Non-linear relationships between latents
Message/Author
 Steven Siciliano posted on Tuesday, June 12, 2012 - 10:09 am
Dr. Muthen,

We have developed a SEM that involves a structural path that we expect to be nonlinear.

Examining the scatter plot between Y1 and Y2 indicates that the relationship likely follows an exponential decay function such as y=a e^xc.

We would like to model this particular structural path using this or a similar function. We see how the XWITH function can be used to introduce squared terms into the model, but are unsure how to proceed with an exponential type of relationship.

Thank you,
 Linda K. Muthen posted on Tuesday, June 12, 2012 - 1:56 pm

 Steven Siciliano posted on Wednesday, July 18, 2012 - 9:45 am
Thanks for the paper. Unfortunately, our data set structure won't work with it. We have made a lot of progress with the modelling but have run into the non-linear issue again and are hoping you can help us.

We are creating a latent variable called ‘Response’ based on 16 continous observed variables. This latent ‘Response’ has a classic dose response features of a logistic type response. We would like to regress our continous latent variable ‘Response’ against the observed toxicant concentration. Models assuming a linear or quadratic (fit using a composite variable) relationship are not adequately capturing the relationship between ‘Response’ and toxicant concentration.
We are thinking that if we can use a logit link between ‘Response’ and toxic concentration, that this would solve our problem. We understand that the logit link is the default for categorical latents. Thus, our question is:
a) Is there some way to specify a logit link between a continous latent and an observed variable?
b) Can we instead specify that our ‘Response’ latent is a categorical variable?
Currently our ‘Response’ latent is specified as follows:
RESPONSE BY x1* x2-x16
RESPONSE@1;
We are trying to model this link:
Response ON x17;

Inspection of the data confirms that the Response/x17 relationship is a beautiful sigmoid curve.
 Linda K. Muthen posted on Wednesday, July 18, 2012 - 11:37 am
I'm afraid this is not possible. The ON statement you show is a linear regression.
 Steven Siciliano posted on Wednesday, July 18, 2012 - 12:29 pm
I'm sorry but I'm not quite clear on your answer. Do you mean that it is not possible to model Response onto X17 as a logistic equation? Or did you mean to say that the 'ON' statement is a linear regression. If the later, we are aware that it is a linear regression.

To confirm: This is no way to specify a logit link between a continous latent and an observed variable?

If not... how can we specify a latent derived from continous data should be treated as a categorical latent.

Thank you,
 Linda K. Muthen posted on Wednesday, July 18, 2012 - 12:39 pm
It is not possible to have a logit link for a continuous variable, latent or observed, regressed on a covariate.

A categorical latent variable is a nominal variable. You cannot use it in this way.
 Steven Siciliano posted on Wednesday, July 18, 2012 - 12:53 pm
Okay, thanks. We thought so but thought it best to make sure.
 Bengt O. Muthen posted on Wednesday, July 18, 2012 - 1:36 pm
When you use a categorical latent variable it is not clear that the categories will correspond to an ordering, but may capture other features of the data.

Talking about mixtures, you may also want to check the papers on our web site under Papers,

Non-Linear SEM with a Semi-Parametric Approach

to see if any of that is useful to you.