Confirmatory Factor Analysis (CFA)
Getting Started
Acitelli dataset consists of 148 married couples, average length of marriage being about 10 years.
Key parameters in CFA
* Factor loadings: the effect of the factor or latent variable on the measure
* Factor variance: how much individuals differ on the factor
* Factor mean: the average score on the factor
* Error variance: variance in the measure not attributable to the factor
* Factor correlation: how similar the factor is across the two members
* Error correlations: correlation on the same measure between the two members of the dyad
Unlike traditional factor analysis, the loadings are unstandardized (but they can be transformed into standardized loadings).
library(lavaan)## This is lavaan 0.5-23.1097## lavaan is BETA software! Please report any bugs.acitellid<- read.csv("acitellid.csv", header=TRUE)Distinguishable Dyads
Create two latent variables, one for husbands and one for wives.
Three indicators of each factor: Other Positivity (seeing the partner positively), Satisfaction, and Tension. To estimate the model with free loadings, there need to be three indicators. To be able to identify the model, i.e., have a unique solution, one measure is known as a marker variable and its loading is fixed to one. For this analysis, Other Positivity is chosen as the marker variable. The loadings for the other two indicators are free. With just two indicators normally the two loading must both be set to one.
With dyadic data, it is advisable to correlate the errors of the same measure across the two members.
Estimate first what is called the configural model to see if a single factor can be used to explain the covariation in the variables. The configural model does not have constraints across the two members. If the configural model is good fitting, we can proceed to estimate the dyadic model.
In lavaan to designate a loading, the code is xfact =~ x1 + x2. The variable x1 is designated as the marker and given a loading of one, and the variable x2 has a loading that is estimated.
Step 1: Configural Model
This model has no dyadic constraints. However, it needs to be determined that the model fits before estimating and testing the dyadic model.
cfadc.model <- "
ClosenessH =~ OtherPos_H + a1*Satisfaction_H + b1*Tension_H
ClosenessW =~ OtherPos_W + a2*Satisfaction_W + b2*Tension_W
ClosenessH ~ 1
ClosenessW ~ 1
OtherPos_H ~ 0*1
OtherPos_W ~ 0*1
Satisfaction_H ~ i11*1
Satisfaction_W ~ i12*1
Tension_H ~ i21*1
Tension_W ~ i22*1
ClosenessH ~~ ClosenessW
OtherPos_H ~~ OtherPos_W
Satisfaction_H ~~ Satisfaction_W
Tension_H ~~ Tension_W
"
cfadcc <- sem(cfadc.model,fixed.x=FALSE, data = acitellid,missing="fiml")
summary(cfadcc, fit.measures = TRUE)## lavaan (0.5-23.1097) converged normally after 115 iterations
