Latent growth curve model: how to include regressors?

[Amonet]

How can I include a regression using Lavaan when I am building a latent growth curve model? 

I have a measurement model with one latent variable (LV) measured at 4 periods, where strong invariance is established. I omit that part of the syntax, for brevity. LV = the latent variable that was measured at four time points, and for which invariance was established.

The covariance among the latent variables is fixed to zero. 

The variance of each latent variable is constrained to be equal over time. 

Model <- '

int =~ 1*LV0 + 1*LV1 + 1*LV3 + 1*LV4

int ~~ int

int ~ 1

sl  =~ 0*LV0 + 1*LV1 + 2*LV3 + 3*LV4

sl ~~ sl 

sl ~ 1

sl ~~ int 

#constrained variance of the LV over time

LV0 ~~ var*LV0

LV1 ~~ var*LV1

LV3 ~~ var*LV3

LV4 ~~ var*LV4

# regression

LV0 ~ X1

LV1 ~ X1

LV3 ~ X1

LV4 ~ X1

'

Is the part in bold correct? I aim to have it in the model as a simple predictor, such as in OLS. E.g.: Y = B0 + B1*X1

Would it make sense to regress: sl ~ X1 if the regressions in the syntax turn out insignificant? 

Thank you for your help!

[Jeremy Miles]

It looks OK to me. (I'm not absolutely sure, without all the syntax). 

I would regress sl on x1 whether is was significant or not - but why not regress int on x1 instead of regressing the indicators. If the effect of X1 is equal on each LV then:

LV0 ~ a * X1

LV1 ~ a * X1

LV3 ~ a * X1

LV4 ~ a * X1

is equivalent to:

sl on x1

(I think).

Jeremy

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[Amonet]

Thank you Jeremy,

I should have provided more context. The reason I did not want to regress the intercept (int) on X1 is because the sample was randomly divided in three treatment groups at baseline.

Also, in my 'real model' I did not regress LV1 on X1, because LV1 was measured at baseline. In other words, my hypothesis is that X1 could not have affected LV1 yet. 

You would keep in the regressed effects even if they for example are all highly insignificant? The latent variance (i.e. the error of LV1, LV2, LV3, LV4) which is called 'var' in the model syntax, was held equal over time and was only 0.02. So there was very little variance in LV left to explain... Because of the large insignificance and because the slope and intercept already explained most of the variance, I removed X1 from the model. Do you agree on this approach, or have suggestions? 

Thanks,

Amonet