Difference between standardized values

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Pierre-Charles Soulié

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Mar 10, 2026, 10:35:23 AM (22 hours ago) Mar 10

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Hi lavaan users,

I have a question regarding differences between a bifactor model and a second-order model estimated using the same covmat.

My covmat is specified to get a bifactor model with all loadings equal to 7.

When I estimated the bifactor on this covmat both my unstandardized and standardized values have a value of 7.

When I estimate a second-order model on the same covmat only the unstandardized values have the expected value of 7.

Why there is this difference with the second-order model ?

Thanks in advance for your helpful replies.

Hereunder is my syntax and the results.

library(lavaan)

lower <- '

1
0.98 1
0.98 0.98 1
0.49 0.49 0.49 1
0.49 0.49 0.49 0.98 1
0.49 0.49 0.49 0.98 0.98 1
0.49 0.49 0.49 0.49 0.49 0.49 1
0.49 0.49 0.49 0.49 0.49 0.49 0.98 1
0.49 0.49 0.49 0.49 0.49 0.49 0.98 0.98 1
0.49 0.49 0.49 0.49 0.49 0.49 0.49 0.49 0.49 1
0.49 0.49 0.49 0.49 0.49 0.49 0.49 0.49 0.49 0.98 1
0.49 0.49 0.49 0.49 0.49 0.49 0.49 0.49 0.49 0.98 0.98 1

'

model.cov <-
getCov(lower, names = c("y1", "y2", "y3", "y4",
"y5", "y6","y7", "y8","y9","y10", "y11", "y12"))

model_bif <- '

g =~ NA*y1 + y2 + y3 + y4 + y5 + y6 + y7 + y8 + y9+ y10 + y11 + y12
S1 =~ NA*y1 + y2 + y3
S2 =~ NA*y4 + y5 + y6
S3 =~ NA*y7 + y8 + y9
S4 =~ NA*y10 + y11 + y12

S1 + S2 + S3 + S4 ~~ 0*g
S1 + S2 + S3 ~~ 0*S4
S1 + S2 ~~ 0*S3
S1 ~~ 0*S2

g ~~ 1*g
S1 ~~ 1*S1
S2 ~~ 1*S2
S3 ~~ 1*S3
S4 ~~ 1*S4'

model_sec <- '

S1 =~ NA*y1 + y2 + y3
S2 =~ NA*y4 + y5 + y6
S3 =~ NA*y7 + y8 + y9
S4 =~ NA*y10 + y11 + y12
g =~ NA*S1 + S2 + S3 + S4

S1 + S2 + S3 + S4 ~~ 0*g
S1 + S2 + S3 ~~ 0*S4
S1 + S2 ~~ 0*S3
S1 ~~ 0*S2

g ~~ 1*g
S1 ~~ 1*S1
S2 ~~ 1*S2
S3 ~~ 1*S3
S4 ~~ 1*S4'

fit_bifactor <- sem(model_bif,
sample.cov = model.cov,
sample.nobs = 2000)
summary(fit_bifactor, fit.measures =TRUE, standardized = TRUE)

fit_second_order <- sem(model_sec,
sample.cov = model.cov,
sample.nobs = 2000)
summary(fit_second_order, fit.measures =TRUE, standardized = TRUE)

lavaan 0.6-21 ended normally after 137 iterations

Estimator ML
Optimization method NLMINB
Number of model parameters 36

Number of observations 2000

Model Test User Model:

Test statistic 0.000
Degrees of freedom 42
P-value (Chi-square) 1.000

Model Test Baseline Model:

Test statistic 56204.876
Degrees of freedom 66
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) -5946.085
Loglikelihood unrestricted model (H1) -5946.085

Akaike (AIC) 11964.171
Bayesian (BIC) 12165.803
Sample-size adjusted Bayesian (SABIC) 12051.429

Root Mean Square Error of Approximation:

RMSEA 0.000
90 Percent confidence interval - lower 0.000
90 Percent confidence interval - upper 0.000
P-value H_0: RMSEA <= 0.050 1.000
P-value H_0: RMSEA >= 0.080 0.000