##
## Number of observations 148
##
## Number of missing patterns 1
##
## Estimator ML
## Minimum Function Test Statistic 2.521
## Degrees of freedom 5
## P-value (Chi-square) 0.773
##
## Model test baseline model:
##
## Minimum Function Test Statistic 293.125
## Degrees of freedom 15
## P-value 0.000
##
## User model versus baseline model:
##
## Comparative Fit Index (CFI) 1.000
## Tucker-Lewis Index (TLI) 1.027
##
## Loglikelihood and Information Criteria:
##
## Loglikelihood user model (H0) -583.372
## Loglikelihood unrestricted model (H1) -582.112
##
## Number of free parameters 22
## Akaike (AIC) 1210.745
## Bayesian (BIC) 1276.684
## Sample-size adjusted Bayesian (BIC) 1207.062
##
## Root Mean Square Error of Approximation:
##
## RMSEA 0.000
## 90 Percent Confidence Interval 0.000 0.077
## P-value RMSEA <= 0.05 0.879
##
## Standardized Root Mean Square Residual:
##
## SRMR 0.018
##
## Parameter Estimates:
##
## Information Observed
## Standard Errors Standard
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## ClosenessH =~
## OtherPs_H 1.000
## Stsfctn_H (a1) 1.591 0.269 5.912 0.000
## Tension_H (b1) -1.558 0.274 -5.692 0.000
## ClosenessW =~
## OtherPs_W 1.000
## Stsfctn_W (a2) 1.649 0.300 5.502 0.000
## Tension_W (b2) -1.825 0.317 -5.759 0.000
##
## Covariances:
## Estimate Std.Err z-value P(>|z|)
## ClosenessH ~~
## ClosenessW 0.054 0.014 3.834 0.000
## .OtherPos_H ~~
## .OtherPos_W -0.002 0.016 -0.140 0.889
## .Satisfaction_H ~~
## .Satisfaction_W 0.011 0.016 0.698 0.485
## .Tension_H ~~
## .Tension_W 0.013 0.027 0.468 0.639
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|)
## ClosnssH 4.281 0.039 110.220 0.000
## ClosnssW 4.246 0.043 99.147 0.000
## .OthrPs_H 0.000
## .OthrPs_W 0.000
## .Stsfct_H (i11) -3.194 1.153 -2.769 0.006
## .Stsfct_W (i12) -3.410 1.274 -2.677 0.007
## .Tensin_H (i21) 9.011 1.174 7.678 0.000
## .Tensin_W (i22) 10.268 1.347 7.620 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .OtherPos_H 0.154 0.020 7.729 0.000
## .Satisfaction_H 0.037 0.021 1.795 0.073
## .Tension_H 0.259 0.037 7.024 0.000
## .OtherPos_W 0.195 0.025 7.690 0.000
## .Satisfaction_W 0.071 0.026 2.773 0.006
## .Tension_W 0.245 0.041 5.956 0.000
## ClosenessH 0.069 0.021 3.313 0.001
## ClosenessW 0.077 0.025 3.096 0.002Chi square is not significant (p = .773) and the RMSEA is equal to zero. Both indicate good fit.
Because this is a good fitting model, it makes sense to proceed to estimate the dyadic model.
Step 2: CFA with Dyadic Equality Constraints
Two sorts of equality constraints are made to ensure that the same factor is being estimated for the two members.
- Equal loadings: For the two non-marker variables, set loadings of the same indicator to be equal across members.
- Equal intercepts: For the two non-marker variables, set intercepts of the same indicator to be equal across members. Set the markers’ intercepts to zero.
Note that if there are k indicators, then there are 2(k -1) constraints over and above the configural model.
Most everyone accepts the idea that the loadings need to be the same across members to claim that the factor is the same. If the intercepts are not the same across members, one should not interpret the mean difference in the latent variables.
