(Syntax) Equivalence second-order model and bifactor model with proportionality constraints
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Pedro Ribeiro
Jul 3, 2023, 2:23:43 AM7/3/23
to lavaan
Hi all,
There is a well-known equivalence between the second-order factor model and the bi-factor model with proportionality constraints:
Yung, Y. F., Thissen, D., & McLeod, L. D. (1999). On the relationship between the higher-order factor model and the hierarchical factor model. Psychometrika, 64, 113-128.
In this website, it is demonstrated how to display this equivalence using SAS syntax:
However, I have been unsure of how to impose these proportionality constraints using lavaan syntax and have not been able to find this information elsewhere. My bifactor model syntax is:
model_bi <- '
general =~ x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9
specific1 =~ x1 + x2 + x3
specific2 =~ x4 + x5 + x6
specific3 =~ x7 + x8 + x9
general ~~ 0*specific1
general ~~ 0*specific2
general ~~ 0*specific3
specific1 ~~ 0*specific2
specific1 ~~ 0*specific3
specific2 ~~ 0*specific3
'
general =~ x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9
specific1 =~ x1 + x2 + x3
specific2 =~ x4 + x5 + x6
specific3 =~ x7 + x8 + x9
general ~~ 0*specific1
general ~~ 0*specific2
general ~~ 0*specific3
specific1 ~~ 0*specific2
specific1 ~~ 0*specific3
specific2 ~~ 0*specific3
'
Does anyone know how to write this bifactor model with proportionality constraints so it is equivalent to the second-order factor model shown below?
second_model <- '
second =~ specific1 + specific2 + specific3
specific1 =~ x1 + x2 + x3
specific2 =~ x4 + x5 + x6
specific3 =~ x7 + x8 + x9
'
second =~ specific1 + specific2 + specific3
specific1 =~ x1 + x2 + x3
specific2 =~ x4 + x5 + x6
specific3 =~ x7 + x8 + x9
'
I tried the syntax below but it did not work:
model_cons <- '
general =~ (p1*X11)*x1 + (p1*X21)*x2 + (p1*X31)*x3 + (p2*X42)*x4 + (p2*X52)*x5 + (p2*X62)*x6 + (p3*X73)*x7 + (p3*X83)*x8 + (p3*X93)*x9
specific1 =~ X11*x1 + X21*x2 + X31*x3
specific2 =~ X42*x4 + X52*x5 + X62*x6
specific3 =~ X73*x7 + X83*x8 + X93*x9
general ~~ 0*specific1
general ~~ 0*specific2
general ~~ 0*specific3
specific1 ~~ 0*specific2
specific1 ~~ 0*specific3
specific2 ~~ 0*specific3
'
general =~ (p1*X11)*x1 + (p1*X21)*x2 + (p1*X31)*x3 + (p2*X42)*x4 + (p2*X52)*x5 + (p2*X62)*x6 + (p3*X73)*x7 + (p3*X83)*x8 + (p3*X93)*x9
specific1 =~ X11*x1 + X21*x2 + X31*x3
specific2 =~ X42*x4 + X52*x5 + X62*x6
specific3 =~ X73*x7 + X83*x8 + X93*x9
general ~~ 0*specific1
general ~~ 0*specific2
general ~~ 0*specific3
specific1 ~~ 0*specific2
specific1 ~~ 0*specific3
specific2 ~~ 0*specific3
'
Thanks in advance, any help is appreciated!
Cheers,
Pedro
Shu Fai Cheung (張樹輝)
Jul 3, 2023, 7:34:04 AM7/3/23
to lavaan
Dear Pedro,
I happened to be examining something similar a while ago. This is what I found in my computer, adapted to your case:
``` r
library(lavaan)
#> This is lavaan 0.6-15
#> lavaan is FREE software! Please report any bugs.
