Exogenous perdictors in Ordinal LGC m...
Message/Author
 Paras Mehta posted on Friday, May 19, 2000 - 2:10 pm
I have an ordinal variable (6-categories) measured at four time points. A linear LGC model fits well. When I introduced an exogenous-predictor of the intercept and slope factors, I discovered a curious phenomenon:
Changing how the metric of the latent y(t) is set changes the beta weight and the corresponding s.e. for the regression of the growth factors on exogenous predictors.
I am changing the metric of y(t) by fixing the delta-factor of one of the ordinal outcomes equal to 1.0:
{a1@1 a2 a3 a4};
vs
{a1 a2@1 a3 a4};

Thanks,
 Linda K. Muthen posted on Friday, May 19, 2000 - 4:00 pm
This is as expected. When you change the metric of the y* variables by changing the delta that is fixed to one, you will see changes in the coefficients and standard errors. Changing the metric of latent variables can have this effect.
 Paras Mehta posted on Saturday, May 20, 2000 - 10:30 am
This has important imlications for including exogenous predictors in an ordinal LGC model. Since, arbitrary-scale changes can influence regression-coefficients, the specific choice of scaling will impact sunstantive interpretations, which is undesirable.

A reasonable alternative is to admit that with ordinal outcomes, the only scale-invariant parameter for the regression-part is the standardized-beta and NOT the unstandardized refression-coefficient.

The only way to estimate scale-free path-coefficients is to ensure that the model is paramatrized such that the growth factors are in a "standardized" form. This way arbitrary scale choices for y(t) will not influence the regression of the growth factors on exogenous predictors.

This would invlove including "dummy" standardized growth factors along with regular unstandardized growth factors where regression of standardized growth factors on the unstandardized ones contain the scale factors that absorb arbitrary-scale changes.

i. Is this correct interpretation of what is going?

ii. Is my solution appropriate?

Thanks,
 bmuthen@ucla.edu posted on Sunday, May 21, 2000 - 4:42 pm
Your input here is helpful. I think this is a research area that could use some more thinking; we don't know enough about these parameterizations. Part of it is rather deep in my view.

It is always a good idea to really understand a given parameterization. To this aim, it might be useful to think of a simple growth model case where the intercept growth factor is defined at time 1 (so y1* has zero loading on the slope factor(s)), i.e. centering at time 1. Assume also that there is a single x covariate. So, here y1* = 1* i + e, where 1 is a fixed unit loading, i is the intercept factor, and e is the residual. And, i = a + g* x + z, where a is the intercept, g is the slope, and z is the residual. With delta fixed at 1 for time 1, Mplus uses the standardization of V(y1*|x) = 1 as in regular probit regression of y1 on x (the variance of y1* is a function of the e and z variances; fixing it to 1 gives the e variance as a remainder). The growth model says that the probit slope for y1 on x is expressed as 1*g, i.e. the raw estimate of g is in the usual probit metric. The estimate says how strongly x influences the systematic part of y1* in this metric. So, here g has a certain meaning due to scaling the two latent variables i and y1* in a certain way. Instead fixing delta=1 for y2* would change this meaning and therefore we should see a different g estimate, s.e., and t value. Centering at time 2, one could instead fix delta2=1 and then g would have a different meaning because it refers to systematic variance at time 2.

Having said that, I agree that there is a value in looking at the standardized solution; this parameterization has yet another meaning. It is interesting to study invariance implications for different parameterizations.

These issues are similar to those arising when trying to decide which of the multiple indicators of a factor to have loading fixed at 1. There was a debate about this some time ago. For example, in CFA of continuous longitudinal data, with a single factor at each timepoint, fixing a different indicator's loading to 1 maintains a constant ratio of factor variances at different timepoints (while absolute variance differences are of course not invariant).
 Blaze Aylmer posted on Wednesday, December 07, 2005 - 10:30 am
Linda/Bengt

I'm looking at a growth model and I'm measuring change in a dv using 5 point likert scale.What is the implication of estimating a latent growth model with likert scale measuring a DV on model fit? As far as I am aware the basic assumption is that the dv in an LGC should be continous?? Is is possible to estimate such a model?

Thanks
 Linda K. Muthen posted on Wednesday, December 07, 2005 - 11:23 am
In Mplus, the outcome in a growth model can be continuous, censored, binary, ordered categorical (ordinal), or a count variable. The model is estimated taking the scale of the outcome into account.