cfad.model <- "
ClosenessH =~ OtherPos_H + a*Satisfaction_H + b*Tension_H
ClosenessW =~ OtherPos_W + a*Satisfaction_W + b*Tension_W
ClosenessH ~ mH*1
ClosenessW ~ mW*1
OtherPos_H ~ 0*1
OtherPos_W ~ 0*1
Satisfaction_H ~ i1*1
Satisfaction_W ~ i1*1
Tension_H ~ i2*1
Tension_W ~ i2*1
ClosenessH ~~ vH*ClosenessH
ClosenessW ~~ vW*ClosenessW
ClosenessH ~~ ClosenessW
OtherPos_H ~~ OtherPos_W
Satisfaction_H ~~ Satisfaction_W
Tension_H ~~ Tension_W
mdiff := mH - mW
vdiff := vH - vW
"
cfad <- sem(cfad.model,fixed.x=FALSE, data = acitellid,missing="fiml")
summary(cfad, fit.measures = TRUE)## lavaan (0.5-23.1097) converged normally after 109 iterations
##
## Number of observations 148
##
## Number of missing patterns 1
##
## Estimator ML
## Minimum Function Test Statistic 8.804
## Degrees of freedom 9
## P-value (Chi-square) 0.456
##
## Model test baseline model:
##
## Minimum Function Test Statistic 293.125
## Degrees of freedom 15
## P-value 0.000
##
## User model versus baseline model:
##
## Comparative Fit Index (CFI) 1.000
## Tucker-Lewis Index (TLI) 1.001
##
## Loglikelihood and Information Criteria:
##
## Loglikelihood user model (H0) -586.514
## Loglikelihood unrestricted model (H1) -582.112
##
## Number of free parameters 18
## Akaike (AIC) 1209.029
## Bayesian (BIC) 1262.978
## Sample-size adjusted Bayesian (BIC) 1206.015
##
## Root Mean Square Error of Approximation:
##
## RMSEA 0.000
## 90 Percent Confidence Interval 0.000 0.091
## P-value RMSEA <= 0.05 0.708
##
## Standardized Root Mean Square Residual:
##
## SRMR 0.034
##
## Parameter Estimates:
##
## Information Observed
## Standard Errors Standard
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## ClosenessH =~
## OtherPos_H 1.000
## Satsfctn_H (a) 1.584 0.204 7.753 0.000
## Tension_H (b) -1.706 0.210 -8.129 0.000
## ClosenessW =~
## OtherPos_W 1.000
## Satsfctn_W (a) 1.584 0.204 7.753 0.000
## Tension_W (b) -1.706 0.210 -8.129 0.000
##
## Covariances:
## Estimate Std.Err z-value P(>|z|)
## ClosenessH ~~
## ClosenessW 0.055 0.014 3.916 0.000
## .OtherPos_H ~~
## .OtherPos_W -0.003 0.016 -0.171 0.864
## .Satisfaction_H ~~
## .Satisfaction_W 0.012 0.015 0.802 0.422
## .Tension_H ~~
## .Tension_W 0.005 0.027 0.188 0.851
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|)
## ClosenssH (mH) 4.280 0.033 131.335 0.000
## ClosenssW (mW) 4.247 0.035 122.610 0.000
## .OtherPs_H 0.000
## .OtherPs_W 0.000
## .Stsfctn_H (i1) -3.153 0.872 -3.615 0.000
## .Stsfctn_W (i1) -3.153 0.872 -3.615 0.000
## .Tension_H (i2) 9.706 0.897 10.821 0.000
## .Tension_W (i2) 9.706 0.897 10.821 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## ClosenssH (vH) 0.066 0.016 4.127 0.000
## ClosenssW (vW) 0.083 0.020 4.156 0.000
## .OtherPs_H 0.154 0.020 7.855 0.000
## .Stsfctn_H 0.044 0.019 2.313 0.021
## .Tension_H 0.256 0.038 6.805 0.000
## .OtherPs_W 0.193 0.025 7.723 0.000
## .Stsfctn_W 0.073 0.024 3.027 0.002
## .Tension_W 0.252 0.040 6.249 0.000
##
## Defined Parameters:
## Estimate Std.Err z-value