Standardized Root Mean Square Residual:

SRMR 0.000

Parameter Estimates:

Standard errors Standard
Information Expected
Information saturated (h1) model Structured

Latent Variables:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
g =~
y1 0.700 0.022 31.797 0.000 0.700 0.700
y2 0.700 0.022 31.797 0.000 0.700 0.700
y3 0.700 0.022 31.797 0.000 0.700 0.700
y4 0.700 0.022 31.797 0.000 0.700 0.700
y5 0.700 0.022 31.797 0.000 0.700 0.700
y6 0.700 0.022 31.797 0.000 0.700 0.700
y7 0.700 0.022 31.797 0.000 0.700 0.700
y8 0.700 0.022 31.797 0.000 0.700 0.700
y9 0.700 0.022 31.797 0.000 0.700 0.700
y10 0.700 0.022 31.797 0.000 0.700 0.700
y11 0.700 0.022 31.797 0.000 0.700 0.700
y12 0.700 0.022 31.797 0.000 0.700 0.700
S1 =~
y1 0.700 0.015 45.191 0.000 0.700 0.700
y2 0.700 0.015 45.191 0.000 0.700 0.700
y3 0.700 0.015 45.191 0.000 0.700 0.700
S2 =~
y4 0.700 0.015 45.191 0.000 0.700 0.700
y5 0.700 0.015 45.191 0.000 0.700 0.700
y6 0.700 0.015 45.191 0.000 0.700 0.700
S3 =~
y7 0.700 0.015 45.191 0.000 0.700 0.700
y8 0.700 0.015 45.191 0.000 0.700 0.700
y9 0.700 0.015 45.191 0.000 0.700 0.700
S4 =~
y10 0.700 0.015 45.191 0.000 0.700 0.700
y11 0.700 0.015 45.191 0.000 0.700 0.700
y12 0.700 0.015 45.191 0.000 0.700 0.700

Covariances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
g ~~
S1 0.000 0.000 0.000
S2 0.000 0.000 0.000
S3 0.000 0.000 0.000
S4 0.000 0.000 0.000
S1 ~~
S4 0.000 0.000 0.000
S2 ~~
S4 0.000 0.000 0.000
S3 ~~
S4 0.000 0.000 0.000
S1 ~~
S3 0.000 0.000 0.000
S2 ~~
S3 0.000 0.000 0.000
S1 ~~
S2 0.000 0.000 0.000

Variances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
g 1.000 1.000 1.000
S1 1.000 1.000 1.000
S2 1.000 1.000 1.000
S3 1.000 1.000 1.000
S4 1.000 1.000 1.000
.y1 0.020 0.001 20.000 0.000 0.020 0.020
.y2 0.020 0.001 20.000 0.000 0.020 0.020
.y3 0.020 0.001 20.000 0.000 0.020 0.020
.y4 0.020 0.001 20.000 0.000 0.020 0.020
.y5 0.020 0.001 20.000 0.000 0.020 0.020
.y6 0.020 0.001 20.000 0.000 0.020 0.020
.y7 0.020 0.001 20.000 0.000 0.020 0.020
.y8 0.020 0.001 20.000 0.000 0.020 0.020
.y9 0.020 0.001 20.000 0.000 0.020 0.020
.y10 0.020 0.001 20.000 0.000 0.020 0.020
.y11 0.020 0.001 20.000 0.000 0.020 0.020
.y12 0.020 0.001 20.000 0.000 0.020 0.020

>
>
> fit_second_order <- sem(model_sec,
+ sample.cov = model.cov,
+ sample.nobs = 2000)
> summary(fit_second_order, fit.measures =TRUE, standardized = TRUE)
lavaan 0.6-21 ended normally after 133 iterations

Estimator ML
Optimization method NLMINB
Number of model parameters 28

Number of observations 2000

Model Test User Model:

Test statistic 0.000
Degrees of freedom 50
P-value (Chi-square) 1.000

Model Test Baseline Model:

