Jon Heron posted on Thursday, April 30, 2015 - 3:39 am
sorry to dredge this up again but I am searching for some clarity in what I find to be contradictory messages.
In Lanza's LTB method paper (2013) we are informed that if a distal outcome is binary, the various options for its inclusion in a (one-step) model are mathematically equivalent.
We are then given the impression in a number of places including Lanza's paper and Mplus webnote 15 that the covariate approach somehow protects the latent variable from distortion.
I'm attempting to align these two pieces of information in my mind.
It seems to me that including distal outcome Y as a covariate will always impact on the estimation of C (hence the popularity of three-step methods) however when Y is not binary, the covariate approach will impact C *less* due to different assumptions. This is demonstrated nicely with your webnote example of a bimodal continuous Y. This has a greater impact on C when treated as an indicator, nevertheless C will still be affected, to a lesser extent, when Y is a covariate.
There are obviously other differences to Y being treated as a dependent variable, e.g. its role in fit indices and the impact of any missing data using ML methods.
many thanks for your thoughts
Jon Heron posted on Thursday, April 30, 2015 - 7:30 am
Clarification: when Y is continuous, its inclusion as a covariate is likely to have less on an impact than its inclusion as a class indicator.
"the covariate approach somehow protects the latent variable from distortion."
do you mean the Lanza method?
Jon Heron posted on Thursday, April 30, 2015 - 11:38 pm
no I am thinking one-step for all of this.
There are places in the recent literature to suggest that one-step with a covariate will not distort C whilst including the same covariate as a class indicator will impact C.
As I understand it, all variables, whether covariate or indicator will effect C. The comments I have seen are actually referring to the case of this additional variable Y being continuous and crucially - the situation where the within-class distribution of Y is not Gaussian. In this instance the indicator method for Y will severely distort the estimation of C, whilst the effect on C of including Y as a covariate will be more moderate.
You should take a look at http://statmodel.com/examples/webnotes/webnote21.pdf Table 6 and 7 summarize the latest. In particular Lanza's method with continuous Y distribution is no longer recommended due to problems that occur when the variance of Y varies across class, see Table 3. The new BCH method or the 3-stage method should work better in that case. The DCAT option for categorical distal works well and is based on Lanza's method and the observation you mention that "one-step with a covariate will not distort C whilst including the same covariate as a class indicator will impact C".
Jon Heron posted on Saturday, May 02, 2015 - 3:50 am
so you agree with that observation?
this is what I am challenging. With a one-step any covariate or outcome has the potential to affect C, however the only time we may have problems - a major distortion of C rather than a subtle tweak - is for continuous outcomes with awkward within-class distributions.