Check multi-factor, higher-order, and bi-factor model

[Hyeseung Koh]

Hello, I have a series of question regarding a confirmatory factor analysis.

The factor was measured using a likert-type scale. 

Please check the notation of the R syntaxes below and let me know whether it is correct or incorrect. When incorrect, please also let me know how to correct. 

I examined the following three models, 

a multi-factor model with a syntax for it; Let's call it #model1

a higher-order model with 2 distinct syntaxes for it; Let's call it #model2 and #model3

a bi-factor model with 2 distinct syntaxes for it; Let's call it #model4 and #model5

#model1

m <- 'factor1 =~ the sum of factor1 items

factor2 =~ the sum of factor2 items

factor3 =~ the sum of factor 3 items

factor1 ~~ factor2 + factor3
factor2 ~~ factor3

factor1 ~~ 1*factor1
factor2 ~~ 1*factor2
factor2~~ 1*factor2'

#model2

m <- 'factor1 =~ the sum of factor1 items

factor2 =~ the sum of factor2 items

factor3 =~ the sum of factor 3 items

factor =~ 1*factor1 + 1*factor2+ 1*factor3

factor ~~ factor'

Reference: https://stats.oarc.ucla.edu/r/seminars/rcfa/#s2b

#model3

I also have an alternative higher-order model

m <- 'factor1 =~ the sum of factor1 items

factor2 =~ the sum of factor2 items

factor3 =~ the sum of factor 3 items

factor =~ NA*factor1 + factor2+ factor3

factor ~~ 1*factor'

Reference: https://blogs.baylor.edu/rlatentvariable/sample-page/r-syntax/#Bi-factor_model

#model4
m  <- 'factor1 =~ NA*item 1 and the sum of the factor1 items 
factor2 =~ NA*item 2 and the sum of the factor2 items
factor3 =~ NA*item 3 and the sum of the factor3 items 
factor =~ NA*item1+ the sum of the factor1-3 items

factor1 =~ 0*factor2+ 0*factor3+ 0*factor
factor2 =~ 0*factor3 + 0*factor
factor3 =~ 0*factor

factor1 ~~ 1* factor1
factor2 ~~ 1* factor2
factor3 ~~ 1* factor3
factor ~~ 1* factor'

Reference: https://www.frontiersin.org/articles/10.3389/fpsyg.2020.01357/full

#model5

I also have an alternative bi-factor model

m  <- 'factor1 =~ the sum of the factor1 items 
factor2 =~ the sum of the factor2 items; each factor loading is fixed by a
factor3 =~ the sum of the factor3 items; each factor loading is fixed by b
factor =~ the sum of the factor1-3 items'

Reference: https://blogs.baylor.edu/rlatentvariable/sample-page/r-syntax/#Import_data-3

fit <- cfa(m, data=name, std.lv=TRUE)
summary(fit, fit.measures=TRUE, standardized=TRUE)

[Daniel Morillo Cuadrado]

Hi Hyeseung, please find some comments below.

--

Daniel Morillo, Ph.D.

GitHub | ORCID

El vie, 1 dic 2023 a las 8:28, Hyeseung Koh (<thedinosa...@gmail.com>) escribió:

#model1

m <- 'factor1 =~ the sum of factor1 items

factor2 =~ the sum of factor2 items

factor3 =~ the sum of factor 3 items

factor1 ~~ factor2 + factor3
factor2 ~~ factor3

factor1 ~~ 1*factor1
factor2 ~~ 1*factor2
factor2~~ 1*factor2'

Correct

 

#model2

m <- 'factor1 =~ the sum of factor1 items

factor2 =~ the sum of factor2 items

factor3 =~ the sum of factor 3 items

factor =~ 1*factor1 + 1*factor2+ 1*factor3

factor ~~ factor'

Reference: https://stats.oarc.ucla.edu/r/seminars/rcfa/#s2b

The reference you take it from gives a specific example where you only have two first-order items. In that case, you need those constraints, but here you have a different case, as you have three first-order factors, so the loadings do not need (or should not, unless you have a good reason to do so) to be constrained to be equal. Therefore, it's better if you estimate them freely and fix the variance of the second-order factor instead, as you do in #model3.