P(>|z|)
## mdiff 0.033 0.022 1.478 0.139
## vdiff -0.017 0.013 -1.293 0.196parameterEstimates(cfad, standardized = TRUE)## lhs op rhs label est se z pvalue
## 1 ClosenessH =~ OtherPos_H 1.000 0.000 NA NA
## 2 ClosenessH =~ Satisfaction_H a 1.584 0.204 7.753 0.000
## 3 ClosenessH =~ Tension_H b -1.706 0.210 -8.129 0.000
## 4 ClosenessW =~ OtherPos_W 1.000 0.000 NA NA
## 5 ClosenessW =~ Satisfaction_W a 1.584 0.204 7.753 0.000
## 6 ClosenessW =~ Tension_W b -1.706 0.210 -8.129 0.000
## 7 ClosenessH ~1 mH 4.280 0.033 131.335 0.000
## 8 ClosenessW ~1 mW 4.247 0.035 122.610 0.000
## 9 OtherPos_H ~1 0.000 0.000 NA NA
## 10 OtherPos_W ~1 0.000 0.000 NA NA
## 11 Satisfaction_H ~1 i1 -3.153 0.872 -3.615 0.000
## 12 Satisfaction_W ~1 i1 -3.153 0.872 -3.615 0.000
## 13 Tension_H ~1 i2 9.706 0.897 10.821 0.000
## 14 Tension_W ~1 i2 9.706 0.897 10.821 0.000
## 15 ClosenessH ~~ ClosenessH vH 0.066 0.016 4.127 0.000
## 16 ClosenessW ~~ ClosenessW vW 0.083 0.020 4.156 0.000
## 17 ClosenessH ~~ ClosenessW 0.055 0.014 3.916 0.000
## 18 OtherPos_H ~~ OtherPos_W -0.003 0.016 -0.171 0.864
## 19 Satisfaction_H ~~ Satisfaction_W 0.012 0.015 0.802 0.422
## 20 Tension_H ~~ Tension_W 0.005 0.027 0.188 0.851
## 21 OtherPos_H ~~ OtherPos_H 0.154 0.020 7.855 0.000
## 22 Satisfaction_H ~~ Satisfaction_H 0.044 0.019 2.313 0.021
## 23 Tension_H ~~ Tension_H 0.256 0.038 6.805 0.000
## 24 OtherPos_W ~~ OtherPos_W 0.193 0.025 7.723 0.000
## 25 Satisfaction_W ~~ Satisfaction_W 0.073 0.024 3.027 0.002
## 26 Tension_W ~~ Tension_W 0.252 0.040 6.249 0.000
## 27 mdiff := mH-mW mdiff 0.033 0.022 1.478 0.139
## 28 vdiff := vH-vW vdiff -0.017 0.013 -1.293 0.196
## ci.lower ci.upper std.lv std.all std.nox
## 1 1.000 1.000 0.257 0.548 0.548
## 2 1.184 1.985 0.407 0.889 0.889
## 3 -2.118 -1.295 -0.439 -0.655 -0.655
## 4 1.000 1.000 0.287 0.547 0.547
## 5 1.184 1.985 0.455 0.860 0.860
## 6 -2.118 -1.295 -0.491 -0.699 -0.699
## 7 4.216 4.344 16.649 16.649 16.649
## 8 4.179 4.315 14.774 14.774 14.774
## 9 0.000 0.000 0.000 0.000 0.000
## 10 0.000 0.000 0.000 0.000 0.000
## 11 -4.863 -1.444 -3.153 -6.883 -6.883
## 12 -4.863 -1.444 -3.153 -5.957 -5.957
## 13 7.948 11.464 9.706 14.496 14.496
## 14 7.948 11.464 9.706 13.832 13.832
## 15 0.035 0.097 1.000 1.000 1.000
## 16 0.044 0.122 1.000 1.000 1.000
## 17 0.028 0.083 0.746 0.746 0.746
## 18 -0.033 0.028 -0.003 -0.015 -0.015
## 19 -0.018 0.042 0.012 0.217 0.217
## 20 -0.048 0.058 0.005 0.020 0.020
## 21 0.115 0.192 0.154 0.699 0.699
## 22 0.007 0.081 0.044 0.210 0.210
## 23 0.182 0.330 0.256 0.571 0.571
## 24 0.144 0.242 0.193 0.700 0.700
## 25 0.026 0.120 0.073 0.260 0.260
## 26 0.173 0.331 0.252 0.511 0.511
## 27 -0.011 0.076 1.875 1.875 1.875
## 28 -0.042 0.009 0.000 0.000 0.000anova(cfad,cfadcc)## Chi Square Difference Test
##
## Df AIC BIC Chisq Chisq diff Df diff Pr(>Chisq)
## cfadcc 5 1210.7 1276.7 2.5208
## cfad 9 1209.0 1263.0 8.8043 6.2835 4 0.179Conclusions
- Fit: chi square Difference = 6.2835 with 4 df, p = .179. The indication is good fit.
