David A. Kenny
December 9, 2013
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Longitudinal Models



Example Dataset
Dumenci, L., & Windle, M.  (1996).  A latent trait-state model of adolescent depression using the center for epidemiologic studies-depression scale.  Multivariate Behavioral Research, 31, 313-330. Download the data.
          Depression with four indicators (CESD)
                  PA: Positive Affect (lack thereof)
                  DA: Depressive Affect
                  SO: Somatic Symptoms
                  IN: Interpersonal Issues
Four times separated by 6 months
433 adolescent females
Age 16.2 at wave 1  

Topics on this page (click to go there)
          Single Indicator Models (at least 3 waves)
                  Autoregressive Model 
                             One Variable
                             Two Variables and Cross-causal Effects
                  STARTS Model (at least 4 waves) – at the end of this webpage 
          Multiple Indicator Models
                  One Construct
                             Saturated Model (at least 2 waves)
                                        Correlated Errors
                                        Invariance of Loadings
                                        Invariance of Error Variances
                                        Invariance of Latent Variances
                                        Invariance of Latent Means and Intercepts
                             Constraints on Latent Covariances
                                        Autoregressive Model (2 waves)
                                        Trait-State-Occasion Model (3 waves) – at the end of this webpage
                                        STARTS (4 waves) – at the end of this webpage  
                   Two Constructs: Cross-Causal Effects
                             Standard Model: Cross-lagged Path Model
                             Time Reversed Analysis
                             Cross-Lagged Panel Correlation (no cross-causal effects)
                    Measuring Change
                            Standard Model
                            Change Score Model

Single Indicator Variable: Autoregressive Model
One Variable


         

   Model
               Each factor has a single indicator (path set to one)
               Autoregressive factor
               Errors uncorrelated
    Identification
               error variances and stabilities identified for middle waves
               if the equal error variance assumption is made, model identified
                            must be at least three waves
                            with three waves, the model is saturated
                            over-identified with four or more waves
                   model identified by instrumental variable estimation
                            prior measure used as an instrumental variable to estimate error variance
   Parameters (PA example)
          Stability
                  unstandardized (can be larger than one)
                             b21 = .84, b32 = .91, b43  = .87
                  standardized (should not be larger than one)
                             b21 = .88, b32 = .96, b43  = .88
          Reliability (squared multiple correlation of the measure)
                    Time 1:  .641
                    Time 2:  .618
                    Time 3:  .596
                    Time 4:  .593
                  Note that there is a slight decline over time.
          Specification Error – χ²(2) = 2.175 (good fit)
                   Alternative models
                              model not autoregressive (free b31 and b42 or b41)
                              error variances not equal (V(E1) = V(E2) ≠ V(E3) = V(E4):  3.82 and 3.85)
   Typical Results
         generally low reliability and high stability because unstable true variance is treated as “error”
         unpublished example from 1977: 4 waves of uric acid
                   reliabilities of about .5
                   stability of .888
         meaning of “error” in this context
                   error as random change
                   that change may not be measurement error in the conventional sense of the term

Two Variables
           Structure
                   Set up single indicator model for both variables.
                   Cross-variable covariances
                             Time 1 latent variables
                             Other times correlated disturbances
                             Contemporaneous error covariances to be nonzero and equal at all t waves.
          Minimum number of waves to be identified: 3
          Submodels
                   Test to determine if cross-causal paths are all zero


  

Selected Models
 
Model

χ²

df

RMSEA

χ²diff

df diff

p

Comparison Model


I

Base Model

10.059

9

0.016

       

II

No Correlated Errors

56.662

10

0.104

46.603

1

<.001

I


III

No SO to DA Paths

19.345

12

0.038

9.286

 3

 .026

I


IV

No DA to SO Paths

10.291

12

0.000

0.232

3

.972

I


Indication of a path from SO to DA which is negative. 