#model3

I also have an alternative higher-order model

m <- 'factor1 =~ the sum of factor1 items

factor2 =~ the sum of factor2 items

factor3 =~ the sum of factor 3 items

factor =~ NA*factor1 + factor2+ factor3

factor ~~ 1*factor'

Reference: https://blogs.baylor.edu/rlatentvariable/sample-page/r-syntax/#Bi-factor_model

Correct

#model4
m  <- 'factor1 =~ NA*item 1 and the sum of the factor1 items 
factor2 =~ NA*item 2 and the sum of the factor2 items
factor3 =~ NA*item 3 and the sum of the factor3 items 
factor =~ NA*item1+ the sum of the factor1-3 items

factor1 =~ 0*factor2+ 0*factor3+ 0*factor
factor2 =~ 0*factor3 + 0*factor
factor3 =~ 0*factor

factor1 ~~ 1* factor1
factor2 ~~ 1* factor2
factor3 ~~ 1* factor3
factor ~~ 1* factor'

Reference: https://www.frontiersin.org/articles/10.3389/fpsyg.2020.01357/full

When you want to specify a bi-factor model, what you actually specify is that the factors are "uncorrelated". What you are specifying in the three `factor#=~'  sentences is that left-side factors do not "load" on the right-side one. The correct specification would be:

factor1 ~~ 0*factor2+ 0*factor3+ 0*factor
factor2 ~~ 0*factor3 + 0*factor
factor3 ~~ 0*factor

#model5

I also have an alternative bi-factor model

m  <- 'factor1 =~ the sum of the factor1 items 
factor2 =~ the sum of the factor2 items; each factor loading is fixed by a
factor3 =~ the sum of the factor3 items; each factor loading is fixed by b
factor =~ the sum of the factor1-3 items'

Reference: https://blogs.baylor.edu/rlatentvariable/sample-page/r-syntax/#Import_data-3

In this case, you are missing the specification that the factors need to be uncorrelated; the specification would be the same as in the previous model:

factor1 ~~ 0*factor2+ 0*factor3+ 0*factor
factor2 ~~ 0*factor3 + 0*factor
factor3 ~~ 0*factor

[Hyeseung Koh]

Hello, Daniel Morillo

I have checked your comment and explanation. 

I have two points to further check.

i) It is the CFA. I am not sure whether fixing the loadings in #model2 could be pre-determined. 

ii) Regarding #model4-5, are the 'by' and 'with' functions in Mplus applicable to lavaan? 

2023년 12월 1일 금요일 오전 3시 45분 26초 UTC-5에 danielmo...@gmail.com님이 작성:

[Daniel Morillo Cuadrado]

Hi again Hyeseung,

El vie, 1 dic 2023 a las 15:36, Hyeseung Koh (<thedinosa...@gmail.com>) escribió:

i) It is the CFA. I am not sure whether fixing the loadings in #model2 could be pre-determined. 

I don't understand your question I'm afraid, sorry. Do you mean that this is a CFA model? The syntax corresponds to a second-order CFA for what I can tell, and applying the changes I suggest is the same as estimating #model3.

 

ii) Regarding #model4-5, are the 'by' and 'with' functions in Mplus applicable to lavaan? 

'by' in lavaan is '=~' and 'with' is ' ~~', your lavaan syntax is correct in that respect.

 

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[Hyeseung Koh]

Hello Daniel Morillo,

It meant the Confirmatory Factor Analysis. The #model1-5 belong to the CFA. Please check my initial posting, my response to your initial response, and your follow-up response prior to this response. Please let me know whether you need further clarification. 

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