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)?
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?
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?
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.
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.
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.
I am using a LCA-derived 6-class nominal variable as a mediator between a binary X and binary Y. I believe I have successfully run a model adapting code from the 8.35 example. I also incorporated a manual 3-step approach to correct for classification error using Matt McLarnon’s recent tutorial: McLarnon & O’Neill. Organizational Research Methods 21.4 (2018): 955-982.
Q1: how would one adjust for confounders? My instinct is to add to the 8.35 code (1) Y on confounder and (2) M on confounder to the overall model statement, e.g.: (1) Y on race2 race3 race4; (2) c on race2 race3 race4;
Q2: how does one interpret the TIE in the case of a nominal M? does this depend on the reference category (of the nominal M)? Maybe to make tangible one could use the Muthen (2011) example, with an air pollution reduction intervention and indirect effects of various non-car transportation (van pool, bus, or light rail)? My attempt: the intervention-control difference in air pollution mediated by the usage of non-car commuting, calculated as the difference in air pollution if all participants experienced the level of light rail usage expected in the intervention group vs. the level of light rail usage expected in the control group?
Q3: is it possible to calculate at CDE in the case of a nominal M? Perhaps code could be suggested for the 8.35 example?
As an update, I'm now more confident that I have adequately controlled for M-Y confounders, using the approach stated in (Q1) in my prior post.
Also for (Q2), I'm quite sure my interpretation for TIE above was incorrect. Because the effect decomposition with nominal M integrates over all levels of the M to derive one set of causal effects, the reference category in the nominal M does NOT matter for interpretation of indirect effects. Thus perhaps an interpretation of the TIE in the Muthen 2011 (Sec. 8) example is more like: the intervention-control difference in air pollution mediated by the usage of non-car commuting, calculated as the difference in air pollution levels if all participants experienced the distribution of non-car usage seen in the intervention group vs. the distribution of non-car usage seen in the control group.
I'm still hoping you can help with my (Q3): can one derive a CDE in the 8.35 example? Let's say that it was meaningful to examine a CDE in which the entire population was measured as being within category 2 of the nominal M. Is this possible?
Thinking more about calculating a CDE in the case of a nominal mediator... I believe that class/exposure-specific predicted outcome estimates are equivalent to setting the probability of membership in that class to 1 (and all else zero), and these could be compared across X=1 and X=0.
In book example 8.35, if we were interested in a CDE with everyone taking the nominal M value of 2, the CDE could be calculated as: (1/(1+exp(beta02-beta12))) - (1/(1+exp(beta02))) with the first term being the expected outcome when X=1, and the second when X=0.
Please let me know if you think any of this sounds off to you. Perhaps the label CDE is not important, but, as you eluded to in a prior response, this seems at minimum useful descriptively.
I do have a point of clarification regarding book example 8.35: this model is using a probit regression to model the binary Y, correct?
If this is true, my understanding of the implications are as follows: (1) They provide very similar estimates. (2) The probit doesn't require a rare outcome assumption. (3) The probit avoids having to use numerical integration, which is required for logistic reg. (4) Odds ratios can be calculated when probit model is used (as in ex. 8.35).
I will have to think about your CDE idea. Ex 8.35 is logit (the full output is given on our book page on our website). The logit-based indirect effect does not assume a rare outcome. The rareness assumption is made by the approximate formulas used by Vanderweele on pages 315-326 with a continuous mediator and a binary outcome. In that case, Mplus does numerical integration over the mediator residual in the logistic case - no rareness assumption there so the effect is exact, not approximate.
Odds ratios can be computed with probit but as is discussed in Chapter 5, they don't give constant odds ratios.