# Bifactor with proportionalitu constraints
model_bf <- '
visual =~ a1*x1 + a2*x2 + a3*x3
textual =~ a4*x4 + a5*x5 + a6*x6
speed =~ a7*x7 + a8*x8 + a9*x9
g =~ b1*x1 + b2*x2 + b3*x3 + b4*x4 + b5*x5 + b6*x6 + b7*x7 + b8*x8 + b9*x9
g ~~ 0*visual + 0*textual + 0*speed
visual ~~ 0*textual + 0*speed
textual ~~0*speed
lr1 := a1 / b1
lr2 := a2 / b2
lr3 := a3 / b3
lr4 := a4 / b4
lr5 := a5 / b5
lr6 := a6 / b6
lr7 := a7 / b7
lr8 := a8 / b8
lr9 := a9 / b9
lr1 == lr2
lr2 == lr3
lr4 == lr5
lr5 == lr6
lr7 == lr8
lr8 == lr9
'
fit_bf <- cfa(model_bf,
HolzingerSwineford1939,
std.lv = TRUE)
parameterEstimates(fit_bf)
#> lhs op rhs label est se z pvalue ci.lower ci.upper
#> 1 visual =~ x1 a1 -0.438 0.194 -2.256 0.024 -0.819 -0.058
#> 2 visual =~ x2 a2 -0.243 0.108 -2.252 0.024 -0.454 -0.031
#> 3 visual =~ x3 a3 -0.320 0.138 -2.325 0.020 -0.589 -0.050
#> 4 textual =~ x4 a4 0.842 0.064 13.251 0.000 0.718 0.967
#> 5 textual =~ x5 a5 0.938 0.071 13.293 0.000 0.799 1.076
#> 6 textual =~ x6 a6 0.780 0.060 13.084 0.000 0.663 0.897
#> 7 speed =~ x7 a7 0.522 0.066 7.908 0.000 0.393 0.651
#> 8 speed =~ x8 a8 0.616 0.067 9.129 0.000 0.484 0.748
#> 9 speed =~ x9 a9 0.564 0.064 8.808 0.000 0.439 0.690
#> 10 g =~ x1 b1 0.786 0.116 6.744 0.000 0.557 1.014
#> 11 g =~ x2 b2 0.435 0.090 4.827 0.000 0.258 0.611
#> 12 g =~ x3 b3 0.573 0.102 5.605 0.000 0.373 0.773
#> 13 g =~ x4 b4 0.520 0.088 5.936 0.000 0.348 0.691
#> 14 g =~ x5 b5 0.578 0.097 5.940 0.000 0.388 0.769
#> 15 g =~ x6 b6 0.481 0.081 5.918 0.000 0.322 0.641
#> 16 g =~ x7 b7 0.334 0.066 5.026 0.000 0.204 0.464
#> 17 g =~ x8 b8 0.394 0.073 5.432 0.000 0.252 0.536
#> 18 g =~ x9 b9 0.361 0.069 5.259 0.000 0.226 0.496
#> 19 visual ~~ g 0.000 0.000 NA NA 0.000 0.000
#> 20 textual ~~ g 0.000 0.000 NA NA 0.000 0.000
#> 21 speed ~~ g 0.000 0.000 NA NA 0.000 0.000
#> 22 visual ~~ textual 0.000 0.000 NA NA 0.000 0.000
#> 23 visual ~~ speed 0.000 0.000 NA NA 0.000 0.000
#> 24 textual ~~ speed 0.000 0.000 NA NA 0.000 0.000
#> 25 x1 ~~ x1 0.549 0.114 4.833 0.000 0.326 0.772
#> 26 x2 ~~ x2 1.134 0.102 11.146 0.000 0.934 1.333
#> 27 x3 ~~ x3 0.844 0.091 9.317 0.000 0.667 1.022
#> 28 x4 ~~ x4 0.371 0.048 7.779 0.000 0.278 0.465
#> 29 x5 ~~ x5 0.446 0.058 7.642 0.000 0.332 0.561
#> 30 x6 ~~ x6 0.356 0.043 8.277 0.000 0.272 0.441
#> 31 x7 ~~ x7 0.799 0.081 9.823 0.000 0.640 0.959
#> 32 x8 ~~ x8 0.488 0.074 6.573 0.000 0.342 0.633
#> 33 x9 ~~ x9 0.566 0.071 8.003 0.000 0.427 0.705
#> 34 visual ~~ visual 1.000 0.000 NA NA 1.000 1.000
#> 35 textual ~~ textual 1.000 0.000 NA NA 1.000 1.000