Multiple Indicator Model (Single Construct)
It is necessary to test invariance in loadings over time
Required for
                    To claim that the latent variable is the “same” variable at each time:
                    Test for
                              Equal factor variance
                              Equal factor means
Almost always in the models there are correlated errors:  errors of the same measure at different times correlated.  Need to allow for such correlations.
Saturated Model: Correlated Errors, Equal Loadings and Error Variances
 
Model

χ²

df

RMSEA

χ² diff

df diff

p

Comparison Model


I

No Correlated Errors

856.729

98

0.135

       

II

Correlated Errors (CE)

107.718

74

0.032

749.010

24

>.001

I


III

CE and Equal Loadings (EL)

123.657

83

0.034

15.938

 9

 .068

II


IV

CE, EL, and Equal Error Variances

143.645

95

0.034

19.998

12

 .067

III


          Conclusions:
                    Correlated errors are definitely needed. 
Equal loadings and equal error variances are plausible.
Saturated Model: Equal Means and Variances
          Equal Variances
                   Allow all factors correlated (CFA: no paths between variables)
                   Force loadings to be equal over time (assuming it is reasonable to do so).
                   Force factor variances to be equal.
                   Optional to force equal error variances (will not be done)
                   Example
                             Chi square difference 9.762, 3 df, p = .021
                             Variances change over time (decline)
                                      25.34, 25.82, 21.72, 20.09
          Equal Factor Means or Intercepts
                   Saturated model
Free intercepts for each indicator except the marker (set it to zero)
                              Free the factor means (or intercept)
                   Intercept Invariance
Set intercepts of the indicator to the same value over time.
          except the marker (set it to zero)
                             Free factor means (or intercepts)
                                        6.03, 5.68, 5.13, 4.79
                                        Decline in mean depression over time
                   Equal Factor Means
                             Set the intercepts to the same value over time.
except the markers (set them to zero)
                             Set factor means equal.
Tests of Equal Means and Variances
 
Model

χ²

df

RMSEA

χ² diff

df diff

p     

Comparison Model


I

Base Model

123.657

83

.034

       

II

Equal Variances

133.419

86

.036

9.762

3

.021

I


III

Equal Intercepts, Unequal Latent Means

157.490

92

.041

33.833

9

<.001

I


IV

Equal Intercepts and Latent Means

182.935

95

.046

25.445

3

<.001

III


          Conclusion: Means and intercepts very different; variances somewhat different.
Autoregressive Model
Need only two waves and set loadings equal (must be plausible)
          Correlated errors
need at least 3 indicators per latent variable to be identified
with two indicators set loadings equal (both to one)
Over-Time Paths
          Autoregressive
                   first-order
                   more complicated
Example (more detail below)

Poor fit
Evidence for a second-order process
     

Estimate

S.E.

C.R.

p


S1

--->

S2

.621

.044

14.125

***


S2

--->

S3

.397

.050

7.920

***


S3

--->

S4

.309

.061

5.091

***


S1

--->

S3

.287

.051

5.634

***


S2

<---

S4

.304

.057

5.321

***


S1

<---

S4

-.013

.056

-.225

.822

 
Model

χ²

df

RMSEA

χ² diff

df diff

p

Comparison Model


I

Saturated Model with Equal Loadings

123.657

83

.034

       

II

Autoregressive

183.736

86

.051

60.079

3

>.001

I


Multiple Indicator Model with Multiple Constructs
          Same measures at each time
          Specification
                   Paths:  Autoregressive: Time t constructs cause time t + 1 constructs                    
                   Correlated errors: Same measure at different times
                   Equality
                             Loadings: same measure, different times
                             Error variances: same measure, different times
                             Paths  
                                      types
                                                stability (e.g., X1 à X2 = X2 à X3)
                                                causal (e.g., X1 à Y2 = X2 à Y3)
                                      trimming:  for 3 or more wave models, avoid dropping paths from one wave and not the other, e.g. trimming X1 to Y2 and keeping X2 to Y3

 Time-Reversed Analysis
From Campbell and Kenny, Primer on Regression Artifacts, Chapter 10
Flip times 1 and 2.
Take the paths in an over-time model and reverse their direction.  This is a nonsensical analysis, but a nonsensical analysis should give nonsense results.
If the results look essential the same as a regular analysis, then the regular analysis might be wrong.  Ideally, the cross-causal effects would not be negative and significant as they were in a regular analysis.

Models of Change
Standard Model
An exogenous variable, X, correlated with the Time 1 construct.
Time 1 and X cause the Time 2 construct with a temporally invariant measurement model.
Use two-wave autoregressive model.
          The two-wave autoregressive model is usually interpreted as measuring change, but it is not change, per se, but a form of residualized change.

Two-Wave Change Model
      Due to Raykov, T. (1992). Structural models for studying correlates and predictors of change. Australian Journal of Psychology, 44, 101-112.
          Two Factors
                   Baseline: Waves one and two measures load on this factor.
                   Change: Only wave two measures load on that factor.
                   Loadings of the same measure on the the different factors set equal.
                   Errors of the same measure correlated over time.  Need at least three measures at each time to be safely identifed; if just two, with both loadings set equal, the model is identified.
                   Note that X causes change, but Baseline does not.  Rather the disturbance in Change is correlated with the disturbance in Baseline.