- Loadings: Values are large, but not too large. Standardized loadings are .548, .889, and -.655 for husbands and .547, .860, and -.699 for wives.
- Factor Means: Men have a slightly higher mean, but the difference is not significant (p = .139)
- Factor Variances: Women have slightly higher variance, but the difference is not significant (p = .196)
- Correlated Errors: None of the three correlated errors are statistically significant. They could be dropped from the model.
From CFA_D app:
https://davidakenny.shinyapps.io/CFA_D/
CFA Distinguishable
Two different models are estimated. Model 1 is the configural model in which a single-factor model is estimated for Men and Women but there are no additional constraints. The configural model is estimated with correlated errors between pairs of the same indicators. Model 2 is the proposed or specified model which places 4 constraints (invariances) on the configural model. The constraints Model 2 makes are: 1) equal loadings across members 2) equal intercepts across members 3) no correlated errors across members. The models are compared using the chi square p values.
The fit of the configural CFA model is not statistically significant (chi-square(5) = 2.52, p = .773), with an RMSEA of 0.000 and an SABIC of 42.838., The fit of the model is acceptable because the chi square test is not statistically significant. The fit of the specified CFA model is not statistically significant (chi-square(9) = 8.80, p = .456), with an RMSEA of 0.000 and an SABIC of 41.791., The fit of the model is acceptable because the chi square test is not statistically significant. The test of the constraints of the specified model compared to the configural model using a chi square difference test is not statistically significant (chi-square(4) = 6.28, p = .179).
Here are the results using the specified model are presented. As it should be, all of the loadings, error variances, and factor variances are statistically significant. The correlation of Closeness across members is .746 and is statistically significant (p < .001). Given the .746 correlation, dyad members are very similar to each other on Closeness. The two factor means are equal to 4.280 and 4.247.
Indistinguishable Dyads
Key differences between Indistinguishable and Distinguishable CFA * Set factor mean equal across members.
* Set factor variances equal across members. * Set measure intercepts equal across members.
* Set error variances equal across members.
Step 1: I-SAT Model
Following, Olsen & Kenny (2006) adjustments need to be made to chi square and the RMSEA. To so, the I-SAT or indistinguishable saturated model must be estimated. (can use the app Dingy to compute this)
This can be used to test to see if it is ok to treat dyads as indistinguishable if they are distinguishable. Or if they are truly indistinguishable, it can be used to adjust for arbitrary assignment to “first” and “second.”
The number of constraints for k variables in the ISAT model.
* k for the means
* k for the variances
* k(k - 1)/2 correlations within a member
* k(k - 1)/2 correlations between members
* k(k + 1) total
Note that for the APIM Indistinguishable Model, the df would be 2 x 3 = 6. For the CFA Model, it would be 3 x 4 = 12.
Step 1: ISAT Model
ISAT.model <- "
OtherPos_H ~ m1*1
OtherPos_W ~ m1*1
Satisfaction_H ~ m2*1
Satisfaction_W ~ m2*1
Tension_H ~ m3*1
Tension_W ~ m3*1
OtherPos_H ~~ OtherPos_W
Satisfaction_H ~~ Satisfaction_W
Tension_H ~~ Tension_W
OtherPos_H ~~ v1* OtherPos_H
Satisfaction_H ~~ v2*Satisfaction_H
Tension_H ~~ v3*Tension_H
OtherPos_W ~~ v1* OtherPos_W
Satisfaction_W ~~ v2*Satisfaction_W
Tension_W ~~ v3*Tension_W
OtherPos_H ~~ c12*Satisfaction_H
OtherPos_H ~~ c13*Tension_H
Satisfaction_H ~~ c23*Tension_H
OtherPos_ W ~~ c12*Satisfaction_W
OtherPos_W ~~ c13*Tension_W
Satisfaction_W ~~ c23*Tension_W
OtherPos_H ~~ cc12*Satisfaction_W
OtherPos_H ~~ cc13*Tension_W
Satisfaction_H ~~ cc23*Tension_W
OtherPos_W ~~ cc12*Satisfaction_H
OtherPos_W ~~ cc13*Tension_H
Satisfaction_W ~~ cc23*Tension_H
"
sem(ISAT.model,fixed.x=FALSE, data = acitellid,missing="fiml")## lavaan (0.5-23.1097) converged normally after 55 iterations
##
## Number of observations 148
##
## Number of missing patterns 1
##
## Estimator ML
## Minimum Function Test Statistic 17.293
## Degrees of freedom 12
## P-value (Chi-square) 0.139Dingy: “The model of complete indistinguishability is called the I-Sat model by Olsen and Kenny (2006; Psychological Methods, 11, 127-141) and that model has a chi square of 17.293 with 12 degrees of freedom.”
