional caution: Increases complexity “Cheap” way to improve fit D29
CFA specification o Special variations: Hierarchical CFA MTMM models D30
E1 1
X1
E2 1
X2
E3
E4
1
X3
1
X4
1
E5 1
1
1
X5
X6
X7
E8 1
E9 1
X8
X9
1
1
VisualSpatial
Verbal
E7
E6
Memory 1
1
1
DVS
DVe
DMe
1
g
D31
Method 1
Method 2
1
Method 3 1
1
X1
X2
1
Trait 1
X3
X4
X5
1
X7
X6
X8
X9
1
Trait 2
Trait 3
D32
E1
E2
1
X1
E3
1
X2
1
E4 1
1
X3
E5
X4
1
X5
1
Trait 1
E6
E7 1
1
X6
E8
X7
E9
1
X8
1
X9
1
Trait 2
Trait 3
D33
CFA specification o
Eid, M., Nussbeck, F. W., Geiser, C., Cole, D. A., Gollwitzer, M., & Lischetzke, T. (2008). Structural equation modeling of multitrait-multimethod data: Different models for different types of methods. Psychological Methods, 13, 230–253.
D34
CFA identification o Necessary: dfM ≥ 0 Scale each latent D35
Scale E
ULI constraint:
D36
Scale factor 1. Reference (marker) variable ULI = 1, unstandardized 2. Standardize factors UVI = 1 3. Effects coding AVE = 1, all same metric D37
E1 1
1
X2
X1
E2
1
E3
E4
1
X3
1
1
X5
X4
E5
E6 1
X6
1
A
B
D38
E1 1
E2 1
X1
X2
1
A
E3
E4
1
X3
1
X4
E5 1
E6 1
X5
X6
B
1
D39
E1 1
X1
E2 1
X2
λ1 λ2
E3 1
X3
λ3
A
λ1 + λ 2 + λ 3 =1 3
D40
λ1 + λ 2 + λ 3 =1 3
λ1 = 3 − λ 2 − λ 3 λ 2 = 3 − λ1 − λ 3 λ 3 = 3 − λ1 − λ 2
D41
CFA identification o Counting parameters: 1. Exog:
Vars. + Covs.
2. Endog: Direct effects D42
CFA identification o Standard models: 1 factor, ≥ 3 indicators ≥ 2 factors, ≥ 2 indicators But… D43
CFA identification o Nonstandard models: No single heuristic Undecidable Ambiguous status D44
TABLE 6.1. Identification Rule 6.6 for Nonstandard Confirmatory Factor Analysis Models with Measurement Error Correlations
For a nonstandard CFA model with measurement error correlations to be identified, all three of the conditions listed next must hold: For each factor, at least one of the following must hold:
(Rule 6.6)
(Rule 6.6a)
1. There are at least three indicators whose errors are uncorrelated with each other. 2. There are at least two indicators whose errors are uncorrelated and either a. the errors of both indicators are not correlated with the error term of a third indicator for a different factor, or b. an equality constraint is imposed on the loadings of the two indicators. For every pair of factors, there are at least two indicators, one from each factor, whose error terms are uncorrelated.
(Rule 6.6b)
For every indicator, there is at least one other indicator (not necessarily of the same factor) with which its error term is not correlated.
(Rule 6.6c)
D45
(c)
1 X1
(d) EX1 1 X2
1
EX2 1 X3
X4
1 A
EX3 1
EX4 1 X1
X2
1 B
EX1 1
EX2 1 X3
EX3 1
EX4
X4
1 A
For each factor, at least one of the following must hold:
B
(Rule 6.6a)
1. There are at least three indicators whose errors are uncorrelated with each other. 2. There are at least two indicators whose errors are uncorrelated and either a. the errors of both indicators are not correlated with the error term of a third indicator for a different factor, or b. an equality constraint is imposed on the loadings of the two indicators.
D46
TABLE 6.2. Identification Rule 6.7 for Multiple Loadings of Complex Indicators in Nonstandard Confirmatory Factor Analysis Models and Rule 6.8 for Error Correlations of Complex Indicators
Factor loadings For every complex indicator in a nonstandard CFA model:
(Rule 6.7)
In order for the multiple factor loadings to be identified, both of the following must hold: 1. Each factor on which the complex indicator loads must satisfy Rule 6.6a for a minimum number of indicators. 2. Every pair of those factors must satisfy Rule 6.6b that each factor has an indicator that does not have an error correlation with a corresponding indicator on the other factor of that pair. Error correlations In order for error correlations that involve complex indicators to be identified, both of the following must hold: 1. Rule 6.7 is satisfied. 2. For each factor on which a complex indicator loads, there must be at least one indicator with a single loading that does not have an error correlation with the complex indicator.
