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 Matt McLarnon posted on Thursday, November 09, 2017 - 1:40 pm
I've been considering Muthen(2011), but is this syntax on the right track for a 4-category nominal mediator (continuous X and Y)?

%OVERALL%
[c#1] (g01); c#1 ON x (g11);
[c#2] (g02); c#2 ON x (g12);
[c#3] (g03); c#3 ON x (g13);
y ON x; y;

!c#1 and c#2 as in Muthen (2011)

%c#3%
[y] (b03); y ON x (b13);
%c#4%
[y] (b04); y ON x (b14);

MODEL CONSTRAINT:
NEW(d0 d1 p10 p11 p20 p21 p30 p31 p40 p41
t11 t10 t01 t00 DE TIE TOT PIE);

d0=exp(g01)+exp(g02)+exp(g03)+1;
d1=exp(g01+g11)+exp(g02+g12)+exp(g03+g13)+1;
!p10-p21 as in Muthen (2011)

p30=exp(g03)/d0;
p31=exp(g03+g13)/d1;
p40=1/d0;
p41=1/d1;

t11=(b01+b11)*p11+(b02+b12)*p21+(b03+b13)*p31+(b04+b14)*p41;
t10=(b01+b11)*p10+(b02+b12)*p20+(b03+b13)*p30+(b04+b14)*p40;
t01=(b01)*p11+(b02)*p21+(b03)*p31+(b04)*p41;
t00=(b01)*p10+(b02)*p20+(b03)*p30+(b04)*p40;

!DE TIE TOT PIE as in Muthen (2011)
 Bengt O. Muthen posted on Friday, November 10, 2017 - 4:39 pm
See the book example Table 8.35 at

http://www.statmodel.com/Mplus_Book_Tables.shtml
 Matt McLarnon posted on Sunday, November 12, 2017 - 12:58 pm
Thank you - both the book and 2011 paper are excellent resources that have been extremely useful in this and other projects

I had considered the 2011 example closer to my purpose because of continuous Y, whereas the example in the book and Table 8.35 focused on a binary Y

1) instead of the above, would there be value added in estimating causal effects for each of the m-1 effects (i.e., a TNIE, PNDE etc. for each of the target categories vs. the referent category of the nominal variable, so for a 4-category nominal variable there would be three TNIEs and sets of causal effects)?

If so,

2) is inclusion of the "exp(g03+g13)" terms to the calculation of d0 and d1 (the denominator from the book/2011 formulae) correct?

3) is inclusion of the "(b04+b14)*p41" terms to the calculation of t11 and t01 (the terms representing the expectation values from the book/2011 formulae) correct?

4) is inclusion of the "(b04)*p41" terms to the calculation of t01 and t00 (expectation values) correct?
 Bengt O. Muthen posted on Sunday, November 12, 2017 - 5:38 pm
The "causal effect" principle is to collapse/integrate over M so I don't know what the m-1 effects would mean.
 Matt McLarnon posted on Monday, November 13, 2017 - 5:38 am
I was thinking that there would be "causal effects" for each comparison, between a target and a referent category, such that it would be meaningful to estimate whether choosing one response option of the nominal mediator vs. a referent category would transmit the effect of X on Y
But this makes sense, thank you.

May I ask then, are calculations for the d0, d1, t11, t10, t01, and t00 from the first post appropriate?

Thank you!
 Bengt O. Muthen posted on Monday, November 13, 2017 - 3:56 pm
You can discuss the various effects from M to Y.

Yes, it looks like you generalized the 3-class input correctly to a 4-class input.
 Matt McLarnon posted on Tuesday, November 14, 2017 - 4:47 am
Many thanks, Dr. Muthén, much appreciated!
 Adam Garber posted on Sunday, November 25, 2018 - 10:43 pm
I have a similar question to the one mentioned above -

Is it possible to estimate class-specific indirect effects in the context of a nominal latent class variable as mediator?
(i.e. conditional mediation)

In Muthen (2011), I have read that the indirect effect is summed over the levels of the mediator. Why is this? It seems like a natural extension to consider how indirect effects vary across levels of the categorical mediator.

4-class example:

t11_1= (b01+b11)*p11;
t11_2= (b02+b12)*p21;
t11_3= (b03+b13)*p31;
t11_4= (b04+b14)*p41;

t10_1= (b01+b11)*p10;
t10_2= (b02+b12)*p20;
t10_3= (b03+b13)*p30;
t10_4= (b04+b14)*p40;

t01_1= (b01)*p11;
t01_2= (b02)*p21;
t01_3= (b03)*p31;
t01_4= (b04)*p41;

t00_1= (b01)*p10;
t00_2= (b02)*p20;
t00_3= (b03)*p30;
t00_4= (b04)*p40;
 Bengt O. Muthen posted on Monday, November 26, 2018 - 1:07 pm
The indirect effects are summed over the nominal mediator because that is the definition of the counterfactually-defined "causal" indirect effect. That same summing (or integration for continuous mediator) is used for all types of M and Y variables. I don't know under which indirect effect definition effects for each category of the mediator would be appropriate - but maybe it is a useful descriptive statistic.
 Adam Garber posted on Monday, November 26, 2018 - 11:10 pm
Thanks Dr. Muthen your response is greatly appreciated.

In the specification of a mediation model with the nominal latent class variable as mediator, separate Y on X regressions are estimated for each class. Therefore, in specific research contexts may the latent class variable be conceived and reported to act both as a mediator and moderator simultaneously? Such an analysis, could report the summed/integrated indirect effect and also report the moderation effect. T. Beauchaine (2005), has suggested such a modeling situation in which a variable concurrently has moderating and mediating effects.
 Bengt O. Muthen posted on Tuesday, November 27, 2018 - 5:06 pm
Yes, separate Y on X regressions for each class represents moderation, that is, an X*M interaction. But you don't need to let Y on X be class varying. If you mention this regression in only the Overall part of the model, the slope will be equal across the classes - no moderation.
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