#> 36 speed ~~ speed 1.000 0.000 NA NA 1.000 1.000
#> 37 g ~~ g 1.000 0.000 NA NA 1.000 1.000
#> 38 lr1 := a1/b1 lr1 -0.558 0.309 -1.809 0.070 -1.163 0.047
#> 39 lr2 := a2/b2 lr2 -0.558 0.309 -1.809 0.070 -1.163 0.047
#> 40 lr3 := a3/b3 lr3 -0.558 0.309 -1.809 0.070 -1.163 0.047
#> 41 lr4 := a4/b4 lr4 1.621 0.338 4.798 0.000 0.959 2.283
#> 42 lr5 := a5/b5 lr5 1.621 0.338 4.798 0.000 0.959 2.283
#> 43 lr6 := a6/b6 lr6 1.621 0.338 4.798 0.000 0.959 2.283
#> 44 lr7 := a7/b7 lr7 1.563 0.348 4.489 0.000 0.881 2.246
#> 45 lr8 := a8/b8 lr8 1.563 0.348 4.489 0.000 0.881 2.246
#> 46 lr9 := a9/b9 lr9 1.563 0.348 4.489 0.000 0.881 2.246
# 2nd Order CFA
model_2nd <- '
visual =~ a1*x1 + a2*x2 + a3*x3
textual =~ a4*x4 + a5*x5 + a6*x6
speed =~ a7*x7 + a8*x8 + a9*x9
g =~ visual + textual + speed
'
fit_2nd <- cfa(model_2nd,
HolzingerSwineford1939,
std.lv = TRUE)
# Check if they are equivalent
fit_bf
#> lavaan 0.6.15 ended normally after 234 iterations
#>
#> Estimator ML
#> Optimization method NLMINB
#> Number of model parameters 27
#>
#> Number of observations 301
#>
#> Model Test User Model:
#>
#> Test statistic 85.306
#> Degrees of freedom 24
#> P-value (Chi-square) 0.000
fit_2nd
#> lavaan 0.6.15 ended normally after 36 iterations
#>
#> Estimator ML
#> Optimization method NLMINB
#> Number of model parameters 21
#>
#> Number of observations 301
#>
#> Model Test User Model:
#>
#> Test statistic 85.306
#> Degrees of freedom 24
#> P-value (Chi-square) 0.000
all.equal(lavInspect(fit_bf, "implied")$cov,
lavInspect(fit_2nd, "implied")$cov,
tolerance = 1e-4)
#> [1] TRUE
semTools::net(fit_bf, fit_2nd)
#>
#> If cell [R, C] is TRUE, the model in row R is nested within column C.
#>
#> If the models also have the same degrees of freedom, they are equivalent.
#>
#> NA indicates the model in column C did not converge when fit to the
#> implied means and covariance matrix from the model in row R.
#>
#> The hidden diagonal is TRUE because any model is equivalent to itself.
#> The upper triangle is hidden because for models with the same degrees
#> of freedom, cell [C, R] == cell [R, C]. For all models with different
#> degrees of freedom, the upper diagonal is all FALSE because models with
#> fewer degrees of freedom (i.e., more parameters) cannot be nested
#> within models with more degrees of freedom (i.e., fewer parameters).
#>
#> fit_bf fit_2nd
#> fit_bf (df = 24)
#> fit_2nd (df = 24) TRUE
```
<sup>Created on 2023-07-03 with [reprex v2.0.2](https://reprex.tidyverse.org)</sup>
library(lavaan)
#> This is lavaan 0.6-15
#> lavaan is FREE software! Please report any bugs.
