Questions about piecewise linear growth model

[xuey qiny]

Hi Professors, 

This is Xueying again. I have a few more questions relating to the piecewise growth model which I have posted in the group several months ago. 

I draw a figure to illustrate my study design, briefly, as I have posted before, I would like to fit a piecewise growth model to assess whether the trajectory of Y variable would be different before and after certain intervention. The population in my study have repeated measurements of Y at six time points (T1-T6), and the data has the three features which might be considered when fitting piecewise model. First, the time-interval among follow-up time points were not exact the same for the adjacent time points, for example, the time-interval for T1 and T2 is 2 years apart, T2-T3 is 1 year apart, T3-T4 is 2 years apart, et al. (shown in Figure). Second, individuals had different numbers of repeated measurements of Y, for example, individual A had Y measurements at T1, T2 and T5, individual B had Y measurements at T3, T4 and T6, et al. Thirdly, individuals might have different intervention starting time points, for example, individual A started the intervention at T1, and individual B started the intervention at T3, however, majority of the population started the intervention at T3/T4.  

I followed the suggestions from previous post (smart idea from Patrick) to fit the piecewise growth model, that is, re-arranging every individual's data, and set the knot (the starting time point of the intervention) at "time zero" for everyone in the study. Therefore, I defined time-zero, time-minus*, time-plus* which were shown in the table below for different scenarios in which individuals had different time point of starting intervention (knot), and I also have considered the different time-intervals when defining time-minus* and time-plus*.  

I used the syntax like this to fit the piecewise growth model: 

intercept=~1*tminus7+1*tminus6+1*tminus5+1*tminus4+1*tminus3+1*tminus2+1*tminus0+1*tzero+1*tplus1+1*tplus2+1*tplus3+1*tplus4+1*tplus5+1*tplus6+1*tplus7

preslp=~(-7)*tminus7+(-6)*tminus6+(-5)*tminus5+(-4)*tminus4+(-3)*tminus3+(-2)*tminus2+(-1)*tminus0+0*tzero+0*tplus1+0*tplus2+0*tplus3+0*tplus4+0*tplus5+0*tplus6+0*tplus7

postslp=~0*tminus7+0*tminus6+0*tminus5+0*tminus4+0*tminus3+0*tminus2+0*tminus0+0*tzero+1*tplus1+2*tplus2+3*tplus3+4*tplus4+5*tplus5+6*tplus6+7*tplus7

So I have two questions for this piecewise model: 

1. As you have suggested before, this model assumed that ***everyone has the same lag between observations before and after the knot, just like standard LGC models assume equal time intervals***, I have tried to fit the model corresponding to the statement as much as possible (definition of time-zero, time-minus, and time-plus), but I still need your help to confirm whether I am doing the correct analysis. 

2. Currently, I used wave year as the time scale (figure above). Considering that age might be associated with Y and I adjusted baseline age (first measument of age as the covariate for latent intercept and slope), however, my colleagues have suggested that whether I could use biological age as the time scale in my analysis. So I would like to know first whether it's correct to use baseline age as covariate in my analysis. Then, I don't know whether it's reasonable to use age as time scale in this piecewise model. If so, I think age at the time point of starting intervention would be very heteregenous in different persons, and in this situation, shall I fit the piecewise growth model with random knot? Then how to fit a piecewise model with random knot? 

I have got lots of help from you and I appreciate you very much. 

Regards

Xueying