          Results will be different from those in the autogressive model (Lord’s Paradox)
          Which one is right?
Use the STARTS model as a conceptual model and ask: What in the Baseline is X correlated with?
                             If X is correlated with all the components, the Standard Autogressive Model is best.
                             If is correlated with the Stable Trait only, then change score model is best.
Cross-Lagged Panel Correlation (CLPC)
A simple idea: the relative size of cross-lagged correlations reveals something about causal preponderance
         Rationale
Instead of assuming causal relationships between variables, assume that covariation is due to unspecified set of third variables.
Assume stationarity: a1 = a2 and b1 = b2


Standard causal model does not really contain a reasonable null model.
It presumes that covariance between measures is created by a third variable that has a zero autocorrelation.
Kenny and Campbell show that CLPC can be viewed as a multiplicative MTMM:
      Stationarity: Equal correlations at each time after adjustment for changes in reliability
      Temporal Erosion: Time lagged correlations less than synchronous correlations (the method correlations between times)
      Synchronous Correlations between Measures (the trait correlations)
      Causal Clues: Deviations between Observed and Predicted Covariances
Few researchers today use CLPC because they want to estimate and test a causal not a non-causal model.               

STARTS and TSO Models
STARTS (or TSE) Model – Univariate Models
      See Kenny, D. A., & Zautra, A.  (2001).  Trait-state models for longitudinal data.  In A. Sayer & L. M. Collins (Eds.), New methods for the analysis of change (pp. 243-263).  Washington, DC:  American Psychological Association. 
         ST (Trait) – Stable Trait: A latent variable that does not change
         ART (State) – Autoregressive Trait: A latent variable that changes slowly
         S (Error) – State: A latent variable that is random
 
         Model
                  stable trait (autoregressive factor of one)
                  autoregressive trait (less than one but greater than zero)
                  state (zero stability)
                            in single indicator model, error variance contained here
          Identification
                              at least 4 waves
                              autoregressive trait cannot be too stable or unstable
                              serious empirical under-identification issues
                                        need many waves (10 or so)
                                        large numbers of participants

                              force a complicated nonlinear constraint
                                        V(ARTt) = V(ARTt-1)(1 – pARTtARTt-12) + V(Ut)
                                        how to?
                                                  use a program that does so (e.g., Mplus or LISREL)
                                                  or iterate
                                                            in first run, fix V(ART1)
                  In next run V(ART1) to estimated V(ARTt) from the prior run
                              can also allow for variances to change over time
                                        paths from ST, ART, and S to the measure no longer set to 1
                                        one wave, marker wave (e.g., wave 1)
                                        other waves the three paths set to the same value (e.g., w2, w3, w4)
that path squared represents how much more variance that wave has relative to the marker wave
Example
            Trouble running
            Model for SO does run with unequal variance over time
            Variances: ST -- 1.892; ART -- 7.202; S -- 7.336
                               AR coefficient: .892                 
More of a theoretical than a practical model     
                        complicated (e.g., non-linear constraint)
                        estimation problems
                        anomalous results
                        e.g., for this data set we could not obtain a solution
                        still can be useful (Donnellan et al. Self Esteem Study)

Multivariate Models
Multiple Indicator STARTS Model


Trait-State-Occasion (TSO) Model
          Cole, Martin, & Steiger (2005) – Psychological Methods, 10, 3-20.
          Ormel & Schaufeli (1991) – JPSP, 60, 288-299.
                    Essentially the STARTS model with no state factors
                              Trait = Stable Trait
                              State = Autoregressive Trait
Suggested Changes
                              Invariance of factor loadings
                              Correlated measurement errors
                                        Stationarity of State variance V(S1)(1 – aa2) = V(Ut)
            Each model can allow for non-stationary variances
                              For each time create a latent variable.
Have the latent variables (T+S for TSO and ST, ART, S for STARTS) cause this latent variable
Fix those paths to be equal with time; for one time set all paths to 1
Parameter measures the change in standard deviation across time