(Rule 6.8)
D47
CFA estimates o Unstandardized: 1. Indicators loadings (B) 2. Factor, error variances 3. Factor, error covariances D48
CFA estimates o Standardized: 1. Indicators loadings (r, b) 2. Proportion unexplained 3. Factor, error correlations D49
CFA estimates o Failure to converge: 1. Data matrix (NPD) 2. Poor start values 3. Small N, 2 ind./factor D50
CFA estimates o Heywood cases (inadmissible): 1. Error variance < 0 2. | r or R2 | > 1.0 3. NPD parameter matrix D51
EMo
EFa 1
1
Father
Mother
EFM
EPr
1
FatherMother
1
Problems
EIn 1
Intimacy
1
1
Family of Origin
Marital Adjustment
D52
Group 2: Wives THETA-DELTA
problems
intimacy
problems -------520.305 (130.844) 3.977 - -
intimacy --------
-27.093 (104.927) -0.258 - -
father
- -
mother
- -
- -
fa_mo
- -
- -
father --------
mother --------
32.147 (29.214) 1.100 9.967 (26.870) 0.371 - -
63.416 (28.138) 2.254 - -
fa_mo --------
97.049 (25.232) 3.846
Squared Multiple Correlations for X - Variables problems -------0.520
intimacy -------1.052
father -------0.821
mother -------0.661
fa_mo -------0.531
D53
CFA estimates o Heywood causes: Identification Poor start values Small N, 2 inds./factor D54
CFA analysis o Testing strategy: 1. Fit 1-factor model 2. Nested under higher-order 3. Compare with χ
2 D
D55
EHM 1
1 Number Recall
Hand Movements
EWO
ENR
EGC
1 Word Order
1 Gestalt Closure
1
1 Spatial Memory
Triangles
EMA
ESM
EPS
1 Matrix Analogies
1 Photo Series
1
1
Sequential Processing
Simultaneous Processing
EHM 1 Hand Movements
ETr
EWO
ENR 1
Number Recall
EGC
1 Word Order
1
1 Gestalt Closure
ETr
Triangles
1 Spatial Memory
EMA
ESM
EPS
1 Matrix Analogies
1 Photo Series
1 General
D56
CFA analysis o Example: 4-factor model: 4 vs. 3 4 vs. 2 4 vs. 1 D57
CFA respecify o Options: 1. Number of factors 2. Indicator-factor match 3. Error correlations D58
CFA respecify o Residual patterns: Result Indicator has low standardized loading on original factor
Correlation residuals High correlation residuals with indicators of another factor
Indicator has reasonably High correlation residuals high standardized loading with indicators of another on original factor factor
Respecification Switch loading of indicator to other factor Allow indicator to also load on the other factor Allow measurement errors to covary
D59
CFA respecify o Wrong number of factors: Discriminant validity Convergent validity D60
CFA respecify o MIs in latent variable models: Approach with caution Nonsensical respecification May not be identified D61
EHM 1 Hand Movements
1 Number Recall
EWO
ENR 1 Word Order
EGC 1
Gestalt Closure
ETr 1
Triangles
EMA
ESM 1
Spatial Memory
EPS
1 Matrix Analogies
1 Photo Series
1
1
Sequential Processing
Simultaneous Processing
Observations = v (v + 1)/2 = 36 Parameters = 17 dfM = 19 D62
Exogenous variables
Direct effects on endogenous variables
Sequential → NR
Variances
Sequential → WO
Seq, Sim
Simultaneous → Tr
Simultaneous → SM
E terms (8)
Simultaneous → MA
Simultaneous → PS
Covariances Total
Seq
Sim
17
D63
Example o o o o o o
Amos EQS lavaan LISREL Mplus Stata D64
title: principles and practice of sem (4th ed.), rex kline two-factor model of the kabc-i, figure 9.7, table 13.1 data: file is "kabc-mplus.dat"; type is stdeviations correlation; nobservations = 200; variable: names are handmov numbrec wordord gesclos triangle spatmem matanalg photser; analysis: type is general; model: Sequent by handmov numbrec wordord; Simul by gesclos triangle spatmem matanalg photser ! first indicator in each list is automatically ! specified as the reference variable output: sampstat modindices(all, 0) residual standardized tech4; ! requests sample data matrix, residual diagnostics, ! modification indexes > 0, all standardized ! solutions (STDYX is reported), and estimated ! correlation matrix for all variables
