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Running head: CONFIRMATORY FACTOR ANALYSIS
Final Project 1: Confirmatory Factor Analysis
Michael J. Walk
University of Baltimore
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Final Project 1: Confirmatory Factor Analysis
PRELIS Syntax
!PRELIS SYNTAX: Can be edited
SY='E:\Class\Multivariate\Homework\Project 1\project1.PSF'
OU MA=KM SM=corr.dat SD=sd.dat XT XM
Correlation Matrix
var12
--------
var13
--------
var21
--------
var22
--------
var23
--------
var11
var12
var13
var21
var22
var23
var31
var32
var33
var11
-------1.000
0.298
0.314
0.172
0.138
0.053
0.347
0.197
0.229
1.000
0.408
0.279
0.376
0.284
0.083
0.160
0.273
1.000
0.148
0.127
0.221
0.283
0.212
0.368
1.000
0.316
0.244
0.430
0.131
0.278
1.000
0.206
0.153
0.173
0.077
1.000
0.205
0.101
0.255
var32
--------
var33
--------
var31
var32
var33
var31
-------1.000
0.201
0.331
1.000
0.241
1.000
var21
-------1.833
var22
-------1.904
var23
-------2.120
Standard Deviations
Standard Deviations
var11
-------2.199
var12
-------1.890
var13
-------2.028
var32
-------1.868
var33
-------1.943
Standard Deviations
var31
-------1.959
Comparison of SPSS and PRELIS Outputs
SPSS and PRELIS were used to calculate the standard deviations and correlations for
var11-33. The outputs from each program were identical. The outputs from SPSS and PRELIS
are included in Appendix A1 and A2, respectively.
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Model Testing
The following research examined the validity of three traits: problem solving (PS),
interpersonal skill (IS), and initiative (IN). Data were collected from 300 cases by using three
methods: group discussion (GD), competitive task (TK), and role-play (RP). There were nine
observed variables: var11, var12, var13, var21, var22, var23, var31, var32, and var33.
Confirmatory factor analyses were performed using LISREL to estimate all measurement
models.
Model 1
The first analysis was performed using the three hypothesized traits as factors; the three
methods were not included in the model. The tested model is presented as Figure 1. LISREL
output is included in Appendix B.
Model 1 did not adequately fit the data, 2(24, N = 300) = 107.75, p < .01, RMSEA =
.11, GFI = .93, and AGFI = .86, suggesting that the three traits were not the only latent variables
influencing the nine observed variables.
Model 2
A second model was tested that included six factors, the three traits as well as the three
methods. Correlations between traits and methods were fixed to zero. The tested model is
presented as Figure 2. LISREL output is included in Appendix C.
Support was found for Model 2, χ2(12, N = 300) = 25.46, p < .05, RMSEA = .06, GFI =
.98, and AGFI = .93. However, there were three problems in the model’s path coefficients. (The
following reported model coefficients are completely standardized.) The factor loading for var31
on role-play was negative (-.11), while all other variable loadings were positive. The coefficient
between role-play and competitive task was negative (-.02), while all other coefficients between
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latent variables were positive. In addition, var12 had a negative error variance (-2.23), while all
other error variances were positive.
Model 2-2
In order to improve upon Model 2, the analysis was performed with the error variance of
var12 constrained to equal the error variance of var32, which was a conceptually similar variable
since both were indicators of interpersonal skill. LISREL output is included in Appendix D.
The model fit was not substantially altered, χ2(13, N = 300) = 26.27, p < .05, RMSEA =
.06, GFI = .98, AGFI = .93. The completely standardized solution of Model 2-2 was examined to
determine if constraining the error variance of var12 to equal the error variance of var32 fixed
the problems present in Model 2. The negative factor loading of var31 on role-play was
improved; the value became 0.0. The negative path coefficient between role-play and
competitive task was no longer negative; however the value was extremely large: 36.97. The
error variance of var12 was not improved; both var12 and var32 had a negative error variance (.40 and -.41, respectively).
Model 3
In order to improve upon Model 2-2, the analysis was performed again; the error variance
of var12 was constrained to equal the error variance of var31, which had the smallest error
variance in Model 1. LISREL output is included in Appendix E.
The model fit remained relatively the same, χ2(13, N = 300) = 26.68, p < .05, RMSEA =
.06, GFI = .98, AGFI = .93. The completely standardized solution of Model 3 was examined to
determine if constraining the error variance of var12 to equal the error variance of var31
improved upon the previously identified problems. No improvement was found. The negative
factor loading was still present (-.66), the negative coefficient between role-play and competitive
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task was still present (-.04), and the negative error variance, although reduced, was still present
(var12 = -.22 and var31 = -.20).