The chi square and its df are saved to measure fit of the indistinguishable CFA model. One subtracts off the ISAT chi square and df. Note in this case, because the dyad members can be distinguished by their gender, the non-significant chi square tells us that it makes sense statistically to treat them as if they were indistinguishable.
Step 2: Run the CFA
cfai.model <- "
ClosenessH =~ OtherPos_H + a*Satisfaction_H + b*Tension_H
ClosenessW =~ OtherPos_W + a*Satisfaction_W + b*Tension_W
ClosenessH ~ m*1
ClosenessW ~ m*1
OtherPos_H ~ 0*1
OtherPos_W ~ 0*1
Satisfaction_H ~ i1*1
Satisfaction_W ~ i1*1
Tension_H ~ i2*1
Tension_W ~ i2*1
ClosenessH ~~ v*ClosenessH
ClosenessW ~~ v*ClosenessW
ClosenessH ~~ ClosenessW
OtherPos_H ~~ OtherPos_W
Satisfaction_H ~~ Satisfaction_W
Tension_H ~~ Tension_W
OtherPos_H ~~ ev1* OtherPos_H
Satisfaction_H ~~ ev2*Satisfaction_H
Tension_H ~~ ev3*Tension_H
OtherPos_W ~~ ev1* OtherPos_W
Satisfaction_W ~~ ev2*Satisfaction_W
Tension_W ~~ ev3*Tension_W
"
cfai <- sem(cfai.model,fixed.x=FALSE, data = acitellid,missing="fiml")
summary(cfai, fit.measures = TRUE)## lavaan (0.5-23.1097) converged normally after 86 iterations
##
## Number of observations 148
##
## Number of missing patterns 1
##
## Estimator ML
## Minimum Function Test Statistic 18.948
## Degrees of freedom 14
## P-value (Chi-square) 0.167
##
## Model test baseline model:
##
## Minimum Function Test Statistic 293.125
## Degrees of freedom 15
## P-value 0.000
##
## User model versus baseline model:
##
## Comparative Fit Index (CFI) 0.982
## Tucker-Lewis Index (TLI) 0.981
##
## Loglikelihood and Information Criteria:
##
## Loglikelihood user model (H0) -591.586
## Loglikelihood unrestricted model (H1) -582.112
##
## Number of free parameters 13
## Akaike (AIC) 1209.172
## Bayesian (BIC) 1248.136
## Sample-size adjusted Bayesian (BIC) 1206.996
##
## Root Mean Square Error of Approximation:
##
## RMSEA 0.049
## 90 Percent Confidence Interval 0.000 0.099
## P-value RMSEA <= 0.05 0.467
##
## Standardized Root Mean Square Residual:
##
## SRMR 0.073
##
## Parameter Estimates:
##
## Information Observed
## Standard Errors Standard
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## ClosenessH =~
## OtherPos_H 1.000
## Satsfctn_H (a) 1.614 0.213 7.570 0.000
## Tension_H (b) -1.715 0.213 -8.052 0.000
## ClosenessW =~
## OtherPos_W 1.000
## Satsfctn_W (a) 1.614 0.213 7.570 0.000
## Tension_W (b) -1.715 0.213 -8.052 0.000
##
## Covariances:
## Estimate Std.Err z-value P(>|z|)
## ClosenessH ~~
## ClosenessW 0.053 0.014 3.810 0.000
## .OtherPos_H ~~
## .OtherPos_W -0.003 0.016 -0.161 0.872
## .Satisfaction_H ~~
## .Satisfaction_W 0.013 0.016 0.837 0.402
## .Tension_H ~~
## .Tension_W 0.004 0.027 0.134 0.893
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|)