Test statistic 56204.876
Degrees of freedom 66
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) -5946.085
Loglikelihood unrestricted model (H1) -5946.085

Akaike (AIC) 11948.171
Bayesian (BIC) 12104.996
Sample-size adjusted Bayesian (SABIC) 12016.038

Root Mean Square Error of Approximation:

RMSEA 0.000
90 Percent confidence interval - lower 0.000
90 Percent confidence interval - upper 0.000
P-value H_0: RMSEA <= 0.050 1.000
P-value H_0: RMSEA >= 0.080 0.000

Standardized Root Mean Square Residual:

SRMR 0.000

Parameter Estimates:

Standard errors Standard
Information Expected
Information saturated (h1) model Structured

Latent Variables:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
S1 =~
y1 0.700 0.015 45.746 0.000 0.990 0.990
y2 0.700 0.015 45.746 0.000 0.990 0.990
y3 0.700 0.015 45.746 0.000 0.990 0.990
S2 =~
y4 0.700 0.015 45.746 0.000 0.990 0.990
y5 0.700 0.015 45.746 0.000 0.990 0.990
y6 0.700 0.015 45.746 0.000 0.990 0.990
S3 =~
y7 0.700 0.015 45.746 0.000 0.990 0.990
y8 0.700 0.015 45.746 0.000 0.990 0.990
y9 0.700 0.015 45.746 0.000 0.990 0.990
S4 =~
y10 0.700 0.015 45.746 0.000 0.990 0.990
y11 0.700 0.015 45.746 0.000 0.990 0.990
y12 0.700 0.015 45.746 0.000 0.990 0.990
g =~
S1 1.000 0.043 23.126 0.000 0.707 0.707
S2 1.000 0.043 23.126 0.000 0.707 0.707
S3 1.000 0.043 23.126 0.000 0.707 0.707
S4 1.000 0.043 23.126 0.000 0.707 0.707

Covariances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.S1 ~~
g 0.000 0.000 0.000
.S2 ~~
g 0.000 0.000 0.000
.S3 ~~
g 0.000 0.000 0.000
.S4 ~~
g 0.000 0.000 0.000
.S1 ~~
.S4 0.000 0.000 0.000
.S2 ~~
.S4 0.000 0.000 0.000
.S3 ~~
.S4 0.000 0.000 0.000
.S1 ~~
.S3 0.000 0.000 0.000
.S2 ~~
.S3 0.000 0.000 0.000
.S1 ~~
.S2 0.000 0.000 0.000

Edward Rigdon

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Mar 10, 2026, 10:58:39 AM (22 hours ago) Mar 10

to lav...@googlegroups.com

What happened to the residual variances for the observed variables in
the second order case? What do you get if you ask for R2 in both
cases?

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Edward Rigdon

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Mar 10, 2026, 11:09:45 PM (10 hours ago) Mar 10

to lav...@googlegroups.com

In the second order model, the "variances" of the four first order
factors are residual variances, not total variances. When you refer to
the variances of the first order factors, you are actually referencing
the part of the variances not explained by the second order factor.
The dot '.' in front of the parameter estimate, in the output,
indicates this. You may need to code using nonlinear
constraints--label your parameters and then apply constraints so that
you actually constrain the first order variables total variances to 1,
not their residual variances.

Here is an example of what I mean. I wanted to constrain total
variances to 1 where there was a structural model with f1 predicting
f2 and f2 predicting f3. I achieved that by constraining the residual
variance of f2, labeled c, to be equal to 1 - the squared path from f1
to f2. Since I was constraining the variance of f1 to 1, this
constraint made the total variance of f2 also 1. And the same for f3.

model.std<-'
f1=~NA*y1+y2+y3
f2=~NA*y4+y5+y6
f3=~NA*y7+y8+y9
f2~a*f1
f3~b*f2
f1~~1*f1
f2~~c*f2
f3~~d*f3
c == 1 - a^2
d == 1 - b^2'

I think you can use a similar approach. Good luck.

On Tue, Mar 10, 2026 at 10:35 AM Pierre-Charles Soulié
<souli...@gmail.com> wrote:
>

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