# Bifactor with proportionalitu constraints
model_bf <- '
visual =~ a1*x1 + a2*x2 + a3*x3
textual =~ a4*x4 + a5*x5 + a6*x6
speed =~ a7*x7 + a8*x8 + a9*x9
g =~ b1*x1 + b2*x2 + b3*x3 + b4*x4 + b5*x5 + b6*x6 + b7*x7 + b8*x8 + b9*x9
g ~~ 0*visual + 0*textual + 0*speed
visual ~~ 0*textual + 0*speed
textual ~~0*speed
lr1 := a1 / b1
lr2 := a2 / b2
lr3 := a3 / b3
lr4 := a4 / b4
lr5 := a5 / b5
lr6 := a6 / b6
lr7 := a7 / b7
lr8 := a8 / b8
lr9 := a9 / b9
lr1 == lr2
lr2 == lr3
lr4 == lr5
lr5 == lr6
lr7 == lr8
lr8 == lr9
'
fit_bf <- cfa(model_bf,
HolzingerSwineford1939,
std.lv = TRUE)
parameterEstimates(fit_bf)
#> lhs op rhs label est se z pvalue ci.lower ci.upper
#> 1 visual =~ x1 a1 -0.438 0.194 -2.256 0.024 -0.819 -0.058
#> 2 visual =~ x2 a2 -0.243 0.108 -2.252 0.024 -0.454 -0.031
#> 3 visual =~ x3 a3 -0.320 0.138 -2.325 0.020 -0.589 -0.050
#> 4 textual =~ x4 a4 0.842 0.064 13.251 0.000 0.718 0.967
#> 5 textual =~ x5 a5 0.938 0.071 13.293 0.000 0.799 1.076
#> 6 textual =~ x6 a6 0.780 0.060 13.084 0.000 0.663 0.897
#> 7 speed =~ x7 a7 0.522 0.066 7.908 0.000 0.393 0.651
#> 8 speed =~ x8 a8 0.616 0.067 9.129 0.000 0.484 0.748
#> 9 speed =~ x9 a9 0.564 0.064 8.808 0.000 0.439 0.690
#> 10 g =~ x1 b1 0.786 0.116 6.744 0.000 0.557 1.014
#> 11 g =~ x2 b2 0.435 0.090 4.827 0.000 0.258 0.611
#> 12 g =~ x3 b3 0.573 0.102 5.605 0.000 0.373 0.773
#> 13 g =~ x4 b4 0.520 0.088 5.936 0.000 0.348 0.691
#> 14 g =~ x5 b5 0.578 0.097 5.940 0.000 0.388 0.769
#> 15 g =~ x6 b6 0.481 0.081 5.918 0.000 0.322 0.641
#> 16 g =~ x7 b7 0.334 0.066 5.026 0.000 0.204 0.464
#> 17 g =~ x8 b8 0.394 0.073 5.432 0.000 0.252 0.536
#> 18 g =~ x9 b9 0.361 0.069 5.259 0.000 0.226 0.496
#> 19 visual ~~ g 0.000 0.000 NA NA 0.000 0.000
#> 20 textual ~~ g 0.000 0.000 NA NA 0.000 0.000
#> 21 speed ~~ g 0.000 0.000 NA NA 0.000 0.000
#> 22 visual ~~ textual 0.000 0.000 NA NA 0.000 0.000
#> 23 visual ~~ speed 0.000 0.000 NA NA 0.000 0.000
#> 24 textual ~~ speed 0.000 0.000 NA NA 0.000 0.000
#> 25 x1 ~~ x1 0.549 0.114 4.833 0.000 0.326 0.772
#> 26 x2 ~~ x2 1.134 0.102 11.146 0.000 0.934 1.333
#> 27 x3 ~~ x3 0.844 0.091 9.317 0.000 0.667 1.022
#> 28 x4 ~~ x4 0.371 0.048 7.779 0.000 0.278 0.465
#> 29 x5 ~~ x5 0.446 0.058 7.642 0.000 0.332 0.561
#> 30 x6 ~~ x6 0.356 0.043 8.277 0.000 0.272 0.441
#> 31 x7 ~~ x7 0.799 0.081 9.823 0.000 0.640 0.959
#> 32 x8 ~~ x8 0.488 0.074 6.573 0.000 0.342 0.633
#> 33 x9 ~~ x9 0.566 0.071 8.003 0.000 0.427 0.705
#> 34 visual ~~ visual 1.000 0.000 NA NA 1.000 1.000
#> 35 textual ~~ textual 1.000 0.000 NA NA 1.000 1.000
#> 36 speed ~~ speed 1.000 0.000 NA NA 1.000 1.000
#> 37 g ~~ g 1.000 0.000 NA NA 1.000 1.000
#> 38 lr1 := a1/b1 lr1 -0.558 0.309 -1.809 0.070 -1.163 0.047