Model 4
The consistent problems found in Models 2, 2-2, and 3 suggested that role-play was not a
valid measure of var31, so a new model was designed without the path between role-play and
var31. In this model, Model 4, the path between role-play and var31 was fixed to zero, and the
error variance of var12 was constrained to equal the error variance of var31.The model is
illustrated in Figure 3. The LISREL matrix language necessary to estimate the proposed paths in
the model is included below the figure and individual commands for each lambda-X are
presented on the corresponding model path. LISREL output is included in Appendix F.
The hypothesized model demonstrated sufficient fit, χ2(14, N = 300) = 31.05, p < .01,
RMSEA = .06, CFI = .97, GFI = .98, AGFI = .93. Also, examination of the completely
standardized solution revealed no problems in the model coefficients. All problematically
negative values were now positive.
Model 4 fit the data well (according to standardized fit indices). All factor loadings were
significant except for the loading of var12 on IS (B = .18, SE = .20, t = .91, ns), indicating that
var12 was not a good measure of interpersonal skill. However, the need to include methods in
the model in order to produce adequate model fit suggests that the variables measured in this
study included not only trait effects but also method effects.
Model 5
In order to examine the discriminate validity of the three traits, a confirmatory factor
analaysis was run with the correlations between the three traits fixed to 1.00. LISREL output is
included in Appendix G.
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The results indicated mediocre model fit, χ2(17, N = 300) = 49.69, p < .01, RMSEA =
.08, GFI = .96, AGFI = .91 A pseudo chi-square test was used to determine if there was a
significant difference between Model 4 (in which coefficients between traits were estimated) and
Model 5 (in which coefficients between traits were fixed to 1.00). The difference in degress of
freedom between Model 5 and Model 4 was 17 – 14 = 3. The difference in chi-square was 49.69
– 31.05 = 18.64. The critical chi-square for df = 3, α = .05, was 7.82. The psuedo chi-square
exceed the critical value; therefore, Model 4 fit the data significantly better than Model 5,
supporting the hypothesis that the three traits (problem solving, interpersonal skill, and initiative)
are different constructs.
Conclusion
Several models were examined through confirmatory factor analyses. There were nine
observed variables and six latent variables—three traits (problem solving, interpersonal skill, and
initiative) and three methods (group discussion, competitive task, and role-play). Indices of fit
for all five models are presented in Table 1. The traits only model (Model 1) lacked adquate data
fit. The traits and methods model (Model 2) was found to adequately fit the data; however, there
were problems in several of the model coefficients (i.e., three of them were negative). Several
attempts were made to fix the problematic coefficients (Models 2-2, 3, and 4), with only Model 4
having both adequate fit and no problematic coefficients. Model 4 was tested against Model 5 (in
which inter-trait correlations were fixed to 1.00) to demonstrate evidence of discriminant validity
for the three traits. Model 4 fit the data significantly better than Model 5, providing evidence of
trait validity. However, Model 4 did not fit the data better than Model 1, suggesting that the
traits, as well as the methods used, influenced participants’ scores on the nine observed variables.
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Table 1
Exercises (Methods) and Traits: Indices of Fit
df
2
RMSEA
GFI
AGFI
Model 1 (traits only)
24
107.75
.11
.93
.86
Model 2 (traits and methods)
12
25.46
.06
.98
.93
Model 3 (with rogue value)
13
26.68
.06
.98
.93
Model 4 (delete rogue LX)
14
31.05
.06
.98
.93
Model 5 (traits fixed to 1)
17
49.69
.08
.96
.91
Model
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Figure 1. Model 1: Nine observed variables influenced by three traits.
var11
PS
var12
var13
var21
IS
var22
var23
var31
IN
var32
var33
Figure 2. Model 2: nine observed variables influenced by three traits and three methods.
var11
GD
var12
PS
var13
var21
TK
var22
IS
var23
var31
RP
var32
var33
IN
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Figure 3. Model 4: Nine observed variables influenced by six latent variables. The LISREL
matrix language associated with each proposed lambda-X is included in the model.
var11
GD
var12
PS
var13
var21
TK
var22
IS
var23
var31
IN
RP
var32
var33
To free lambda-X’s:
FR LX(1,1) LX(4,1) LX(7,1) LX(2,2) LX(5,2) LX(8,2) LX(3,3) LX(6,3) LX(9,3) LX(1,4)
LX(2,4) LX(3,4) LX(4,5) LX(5,5) LX(6,5) LX(8,6) LX(9,6)
To fix lambda-X of RP and var31:
FI LX(7,6)
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