 
Mplus Setup and Output for STARTS
 TITLE: Mplus run
        Multivariate STARTS Model for Windle Data – Females
        Note nonlinear constraints
        Note Stable Trait Variance is estimated to be zero    
  DATA: FILE is C:\MyDocuments\MyData\Windle\wincov.txt;
  type is covariance;
  nobservations=433;
  VARIABLE:  NAMES ARE da1 pa1 so1 in1 da2 pa2 so2 in2
              da3 pa3 so3 in3  da4 pa4 so4 in4;
             UseVAR = da1-in4;
  analysis:
  iterations = 1000;
  model:
  dep1 by Da1@1;dep1 by pa1 (b); dep1 by so1 (c); dep1 by in1 (d);
  dep2 by da2@1;dep2 by pa2 (b); dep2 by so2 (c); dep2 by in2 (d);
  dep3 by da3@1;dep3 by pa3 (b); dep3 by so3 (c); dep3 by in3 (d);
  dep4 by da4@1;dep4 by pa4 (b); dep4 by so4 (c); dep4 by in4 (d);
         STrait by DEP1@1; STrait by DEP2@1;
         STrait by DEP3@1 ; STrait by DEP4@1;
         State1 by DEP1@1; State2 by DEP2@1;
         State3 by DEP3@1 ; State4 by DEP4@1;
  dep1@0.0;dep2@0.0;dep3@0.0;dep4@0.0;
         ART1 by DEP1@1; ART2 by DEP2@1 ;
         ART3 by DEP3@1 ; ART4 by DEP4@1 ;
         ART2 on ART1*.5 (b1); ART3 on ART2 (b1); ART4 on ART3 (b1);
         ART1 ART2 ART3 ART4  with  STrait@0;
  STrait (stv);
  state1 (ev);state2 (ev);state3 (ev);state4 (ev);
                  art1 (v);
                  art2 (vv);
                  art3 (vv);
                  art4 (vv);
         State1 with State2@0; State1 with State3@0;State1 with State4@0;
         State2 with State3@0; State2 with State4@0; State3 with State4@0;
         ART1 ART2 ART3 ART4  with State1@0;
         ART1 ART2 ART3 ART4  with State2@0;
         ART1 ART2 ART3 ART4  with State3@0;
         ART1 ART2 ART3 ART4  with State4@0;
         state1 state2 state3 state4 with strait@0;
da1 with da2;da1 with da3;da1 with da4;da2 with da3;da2 with da4;da3 with da4;
pa1 with pa2;pa1 with pa3;pa1 with pa4;pa2 with pa3;pa2 with pa4;pa3 with pa4;
in1 with in2;in1 with in3;in1 with in4;in2 with in3;in2 with in4;in3 with in4;
so1 with so2;so1 with so3;so1 with so4;so2 with so3;so2 with so4;so3 with so4;
        model constraint:
            vv= v - v*b1*b1;
            stv>0;
            ev > 0;
            vv > 0;
  output:  sampstat stand;
INPUT READING TERMINATED NORMALLY
START Model for Windle Data
SUMMARY OF ANALYSIS
Number of groups                                                 1
Number of observations                                         433
Number of dependent variables                                   16
Number of independent variables                                  0
Number of continuous latent variables                           13
Observed dependent variables
  Continuous
   DA1         PA1         SO1         IN1         DA2         PA2
   SO2         IN2         DA3         PA3         SO3         IN3
   DA4         PA4         SO4         IN4
Continuous latent variables
   DEP1        DEP2        DEP3        DEP4        STRAIT      STATE1
   STATE2      STATE3      STATE4      ART1        ART2        ART3
   ART4
Estimator                                                       ML
Information matrix                                        EXPECTED
Maximum number of iterations                                  1000
Convergence criterion                                    0.500D-04
Maximum number of steepest descent iterations                   20
Input data file(s)
  C:\MyDocuments\MyData\Windle\wincov.txt
Input data format  FREE
THE MODEL ESTIMATION TERMINATED NORMALLY
TESTS OF MODEL FIT
Chi-Square Test of Model Fit
          Value                            144.574
          Degrees of Freedom                    89
          P-Value                           0.0002
Chi-Square Test of Model Fit for the Baseline Model
          Value                           4170.345
          Degrees of Freedom                   120
          P-Value                           0.0000
CFI/TLI
          CFI                                0.986
          TLI                                0.982
Loglikelihood
          H0 Value                      -15642.290
          H1 Value                      -15570.003
Information Criteria
          Number of Free Parameters             47
          Akaike (AIC)                   31378.580
          Bayesian (BIC)                 31569.905
          Sample-Size Adjusted BIC       31420.753
            (n* = (n + 2) / 24)
RMSEA (Root Mean Square Error Of Approximation)
          Estimate                           0.038
          90 Percent C.I.                    0.026  0.049
          Probability RMSEA <= .05           0.964
