standardized solutions and standardized estimates with partial invariance

[Silvio Maltagliati]

Hello, 

I am computing a multigroup SEM with partial invariance. Some intercepts are allowed to vary across groups. I would like to report the unstandardized coefficients (and their p-values), as well as the standardized coefficients of the paths I have entered. I have already read with much attention the following two discussions : https://groups.google.com/g/lavaan/c/aI6dXkbStaQ and https://groups.google.com/g/lavaan/c/lwCZIH-Unyk/m/fDdPfbAxAgAJ 

So far, I have successfully constrained paths across groups: unstandardized coefficients are equal across groups. However, standardized coefficients do differ by group, which I first found surprising. 

Do you think that this output is normal, given the partial invariance approach? I guess that standardized coefficients are different because they are depending on parameters that are allowed to vary across groups?

How would you recommend to report this please? 

Thank you very much for your help and support, 

Silvio 

[Christian Arnold]

Hi Silvio,

during standardization Sx and Sy enter the arena. Here is a link for details: https://www.sciencedirect.com/topics/mathematics/standardized-regression-coefficient

Or with lavaan:

pop.model <- "y ~ 0.6 * x"

data <- simulateData(pop.model, sample.nobs = 10000)

model <- "y ~ x"

fit <- sem(model, data)

b  <- lavInspect(fit, "est")$beta["y", "x"]

sx <- sqrt(lavInspect(fit, "cov.ov")["x", "x"])

sy <- sqrt(lavInspect(fit, "cov.ov")["y", "y"])

b * sx / sy

As long as the variances of x and y are not equal across groups, the standardized betas differ. Your results are indeed "normal".

Best 

Christian

---

Von: lav...@googlegroups.com <lav...@googlegroups.com> im Auftrag von Silvio Maltagliati <silvio.ma...@gmail.com>
Gesendet: Samstag, 17. September 2022 05:10
An: lavaan <lav...@googlegroups.com>
Betreff: standardized solutions and standardized estimates with partial invariance

 

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[Silvio Maltagliati]

Dear Christian, 

thank you so much for your help! That's crystal clear. 

I think that an "naive" reader may be really confused by this finding (i.e., one unstandardized coefficient by group, but different standardized coefficients across groups). Would you know any paper that report such kind of analyses? 

Accordingly, would you recommend, to constrain residual covariances across countries (after checking for invariance)? 

Thanks again for your precious insights, 

Silvio 

[Silvio Maltagliati]

I meant "constrain residual variance" across groups, sorry 

[Christian Arnold]

Hi Silvio,

You wrote: 'I think that an "naive" reader may be really confused by this finding (i.e., one unstandardized coefficient by group, but different standardized coefficients across groups). Would you know any paper that report such kind of analyses?'

It is possible that "naive" readers will be confused. You can attach a note to your table and explain why you report multiple standardized betas. Maybe this will have a learning effect on the "naive" readers? Frankly, I don't usually pay attention to the standardized betas. You are likely to find articles fairly quickly where only a single standardized beta per group and regression coefficient is reported, even if only the nonstandardized betas were constrained.

You wrote: 'Accordingly, would you recommend, to constrain residual covariances across countries (after checking for invariance)?'

That depends on your epistemological interest that you are pursuing with your studies. Honestly, I don't know what exactly you are aiming for. If you are referring to a CFA and the factor loadings, and you want the standardized factor loadings to be identical across groups, then you need to use the following constraints: loadings, lv.variances, residuals

Example:

pop.model <- "

f1 =~ 1.0 * v1 + 0.9 * v2 + 0.9 * v3

f2 =~ 1.0 * v4 + 1.0 * v5 + 0.9 * v6

f3 =~ 1.0 * v7 + 0.9 * v8 + 1.0 * v9

f2 ~ 0.6 * f1

f3 ~ 0.3 * f1 + 0.5 * f2"

n <- 10000

set.seed(1)

data <- cbind(

  rbind(

    simulateData(pop.model, sample.nobs = n),

    simulateData(pop.model, sample.nobs = n)  

  ),

  g = rep(1 : 2, each = n)

)

diff <- function(fit, op) {

  ssl <- standardizedSolution(fit)

  round(

    ssl[ssl$op == op & ssl$group == 2, "est.std"] - ssl[ssl$op == op & ssl$group == 1, "est.std"],

    6

  )

}

m.cfa <- "

f1 =~ v1 + v2 + v3

f2 =~ v4 + v5 + v6

f3 =~ v7 + v8 + v9"

fit <- cfa(m.cfa, data, group = "g", group.equal = c("loadings", "lv.variances", "residuals"))

diff(fit, "=~")

If we are talking about the regressions in the structural model, this is sufficient (you will also find 2 examples that do not work because either a regression coefficient or a variance was assumed to be invariant):

m.sem <- paste(

  m.cfa,

  "f2 ~ f1",

  "f3 ~ f1 + f2",

   sep = "\n"

)

fit <- sem(m.sem, data, group = "g", group.equal = c("loadings", "lv.variances", "regressions"))

diff(fit, "~")

fit <- sem(m.sem, data, group = "g", group.equal = c("loadings", "regressions", "lv.variances"), group.partial = "f2~f1")

diff(fit, "~")

fit <- sem(m.sem, data, group = "g", group.equal = c("loadings", "regressions", "lv.variances"), group.partial = "f1~~f1")

diff(fit, "~")

HTH

Christian

---

Von: lav...@googlegroups.com <lav...@googlegroups.com> im Auftrag von Silvio Maltagliati <silvio.ma...@gmail.com>

Gesendet: Samstag, 17. September 2022 15:04
An: lavaan <lav...@googlegroups.com>

Betreff: Re: standardized solutions and standardized estimates with partial invariance

 

To view this discussion on the web visit https://groups.google.com/d/msgid/lavaan/9fa88709-908e-4c99-b4af-d378791b6567n%40googlegroups.com.

[Silvio Maltagliati]

Dear Christian, 

sorry for my late reply, I was playing around with your code and was trying to implement this approach with my dataset. Thank you so much for your precious help, that is great to benefit from such advices. 

I am interested in 1- measuring invariance in measures across different groups (countries in my case), 2- test a global constrained model and 3- compute multigroup analyses in order to examine whether paths differ across countries.

As such, I believe that, as you suggest it, I do not need to report effect sizes (i.e., standardized beta coefficients and explained variance) for my constrained model, it may confuse the reader to get different estimates across countries. 

Still, I can provide this information in multigroup analyses, especially as I found that multiple paths actually differed across countries. 

Hoping that it will provide you a better perspective on my objectives ! 

Thanks again, 

warm regards,

Silvio 

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