#> 39 lr2 := a2/b2 lr2 -0.558 0.309 -1.809 0.070 -1.163 0.047
#> 40 lr3 := a3/b3 lr3 -0.558 0.309 -1.809 0.070 -1.163 0.047
#> 41 lr4 := a4/b4 lr4 1.621 0.338 4.798 0.000 0.959 2.283
#> 42 lr5 := a5/b5 lr5 1.621 0.338 4.798 0.000 0.959 2.283
#> 43 lr6 := a6/b6 lr6 1.621 0.338 4.798 0.000 0.959 2.283
#> 44 lr7 := a7/b7 lr7 1.563 0.348 4.489 0.000 0.881 2.246
#> 45 lr8 := a8/b8 lr8 1.563 0.348 4.489 0.000 0.881 2.246
#> 46 lr9 := a9/b9 lr9 1.563 0.348 4.489 0.000 0.881 2.246
# 2nd Order CFA
model_2nd <- '
visual =~ a1*x1 + a2*x2 + a3*x3
textual =~ a4*x4 + a5*x5 + a6*x6
speed =~ a7*x7 + a8*x8 + a9*x9
g =~ visual + textual + speed
'
fit_2nd <- cfa(model_2nd,
HolzingerSwineford1939,
std.lv = TRUE)
# Check if they are equivalent
fit_bf
#> lavaan 0.6.15 ended normally after 234 iterations
#>
#> Estimator ML
#> Optimization method NLMINB
#> Number of model parameters 27
#>
#> Number of observations 301
#>
#> Model Test User Model:
#>
#> Test statistic 85.306
#> Degrees of freedom 24
#> P-value (Chi-square) 0.000
fit_2nd
#> lavaan 0.6.15 ended normally after 36 iterations
#>
#> Estimator ML
#> Optimization method NLMINB
#> Number of model parameters 21
#>
#> Number of observations 301
#>
#> Model Test User Model:
#>
#> Test statistic 85.306
#> Degrees of freedom 24
#> P-value (Chi-square) 0.000
all.equal(lavInspect(fit_bf, "implied")$cov,
lavInspect(fit_2nd, "implied")$cov,
tolerance = 1e-4)
#> [1] TRUE
semTools::net(fit_bf, fit_2nd)
#>
#> If cell [R, C] is TRUE, the model in row R is nested within column C.
#>
#> If the models also have the same degrees of freedom, they are equivalent.
#>
#> NA indicates the model in column C did not converge when fit to the
#> implied means and covariance matrix from the model in row R.
#>
#> The hidden diagonal is TRUE because any model is equivalent to itself.
#> The upper triangle is hidden because for models with the same degrees
#> of freedom, cell [C, R] == cell [R, C]. For all models with different
#> degrees of freedom, the upper diagonal is all FALSE because models with
#> fewer degrees of freedom (i.e., more parameters) cannot be nested
#> within models with more degrees of freedom (i.e., fewer parameters).
#>
#> fit_bf fit_2nd
#> fit_bf (df = 24)
#> fit_2nd (df = 24) TRUE
```
<sup>Created on 2023-07-03 with [reprex v2.0.2](https://reprex.tidyverse.org)</sup>
I did that when exploring something. I haven't fully tested it. You can take a look and see whether this is what you need.
Hope this helps.
-- Shu Fai
Pedro Ribeiro
Jul 3, 2023, 7:43:23 PM7/3/23
to lavaan
Dear Shu Fai,
Thanks so much, I adapted the syntax and it worked perfectly!
Cheers,
Pedro
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