SRMR (Standardized Root Mean Square Residual)
          Value                              0.071
MODEL RESULTS
                   Estimates     S.E.  Est./S.E.    Std     StdYX
 DEP1     BY
    DA1                1.000    0.000      0.000    4.795    0.955
    PA1                0.410    0.016     25.483    1.968    0.625
    SO1                0.616    0.021     29.346    2.955    0.718
    IN1                0.174    0.008     22.547    0.837    0.551
 DEP2     BY
    DA2                1.000    0.000      0.000    4.795    0.966
    PA2                0.410    0.016     25.483    1.968    0.645
    SO2                0.616    0.021     29.346    2.955    0.727
    IN2                0.174    0.008     22.547    0.837    0.588
 DEP3     BY
    DA3                1.000    0.000      0.000    4.795    0.953
    PA3                0.410    0.016     25.483    1.968    0.652
    SO3                0.616    0.021     29.346    2.955    0.738
    IN3                0.174    0.008     22.547    0.837    0.563
 DEP4     BY
    DA4                1.000    0.000      0.000    4.795    0.954
    PA4                0.410    0.016     25.483    1.968    0.648
    SO4                0.616    0.021     29.346    2.955    0.746
    IN4                0.174    0.008     22.547    0.837    0.610
 STRAIT   BY
    DEP1               1.000    0.000      0.000    0.003    0.003
    DEP2               1.000    0.000      0.000    0.003    0.003
    DEP3               1.000    0.000      0.000    0.003    0.003
    DEP4               1.000    0.000      0.000    0.003    0.003
 STATE1   BY
    DEP1               1.000    0.000      0.000    0.561    0.561
 STATE2   BY
    DEP2               1.000    0.000      0.000    0.561    0.561
 STATE3   BY
    DEP3               1.000    0.000      0.000    0.561    0.561
 STATE4   BY
    DEP4               1.000    0.000      0.000    0.561    0.561
 ART1     BY
    DEP1               1.000    0.000      0.000    0.828    0.828
 ART2     BY
    DEP2               1.000    0.000      0.000    0.828    0.828
 ART3     BY
    DEP3               1.000    0.000      0.000    0.828    0.828
 ART4     BY
    DEP4               1.000    0.000      0.000    0.828    0.828
 ART2     ON
    ART1               0.868    0.032     26.853    0.868    0.868
 ART3     ON
    ART2               0.868    0.032     26.853    0.868    0.868
 ART4     ON
    ART3               0.868    0.032     26.853    0.868    0.868
 ART1     WITH
    STRAIT             0.000    0.000      0.000    0.000    0.000
    STATE1             0.000    0.000      0.000    0.000    0.000
    STATE2             0.000    0.000      0.000    0.000    0.000
    STATE3             0.000    0.000      0.000    0.000    0.000
    STATE4             0.000    0.000      0.000    0.000    0.000
 ART2     WITH
    STRAIT             0.000    0.000      0.000    0.000    0.000
    STATE1             0.000    0.000      0.000    0.000    0.000
    STATE2             0.000    0.000      0.000    0.000    0.000
    STATE3             0.000    0.000      0.000    0.000    0.000
    STATE4             0.000    0.000      0.000    0.000    0.000
 ART3     WITH
    STRAIT             0.000    0.000      0.000    0.000    0.000
    STATE1             0.000    0.000      0.000    0.000    0.000
    STATE2             0.000    0.000      0.000    0.000    0.000
    STATE3             0.000    0.000      0.000    0.000    0.000
    STATE4             0.000    0.000      0.000    0.000    0.000
 ART4     WITH
    STRAIT             0.000    0.000      0.000    0.000    0.000
    STATE1             0.000    0.000      0.000    0.000    0.000
    STATE2             0.000    0.000      0.000    0.000    0.000
    STATE3             0.000    0.000      0.000    0.000    0.000
    STATE4             0.000    0.000      0.000    0.000    0.000
 STATE1   WITH
    STATE2             0.000    0.000      0.000    0.000    0.000
    STATE3             0.000    0.000      0.000    0.000    0.000
    STATE4             0.000    0.000      0.000    0.000    0.000
    STRAIT             0.000    0.000      0.000    0.000    0.000
 STATE2   WITH
    STATE3             0.000    0.000      0.000    0.000    0.000
    STATE4             0.000    0.000      0.000    0.000    0.000
    STRAIT             0.000    0.000      0.000    0.000    0.000
 STATE3   WITH
    STATE4             0.000    0.000      0.000    0.000    0.000
    STRAIT             0.000    0.000      0.000    0.000    0.000
 STATE4   WITH