## ClosenssH (m) 4.264 0.032 134.329 0.000
## ClosenssW (m) 4.264 0.032 134.329 0.000
## .OtherPs_H 0.000
## .OtherPs_W 0.000
## .Stsfctn_H (i1) -3.276 0.910 -3.600 0.000
## .Stsfctn_W (i1) -3.276 0.910 -3.600 0.000
## .Tension_H (i2) 9.744 0.910 10.712 0.000
## .Tension_W (i2) 9.744 0.910 10.712 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## ClosnssH (v) 0.073 0.017 4.324 0.000
## ClosnssW (v) 0.073 0.017 4.324 0.000
## .OthrPs_H (ev1) 0.175 0.016 10.773 0.000
## .Stsfct_H (ev2) 0.056 0.019 2.909 0.004
## .Tensin_H (ev3) 0.257 0.030 8.580 0.000
## .OthrPs_W (ev1) 0.175 0.016 10.773 0.000
## .Stsfct_W (ev2) 0.056 0.019 2.909 0.004
## .Tensin_W (ev3) 0.257 0.030 8.580 0.000parameterEstimates(cfai, standardized = TRUE)## lhs op rhs label est se z pvalue
## 1 ClosenessH =~ OtherPos_H 1.000 0.000 NA NA
## 2 ClosenessH =~ Satisfaction_H a 1.614 0.213 7.570 0.000
## 3 ClosenessH =~ Tension_H b -1.715 0.213 -8.052 0.000
## 4 ClosenessW =~ OtherPos_W 1.000 0.000 NA NA
## 5 ClosenessW =~ Satisfaction_W a 1.614 0.213 7.570 0.000
## 6 ClosenessW =~ Tension_W b -1.715 0.213 -8.052 0.000
## 7 ClosenessH ~1 m 4.264 0.032 134.329 0.000
## 8 ClosenessW ~1 m 4.264 0.032 134.329 0.000
## 9 OtherPos_H ~1 0.000 0.000 NA NA
## 10 OtherPos_W ~1 0.000 0.000 NA NA
## 11 Satisfaction_H ~1 i1 -3.276 0.910 -3.600 0.000
## 12 Satisfaction_W ~1 i1 -3.276 0.910 -3.600 0.000
## 13 Tension_H ~1 i2 9.744 0.910 10.712 0.000
## 14 Tension_W ~1 i2 9.744 0.910 10.712 0.000
## 15 ClosenessH ~~ ClosenessH v 0.073 0.017 4.324 0.000
## 16 ClosenessW ~~ ClosenessW v 0.073 0.017 4.324 0.000
## 17 ClosenessH ~~ ClosenessW 0.053 0.014 3.810 0.000
## 18 OtherPos_H ~~ OtherPos_W -0.003 0.016 -0.161 0.872
## 19 Satisfaction_H ~~ Satisfaction_W 0.013 0.016 0.837 0.402
## 20 Tension_H ~~ Tension_W 0.004 0.027 0.134 0.893
## 21 OtherPos_H ~~ OtherPos_H ev1 0.175 0.016 10.773 0.000
## 22 Satisfaction_H ~~ Satisfaction_H ev2 0.056 0.019 2.909 0.004
## 23 Tension_H ~~ Tension_H ev3 0.257 0.030 8.580 0.000
## 24 OtherPos_W ~~ OtherPos_W ev1 0.175 0.016 10.773 0.000
## 25 Satisfaction_W ~~ Satisfaction_W ev2 0.056 0.019 2.909 0.004
## 26 Tension_W ~~ Tension_W ev3 0.257 0.030 8.580 0.000
## ci.lower ci.upper std.lv std.all std.nox
## 1 1.000 1.000 0.270 0.542 0.542
## 2 1.196 2.032 0.436 0.879 0.879
## 3 -2.133 -1.298 -0.463 -0.675 -0.675
## 4 1.000 1.000 0.270 0.542 0.542
## 5 1.196 2.032 0.436 0.879 0.879
## 6 -2.133 -1.298 -0.463 -0.675 -0.675
## 7 4.201 4.326 15.795 15.795 15.795
## 8 4.201 4.326 15.795 15.795 15.795
## 9 0.000 0.000 0.000 0.000 0.000
## 10 0.000 0.000 0.000 0.000 0.000
## 11 -5.059 -1.492 -3.276 -6.610 -6.610
## 12 -5.059 -1.492 -3.276 -6.610 -6.610
