effect size of latent mean differences

[Kathrin Hahn]

Hi,

I want to calculate effect sizes of latent mean differences. Factor variances have been shown to be equal in both groups. For my measurement invariance analysis I used fixed variances for model identification (instead of fixing factor loadings). 

But for effect size I need the factor variances, therefore I computed the same model but with fixed factor loadings instead of fixed factor variances. This resulted in exactly the same fit indizes and so on, but I received different latent means. 

1) Is that normal and why is that? Or did I make a mistake?

2) Can I use this latent mean differences and the (to eqaulity restrcited) variance to calculate effect size? 

######### fixed factor variances

model.variance <- '

# Factor loadings

negm =~ load1*m1+ load3*m3+ load4*m4+ load6*m6+ load7*m7+ load8*m8

negw =~ load1*w1+ load3*w3+ load4*w4+ load6*w6+ load7*w7+ load8*w8

posm =~ load2*m2+ load5*m5+ load9*m9+ load10*m10

posw =~ load2*w2+ load5*w5+ load9*w9+ load10*w10

#intercepts

m1 ~ int11*1

m2 ~ int2*1

m3 ~ int3*1

m4 ~ int4*1

m5 ~ int5*1

m6 ~ int61*1

m7 ~ int7*1

m8 ~ int8*1

m9 ~ int9*1

m10 ~ int10*1

w1 ~ int12*1

w2 ~ int2*1

w3 ~ int3*1

w4 ~ int4*1

w5 ~ int5*1

w6 ~ int62*1

w7 ~ int7*1

w8 ~ int8*1

w9 ~ int9*1

w10 ~ int10*1

# Factor variances

negm ~~ 1*negm

posm ~~ 1*posm

negw ~~ 1*negw

posw ~~ 1*posw

# Factor means

negm ~ NA*1

posm ~ NA*1

#correlated error terms

m1 ~~ w1

m2 ~~ w2

m3 ~~ w3

m4 ~~ w4

m5 ~~ w5

m6 ~~ w6

m7 ~~ w7

m8 ~~ w8

m9 ~~ w9

m10 ~~ w10

'

variance <- cfa(model.variance, data=datw, estimator="MLM", std.lv=TRUE)

summary(variance, fit.measures=TRUE)

######### fixed factor loadings, egaul variances in both groups

model.variance <- '

# Factor loadings

negm =~ load1*m1+ load3*m3+ load4*m4+ load6*m6+ load7*m7+ load8*m8

negw =~ load1*w1+ load3*w3+ load4*w4+ load6*w6+ load7*w7+ load8*w8

posm =~ load2*m2+ load5*m5+ load9*m9+ load10*m10

posw =~ load2*w2+ load5*w5+ load9*w9+ load10*w10

#intercepts

m1 ~ int11*1

m2 ~ int2*1

m3 ~ int3*1

m4 ~ int4*1

m5 ~ int5*1

m6 ~ int61*1

m7 ~ int7*1

m8 ~ int8*1

m9 ~ int9*1

m10 ~ int10*1

w1 ~ int12*1

w2 ~ int2*1

w3 ~ int3*1

w4 ~ int4*1

w5 ~ int5*1

w6 ~ int62*1

w7 ~ int7*1

w8 ~ int8*1

w9 ~ int9*1

w10 ~ int10*1

# Factor variances

negm ~~ a*negm

posm ~~ b*posm

negw ~~ a*negw

posw ~~ b*posw

# Factor means

negm ~ NA*1

posm ~ NA*1

#correlated error terms

m1 ~~ w1

m2 ~~ w2

m3 ~~ w3

m4 ~~ w4

m5 ~~ w5

m6 ~~ w6

m7 ~~ w7

m8 ~~ w8

m9 ~~ w9

m10 ~~ w10

'

variance <- cfa(model.variance, data=datw, estimator="MLM")

summary(variance, fit.measures=TRUE)

[Terrence Jorgensen]

I want to calculate effect sizes of latent mean differences. Factor variances have been shown to be equal in both groups. For my measurement invariance analysis I used fixed variances for model identification (instead of fixing factor loadings). 

If the effect size you want is Cohen's D, then you have it easy.  If latent variances are fixed to 1 in both groups, then differences between estimated latent means are already standardized mean differences.  Assuming the latent mean is fixed to 0 in the first group, any other group's estimated latent mean is the standardized mean difference of that group from Group 1 (because any mean minus zero is the same number).

But for effect size I need the factor variances, therefore I computed the same model but with fixed factor loadings instead of fixed factor variances.

This is not necessary.  Fixing the latent variances does not mean that the variances do not exist; it means that they are equal to 1, which makes calculations easier because the denominator of Cohen's D is 1 (so you only need the numerator, which is the mean difference).

Terrence D. Jorgensen

Postdoctoral Researcher, Methods and Statistics

Research Institute for Child Development and Education, the University of Amsterdam

UvA web page: http://www.uva.nl/profile/t.d.jorgensen

[Maria Magdalena Kwiatkowska]

Dear Terrence,

Many thanks for your explanation!

Is there any reference to the theorem that this deifference between latent means can be treated equally with Cohen's d?

Best,

Maria

[Terrence Jorgensen]

Is there any reference to the theorem that this deifference between latent means can be treated equally with Cohen's d?

Cohen's d is only defined for the 2-group case, and even that is not the same as a standardized mean difference.  Cohen's d expresses the difference in units of residual SD, not the total SD of the variable (which is of course larger when there is a group mean difference).  But in your case (multigroup CFA), you just have exogenous factors, and the constrained variance across groups would be the square of the pooled residual SD used in the Cohen's d calculation.  You can find details and references here (also points to a semTools function you can use):

https://stats.stackexchange.com/questions/223729/estimating-effect-size-for-mean-difference-cohens-d-in-a-structural-equation

Terrence D. Jorgensen

Assistant Professor, Methods and Statistics

Research Institute for Child Development and Education, the University of Amsterdam

http://www.uva.nl/profile/t.d.jorgensen

[Erik O'Donnell]

Dear Terrence,

Assuming a large sample size (thousands) and measurement invariance up to the means, is there any reason not to use the scores generated by the lavPredict function with the grouping variable to calculate effect sizes?

Best,

Erik