    STRAIT             0.000    0.000      0.000    0.000    0.000
 DA1      WITH
    DA2               -0.813    0.522    -1.557   -0.813   -0.033
    DA3               -0.349    0.527    -0.663   -0.349   -0.014
    DA4               -0.563    0.510    -1.105   -0.563   -0.022
 DA2      WITH
    DA3                0.270    0.518      0.521    0.270    0.011
    DA4               -0.187    0.497     -0.377   -0.187   -0.008
 DA3      WITH
    DA4               -0.550    0.516     -1.067   -0.550   -0.022
 PA1      WITH
    PA2                2.469    0.317      7.792    2.469    0.257
    PA3                2.068    0.306      6.746    2.068    0.218
    PA4                2.007    0.307      6.537    2.007    0.210
 PA2      WITH
    PA3                2.270    0.296      7.663    2.270    0.247
    PA4                2.409    0.300      8.032    2.409    0.260
 PA3      WITH
    PA4                2.481    0.298      8.311    2.481    0.271
 IN1      WITH
    IN2                0.551    0.078      7.109    0.551    0.255
    IN3                0.510    0.082      6.253    0.510    0.226
    IN4                0.275    0.070      3.903    0.275    0.132
 IN2      WITH
    IN3                0.430    0.074      5.827    0.430    0.203
    IN4                0.418    0.066      6.324    0.418    0.214
 IN3      WITH
    IN4                0.418    0.070      5.939    0.418    0.205
 SO1      WITH
    SO2                3.045    0.463      6.581    3.045    0.182
    SO3                3.243    0.456      7.117    3.243    0.197
    SO4                3.063    0.442      6.929    3.063    0.188
 SO2      WITH
    SO3                2.745    0.438      6.261    2.745    0.169
    SO4                2.758    0.428      6.437    2.758    0.171
 SO3      WITH
    SO4                3.086    0.428      7.211    3.086    0.195
 Variances
    STRAIT             0.000    0.063      0.003    1.000    1.000
    STATE1             7.227    0.844      8.561    1.000    1.000
    STATE2             7.227    0.844      8.561    1.000    1.000
    STATE3             7.227    0.844      8.561    1.000    1.000
    STATE4             7.227    0.844      8.561    1.000    1.000
    ART1              15.763    1.319     11.954    1.000    1.000
 Residual Variances
    DA1                2.194    0.787      2.787    2.194    0.087
    PA1                6.051    0.439     13.785    6.051    0.610
    SO1                8.202    0.640     12.811    8.202    0.484
    IN1                1.606    0.113     14.160    1.606    0.697
    DA2                1.631    0.717      2.275    1.631    0.066
    PA2                5.430    0.394     13.788    5.430    0.584
    SO2                7.784    0.601     12.946    7.784    0.471
    IN2                1.322    0.094     14.106    1.322    0.654
    DA3                2.316    0.749      3.094    2.316    0.092
    PA3                5.228    0.384     13.625    5.228    0.574
    SO3                7.291    0.578     12.611    7.291    0.455
    IN3                1.512    0.107     14.128    1.512    0.684
    DA4                2.254    0.726      3.103    2.254    0.089
    PA4                5.338    0.390     13.703    5.338    0.580
    SO4                6.976    0.555     12.571    6.976    0.444
    IN4                1.182    0.085     13.943    1.182    0.628
    DEP1               0.000    0.000      0.000    0.000    0.000
    DEP2               0.000    0.000      0.000    0.000    0.000
    DEP3               0.000    0.000      0.000    0.000    0.000
    DEP4               0.000    0.000      0.000    0.000    0.000
    ART2               3.881    0.980      3.960    0.246    0.246
    ART3               3.881    0.980      3.960    0.246    0.246
    ART4               3.881    0.980      3.960    0.246    0.246
R-SQUARE
    Observed
    Variable  R-Square
    DA1          0.913
    PA1          0.390


    SO1          0.516


    IN1          0.303


    DA2          0.934


    PA2          0.416


    SO2          0.529


    IN2          0.346


    DA3          0.908


    PA3          0.426


    SO3          0.545


    IN3          0.316


    DA4          0.911


    PA4          0.420


    SO4          0.556


    IN4          0.372


     Latent


    Variable  R-Square


    DEP1         1.000


    DEP2         1.000


    DEP3         1.000


    DEP4         1.000


    ART2         0.754


    ART3         0.754


    ART4         0.754


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