## 13 7.961 11.526 9.744 14.199 14.199
## 14 7.961 11.526 9.744 14.199 14.199
## 15 0.040 0.106 1.000 1.000 1.000
## 16 0.040 0.106 1.000 1.000 1.000
## 17 0.026 0.080 0.728 0.728 0.728
## 18 -0.033 0.028 -0.003 -0.015 -0.015
## 19 -0.018 0.044 0.013 0.234 0.234
## 20 -0.050 0.057 0.004 0.014 0.014
## 21 0.143 0.207 0.175 0.706 0.706
## 22 0.018 0.093 0.056 0.227 0.227
## 23 0.198 0.315 0.257 0.545 0.545
## 24 0.143 0.207 0.175 0.706 0.706
## 25 0.018 0.093 0.056 0.227 0.227
## 26 0.198 0.315 0.257 0.545 0.545Conclusions * Healthy loadings
* No Heywood cases (negative error variances)
* Strong correlation between the two factors
* Weak correlated errors (could drop)
From CFA_I app:
CFA Distinguishable
The dyad members are treated as if they were indistinguishable. The test of distinguishability is not statistically significant (chi-square(12) = 17.29, p = .139), with an RMSEA of 0.055. Thus, the data are consistent with the hypothesis that members are indistinguishable. If dyad members are truly indistinguishable, these test only the non-random assignment of scores to Person 1 and Person 2. If, however, dyad members are distinguishable, these tests evaluate the assumption of indistinguishability. All chi square tests and RMSEAs are adjusted by this chi square and degrees of freedom, as described by Olsen and Kenny (2006).
The user has requested that the model with correlated errors be displayed in the table and figure. Table 2 provides the factor loadings, the error variances, and intercepts for each indicator of that model. Note that the modification indices for this model are contained in the computer output. As it should be, all of the loadings, error variances, and factor variances are statistically significant. The fit of the CFA model with correlated errors is not statistically significant (chi-square(2) = 1.66, p = .437), with an RMSEA of 0.000. Thus, the fit of the model with correlated errors is acceptable. The fit of the CFA model without correlated errors is not statistically significant (chi-square(5) = 2.59, p = .762), with an RMSEA of 0.000. Thus, the fit of the model without correlated errors is acceptable.
The correlation of Closeness across members is .728 and is statistically significant (p < .001). Given the .728 correlation, dyad members are very similar to each other on Closeness. The two factor means are equal to 4.264. Of the 3 correlated errors between the same variable from the two members of the dyad, none are statistically significant and 2 are positive. The overall test of the correlated errors using a chi square difference test is not statistically significant (chi-square(3) = 0.94, p = .816). Thus, correlated errors are not needed in the model.