D65
3.40 2.40 2.90 2.70 2.70 4.20 2.80 3.00 1.00 .39 1.00 .35 .67 1.00 .21 .11 .16 1.00 .32 .27 .29 .38 1.00 .40 .29 .28 .30 .47 1.00 .39 .32 .30 .31 .42 .41 1.00 .39 .29 .37 .42 .58 .51 .42 1.00
D66
CFA indicators o Indicators: Scale: Default ML Likert:
Other method D67
CFA indicators o Item distributions: 1. Binary (e.g., T / F) 2. Likert (3-6) 3. Likert (≥ 7) D68
CFA indicators o Estimation options: 1. Corrected ML: a. Robust SEs b. Santorra-Bentler D69
CFA indicators o Estimation options: 2. Robust WLS: a. Item thresholds b. Latent response variable D70
CFA indicators o Threshold: Location on latent dimension Differentiates categories Estimated as z D71
Example: 1 = disagree 2 = not sure 3 = agree
−1.62
1.15
D72
X3
X2
X1
EX * 1
X1*
EX *
EX *
1
1
X 2*
2
1
3
X 3*
A
D73
CFA indicators o Latent response variables: Sample polychoric Predicted polychoric Correlation residuals D74
CFA indicators o Estimation options: 3. ML + numerical integration a. ↑ computation b. Markov chain Monte Carlo D75
D76
CFA indicators o Estimation options: 4. IRT, ICC a. Difficulty, discrimination b. Logit, probit link D77
1.0
Probability of Correct Response
.9 .8 .7
.6
ICC
.5
difficulty
.4
tangent line
.3 .2 .1 0 −3.0
−2.0
−1.0
0
1.0
2.0
3.0
Latent Ability (θ)
D78
CFA indicators o Estimation options: 5. Bootstrapping: a. Very biased small N b. Not as developed D79
CFA indicators o Estimation options: 6. Create parcels: a. Homogenous item set b. Total score D80
It 1
It 2
●●●
It 33
1
It 34
It 35
It 66
1
Pr 2 (It 12–It 22)
1
It 68
B
Pr 3 (It 23–It 33)
Pr 4 (It 34–It 44)
Pr 5 (It 45–It 55)
1
A
It 67
●●●
It 99
1
A
Pr 1 (It 1–It 11)
●●●
C
Pr 6 (It 26–It 66)
Pr 7 (It 67–It 77)
Pr 8 (It 78–It 88)
Pr 9 (It 89–It 99)
1
B
C
D81
Cautions about parcels 1. Assumes unidimensional 2. Ways to parcel 3. Mask multidimensionality
D82
CFA indicators o
Edwards, M. C., Wirth, R. J., Houts, C. R., & Xi, N. (2012). Categorical data in the structural equation modeling framework. In R. Hoyle (Ed.), Handbook of structural equation modeling (pp. 195–208). New York: Guilford Press.
o
Wirth, R. J., & Edwards, M. C. (2007). Item factor analysis: Current approaches and future directions. Psychological Methods, 12, 58–79.
D83
CFA indicators o
Bernstein, I. H., & Teng, G. (1989). Factoring items and factoring scales are different: Spurious evidence for multidimensionality due to item categorization. Psychological Bulletin, 105, 467–477.
o
Bandalos, D. L., & Finney, S. J. (2001). Item parceling issues in structural equation modeling. In G. A. Marcoulides and R. E. Schumaker (Eds.), New developments and techniques in structural equation modeling (pp. 269–296). Mahwah, NJ: Erlbaum.
D84
Exploratory SEM o CFA-EFA-SR hybrid o Restricted + unrestricted o EFA part is rotated D85
E X1
1
X1
1
1 Y1
E X2 1 E X3 1
X2
EY1 Y2
1
EY2 Y3
1
E X5
1
E X6
1
1
EY4
1
Y5
Y4
EY5
1
EY6
Y6
A 1
1
X3 C
E X4
1
EY3
X4
F
1 DC
X5
1
DF
B
X6
D86
Exploratory SEM o Marsh, H. W., Morin, A. J. S., Parker, P. D., & Kaur, G. (2014). Exploratory structural equation modeling: Integration of the best features of exploratory and confirmatory factor analysis. Annual Review of Clinical Psychology, 10, 85– 110.
D87
D88
